PERFORMANCE MODELLING OF
COMPUTER SYSTEMS AND COMPUTER
NETWORKS
(part 1)
Ramon Puigjaner
Universitat de les Illes Balears
Palma, Spain
putxi@uib.es
Facultade de Informatica. A Coruña. Junio 2005
OUTLINE
 INTRODUCTION
 CONCEPT OF QUEUE
 CONCEPT OF QUEUEING NETWORK
 NUMERICAL TECHNIQUES
 EXACT ANALYTICAL SOLUTIONS
 APPROXIMATE ANALYTICAL
SOLUTIONS
 SIMULATION TECHNIQUES
Facultade de Informatica. A Coruña. Junio 2005
2
INTRODUCTION
 What
is the performance of a Computer
Network?
 Performance
is how a software is using a
hardware when they are serving a load.
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INTRODUCTION
 This
definition considers the three elements
intervening in a system:
The load that is externally defined.
 The hardware to be used.
 The basic software that controls the hardware.

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INTRODUCTION: Performance measures
 The performance of a Computer Network is not
a unique value but a set of them to take into
account the heterogeneous composition of such
kind of systems.

External performance measures
o response time
o throughput (flow through the system)
o loss rate
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INTRODUCTION: Performance measures

Internal performance measures
o mean queue length
o device utilisation (% of busy time)
o overlap
o overhead (operating system utilisation, paging, etc.)
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INTRODUCTION: Performance tools
 Measuring
Monitors
 Logs
 Hardware probes
 Software probes

 Modelling
 Benchmarking
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INTRODUCTION: Performance tools
 Measuring
 Modelling
Queuing networks
 Petri nets
 Markov chains

 Benchmarking

Workload modelling
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INTRODUCTION: Measuring
 Measuring is the technique to be used when
system is installed and running. It is used to
verify whether the performance requirements
are met or not.
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INTRODUCTION: Modelling
A
model is an abstract mathematical
representation of the system behaviour in steady
state. It is the appropriate technique when the
computer network, partially or totally, does not
exist. Main existing techniques are:

Petri nets
o Better suited to represent synchronisation mechanisms
o Solving techniques may be either numerical (based on
Markov chains) or simulation.
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INTRODUCTION: Modelling

Queuing networks
o Better suited to represent customer-server mechanisms
o Solving techniques may be either analytical (closed
form formulae) or numerical (based on Markov chains)
or simulation.

Markov chains
o High abstraction level
o Solving techniques are most frequently numerical.
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OUTLINE
 INTRODUCTION
 CONCEPT OF QUEUE
 CONCEPT OF QUEUEING NETWORK
 NUMERICAL TECHNIQUES
 EXACT ANALYTICAL SOLUTIONS
 APPROXIMATE ANALYTICAL
SOLUTIONS
 SIMULATION TECHNIQUES
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CONCEPT OF QUEUE
 Queue: A customer that arrives and finds the
server busy joins the queue
 Service mechanism: It consists of one or more
servers that give service to the customers from
the queue
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CONCEPT OF QUEUE
 Customer source characteristics
finite or infinite
 distribution
of inter-arrival
consecutive customer arrivals
 customer service request

times
between
 Service station characteristics
queue number and capacity
 server number
 server capacity
 service discipline
 queue policy

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CONCEPT OF QUEUE
 Single queue with single server
 Single
queue with single server with state
dependent capacity
m(k)
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CONCEPT OF QUEUE
 Single queue with multiple servers
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CONCEPT OF QUEUE
 Multi-server with no queue
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CONCEPT OF QUEUE
 Infinite server
.
.
.
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OUTLINE
 INTRODUCTION
 CONCEPT OF QUEUE
 CONCEPT OF QUEUEING NETWORK
 NUMERICAL TECHNIQUES
 EXACT ANALYTICAL SOLUTIONS
 APPROXIMATE ANALYTICAL
SOLUTIONS
 SIMULATION TECHNIQUES
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CONCEPT OF QUEUEING NETWORK
A
queuing network is nothing else but a
collection of single queues, which are arbitrarily
interconnected.
 A queuing network is an oriented graph that has
in each node a server of some type.
 The time in traversing the network is spent in
the nodes and the arcs are traversed in a null
time.
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CONCEPT OF QUEUEING NETWORK
 Queuing
networks may be either open or
closed.
In an open queuing network, customers arrive from
outside, circulate through the nodes, and finally
they depart from the network.
 In a closed queuing network, there is a fixed
number of customers constantly circulating through
the nodes. Neither departures from the network nor
arrivals to the network are allowed.
 It is possible to have a queuing network which is
both open and closed. Such a network is known as
a mixed network.

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EXAMPLES OF OPEN QUEUING
NETWORKS
 Tandem configuration
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EXAMPLES OF OPEN QUEUING
NETWORKS
 Tree-like configuration
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EXAMPLES OF OPEN QUEUING
NETWORKS
 Tree-like configuration
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EXAMPLES OF CLOSED QUEUING
NETWORKS
 Cyclic network (closed tandem configuration)
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EXAMPLES OF CLOSED QUEUING
NETWORKS
 Arbitrary configuration
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EXAMPLES OF CLOSED QUEUING
NETWORKS
 Central server model
Disk 1
Disk 2
CPU
Arrivals
Disk 3
Disk 4
Exits
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EXAMPLES OF CLOSED QUEUING
NETWORKS
 Central server model
Disk 1
Disk 2
Arrivals
CPU
Disk 3
Disk 4
Exits
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EXAMPLES OF CLOSED QUEUING
NETWORKS
 Central server model
Disk 1
Terminals
Disk 2
Arrivals
CPU
Disk 3
Disk 4
Exits
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EXAMPLES OF MIXED QUEUING
NETWORKS
Conversational
tasks
Terminals
Transactions
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Central
system
30
CONCEPT OF QUEUEING NETWORK
 Observations

Each node can have any of the single-node
characteristics described above.

In order to specify the queuing network we need to
provide information concerning the routing; that is
to specify how a customer chooses the next node
when it leaves the current node. This routing can be
deterministic, probabilistic, function of the state,
etc.
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CONCEPT OF QUEUEING NETWORK
 How to set-up a queuing network model?

The notion of customer
o Typically a customer may be a piece of software in a
computer system, an information packet in a packetswitched environment, a phone call in a circuitswitched environment, etc.
o Customer classes will be defined if there are
differences in the resource consumption or in the
routing across the network
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CONCEPT OF QUEUEING NETWORK
 How to set-up a queuing network model?

The notion of node
o A node is a service mechanism that may be a hardware
component or a piece of software or a combination of
both, e.g. a CPU, a disk, a memory module, a bus, a
trunk, a switching node, etc.
o Each service mechanism has a buffer (the queue),
where customers wait until they are served. The buffer
capacity is finite; that is, they can accommodate a finite
number of customers. However, if a finite buffer has
low probability of being full, then it can be assumed as
infinite.
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CONCEPT OF QUEUEING NETWORK
 How to set-up a queuing network model?

Collecting information
o Once customers and server have been identified, it is
necessary to characterise service time distributions at
each node, routing probabilities and inter-arrival time
distributions.
o In many cases, this information can be compiled from
raw data (technical information, measurements, etc.); in
other cases it is based on an educated guess.
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CONCEPT OF QUEUEING NETWORK
 Solution techniques for queuing networks
To study the steady state behavior of a network
the following techniques can be used:

Analytic solutions

Numerical techniques

Simulation techniques
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OUTLINE
 INTRODUCTION
 CONCEPT OF QUEUE
 CONCEPT OF QUEUEING NETWORK
 NUMERICAL TECHNIQUES
 EXACT ANALYTICAL SOLUTIONS
 APPROXIMATE
ANALYTICAL
SOLUTIONS
 SIMULATION TECHNIQUES
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NUMERICAL TECHNIQUES
 The
behaviour of a queuing network can be
described in terms of linear equations (known
as the steady-state Kolmogorov equations).
These equations can be solved numerically to
obtain the solution.
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NUMERICAL TECHNIQUES
 To highlight this approach, let us consider the
following two-node closed queuing network
µ1
µ2
 Let us assume that:
there are 5 customers in the system.
 µ1 and µ2 are the service rates.
 both services are exponentially distributed.

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NUMERICAL TECHNIQUES
 The state of the system is described by (n1, n2),
that there are the number of customers in each
queue.
 The numerical analysis approach involves the
following steps:
Generation of all feasible states.
 Setting-up the rate matrix.
 Solving the steady state equations.

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NUMERICAL TECHNIQUES
 Generation of all feasible states.
The states for our example are:
(5,0) (4,1) (3,2) (2,3) (1,4) (0,5)
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NUMERICAL TECHNIQUES
 Setting-up the rate matrix.

This matrix which contains all the transitions and
their associated rates between each pair of states.
(5,0) (4,1) (3,2) (2,3) (1,4) (0,5)
(5,0)
*
µ1
(4,1)
µ2 *
µ1
(3,2)
µ2 *
µ1
(2,3)
µ2 *
µ1
(1,4)
µ2 *
µ1
(0,5)
µ2 *
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NUMERICAL TECHNIQUES
 Setting-up the rate matrix.

Let us refer to the this matrix as Q.

All blanks are assumed to be zero.

Each diagonal element marked with * is equal to
the negative sum of the off-diagonal elements of
the same row.
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NUMERICAL TECHNIQUES
 Solving the steady state equations.

Let p(n1, n2) be the steady-state probability that the
system is in state (n1, n2) and P the row vector of
these probabilities. To determine them we must
solve the following system of equations:
PxQ=0
together with the condition
pn , n   1



1
2
 n1 ,n2
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NUMERICAL TECHNIQUES
 Solving the steady state equations.

From the knowledge of these probabilities we can
determine performance measures such as:
o Server utilisation:
r1 = p(5,0) + p(4,1) + p(3,2) + p(2,3) + p(1,4):
r2 = p(4,1) + p(3,2) + p(2,3) + p(1,4) + p(0,5)
o Throughputs:
l1 = r1 x µ1
l2 = r2 x µ2
o Queue lengths:
N1 = 5p(5,0) + 4p(4,1) + 3p(3,2) + 2p(2,3) + p(1,4)
N2 = p(4,1) + 2p(3,2) + 3p(2,3) + 4p(1,4) + 5p(0,5)
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NUMERICAL TECHNIQUES
 Solving the steady state equations.

Advantages/disadvantages
o There are packages, like QNAP2, that automatically
set-up the rate matrix Q, solve it to find the P vector
and give the performance results. Other packages give
the vector P if the user is able to create the matrix Q.
o This numerical technique gives the exact solution.
There are also approximated solutions in some cases in
order to reduce the amount of computation.
o The approach is limited to cases where the number of
states is not very large.
o In queuing networks, quite often, the rate matrix is
sparse. In this cases, one can analyse larger systems by
using compact storage techniques.
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OUTLINE
 INTRODUCTION
 CONCEPT OF QUEUE
 CONCEPT OF QUEUEING NETWORK
 NUMERICAL TECHNIQUES
 EXACT ANALYTICAL SOLUTIONS
 APPROXIMATE ANALYTICAL
SOLUTIONS
 SIMULATION TECHNIQUES
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EXACT ANALYTICAL SOLUTIONS
 An
analytical solution means that we can
obtain the probabilities of the steady steady by
the application of a closed formula.
 This formula will obviously be a function of the
parameters of the system.
 Quite often an analytic solution is so
complicated that we can not evaluate it "on the
back of an envelope". In fact, one might need to
write a fairly sophisticated program.
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EXACT ANALYTICAL SOLUTIONS
A
certain class of queuing networks has an
analytic solution, known as a product-form
solution because the steady state probability has
the form of the product of the state probabilities
of each node.
 Its solution can be easily evaluated.
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EXACT ANALYTICAL SOLUTIONS
 Product-form networks have been proved to be
very useful in computer and communication
systems performance modelling.
 Also, there are a lot of queuing networks which
do not have product-form solutions. These
networks are analysed approximately.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem

It is the general theorem concerning queuing
networks with product-form solutions.
 Let us consider a BCMP queuing network with:
N nodes arbitrarily linked.
 Multiple classes of customers
 Probabilistic routing
 External arrivals with state-dependent rates
 Different service mechanisms

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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers

Customers are grouped in different classes. Each
class has its own service characteristics at each
node and its own routing probabilities. A class may
be open or closed.

Thus, a BCMP network, in its most general form,
can be seen as consisting of several open classes
and closed classes of customers.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers

It is possible that upon departure from a node a
customer may change of class. A superclass or a
subchain is the set of classes among those the
customers can change.

The use of classes of customers provides the
modeller with a lot of modelling flexibility.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers
Example 1. A queuing network model of a
multiprogramming system
Disk 1
Disk 2
Arrivals
CPU
Disk 3
Disk 4
Exits
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers
Example 1. A queuing network model of a
multiprogramming system

The total number of customers constantly
circulating through the system reflects the degree of
multiprogramming. Implicitly it is assumed that
when a job completes its execution and departs
from the system, another job takes its place. That is,
there is always at least one job waiting to get into
the multiprogramming environment. This rather
simplistic model captures the main features of a
multiprogramming system.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers
Example 1. A queuing network model of a
multiprogramming system

Let us see, how can make this model more useful
by introducing classes. We can introduce different
classes for different types of jobs, i. e.:
o Class 1: Interactive jobs
o Class 2: Short batch jobs
o Class 3: Medium batch jobs
o Class 4: Long batch jobs

Also, in a multiprogramming system, a process
originally classified in some class may be changed
to another one.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers
Example 1. A queuing network model of a
multiprogramming system
Features as service requirements, visit rates to each
node, class change, etc. are captured through the
use of classes.
 However, the BCMP theorem is limited as it does
not allow other features, such as priorities among
classes.

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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers
Example 2. Queuing network of a packetswitching system
2
1
4
3
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers
Example 2. Queuing network of a packetswitching system

A packet is represented by a customer in the
queuing network and each logical end-to-end
connection is represented by a class. This allows us
to assign a different routing to each class, and, if
need be, different service times at each node.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers
Example 2. Queuing network of a packetswitching system

So, we consider the following classes:
o Class 1: packets arrive at node 1, go to node 2, then to
node 3 and then they depart from the system.
o Class 2: packets arrive at node 1, go to node 3, then to
node 4 and then they depart from the system.
o Class 3: packets arrive at node 2, go to node 3 and then
they depart from the system.
o etc.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Arrival processes

If we have open classes of customers, one needs to
specify how these customers arrive from outside. In
general, the rate of arrivals is allowed to be statedependent, i. e. it can be an arbitrary function of the
number of customers in the system.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Arrival processes

Single arrival stream
o All external arrivals come from a single stream. When
a customer arrives to the network, it joins node i as
class r with probability pi,r.
o The inter-arrival times must be exponentially
distributed. The rate of arrivals may be constant or it
may be dependent upon the total number of customers
in the network.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Arrival processes

One arrival stream per class
o Each open class has its own arrival stream. A new
arrival of class r joins node i with probability pi.
o The
inter-arrival
times
must
be
exponentially
distributed. The rate of class r arrivals may be constant
or it may be dependent upon the total number of class r
customers in the network.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers
Example 2. Queuing network of a packetswitching system
Window-flow control allows only up to a prespecified number of packets in the system. Any
additional packets are forced to wait in an input
queue.
 In order to model a sliding-window flow-control
scheme, we need to model the input queue.
However, the BCMP theorem does not provide
such features.

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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers
Example 2. Queuing network of a packetswitching system

We can consider in some way the window-flow
control by making the arrival process of customers
state-dependent. That is, arrivals will occur as long
as the total number of customers of some class is
less than some threshold. When it becomes equal to
this value, the arrival stream will be turned off. The
arrival stream will start again when a customer of
the considered class departs from the network.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers
Example 2. Queuing network of a packetswitching system

In this way, we make sure that the total number of
customers of each class does not exceed its
threshold.

However, this is done by introducing the erroneous
assumption that no arrivals occur during the time
the window is full.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Service mechanisms

Type 1
o State dependent exponentially distributed service times.
o Class independent service time distribution
o FIFO irrespective of classes.
o Single server

Type 2
o Class and state dependent Coxian distributed service
times.
o Processor sharing (PS) discipline (or RR, round robin)
o Single servers
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Service mechanisms

Type 3
o Class-dependent Coxian distributed service times
o Infinite servers

Type 4
o Class and state dependent Coxian distributed service
times
o Pre-emptive server LIFO queue (PI)
o Single server
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Classes of customers
Example 1. A queuing network model of a
multiprogramming system

In this model it makes sense to assume that the
CPU node is a type 2 node, i. e. customer are
processor-shared, while the peripheral (disks 1 to 4)
are type 1. If we assume an interactive system we
would represent the terminals by means of a type 3
node.
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem

Assumption that the system reaches a steady state with
state probabilities p(S)

Balance equations:
pS exit rate from S  
   pS 'rate of going from S' to S
S '

Normalising equation
 p S   1
S
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem
N
1
p S   d S  g i S i 
G
i 1

G normalisation constant to obtain the addition the
probabilities of all states is equal to 1.
o If the system is closed, the number of states is finite and
the problem is numerical
o If the system is open, the number of states is infinite and
the problem is analytical
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem

d(S) is a function that if the network is closed its value
is 1 and if the network is open its value is
M  S 1
 i 
i 0
if the arrival rate depends on M(S),
K M  S , Ek 1
   i 
k
k 1
i 0
if the arrival rate to each subchain depends on M(S,Ek)
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71
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem

The expressions of gi(Si) are
o If the station is of type 1

1 mc  1
g i S i   mi ! 
eic 
 c 1 mic !  m i
C



mc
o If the station is of type 2 or 4
1  eic 


g i S i   mi !
c 1 mic ! m ic 
C
Facultade de Informatica. A Coruña. Junio 2005
mc
72
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem

The expressions of gi(Si) are
o If the station is of type 3
1  eic 


g i S i   
c 1 mic ! m ic 
C
Facultade de Informatica. A Coruña. Junio 2005
mc
73
EXACT ANALYTICAL SOLUTIONS
 The
BCMP theorem: Solving of a BCMP
queuing network

The expression that gives the probability that the
system is in a precise state is quite complicated.

However it is still possible to write down the
solution in form of product of terms, each term
consisting of parameters related to the node.
Facultade de Informatica. A Coruña. Junio 2005
74
EXACT ANALYTICAL SOLUTIONS
 The
BCMP theorem: Solving of a BCMP
queuing network

In the case of closed networks, as the number of
states is finite, it is necessary to compute a
normalising constant.
o There are various algorithms to do that, such as the
convolution algorithm and the mean value analysis.
o These algorithms are available through various network
analysers, such as QNAP2, BEST-1 and RESQ. The
user simply specifies the network characteristics, and
the package produces the solution.
Facultade de Informatica. A Coruña. Junio 2005
75
EXACT ANALYTICAL SOLUTIONS
 The
BCMP theorem: Solving of a BCMP
queuing network

In the case of open networks, as the number of
states is infinite, it is necessary to compute the
result of a series.
o This computation is only possible for specific
combinations of node characteristics.
o As for closed networks, these algorithms are available
through various network analysers, such as QNAP2,
BEST-1 and RESQ. The user simply specifies the
network characteristics, and the package produces the
solution.
Facultade de Informatica. A Coruña. Junio 2005
76
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Case studies
Transactional system

The case proposed is a throughput input system
with two types of transactions (messages arriving to
the system and requiring some process) that have
different arrival frequency, CPU consumption and
profile (number of accesses to the disks), but the
same mean service time to each disk (but different
from disk to disk).
Facultade de Informatica. A Coruña. Junio 2005
77
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Case studies
Transactional system

We assume that these transactions are executed
concurrently on the computer system and that the
conflicts in the its execution are due to the access to
the same servers (CPU and disks) but not to any
kind of synchronisation or use of critical objects.
Facultade de Informatica. A Coruña. Junio 2005
78
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Case studies
Transactional system
Disk 1
Disk 2
CPU
Arrivals
Disk 3
Disk 4
Exits
Facultade de Informatica. A Coruña. Junio 2005
79
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Transactional system
1 /DECLARE/ QUEUE CPU,DISC(4),ENTRADA1,ENTRADA2;
2
REAL PROF1(4)=(2,1.5,1,0.5);
3
REAL PROF2(4)=(1.5,2,3,3.5);
4
REAL TR1,TR2;
5
CLASS C1,C2;
6
INTEGER I;
7 /STATION/ NAME=CPU;
8
SCHED=PS;
9
SERVICE(C1)=CST(8.52);
10
SERVICE(C2)=CST(12.);
11
TRANSIT(C1)=DISC,PROF1,OUT,1;
12
TRANSIT(C2)=DISC,PROF2,OUT,1;
Facultade de Informatica. A Coruña. Junio 2005
80
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Transactional system
13
14
15
16
17
18
19
20
21
22
/STATION/ NAME=DISC;
TRANSIT=CPU;
/STATION/ NAME=DISC(1);
SERVICE=EXP(23.);
/STATION/ NAME=DISC(2);
SERVICE=EXP(22.);
/STATION/ NAME=DISC(3);
SERVICE=EXP(21.);
/STATION/ NAME=DISC(4);
SERVICE=EXP(20.);
Facultade de Informatica. A Coruña. Junio 2005
81
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Transactional system
23 /STATION/ NAME=ENTRADA1;
24
TYPE=SOURCE;
25
SERVICE=EXP(1000./7.);
26
TRANSIT=CPU,C1;
27 /STATION/ NAME=ENTRADA2;
28
TYPE=SOURCE;
29
SERVICE=EXP(1000./3.);
30
TRANSIT=CPU,C2;
31 /CONTROL/ CLASS=ALL QUEUE;
Facultade de Informatica. A Coruña. Junio 2005
82
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Transactional system
32 /EXEC/
33
34
35
36
37
38
39
40
41
42
43
44
45
46
BEGIN
PRINT;
SOLVE;
TR1:=MCUSTNB(CPU,C1);
TR2:=MCUSTNB(CPU,C2);
FOR I:= 1 STEP 1 UNTIL 4 DO
BEGIN
TR1:=TR1+MCUSTNB(DISC(I),C1);
TR2:=TR2+MCUSTNB(DISC(I),C2);
END;
TR1:=TR1/MTHRUPUT(ENTRADA1);
TR2:=TR2/MTHRUPUT(ENTRADA2);
PRINT("RESPONSE TIME OF CLASS C1 =",TR1);
PRINT("RESPONSE TIME OF CLASS C2 =",TR2);
END;
Facultade de Informatica. A Coruña. Junio 2005
83
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Transactional system
- CONVOLUTION METHOD ("CONVOL")
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
*
*
*
*
*
*
*
* CPU
* 10.05
*0.7538
* 3.062
* 40.83
*0.7500E-01*
*(C1
)* 8.520
*0.3578
* 1.454
* 34.61
*0.4200E-01*
*(C2
)* 12.00
*0.3960
* 1.609
* 48.75
*0.3300E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.4255
*0.7406
* 40.03
*0.1850E-01*
*(C1
)* 23.00
*0.3220
*0.5605
* 40.03
*0.1400E-01*
*(C2
)* 23.00
*0.1035
*0.1802
* 40.03
*0.4500E-02*
*
*
*
*
*
*
*
* DISC
2 * 22.00
*0.3630
*0.5699
* 34.54
*0.1650E-01*
*(C1
)* 22.00
*0.2310
*0.3626
* 34.54
*0.1050E-01*
*(C2
)* 22.00
*0.1320
*0.2072
* 34.54
*0.6000E-02*
*
*
*
*
*
*
*
Facultade de Informatica. A Coruña. Junio 2005
84
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Transactional system
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* DISC
3 * 21.00
*0.3360
*0.5060
* 31.63
*0.1600E-01*
*(C1
)* 21.00
*0.1470
*0.2214
* 31.63
*0.7000E-02*
*(C2
)* 21.00
*0.1890
*0.2846
* 31.63
*0.9000E-02*
*
*
*
*
*
*
*
* DISC
4 * 20.00
*0.2800
*0.3889
* 27.78
*0.1400E-01*
*(C1
)* 20.00
*0.7000E-01*0.9722E-01* 27.78
*0.3500E-02*
*(C2
)* 20.00
*0.2100
*0.2917
* 27.78
*0.1050E-01*
*
*
*
*
*
*
*
* ENTRADA1 * 142.9
* 1.000
* 1.000
* 142.9
*0.7000E-02*
*
*
*
*
*
*
*
* ENTRADA2 * 333.3
* 1.000
* 1.000
* 333.3
*0.3000E-02*
*******************************************************************
RESPONSE TIME OF CLASS C1 =
385.1
RESPONSE TIME OF CLASS C2 =
857.5
47 /END/
Facultade de Informatica. A Coruña. Junio 2005
85
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Case studies
Conversational system

We assume that these programs are executed
concurrently on the computer system and that the
conflicts in the its execution are due to the access to
the same servers (CPU and disks) but not to any
kind of synchronisation or use of critical objects.
Also we assume that the human behaviour in front
of the terminal is different for each class.
Facultade de Informatica. A Coruña. Junio 2005
86
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Case studies
Conversational system
Disk 1
Terminals
Disk 2
Arrivals
CPU
Disk 3
Disk 4
Exits
Facultade de Informatica. A Coruña. Junio 2005
87
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
1 /DECLARE/ QUEUE CPU,DISC(4),TERMINAL;
2
REAL PROB1(4)=(2,1.5,1,0.5);
3
REAL PROB2(4)=(1.5,2,3,3.5);
4
REAL TR1,TR2;
5
CLASS C1,C2;
6
INTEGER I,N;
7 /STATION/ NAME=CPU;
8
SCHED=PS;
9
SERVICE(C1)=CST(8.52);
10
SERVICE(C2)=CST(12.);
11
TRANSIT(C1)=DISC,PROB1,TERMINAL,C1,0.6,TERMINAL,C2,0
==> .4;
12
TRANSIT(C2)=DISC,PROB2,TERMINAL,C1,0.6,TERMINAL,C2,0
==> .4;
Facultade de Informatica. A Coruña. Junio 2005
88
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
/STATION/ NAME=DISC;
TRANSIT=CPU;
/STATION/ NAME=DISC(1);
SERVICE=EXP(23.);
/STATION/ NAME=DISC(2);
SERVICE=EXP(22.);
/STATION/ NAME=DISC(3);
SERVICE=EXP(21.);
/STATION/ NAME=DISC(4);
SERVICE=EXP(20.);
/STATION/ NAME=TERMINAL;
TYPE=INFINITE;
INIT(C1)=N;
SERVICE(C1)=EXP(30000.);
SERVICE(C2)=EXP(60000.);
TRANSIT=CPU;
/CONTROL/ CLASS=ALL QUEUE;
Facultade de Informatica. A Coruña. Junio 2005
89
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
30 /EXEC/
FOR N:=150 STEP 150 UNTIL 750 DO
31
BEGIN
32
PRINT;
33
PRINT("NOMBRE D’USUARIS =",N);
34
SOLVE;
35
TR1:=MCUSTNB(CPU,C1);
36
TR2:=MCUSTNB(CPU,C2);
37
FOR I:= 1 STEP 1 UNTIL 4 DO
38
BEGIN
39
TR1:=TR1+MCUSTNB(DISC(I),C1);
40
TR2:=TR2+MCUSTNB(DISC(I),C2);
41
END;
42
TR1:=TR1/MTHRUPUT(TERMINAL,C1);
43
TR2:=TR2/MTHRUPUT(TERMINAL,C2);;
44
PRINT("RESPONSE TIME OF CLASS C1 =",TR1);
45
PRINT("RESPONSE TIME OF CLASS C2 =",TR2);
46
END;
Facultade de Informatica. A Coruña. Junio 2005
90
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
NOMBRE D’USUARIS =
150
- MEAN VALUE ANALYSIS ("MVA") *******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
*
*
*
*
*
*
*
* CPU
* 10.43
*0.2960
*0.4189
* 14.76
*0.2837E-01*
*(C1
)* 8.520
*0.1088
*0.1539
* 12.06
*0.1277E-01*
*(C2
)* 12.00
*0.1873
*0.2650
* 16.98
*0.1560E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.1468
*0.1719
* 26.92
*0.6384E-02*
*(C1
)* 23.00
*0.9790E-01*0.1146
* 26.92
*0.4257E-02*
*(C2
)* 23.00
*0.4894E-01*0.5729E-01* 26.92
*0.2128E-02*
*
*
*
*
*
*
*
* DISC
2 * 22.00
*0.1326
*0.1528
* 25.33
*0.6029E-02*
*(C1
)* 22.00
*0.7023E-01*0.8088E-01* 25.33
*0.3192E-02*
*(C2
)* 22.00
*0.6242E-01*0.7188E-01* 25.33
*0.2837E-02*
Facultade de Informatica. A Coruña. Junio 2005
91
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* DISC
3 * 21.00
*0.1341
*0.1546
* 24.22
*0.6384E-02*
*(C1
)* 21.00
*0.4469E-01*0.5155E-01* 24.22
*0.2128E-02*
*(C2
)* 21.00
*0.8937E-01*0.1031
* 24.22
*0.4256E-02*
*
*
*
*
*
*
*
* DISC
4 * 20.00
*0.1206
*0.1370
* 22.72
*0.6029E-02*
*(C1
)* 20.00
*0.2128E-01*0.2418E-01* 22.72
*0.1064E-02*
*(C2
)* 20.00
*0.9930E-01*0.1128
* 22.72
*0.4965E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 149.0
*0.4200E+05*0.3547E-02*
*(C1
)*0.3000E+05*0.0000E+00* 63.84
*0.3000E+05*0.2128E-02*
*(C2
)*0.6000E+05*0.0000E+00* 85.12
*0.6000E+05*0.1419E-02*
*******************************************************************
RESPONSE TIME OF CLASS C1 =
199.8
RESPONSE TIME OF CLASS C2 =
430.0
Facultade de Informatica. A Coruña. Junio 2005
92
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
NOMBRE D’USUARIS =
300
- MEAN VALUE ANALYSIS ("MVA") *******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
*
*
*
*
*
*
*
* CPU
* 10.43
*0.5905
* 1.426
* 25.19
*0.5659E-01*
*(C1
)* 8.520
*0.2170
*0.5239
* 20.57
*0.2547E-01*
*(C2
)* 12.00
*0.3735
*0.9017
* 28.97
*0.3112E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.2929
*0.4134
* 32.46
*0.1273E-01*
*(C1
)* 23.00
*0.1953
*0.2756
* 32.46
*0.8490E-02*
*(C2
)* 23.00
*0.9761E-01*0.1378
* 32.46
*0.4244E-02*
*
*
*
*
*
*
*
* DISC
2 * 22.00
*0.2646
*0.3592
* 29.87
*0.1203E-01*
*(C1
)* 22.00
*0.1401
*0.1902
* 29.87
*0.6367E-02*
*(C2
)* 22.00
*0.1245
*0.1690
* 29.87
*0.5659E-02*
Facultade de Informatica. A Coruña. Junio 2005
93
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* DISC
3 * 21.00
*0.2674
*0.3644
* 28.62
*0.1273E-01*
*(C1
)* 21.00
*0.8914E-01*0.1215
* 28.62
*0.4245E-02*
*(C2
)* 21.00
*0.1783
*0.2429
* 28.62
*0.8488E-02*
*
*
*
*
*
*
*
* DISC
4 * 20.00
*0.2405
*0.3162
* 26.30
*0.1203E-01*
*(C1
)* 20.00
*0.4245E-01*0.5582E-01* 26.30
*0.2122E-02*
*(C2
)* 20.00
*0.1981
*0.2604
* 26.30
*0.9903E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 297.1
*0.4200E+05*0.7074E-02*
*(C1
)*0.3000E+05*0.0000E+00* 127.3
*0.3000E+05*0.4245E-02*
*(C2
)*0.6000E+05*0.0000E+00* 169.8
*0.6000E+05*0.2830E-02*
*******************************************************************
RESPONSE TIME OF CLASS C1 =
274.9
RESPONSE TIME OF CLASS C2 =
605.0
Facultade de Informatica. A Coruña. Junio 2005
94
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
NOMBRE D’USUARIS =
450
- MEAN VALUE ANALYSIS ("MVA") *******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
*
*
*
*
*
*
*
* CPU
* 10.43
*0.8763
* 6.438
* 76.66
*0.8399E-01*
*(C1
)* 8.520
*0.3220
* 2.366
* 62.60
*0.3780E-01*
*(C2
)* 12.00
*0.5543
* 4.072
* 88.16
*0.4619E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.4347
*0.7667
* 40.57
*0.1890E-01*
*(C1
)* 23.00
*0.2898
*0.5112
* 40.57
*0.1260E-01*
*(C2
)* 23.00
*0.1449
*0.2555
* 40.57
*0.6298E-02*
*
*
*
*
*
*
*
* DISC
2 * 22.00
*0.3926
*0.6451
* 36.14
*0.1785E-01*
*(C1
)* 22.00
*0.2079
*0.3415
* 36.14
*0.9449E-02*
*(C2
)* 22.00
*0.1848
*0.3035
Facultade de Informatica. A Coruña. Junio 2005
* 36.14
*0.8398E-02*
95
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* DISC
3 * 21.00
*0.3968
*0.6564
* 34.74
*0.1890E-01*
*(C1
)* 21.00
*0.1323
*0.2188
* 34.74
*0.6300E-02*
*(C2
)* 21.00
*0.2645
*0.4376
* 34.74
*0.1260E-01*
*
*
*
*
*
*
*
* DISC
4 * 20.00
*0.3569
*0.5541
* 31.05
*0.1785E-01*
*(C1
)* 20.00
*0.6300E-01*0.9779E-01* 31.05
*0.3150E-02*
*(C2
)* 20.00
*0.2939
*0.4563
* 31.05
*0.1470E-01*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 440.9
*0.4200E+05*0.1050E-01*
*(C1
)*0.3000E+05*0.0000E+00* 189.0
*0.3000E+05*0.6299E-02*
*(C2
)*0.6000E+05*0.0000E+00* 252.0
*0.6000E+05*0.4199E-02*
*******************************************************************
RESPONSE TIME OF CLASS C1 =
561.2
RESPONSE TIME OF CLASS C2 =
1316.
Facultade de Informatica. A Coruña. Junio 2005
96
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
NOMBRE D’USUARIS =
600
- MEAN VALUE ANALYSIS ("MVA") *******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
*
*
*
*
*
*
*
* CPU
* 10.43
* 1.000
* 93.51
* 975.7
*0.9584E-01*
*(C1
)* 8.520
*0.3675
* 34.37
* 796.7
*0.4313E-01*
*(C2
)* 12.00
*0.6325
* 59.15
* 1122.
*0.5271E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.4960
*0.9841
* 45.63
*0.2157E-01*
*(C1
)* 23.00
*0.3307
*0.6561
* 45.63
*0.1438E-01*
*(C2
)* 23.00
*0.1653
*0.3280
* 45.63
*0.7187E-02*
*
*
*
*
*
*
*
* DISC
2 * 22.00
*0.4481
*0.8118
* 39.86
*0.2037E-01*
*(C1
)* 22.00
*0.2372
*0.4298
* 39.86
*0.1078E-01*
*(C2
)* 22.00
*0.2108
*0.3820
Facultade de Informatica. A Coruña. Junio 2005
* 39.86
*0.9583E-02*
97
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* DISC
3 * 21.00
*0.4528
*0.8276
* 38.38
*0.2156E-01*
*(C1
)* 21.00
*0.1510
*0.2759
* 38.38
*0.7189E-02*
*(C2
)* 21.00
*0.3019
*0.5517
* 38.38
*0.1437E-01*
*
*
*
*
*
*
*
* DISC
4 * 20.00
*0.4073
*0.6872
* 33.74
*0.2037E-01*
*(C1
)* 20.00
*0.7189E-01*0.1213
* 33.74
*0.3594E-02*
*(C2
)* 20.00
*0.3354
*0.5659
* 33.74
*0.1677E-01*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 503.2
*0.4200E+05*0.1198E-01*
*(C1
)*0.3000E+05*0.0000E+00* 215.7
*0.3000E+05*0.7188E-02*
*(C2
)*0.6000E+05*0.0000E+00* 287.5
*0.6000E+05*0.4792E-02*
*******************************************************************
RESPONSE TIME OF CLASS C1 =
4987.
RESPONSE TIME OF CLASS C2 = 0.1272E+05
Facultade de Informatica. A Coruña. Junio 2005
98
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
NOMBRE D’USUARIS =
750
- MEAN VALUE ANALYSIS ("MVA") *******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
*
*
*
*
*
*
*
* CPU
* 10.43
* 1.000
* 243.5
* 2541.
*0.9584E-01*
*(C1
)* 8.520
*0.3675
* 89.49
* 2075.
*0.4313E-01*
*(C2
)* 12.00
*0.6325
* 154.0
* 2922.
*0.5271E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.4960
*0.9841
* 45.64
*0.2157E-01*
*(C1
)* 23.00
*0.3307
*0.6561
* 45.64
*0.1438E-01*
*(C2
)* 23.00
*0.1653
*0.3280
* 45.64
*0.7187E-02*
*
*
*
*
*
*
*
* DISC
2 * 22.00
*0.4481
*0.8118
* 39.86
*0.2037E-01*
*(C1
)* 22.00
*0.2372
*0.4298
* 39.86
*0.1078E-01*
*(C2
)* 22.00
*0.2108
*0.3820
Facultade de Informatica. A Coruña. Junio 2005
* 39.86
*0.9583E-02*
99
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Conversational system
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* DISC
3 * 21.00
*0.4528
*0.8276
* 38.38
*0.2156E-01*
*(C1
)* 21.00
*0.1510
*0.2759
* 38.38
*0.7189E-02*
*(C2
)* 21.00
*0.3019
*0.5517
* 38.38
*0.1437E-01*
*
*
*
*
*
*
*
* DISC
4 * 20.00
*0.4073
*0.6872
* 33.74
*0.2037E-01*
*(C1
)* 20.00
*0.7189E-01*0.1213
* 33.74
*0.3594E-02*
*(C2
)* 20.00
*0.3354
*0.5659
* 33.74
*0.1677E-01*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 503.2
*0.4200E+05*0.1198E-01*
*(C1
)*0.3000E+05*0.0000E+00* 215.7
*0.3000E+05*0.7188E-02*
*(C2
)*0.6000E+05*0.0000E+00* 287.5
*0.6000E+05*0.4792E-02*
*******************************************************************
RESPONSE TIME OF CLASS C1 = 0.1266E+05
RESPONSE TIME OF CLASS C2 = 0.3252E+05
47 /END/
Facultade de Informatica. A Coruña. Junio 2005
100
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Case studies
Communication network

There are four sources of variable length messages.
Routing across the net is fixed. All the lines have
the same capacity and the transmission is full
duplex. We consider negligible the time spent in
each node for protocol verifications, routing, etc.
The end-to-end traffic is known and we want to
determine the response time also end-to-end.
Facultade de Informatica. A Coruña. Junio 2005
101
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
7
D
6
8
A
5
1
C
4
2
B
Facultade de Informatica. A Coruña. Junio 2005
3
102
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
to
from
A
B
C
D
A
75
80
100
B
60
120
150
Facultade de Informatica. A Coruña. Junio 2005
C
80
50
D
100
25
40
50
103
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
A-1-2-4-3-B
A-1-2-4-6-5-C
A-1-2-8-7-D
B-3-4-2-1-A
B-3-4-6-5-C
B-3-4-8-7-D
C-5-6-8-2-1-A
C-5-6-4-3-B
C-5-6-8-7-D
D-7-8-2-1-A
D-7-8-4-3-B
D-7-8-6-5-C
Facultade de Informatica. A Coruña. Junio 2005
104
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
1 /DECLARE/ QUEUE GEN,LINIA(8,8);
2
REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD,
==> TRDA,TRDB,TRDC;
3
INTEGER I;
4
CLASS CLAB,CLAC,CLAD,CLBA,CLBC,CLBD,CLCA,CLCB,CLCD
==> ,CLDA,CLDB,CLDC;
5
Facultade de Informatica. A Coruña. Junio 2005
105
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
6 /STATION/ NAME = GEN;
7
TYPE = SOURCE;
8
SERVICE = EXP(60000./930.);
9
TRANSIT = LINIA(1,2),CLAB,60,LINIA(1,2),CLAC,80,L
==> INIA(1,2),CLAD,100,
10
LINIA(3,4),CLBA,75,LINIA(3,4),CLBC,50,L
==> INIA(3,4),CLBD,25,
11
LINIA(5,6),CLCA,80,LINIA(5,6),CLCB,120,
==> LINIA(5,6),CLCD,40,
12
LINIA(7,8),CLDA,100,LINIA(7,8),CLDB,150
==> ,LINIA(7,8),CLDC,50;
13
Facultade de Informatica. A Coruña. Junio 2005
106
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
/STATION/ NAME = LINIA;
SERVICE = EXP(256.*8./64.);
/STATION/ NAME = LINIA(1,2);
TRANSIT(CLAB) = LINIA(2,4);
TRANSIT(CLAC) = LINIA(2,4);
TRANSIT(CLAD) = LINIA(2,8);
/STATION/ NAME = LINIA(2,1);
TRANSIT = OUT;
/STATION/ NAME = LINIA(2,4);
TRANSIT(CLAB) = LINIA(4,3);
TRANSIT(CLAC) = LINIA(4,6);
/STATION/ NAME = LINIA(2,8);
TRANSIT(CLAD) = LINIA(8,7);
Facultade de Informatica. A Coruña. Junio 2005
107
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
/STATION/ NAME = LINIA(3,4);
TRANSIT(CLBA) = LINIA(4,2);
TRANSIT(CLBC) = LINIA(4,6);
TRANSIT(CLBD) = LINIA(4,8);
/STATION/ NAME = LINIA(4,2);
TRANSIT(CLBA) = LINIA(2,1);
/STATION/ NAME = LINIA(4,3);
TRANSIT = OUT;
/STATION/ NAME = LINIA(4,6);
TRANSIT(CLAC) = LINIA(6,5);
TRANSIT(CLBC) = LINIA(6,5);
/STATION/ NAME = LINIA(4,8);
TRANSIT(CLBD) = LINIA(8,7);
Facultade de Informatica. A Coruña. Junio 2005
108
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
50 /STATION/ NAME = LINIA(5,6);
51
TRANSIT(CLCA) = LINIA(6,8);
52
TRANSIT(CLCB) = LINIA(6,4);
53
TRANSIT(CLCD) = LINIA(6,8);
54
55 /STATION/ NAME = LINIA(6,4);
56
TRANSIT(CLCB) = LINIA(4,3);
57
58 /STATION/ NAME = LINIA(6,5);
59
TRANSIT = OUT;
60
61 /STATION/ NAME = LINIA(6,8);
62
TRANSIT(CLCA) = LINIA(8,2);
63
TRANSIT(CLCD) = LINIA(8,7);
64
65 /STATION/ NAME = LINIA(7,8);
66
TRANSIT(CLDA) = LINIA(8,2);
67
TRANSIT(CLDB) = LINIA(8,4);
68
TRANSIT(CLDC) = LINIA(8,6);
69
Facultade de Informatica. A Coruña. Junio 2005
109
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
70
71
72
73
74
75
76
77
78
79
80
81
82
/STATION/ NAME = LINIA(8,2);
TRANSIT(CLCA) = LINIA(2,1);
TRANSIT(CLDA) = LINIA(2,1);
/STATION/ NAME = LINIA(8,4);
TRANSIT(CLDB) = LINIA(4,3);
/STATION/ NAME = LINIA(8,6);
TRANSIT(CLDC) = LINIA(6,5);
/STATION/ NAME = LINIA(8,7);
TRANSIT = OUT;
Facultade de Informatica. A Coruña. Junio 2005
110
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
83 /EXEC/
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
BEGIN
NETWORK(GEN,
LINIA(1,2),LINIA(2,1),LINIA(2,4),LINIA(2,8),
LINIA(3,4),LINIA(4,2),LINIA(4,3),LINIA(4,6),
LINIA(4,8),LINIA(5,6),LINIA(6,4),LINIA(6,5),
LINIA(6,8),LINIA(7,8),LINIA(8,2),LINIA(8,4),
LINIA(8,6),LINIA(8,7));
PRINT;
SOLVE;
TRAB:=MRESPONSE(LINIA(1,2))+
MRESPONSE(LINIA(2,4))+MRESPONSE(LINIA(4,3));
TRAC:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+
MRESPONSE(LINIA(4,6))+MRESPONSE(LINIA(6,5));
TRAD:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,8))+
MRESPONSE(LINIA(8,7));
TRBA:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,2))+
MRESPONSE(LINIA(2,1));
Facultade de Informatica. A Coruña. Junio 2005
111
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
TRBC:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,6))+
MRESPONSE(LINIA(6,5));
TRBD:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,8))+
MRESPONSE(LINIA(8,7));
TRCA:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+
MRESPONSE(LINIA(8,2))+MRESPONSE(LINIA(2,1));
TRCB:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,4))+
MRESPONSE(LINIA(4,3));
TRCD:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+
MRESPONSE(LINIA(8,7));
TRDA:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,2))+
MRESPONSE(LINIA(2,1));
TRDB:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,4))+
MRESPONSE(LINIA(4,3));
TRDC:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,6))+
MRESPONSE(LINIA(6,5));
Facultade de Informatica. A Coruña. Junio 2005
112
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
116
117
118
119
120
PRINT(TRAB,TRAC,TRAD);
PRINT(TRBA,TRBC,TRBD);
PRINT(TRCA,TRCB,TRCD);
PRINT(TRDA,TRDB,TRDC);
END;
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
- CONVOLUTION METHOD ("CONVOL") *******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
*
*
*
*
*
*
*
* GEN
* 64.52
* 1.000
* 1.000
* 64.52
*0.1550E-01*
*
*
*
*
*
*
*
* LINIA 2 * 32.00
*0.1280
*0.1468
* 36.70
*0.4000E-02*
*
*
*
*
*
*
*
* LINIA 9 * 32.00
*0.1360
*0.1574
* 37.04
*0.4250E-02*
*
*
*
*
*
*
*
* LINIA 12 * 32.00
*0.7467E-01*0.8069E-01* 34.58
*0.2333E-02*
*
*
*
*
*
*
*
* LINIA 16 * 32.00
*0.5333E-01*0.5634E-01* 33.80
*0.1667E-02*
*
*
*
*
*
*
*
* LINIA 20 * 32.00
*0.8000E-01*0.8696E-01* 34.78
*0.2500E-02*
*
*
*
*
*
*
*
* LINIA 26 * 32.00
*0.4000E-01*0.4167E-01* 33.33
*0.1250E-02*
*
*
*
*
*
*
*
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114
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* LINIA 27 * 32.00
*0.1760
*0.2136
* 38.83
*0.5500E-02*
*
*
*
*
*
*
*
* LINIA 30 * 32.00
*0.6933E-01*0.7450E-01* 34.38
*0.2167E-02*
*
*
*
*
*
*
*
* LINIA 32 * 32.00
*0.1333E-01*0.1351E-01* 32.43
*0.4167E-03*
*
*
*
*
*
*
*
* LINIA 38 * 32.00
*0.1280
*0.1468
* 36.70
*0.4000E-02*
*
*
*
*
*
*
*
* LINIA 44 * 32.00
*0.6400E-01*0.6838E-01* 34.19
*0.2000E-02*
*
*
*
*
*
*
*
* LINIA 45 * 32.00
*0.9600E-01*0.1062
* 35.40
*0.3000E-02*
*
*
*
*
*
*
*
* LINIA 48 * 32.00
*0.6400E-01*0.6838E-01* 34.19
*0.2000E-02*
*
*
*
*
*
*
*
* LINIA 56 * 32.00
*0.1600
*0.1905
* 38.10
*0.5000E-02*
*
*
*
*
*
*
*
* LINIA 58 * 32.00
*0.9600E-01*0.1062
* 35.40
*0.3000E-02*
*
*
*
*
*
*
*
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115
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* LINIA 60 * 32.00
*0.8000E-01*0.8696E-01* 34.78
*0.2500E-02*
*
*
*
*
*
*
*
* LINIA 62 * 32.00
*0.2667E-01*0.2740E-01* 32.88
*0.8333E-03*
*
*
*
*
*
*
*
* LINIA 63 * 32.00
*0.8800E-01*0.9649E-01* 35.09
*0.2750E-02*
*
*
*
*
*
*
*
*******************************************************************
110.1
105.2
143.3
110.5
141.1
104.6
109.7
111.7
105.6
102.3
106.0
106.4
Facultade de Informatica. A Coruña. Junio 2005
116
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Case studies
Communication network considering the nodes

This case is identical to the previous one but
considering the process at each node.
Facultade de Informatica. A Coruña. Junio 2005
117
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
1 /DECLARE/ QUEUE GEN,LINIA(8,8),CPU(8);
2
REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD,TRD
==> A,TRDB,TRDC;
3
INTEGER I;
4
CLASS CLAB,CLAC,CLAD,CLBA,CLBC,CLBD,CLCA,CLCB,CLCD,CL
==> DA,CLDB,CLDC;
5
6 /STATION/ NAME = GEN;
7
TYPE = SOURCE;
8
SERVICE = EXP(60000./930.);
9
TRANSIT = CPU(1),CLAB,60,CPU(1),CLAC,80,CPU(1),CLAD,1
==> 00,
10
CPU(3),CLBA,75,CPU(3),CLBC,50,CPU(3),CLBD,2
==> 5,
11
CPU(5),CLCA,80,CPU(5),CLCB,120,CPU(5),CLCD,
==> 40,
12
CPU(7),CLDA,100,CPU(7),CLDB,150,CPU(7),CLDC
==> ,50,
13
Facultade de Informatica. A Coruña. Junio 2005
118
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
/STATION/ NAME = LINIA;
SERVICE = EXP(256.*8./64.);
/STATION/ NAME = LINIA(1,2);
TRANSIT = CPU(2);
/STATION/ NAME = LINIA(2,1);
TRANSIT = CPU(1);
/STATION/ NAME = LINIA(2,4);
TRANSIT = CPU(4);
/STATION/ NAME = LINIA(2,8);
TRANSIT = CPU(8);
/STATION/ NAME = LINIA(3,4);
TRANSIT = CPU(4);
Facultade de Informatica. A Coruña. Junio 2005
119
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
/STATION/ NAME = LINIA(4,2);
TRANSIT = CPU(2);
/STATION/ NAME = LINIA(4,3);
TRANSIT = CPU(3);
/STATION/ NAME = LINIA(4,6);
TRANSIT = CPU(6);
/STATION/ NAME = LINIA(4,8);
TRANSIT = CPU(8);
/STATION/ NAME = LINIA(5,6);
TRANSIT = CPU(6);
/STATION/ NAME = LINIA(6,5);
TRANSIT = CPU(5);
Facultade de Informatica. A Coruña. Junio 2005
120
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
/STATION/ NAME = LINIA(6,4);
TRANSIT = CPU(4);
/STATION/ NAME = LINIA(6,8);
TRANSIT = CPU(8);
/STATION/ NAME = LINIA(7,8);
TRANSIT = CPU(8);
/STATION/ NAME = LINIA(8,2);
TRANSIT = CPU(2);
/STATION/ NAME = LINIA(8,7);
TRANSIT = CPU(7);
/STATION/ NAME = LINIA(8,6);
TRANSIT = CPU(6);
Facultade de Informatica. A Coruña. Junio 2005
121
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
/STATION/ NAME = LINIA(8,4);
TRANSIT = CPU(4);
/STATION/ NAME = CPU;
SERVICE = EXP(1.);
/STATION/ NAME = CPU(1);
TRANSIT(CLAB,CLAC,CLAD) = LINIA(1,2);
TRANSIT(CLBA,CLCA,CLDA) = OUT;
/STATION/ NAME = CPU(2);
TRANSIT(CLAB) =
TRANSIT(CLAC) =
TRANSIT(CLAD) =
TRANSIT(CLBA) =
TRANSIT(CLCA) =
TRANSIT(CLDA) =
LINIA(2,4);
LINIA(2,4);
LINIA(2,8);
LINIA(2,1);
LINIA(2,1);
LINIA(2,1);
Facultade de Informatica. A Coruña. Junio 2005
122
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
86 /STATION/ NAME = CPU(3);
87
TRANSIT(CLBA,CLBC,CLBD) = LINIA(3,4);
88
TRANSIT(CLAB,CLCB,CLDB) = OUT;
89
90 /STATION/ NAME = CPU(4);
91
TRANSIT(CLAB) = LINIA(4,3);
92
TRANSIT(CLAC) = LINIA(4,6);
93
TRANSIT(CLBA) = LINIA(4,2);
94
TRANSIT(CLBC) = LINIA(4,6);
95
TRANSIT(CLBD) = LINIA(4,8);
96
TRANSIT(CLCB) = LINIA(4,3);
97
TRANSIT(CLDB) = LINIA(4,3);
98
99 /STATION/ NAME = CPU(5);
100
TRANSIT(CLCA,CLCB,CLCD) = LINIA(5,6);
101
TRANSIT(CLAC,CLBC,CLDC) = OUT;
102
Facultade de Informatica. A Coruña. Junio 2005
123
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
103 /STATION/ NAME = CPU(6);
104
TRANSIT(CLAC) = LINIA(6,5);
105
TRANSIT(CLBC) = LINIA(6,5);
106
TRANSIT(CLCA) = LINIA(6,8);
107
TRANSIT(CLCB) = LINIA(6,4);
108
TRANSIT(CLCD) = LINIA(6,8);
109
TRANSIT(CLDC) = LINIA(6,5);
110
111 /STATION/ NAME = CPU(7);
112
TRANSIT(CLDA,CLDB,CLDC) = LINIA(7,8);
113
TRANSIT(CLAD,CLBD,CLCD) = OUT;
114
115 /STATION/ NAME = CPU(8);
116
TRANSIT(CLAD) = LINIA(8,7);
117
TRANSIT(CLBD) = LINIA(8,7);
118
TRANSIT(CLCA) = LINIA(8,2);
119
TRANSIT(CLCD) = LINIA(8,7);
120
TRANSIT(CLDA) = LINIA(8,2);
121
TRANSIT(CLDB) = LINIA(8,4);
122
TRANSIT(CLDC) = LINIA(8,6);
123
Facultade de Informatica. A Coruña. Junio 2005
124
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
124 /CONTROL/ CLASS = ALL QUEUE;
125 /EXEC/
BEGIN
126
NETWORK(GEN,
127
CPU(1 STEP 1 UNTIL 8),
128
LINIA(1,2),LINIA(2,1),LINIA(2,4),LINIA(2,8),
129
LINIA(3,4),LINIA(4,2),LINIA(4,3),LINIA(4,6),
130
LINIA(4,8),LINIA(5,6),LINIA(6,4),LINIA(6,5),
131
LINIA(6,8),LINIA(7,8),LINIA(8,2),LINIA(8,4),
132
LINIA(8,6),LINIA(8,7));
133
PRINT;
134
SOLVE;
135
TRAB := MRESPONSE(CPU(1))+MRESPONSE(LINIA(1,2))+
136
MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,4))+
137
MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,3))+
138
MRESPONSE(CPU(3));
139
TRAC := MRESPONSE(CPU(1))+MRESPONSE(LINIA(1,2))+
140
MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,4))+
141
MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,6))+
142
MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,5))+
143
MRESPONSE(CPU(5));
Facultade de Informatica. A Coruña. Junio 2005
125
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
TRAD := MRESPONSE(CPU(1))+MRESPONSE(LINIA(1,2))+
MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,8))+
MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,7))+
MRESPONSE(CPU(7));
TRBA := MRESPONSE(CPU(3))+MRESPONSE(LINIA(3,4))+
MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,2))+
MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,1))+
MRESPONSE(CPU(1));
TRBC := MRESPONSE(CPU(3))+MRESPONSE(LINIA(3,4))+
MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,6))+
MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,5))+
MRESPONSE(CPU(5));
TRBD := MRESPONSE(CPU(3))+MRESPONSE(LINIA(3,4))+
MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,8))+
MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,7))+
MRESPONSE(CPU(7));
TRCA := MRESPONSE(CPU(5))+MRESPONSE(LINIA(5,6))+
MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,8))+
MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,2))+
MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,1))+
MRESPONSE(CPU(1));
Facultade de Informatica. A Coruña. Junio 2005
126
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
TRCB := MRESPONSE(CPU(5))+MRESPONSE(LINIA(5,6))+
MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,4))+
MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,3))+
MRESPONSE(CPU(3));
TRCD := MRESPONSE(CPU(5))+MRESPONSE(LINIA(5,6))+
MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,8))+
MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,7))+
MRESPONSE(CPU(7));
TRDA := MRESPONSE(CPU(7))+MRESPONSE(LINIA(7,8))+
MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,2))+
MRESPONSE(CPU(2))+MRESPONSE(LINIA(2,1))+
MRESPONSE(CPU(1));
TRDB := MRESPONSE(CPU(7))+MRESPONSE(LINIA(7,8))+
MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,4))+
MRESPONSE(CPU(4))+MRESPONSE(LINIA(4,3))+
MRESPONSE(CPU(3));
Facultade de Informatica. A Coruña. Junio 2005
127
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
181
MRESPONSE(CPU(7))+MRESPONSE(LINIA(7,8))+
182
MRESPONSE(CPU(8))+MRESPONSE(LINIA(8,6))+
183
MRESPONSE(CPU(6))+MRESPONSE(LINIA(6,5))+
184
MRESPONSE(CPU(5));
185
PRINT(TRAB,TRAC,TRAD);
186
PRINT(TRBA,TRBC,TRBD);
187
PRINT(TRCA,TRCB,TRCD);
188
PRINT(TRDA,TRDB,TRDC);
189
END;
Facultade de Informatica. A Coruña. Junio 2005
TRDC
:=
128
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
- CONVOLUTION METHOD ("CONVOL") *******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* GEN
* 64.52
* 1.000
* 1.000
* 64.52
*0.1550E-01*
*
*
*
*
*
*
*
* LINIA 2 * 32.00
*0.1280
*0.1468
* 36.70
*0.4000E-02*
*(CLAB
)* 32.00
*0.3200E-01*0.3670E-01* 36.70
*0.1000E-02*
*(CLAC
)* 32.00
*0.4267E-01*0.4893E-01* 36.70
*0.1333E-02*
*(CLAD
)* 32.00
*0.5333E-01*0.6116E-01* 36.70
*0.1667E-02*
*
*
*
*
*
*
*
* LINIA 9 * 32.00
*0.1360
*0.1574
* 37.04
*0.4250E-02*
*(CLBA
)* 32.00
*0.4000E-01*0.4630E-01* 37.04
*0.1250E-02*
*(CLCA
)* 32.00
*0.4267E-01*0.4938E-01* 37.04
*0.1333E-02*
*(CLDA
)* 32.00
*0.5333E-01*0.6173E-01* 37.04
*0.1667E-02*
*
*
*
*
*
*
*
* LINIA 12 * 32.00
*0.7467E-01*0.8069E-01* 34.58
*0.2333E-02*
*(CLAB
)* 32.00
*0.3200E-01*0.3458E-01* 34.58
*0.1000E-02*
*(CLAC
)* 32.00
*0.4267E-01*0.4611E-01* 34.58
*0.1333E-02*
*
*
*
*
*
*
*
Facultade de Informatica. A Coruña. Junio 2005
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* LINIA 16 * 32.00
*0.5333E-01*0.5634E-01* 33.80
*0.1667E-02*
*(CLAD
)* 32.00
*0.5333E-01*0.5634E-01* 33.80
*0.1667E-02*
*
*
*
*
*
*
*
* LINIA 20 * 32.00
*0.8000E-01*0.8696E-01* 34.78
*0.2500E-02*
*(CLBA
)* 32.00
*0.4000E-01*0.4348E-01* 34.78
*0.1250E-02*
*(CLBC
)* 32.00
*0.2667E-01*0.2899E-01* 34.78
*0.8333E-03*
*(CLBD
)* 32.00
*0.1333E-01*0.1449E-01* 34.78
*0.4167E-03*
*
*
*
*
*
*
*
* LINIA 26 * 32.00
*0.4000E-01*0.4167E-01* 33.33
*0.1250E-02*
*(CLBA
)* 32.00
*0.4000E-01*0.4167E-01* 33.33
*0.1250E-02*
*
*
*
*
*
*
*
* LINIA 27 * 32.00
*0.1760
*0.2136
* 38.83
*0.5500E-02*
*(CLAB
)* 32.00
*0.3200E-01*0.3883E-01* 38.83
*0.1000E-02*
*(CLCB
)* 32.00
*0.6400E-01*0.7767E-01* 38.83
*0.2000E-02*
*(CLDB
)* 32.00
*0.8000E-01*0.9709E-01* 38.83
*0.2500E-02*
*
*
*
*
*
*
*
Facultade de Informatica. A Coruña. Junio 2005
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* LINIA 30 * 32.00
*0.6933E-01*0.7450E-01* 34.38
*0.2167E-02*
*(CLAC
)* 32.00
*0.4267E-01*0.4585E-01* 34.38
*0.1333E-02*
*(CLBC
)* 32.00
*0.2667E-01*0.2865E-01* 34.38
*0.8333E-03*
*
*
*
*
*
*
*
* LINIA 32 * 32.00
*0.1333E-01*0.1351E-01* 32.43
*0.4167E-03*
*(CLBD
)* 32.00
*0.1333E-01*0.1351E-01* 32.43
*0.4167E-03*
*
*
*
*
*
*
*
* LINIA 38 * 32.00
*0.1280
*0.1468
* 36.70
*0.4000E-02*
*(CLCA
)* 32.00
*0.4267E-01*0.4893E-01* 36.70
*0.1333E-02*
*(CLCB
)* 32.00
*0.6400E-01*0.7339E-01* 36.70
*0.2000E-02*
*(CLCD
)* 32.00
*0.2133E-01*0.2446E-01* 36.70
*0.6667E-03*
*
*
*
*
*
*
*
* LINIA 44 * 32.00
*0.6400E-01*0.6838E-01* 34.19
*0.2000E-02*
*(CLCB
)* 32.00
*0.6400E-01*0.6838E-01* 34.19
*0.2000E-02*
*
*
*
*
*
*
*
Facultade de Informatica. A Coruña. Junio 2005
131
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* LINIA 45 * 32.00
*0.9600E-01*0.1062
* 35.40
*0.3000E-02*
*(CLAC
)* 32.00
*0.4267E-01*0.4720E-01* 35.40
*0.1333E-02*
*(CLBC
)* 32.00
*0.2667E-01*0.2950E-01* 35.40
*0.8333E-03*
*(CLDC
)* 32.00
*0.2667E-01*0.2950E-01* 35.40
*0.8333E-03*
*
*
*
*
*
*
*
* LINIA 48 * 32.00
*0.6400E-01*0.6838E-01* 34.19
*0.2000E-02*
*(CLCA
)* 32.00
*0.4267E-01*0.4558E-01* 34.19
*0.1333E-02*
*(CLCD
)* 32.00
*0.2133E-01*0.2279E-01* 34.19
*0.6667E-03*
*
*
*
*
*
*
*
* LINIA 56 * 32.00
*0.1600
*0.1905
* 38.10
*0.5000E-02*
*(CLDA
)* 32.00
*0.5333E-01*0.6349E-01* 38.10
*0.1667E-02*
*(CLDB
)* 32.00
*0.8000E-01*0.9524E-01* 38.10
*0.2500E-02*
*(CLDC
)* 32.00
*0.2667E-01*0.3175E-01* 38.10
*0.8333E-03*
*
*
*
*
*
*
*
* LINIA 58 * 32.00
*0.9600E-01*0.1062
* 35.40
*0.3000E-02*
*(CLCA
)* 32.00
*0.4267E-01*0.4720E-01* 35.40
*0.1333E-02*
*(CLDA
)* 32.00
*0.5333E-01*0.5900E-01* 35.40
*0.1667E-02*
*
*
*
*
*
*
*
Facultade de Informatica. A Coruña. Junio 2005
132
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* LINIA 60 * 32.00
*0.8000E-01*0.8696E-01* 34.78
*0.2500E-02*
*(CLDB
)* 32.00
*0.8000E-01*0.8696E-01* 34.78
*0.2500E-02*
*
*
*
*
*
*
*
* LINIA 62 * 32.00
*0.2667E-01*0.2740E-01* 32.88
*0.8333E-03*
*(CLDC
)* 32.00
*0.2667E-01*0.2740E-01* 32.88
*0.8333E-03*
*
*
*
*
*
*
*
* LINIA 63 * 32.00
*0.8800E-01*0.9649E-01* 35.09
*0.2750E-02*
*(CLAD
)* 32.00
*0.5333E-01*0.5848E-01* 35.09
*0.1667E-02*
*(CLBD
)* 32.00
*0.1333E-01*0.1462E-01* 35.09
*0.4167E-03*
*(CLCD
)* 32.00
*0.2133E-01*0.2339E-01* 35.09
*0.6667E-03*
*
*
*
*
*
*
*
* CPU
1 * 1.000
*0.8250E-02*0.8319E-02* 1.008
*0.8250E-02*
*(CLAB
)* 1.000
*0.1000E-02*0.1008E-02* 1.008
*0.1000E-02*
*(CLAC
)* 1.000
*0.1333E-02*0.1344E-02* 1.008
*0.1333E-02*
*(CLAD
)* 1.000
*0.1667E-02*0.1681E-02* 1.008
*0.1667E-02*
*(CLBA
)* 1.000
*0.1250E-02*0.1260E-02* 1.008
*0.1250E-02*
*(CLCA
)* 1.000
*0.1333E-02*0.1344E-02* 1.008
*0.1333E-02*
*(CLDA
)* 1.000
*0.1667E-02*0.1681E-02* 1.008
*0.1667E-02*
Facultade de Informatica. A Coruña. Junio 2005
133
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
2 * 1.000
*0.8250E-02*0.8319E-02* 1.008
*0.8250E-02*
*(CLAB
)* 1.000
*0.1000E-02*0.1008E-02* 1.008
*0.1000E-02*
*(CLAC
)* 1.000
*0.1333E-02*0.1344E-02* 1.008
*0.1333E-02*
*(CLAD
)* 1.000
*0.1667E-02*0.1681E-02* 1.008
*0.1667E-02*
*(CLBA
)* 1.000
*0.1250E-02*0.1260E-02* 1.008
*0.1250E-02*
*(CLCA
)* 1.000
*0.1333E-02*0.1344E-02* 1.008
*0.1333E-02*
*(CLDA
)* 1.000
*0.1667E-02*0.1681E-02* 1.008
*0.1667E-02*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* CPU
3 * 1.000
*0.8000E-02*0.8065E-02* 1.008
*0.8000E-02*
*(CLAB
)* 1.000
*0.1000E-02*0.1008E-02* 1.008
*0.1000E-02*
*(CLBA
)* 1.000
*0.1250E-02*0.1260E-02* 1.008
*0.1250E-02*
*(CLBC
)* 1.000
*0.8333E-03*0.8401E-03* 1.008
*0.8333E-03*
*(CLBD
)* 1.000
*0.4167E-03*0.4200E-03* 1.008
*0.4167E-03*
*(CLCB
)* 1.000
*0.2000E-02*0.2016E-02* 1.008
*0.2000E-02*
*(CLDB
)* 1.000
*0.2500E-02*0.2520E-02* 1.008
*0.2500E-02*
Facultade de Informatica. A Coruña. Junio 2005
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
4 * 1.000
*0.9333E-02*0.9421E-02* 1.009
*0.9333E-02*
*(CLAB
)* 1.000
*0.1000E-02*0.1009E-02* 1.009
*0.1000E-02*
*(CLAC
)* 1.000
*0.1333E-02*0.1346E-02* 1.009
*0.1333E-02*
*(CLBA
)* 1.000
*0.1250E-02*0.1262E-02* 1.009
*0.1250E-02*
*(CLBC
)* 1.000
*0.8333E-03*0.8412E-03* 1.009
*0.8333E-03*
*(CLBD
)* 1.000
*0.4167E-03*0.4206E-03* 1.009
*0.4167E-03*
*(CLCB
)* 1.000
*0.2000E-02*0.2019E-02* 1.009
*0.2000E-02*
*(CLDB
)* 1.000
*0.2500E-02*0.2524E-02* 1.009
*0.2500E-02*
*
*
*
*
*
*
*
* CPU
5 * 1.000
*0.7000E-02*0.7049E-02* 1.007
*0.7000E-02*
*(CLAC
)* 1.000
*0.1333E-02*0.1343E-02* 1.007
*0.1333E-02*
*(CLBC
)* 1.000
*0.8333E-03*0.8392E-03* 1.007
*0.8333E-03*
*(CLCA
)* 1.000
*0.1333E-02*0.1343E-02* 1.007
*0.1333E-02*
*(CLCB
)* 1.000
*0.2000E-02*0.2014E-02* 1.007
*0.2000E-02*
*(CLCD
)* 1.000
*0.6667E-03*0.6714E-03* 1.007
*0.6667E-03*
*(CLDC
)* 1.000
*0.8333E-03*0.8392E-03* 1.007
*0.8333E-03*
Facultade de Informatica. A Coruña. Junio 2005
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EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
6 * 1.000
*0.7000E-02*0.7049E-02* 1.007
*0.7000E-02*
*(CLAC
)* 1.000
*0.1333E-02*0.1343E-02* 1.007
*0.1333E-02*
*(CLBC
)* 1.000
*0.8333E-03*0.8392E-03* 1.007
*0.8333E-03*
*(CLCA
)* 1.000
*0.1333E-02*0.1343E-02* 1.007
*0.1333E-02*
*(CLCB
)* 1.000
*0.2000E-02*0.2014E-02* 1.007
*0.2000E-02*
*(CLCD
)* 1.000
*0.6667E-03*0.6714E-03* 1.007
*0.6667E-03*
*(CLDC
)* 1.000
*0.8333E-03*0.8392E-03* 1.007
*0.8333E-03*
*
*
*
*
*
*
*
* CPU
7 * 1.000
*0.7750E-02*0.7811E-02* 1.008
*0.7750E-02*
*(CLAD
)* 1.000
*0.1667E-02*0.1680E-02* 1.008
*0.1667E-02*
*(CLBD
)* 1.000
*0.4167E-03*0.4199E-03* 1.008
*0.4167E-03*
*(CLCD
)* 1.000
*0.6667E-03*0.6719E-03* 1.008
*0.6667E-03*
*(CLDA
)* 1.000
*0.1667E-02*0.1680E-02* 1.008
*0.1667E-02*
*(CLDB
)* 1.000
*0.2500E-02*0.2520E-02* 1.008
*0.2500E-02*
*(CLDC
)* 1.000
*0.8333E-03*0.8398E-03* 1.008
*0.8333E-03*
Facultade de Informatica. A Coruña. Junio 2005
136
EXACT ANALYTICAL SOLUTIONS
 The BCMP theorem: Communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
8 * 1.000
*0.9083E-02*0.9167E-02* 1.009
*0.9083E-02*
*(CLAD
)* 1.000
*0.1667E-02*0.1682E-02* 1.009
*0.1667E-02*
*(CLBD
)* 1.000
*0.4167E-03*0.4205E-03* 1.009
*0.4167E-03*
*(CLCA
)* 1.000
*0.1333E-02*0.1346E-02* 1.009
*0.1333E-02*
*(CLCD
)* 1.000
*0.6667E-03*0.6728E-03* 1.009
*0.6667E-03*
*(CLDA
)* 1.000
*0.1667E-02*0.1682E-02* 1.009
*0.1667E-02*
*(CLDB
)* 1.000
*0.2500E-02*0.2523E-02* 1.009
*0.2500E-02*
*(CLDC
)* 1.000
*0.8333E-03*0.8410E-03* 1.009
*0.8333E-03*
*******************************************************************
114.1
109.2
148.4
114.6
146.1
108.6
113.8
115.7
109.6
106.3
110.0
110.4
190
191 /END/
Facultade de Informatica. A Coruña. Junio 2005
137
OUTLINE
 INTRODUCTION
 CONCEPT OF QUEUE
 CONCEPT OF QUEUEING NETWORK
 NUMERICAL TECHNIQUES
 EXACT ANALYTICAL SOLUTIONS
 APPROXIMATE ANALYTICAL
SOLUTIONS
 SIMULATION TECHNIQUES
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APPROXIMATE ANALYTICAL SOLUTIONS
 Approximate
solutions for queuing networks
are useful for tackling problems not covered by
the product-form solutions as, for instance,
priorities
 class dependent or non-exponential service times at
FIFO stations
 finite buffers
 simultaneous resource possession

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APPROXIMATE ANALYTICAL SOLUTIONS
 There are a lot of different types of approximations in
the literature. Basically, they can be grouped in the
following classes of methods:

decomposition-aggregation

diffusion

iterative
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation methods

The idea behind decomposition is to break up the queuing
network into smaller subsystems, so that each subsystem can be
easily analysed in isolation, and then put together these partial
solutions, in order to obtain the solution of the queuing
network.

What is difficult to do is to find a good way to decompose the
network under study into smaller more manageable subsystems
and then put the solution together for the whole network.
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation methods

Various decomposition procedures have been developed
specifically for analysing queuing networks with particular
features.

Criteria for decomposing can be:
o Parts with different timing behaviour
o Norton theorem: inspired in the Norton theorem for electric circuits
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Norton theorem

BCMP network with a non-BCMP node
m2
m1
m4
m3
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Norton theorem

BCMP network with a non-BCMP node
o Let us assume that one node violate the BCMP assumptions. In our
particular case, node 3 is the culprit.
o Let us assume that we know all the parameters of the network, i. e. the
service times, routing probabilities, etc. Also, for simplicity, we
assume a single class of customers. Let N be the total number of
customers in it.
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Norton theorem

Decomposition step
o "Short-out" node 3 and analyse the queuing network as if this node did
not exist. By "short-out" we mean simply that node 3 is removed
without changing the incoming and outgoing flows.
o Now, the resulting network is a BCMP network and it can be easily
solved. We study this network to compute the throughput along the
"short".
o We do this computation assuming that there are k customers in the
"shorted" network, where k = 1, 2, ..., N. Let (k) be the throughput
along the "short" when there are k customers in it.
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Norton theorem
m2
m1
m4
(k)
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Norton theorem

Aggregation step
o Now, the original network can be reduced to node 3 and another node,
known as the composite node, which represents approximately the
shorted-out network. The service rate of the composite node is (k),
where k is the number of customers in the composite node (statedependent service). The total number of customers in the network is
still N.
Facultade de Informatica. A Coruña. Junio 2005
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Norton theorem
(k)
m3
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Norton theorem

Aggregation step
o In general, this network is not a BCMP network because of node 3.
So, we have to be able to analyse this aggregate system by either a
numerical approach, or another approximate method or a simulation
method. However, in any case as the system has only two nodes its
analysis will be easier.
o Let us assume that we can obtain the queue length distribution of
nodes 3 and composite. Then, the queue length distribution of node 3
is an approximation to the queue length distribution of node 3 in the
original network.
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Norton theorem

Aggregation step
o What about the queue length distributions of nodes 1, 2
and 4? They can be obtained by combining the results
obtained in step 1 with the queue length distribution of the
composite node obtained in step 2. To show how to obtain
them, let us assume that we want to obtain the queue
length of node 1.
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Norton theorem

Aggregation step
o Let pc(k) be the probability that there are n customers in the composite
node, where k = 1, 2, ..., N, and q1(n|k) be the probability that there are
n customers in the node 1 when there are k customers in the shorted
network. Then
N
q1 n  
 q n | k  p k , for n  1, 2, ..., N
1
c
k 1
o It is also possible to proceed in a similar way when instead of just one
non-BCMP node we have a non-BCMP sub-network
Facultade de Informatica. A Coruña. Junio 2005
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
Disk 1
Disk 2
CPU
Disk 3
Terminals
Memory
management
Facultade de Informatica. A Coruña. Junio 2005
Disk 4
152
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
Disk 1
Disk 2
CPU
Disk 3
Disk 4
Facultade de Informatica. A Coruña. Junio 2005
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
1 /DECLARE/ QUEUE CPU,DISC(4),TERMINAL,SC;
2
REAL PROB1(4)=(2,1.5,1,0.5);
3
REAL PROB2(4)=(1.5,2,3,3.5);
4
REAL TR,CAP(20);
5
REAL TR1,TR2;
6
CLASS C1,C2;
7
INTEGER I,N,M;
8 /STATION/ NAME=CPU;
9
SCHED=PS;
10
INIT(C1)=N;
11
SERVICE(C1)=CST(8.52);
12
SERVICE(C2)=CST(12.);
13
TRANSIT(C1)=DISC,PROB1,CPU,C1,0.6,CPU,C2,0.4;
14
TRANSIT(C2)=DISC,PROB2,CPU,C1,0.6,CPU,C2,0.4;
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
15
16
17
18
19
20
21
22
23
24
/STATION/ NAME=DISC;
TRANSIT=CPU;
/STATION/ NAME=DISC(1);
SERVICE=EXP(23.);
/STATION/ NAME=DISC(2);
SERVICE=EXP(22.);
/STATION/ NAME=DISC(3);
SERVICE=EXP(21.);
/STATION/ NAME=DISC(4);
SERVICE=EXP(20.);
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
25 /EXEC/
BEGIN
26
NETWORK(CPU,DISC);
27
FOR N := 1 STEP 1 UNTIL 20 DO
28
BEGIN
29
PRINT;
30
PRINT("FACTOR DE MULTIPROGRAMACIO =",N);
31
SOLVE;
32
CAP(N) := MTHRUPUT(CPU);
33
FOR I:=1 STEP 1 UNTIL 4 DO CAP(N):=CAP(N)-MTHRUP
==> UT(DISC(I));
34
END;
35
FOR N := 1 STEP 1 UNTIL 20 DO PRINT(N,CAP(N));
36
END;
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
FACTOR DE MULTIPROGRAMACIO =
1
- MEAN VALUE ANALYSIS ("MVA") *******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.3566
*0.3566
* 10.43
*0.3418E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.1769
*0.1769
* 23.00
*0.7690E-02*
*
*
*
*
*
*
*
* DISC
2 * 22.00
*0.1598
*0.1598
* 22.00
*0.7263E-02*
*
*
*
*
*
*
*
* DISC
3 * 21.00
*0.1615
*0.1615
* 21.00
*0.7690E-02*
*
*
*
*
*
*
*
* DISC
4 * 20.00
*0.1453
*0.1453
* 20.00
*0.7263E-02*
*******************************************************************
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
FACTOR DE MULTIPROGRAMACIO =
5
- MEAN VALUE ANALYSIS ("MVA") *******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.8852
* 2.552
* 30.08
*0.8484E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.4390
*0.7037
* 36.87
*0.1909E-01*
*
*
*
*
*
*
*
* DISC
2 * 22.00
*0.3966
*0.6041
* 33.51
*0.1803E-01*
*
*
*
*
*
*
*
* DISC
3 * 21.00
*0.4009
*0.6137
* 32.15
*0.1909E-01*
*
*
*
*
*
*
*
* DISC
4 * 20.00
*0.3606
*0.5263
* 29.19
*0.1803E-01*
*******************************************************************
Facultade de Informatica. A Coruña. Junio 2005
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
FACTOR DE MULTIPROGRAMACIO =
10
- MEAN VALUE ANALYSIS ("MVA") *******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.9905
* 6.801
* 71.65
*0.9493E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.4913
*0.9443
* 44.21
*0.2136E-01*
*
*
*
*
*
*
*
* DISC
2 * 22.00
*0.4438
*0.7855
* 38.94
*0.2017E-01*
*
*
*
*
*
*
*
* DISC
3 * 21.00
*0.4485
*0.8003
* 37.47
*0.2136E-01*
*
*
*
*
*
*
*
* DISC
4 * 20.00
*0.4034
*0.6685
* 33.14
*0.2017E-01*
*******************************************************************
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
FACTOR DE MULTIPROGRAMACIO =
15
- MEAN VALUE ANALYSIS ("MVA") *******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.9995
* 11.70
* 122.1
*0.9579E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.4957
*0.9807
* 45.50
*0.2155E-01*
*
*
*
*
*
*
*
* DISC
2 * 22.00
*0.4478
*0.8098
* 39.78
*0.2036E-01*
*
*
*
*
*
*
*
* DISC
3 * 21.00
*0.4526
*0.8256
* 38.31
*0.2155E-01*
*
*
*
*
*
*
*
* DISC
4 * 20.00
*0.4071
*0.6860
* 33.70
*0.2036E-01*
*******************************************************************
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
FACTOR DE MULTIPROGRAMACIO =
20
- MEAN VALUE ANALYSIS ("MVA") *******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
* 1.000
* 16.69
* 174.1
*0.9584E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.4960
*0.9838
* 45.62
*0.2156E-01*
*
*
*
*
*
*
*
* DISC
2 * 22.00
*0.4480
*0.8117
* 39.85
*0.2037E-01*
*
*
*
*
*
*
*
* DISC
3 * 21.00
*0.4528
*0.8275
* 38.38
*0.2156E-01*
*
*
*
*
*
*
*
* DISC
4 * 20.00
*0.4073
*0.6872
* 33.74
*0.2037E-01*
*******************************************************************
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161
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
1
0.4272E-02
11
0.1191E-01
2
0.6940E-02
12
0.1194E-01
3
0.8680E-02
13
0.1196E-01
4
0.9835E-02
14
0.1197E-01
5
0.1060E-01
15
0.1197E-01
6
0.1111E-01
16
0.1198E-01
7
0.1144E-01
17
0.1198E-01
8
0.1165E-01
18
0.1198E-01
9
0.1179E-01
19
0.1198E-01
10
0.1187E-01
20
0.1198E-01
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
Terminals
Facultade de Informatica. A Coruña. Junio 2005
163
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
37
38 /STATION/ NAME=TERMINAL;
39
TYPE=INFINITE;
40
INIT(C1)=N;
41
SERVICE(C1)=EXP(30000.);
42
SERVICE(C2)=EXP(60000.);
43
TRANSIT=SC;
44 /STATION/ NAME=SC;
45
SERVICE=EXP(1.);
46
RATE=CAP(1 STEP 1 UNTIL M);
47
TRANSIT=TERMINAL,C1,0.6,TERMINAL,C2;
48 /CONTROL/ OPTION=NRESULT;
Facultade de Informatica. A Coruña. Junio 2005
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
49 /EXEC/
BEGIN
50
NETWORK(TERMINAL,SC);
51
PRINT("
TERM
FM
PRODUCTIVITAT
==> RESPOSTA");
52
FOR N := 150 STEP 150 UNTIL 750 DO
53
FOR M := 1 STEP 1 UNTIL 20 DO
54
BEGIN
55
SOLVE;
56
PRINT(N,M,MTHRUPUT(SC),MRESPONSE(SC));
57
END;
58
END;
Facultade de Informatica. A Coruña. Junio 2005
165
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
TERM
150
150
150
150
150
150
150
150
…….
150
FM PRODUCTIVITAT
1
0.3478E-02
2
0.3541E-02
3
0.3546E-02
4
0.3547E-02
5
0.3547E-02
6
0.3547E-02
7
0.3547E-02
8
0.3547E-02
20
0.3547E-02
Facultade de Informatica. A Coruña. Junio 2005
RESPOSTA
1124.
358.5
304.1
294.4
292.4
292.0
291.9
291.9
291.9
166
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
TERM
300
300
300
300
300
300
300
300
300
300
300
300
……….
300
FM PRODUCTIVITAT RESPOSTA
1
0.4272E-02 0.2822E+05
2
0.6735E-02
2545.
3
0.7031E-02
668.5
4
0.7062E-02
482.4
5
0.7070E-02
433.6
6
0.7073E-02
417.0
7
0.7074E-02
410.8
8
0.7074E-02
408.4
9
0.7074E-02
407.5
10
0.7074E-02
407.2
11
0.7074E-02
407.0
12
0.7074E-02
407.0
20
0.7074E-02
407.0
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167
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
TERM
450
450
450
450
450
450
450
450
450
450
450
450
450
450
450
450
450
450
450
450
FM PRODUCTIVITAT RESPOSTA
1
0.4272E-02 0.6334E+05
2
0.6940E-02 0.2284E+05
3
0.8679E-02
9847.
4
0.9792E-02
3956.
5
0.1026E-01
1872.
6
0.1040E-01
1270.
7
0.1045E-01
1052.
8
0.1048E-01
956.3
9
0.1049E-01
910.5
10
0.1049E-01
887.4
11
0.1050E-01
875.5
12
0.1050E-01
869.4
13
0.1050E-01
866.3
14
0.1050E-01
864.7
15
0.1050E-01
864.0
16
0.1050E-01
863.6
17
0.1050E-01
863.4
18
0.1050E-01
863.3
19
0.1050E-01
863.2
20
0.1050E-01
863.2
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
TERM
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
FM PRODUCTIVITAT RESPOSTA
1
0.4272E-02 0.9845E+05
2
0.6940E-02 0.4446E+05
3
0.8679E-02 0.2713E+05
4
0.9835E-02 0.1901E+05
5
0.1060E-01 0.1458E+05
6
0.1111E-01 0.1199E+05
7
0.1144E-01 0.1043E+05
8
0.1165E-01
9486.
9
0.1179E-01
8910.
10
0.1187E-01
8565.
11
0.1191E-01
8360.
12
0.1194E-01
8241.
13
0.1196E-01
8172.
14
0.1197E-01
8133.
15
0.1197E-01
8111.
16
0.1198E-01
8099.
17
0.1198E-01
8092.
18
0.1198E-01
8088.
19
0.1198E-01
8086.
20
0.1198E-01
8085.
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APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Multiprogramming system
TERM
750
750
750
750
750
750
750
750
750
750
750
750
750
750
750
750
750
750
750
750
59 /END/
FM PRODUCTIVITAT RESPOSTA
1
0.4272E-02 0.1336E+06
2
0.6940E-02 0.6607E+05
3
0.8679E-02 0.4441E+05
4
0.9835E-02 0.3426E+05
5
0.1060E-01 0.2873E+05
6
0.1111E-01 0.2549E+05
7
0.1144E-01 0.2354E+05
8
0.1165E-01 0.2236E+05
9
0.1179E-01 0.2164E+05
10
0.1187E-01 0.2121E+05
11
0.1191E-01 0.2095E+05
12
0.1194E-01 0.2080E+05
13
0.1196E-01 0.2071E+05
14
0.1197E-01 0.2067E+05
15
0.1197E-01 0.2064E+05
16
0.1198E-01 0.2062E+05
17
0.1198E-01 0.2061E+05
18
0.1198E-01 0.2061E+05
19
0.1198E-01 0.2061E+05
20
0.1198E-01 0.2061E+05
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170
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Sliding window flow control
mechanisms

f(S)
tj
tf
Network 1
e(S)
Network 2
Facultade de Informatica. A Coruña. Junio 2005
171
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Sliding window flow control
mechanisms

Step 1: Decomposition
Network 1
Network 2
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172
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Sliding window flow control
mechanisms

Step 1: Decomposition
o We analyze this queuing network to obtain its throughput when there
are k customers where k = 1, 2, …, C.
o Let (k) be the throughput we obtain when there are k customers in the
network. The final result of this step is a set of values (1), (2), …,
(C).
Facultade de Informatica. A Coruña. Junio 2005
173
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Sliding window flow control
mechanisms

Step 2: Aggregation
tj
f(S)

g(k)
e(S)
Facultade de Informatica. A Coruña. Junio 2005
174
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Sliding window flow control
mechanisms

Step 2: Aggregation
o The arrival process at queue f(S) is Poisson distributed of mean 
o There are C tokens.
o The inter-arrival times at queue e(S) are exponentially distributed with a
rate g(k), where k is the number of outstanding tokens, i. e. C - k is the
number of tokens in queue e(S).
o We set g(k) = (k), for k = 1, 2, …, C.
o The state of this system can be described by the tuple (i, j), where i is
the number of customers in queue f(S) and j is the number of tokens in
queue e(S).
Facultade de Informatica. A Coruña. Junio 2005
175
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Sliding window flow control
mechanisms

Step 2: Aggregation
g(C)
g(C)
g(C)
i,0

1,0


g(C)
0,0

g(C-1)
0,1

Facultade de Informatica. A Coruña. Junio 2005
g(C-2)
g(C-j)
0,2

g(1)
0,j


0,C


176
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Sliding window flow control
mechanisms

Step 2: Aggregation
o This system is identical to an M/M/1 queue with an arrival rate  and a
state dependent service rate g(nq) if nq  C, and g(C) if nq > C, where nq
is the number of customers in this M/M/1 queue. The random variables
i and j are related to nq as follows:
i = max(0, nq - C)
j = max(0, C - nq)
o The solution of this system can be obtained by a direct application of
classical results.
Facultade de Informatica. A Coruña. Junio 2005
177
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Sliding window flow control
mechanisms

Step 2: Aggregation
pi ,0  ri p0,0
 j 
p0, j   j p0,0

where
r = /g(C)
 j 1

g C  k 
 j   
 k 0
1

Facultade de Informatica. A Coruña. Junio 2005
j0
j0
178
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Sliding window flow control
mechanisms

Step 2: Aggregation
o The probability p(0, 0) is chosen so that the addition of the state
probabilities is equal to 1:
p0,01 
Facultade de Informatica. A Coruña. Junio 2005
1

1 r
C

j 1
 j 
j
179
APPROXIMATE ANALYTICAL SOLUTIONS
 Decomposition-aggregation: Sliding window flow control
mechanisms

Step 2: Aggregation
o We obtain the following marginal probabilities for each queue (index 1
is for queue f(S) and 2 for queue e(S)):
1  r1  p0 ,0 
p1 0  
1 r
p1 i   ri p0 ,0 
i0
p0 ,0 
p2 0  
1 r
 j 
p2  j   j p0 ,0  0  j  C

Facultade de Informatica. A Coruña. Junio 2005
180
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion Method

Based on the assumption that very probably the queues are never
empty. Under this hypothesis for each queue:
o the queue length discrete distribution is studied
o this discrete distribution is replaced by a continuous one with the same
first two moments (first approximation)
o this continuous probability distribution is described by a diffusion
equation
o this equation is solved with the appropriate contour conditions for the
steady state
o the continuous probability distribution of the queue length is discretised
by means of some heuristic criteria (second approximation)
Facultade de Informatica. A Coruña. Junio 2005
181
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion Method

To solve a queuing network it is assumed that it will have a
product-form.

If the system is open it is possible to determine for each node the
characteristics of the inter-arrival time distribution and those of
the service time are assumed to be known.

From the state probabilities, the performance measures can be
computed.
Facultade de Informatica. A Coruña. Junio 2005
182
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
7
D
6
8
A
5
1
C
4
2
B
Facultade de Informatica. A Coruña. Junio 2005
3
183
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
1 /DECLARE/ QUEUE GEN,LINIA(8,8);
2
REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD,
==> TRDA,TRDB,TRDC;
3
INTEGER I;
4
CLASS CLAB,CLAC,CLAD,CLBA,CLBC,CLBD,CLCA,CLCB,CLCD
==> ,CLDA,CLDB,CLDC;
5
Facultade de Informatica. A Coruña. Junio 2005
184
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
6 /STATION/ NAME = GEN;
7
TYPE = SOURCE;
8
SERVICE = EXP(60000./930.);
9
TRANSIT = LINIA(1,2),CLAB,60,LINIA(1,2),CLAC,80,L
==> INIA(1,2),CLAD,100,
10
LINIA(3,4),CLBA,75,LINIA(3,4),CLBC,50,L
==> INIA(3,4),CLBD,25,
11
LINIA(5,6),CLCA,80,LINIA(5,6),CLCB,120,
==> LINIA(5,6),CLCD,40,
12
LINIA(7,8),CLDA,100,LINIA(7,8),CLDB,150
==> ,LINIA(7,8),CLDC,50;
13
Facultade de Informatica. A Coruña. Junio 2005
185
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
/STATION/ NAME = LINIA;
SERVICE = CST(256.*8./64.);
/STATION/ NAME = LINIA(1,2);
TRANSIT(CLAB) = LINIA(2,4);
TRANSIT(CLAC) = LINIA(2,4);
TRANSIT(CLAD) = LINIA(2,8);
/STATION/ NAME = LINIA(2,1);
TRANSIT = OUT;
/STATION/ NAME = LINIA(2,4);
TRANSIT(CLAB) = LINIA(4,3);
TRANSIT(CLAC) = LINIA(4,6);
/STATION/ NAME = LINIA(2,8);
TRANSIT(CLAD) = LINIA(8,7);
Facultade de Informatica. A Coruña. Junio 2005
186
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
/STATION/ NAME = LINIA(3,4);
TRANSIT(CLBA) = LINIA(4,2);
TRANSIT(CLBC) = LINIA(4,6);
TRANSIT(CLBD) = LINIA(4,8);
/STATION/ NAME = LINIA(4,2);
TRANSIT(CLBA) = LINIA(2,1);
/STATION/ NAME = LINIA(4,3);
TRANSIT = OUT;
/STATION/ NAME = LINIA(4,6);
TRANSIT(CLAC) = LINIA(6,5);
TRANSIT(CLBC) = LINIA(6,5);
/STATION/ NAME = LINIA(4,8);
TRANSIT(CLBD) = LINIA(8,7);
Facultade de Informatica. A Coruña. Junio 2005
187
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
50 /STATION/ NAME = LINIA(5,6);
51
TRANSIT(CLCA) = LINIA(6,8);
52
TRANSIT(CLCB) = LINIA(6,4);
53
TRANSIT(CLCD) = LINIA(6,8);
54
55 /STATION/ NAME = LINIA(6,4);
56
TRANSIT(CLCB) = LINIA(4,3);
57
58 /STATION/ NAME = LINIA(6,5);
59
TRANSIT = OUT;
60
61 /STATION/ NAME = LINIA(6,8);
62
TRANSIT(CLCA) = LINIA(8,2);
63
TRANSIT(CLCD) = LINIA(8,7);
64
65 /STATION/ NAME = LINIA(7,8);
66
TRANSIT(CLDA) = LINIA(8,2);
67
TRANSIT(CLDB) = LINIA(8,4);
68
TRANSIT(CLDC) = LINIA(8,6);
69
Facultade de Informatica. A Coruña. Junio 2005
188
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
70
71
72
73
74
75
76
77
78
79
80
81
82
/STATION/ NAME = LINIA(8,2);
TRANSIT(CLCA) = LINIA(2,1);
TRANSIT(CLDA) = LINIA(2,1);
/STATION/ NAME = LINIA(8,4);
TRANSIT(CLDB) = LINIA(4,3);
/STATION/ NAME = LINIA(8,6);
TRANSIT(CLDC) = LINIA(6,5);
/STATION/ NAME = LINIA(8,7);
TRANSIT = OUT;
Facultade de Informatica. A Coruña. Junio 2005
189
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
83 /EXEC/
BEGIN
84
NETWORK(GEN,
85
LINIA(1,2),LINIA(2,1),LINIA(2,4),LINIA(2,8),
86
LINIA(3,4),LINIA(4,2),LINIA(4,3),LINIA(4,6),
87
LINIA(4,8),LINIA(5,6),LINIA(6,4),LINIA(6,5),
88
LINIA(6,8),LINIA(7,8),LINIA(8,2),LINIA(8,4),
89
LINIA(8,6),LINIA(8,7));
90
PRINT;
91
SOLVE;
92
TRAB:=MRESPONSE(LINIA(1,2))+
93
MRESPONSE(LINIA(2,4))+MRESPONSE(LINIA(4,3));
94
TRAC:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+
95
MRESPONSE(LINIA(4,6))+MRESPONSE(LINIA(6,5));
96
TRAD:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,8))+
97
MRESPONSE(LINIA(8,7));
98
TRBA:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,2))+
99
MRESPONSE(LINIA(2,1));
Facultade de Informatica. A Coruña. Junio 2005
190
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
TRBC:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,6))+
MRESPONSE(LINIA(6,5));
TRBD:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,8))+
MRESPONSE(LINIA(8,7));
TRCA:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+
MRESPONSE(LINIA(8,2))+MRESPONSE(LINIA(2,1));
TRCB:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,4))+
MRESPONSE(LINIA(4,3));
TRCD:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+
MRESPONSE(LINIA(8,7));
TRDA:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,2))+
MRESPONSE(LINIA(2,1));
TRDB:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,4))+
MRESPONSE(LINIA(4,3));
TRDC:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,6))+
MRESPONSE(LINIA(6,5));
Facultade de Informatica. A Coruña. Junio 2005
191
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
116
117
118
119
120
PRINT(TRAB,TRAC,TRAD);
PRINT(TRBA,TRBC,TRBD);
PRINT(TRCA,TRCB,TRCD);
PRINT(TRDA,TRDB,TRDC);
END;
Facultade de Informatica. A Coruña. Junio 2005
192
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
- APPROXIMATE DIFFUSIONS METHOD ("DIFFU") *******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
*
*
*
*
*
*
*
* GEN
* 64.52
* 1.000
* 1.000
* 64.52
*0.1550E-01*
*
*
*
*
*
*
*
* LINIA 2 * 32.00
*0.1280
*0.1374
* 34.35
*0.4000E-02*
*
*
*
*
*
*
*
* LINIA 9 * 32.00
*0.1360
*0.1466
* 34.48
*0.4250E-02*
*
*
*
*
*
*
*
* LINIA 12 * 32.00
*0.7467E-01*0.7765E-01* 33.28
*0.2333E-02*
*
*
*
*
*
*
*
* LINIA 16 * 32.00
*0.5333E-01*0.5482E-01* 32.89
*0.1667E-02*
*
*
*
*
*
*
*
* LINIA 20 * 32.00
*0.8000E-01*0.8348E-01* 33.39
*0.2500E-02*
*
*
*
*
*
*
*
* LINIA 26 * 32.00
*0.4000E-01*0.4083E-01* 32.66
*0.1250E-02*
*
*
*
*
*
*
*
Facultade de Informatica. A Coruña. Junio 2005
193
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* LINIA 27 * 32.00
*0.1760
*0.1945
* 35.37
*0.5500E-02*
*
*
*
*
*
*
*
* LINIA 30 * 32.00
*0.6933E-01*0.7190E-01* 33.18
*0.2167E-02*
*
*
*
*
*
*
*
* LINIA 32 * 32.00
*0.1333E-01*0.1342E-01* 32.22
*0.4167E-03*
*
*
*
*
*
*
*
* LINIA 38 * 32.00
*0.1280
*0.1374
* 34.35
*0.4000E-02*
*
*
*
*
*
*
*
* LINIA 44 * 32.00
*0.6400E-01*0.6617E-01* 33.08
*0.2000E-02*
*
*
*
*
*
*
*
* LINIA 45 * 32.00
*0.9600E-01*0.1011
* 33.68
*0.3000E-02*
*
*
*
*
*
*
*
* LINIA 48 * 32.00
*0.6400E-01*0.6617E-01* 33.08
*0.2000E-02*
*
*
*
*
*
*
*
* LINIA 56 * 32.00
*0.1600
*0.1752
* 35.05
*0.5000E-02*
*
*
*
*
*
*
*
* LINIA 58 * 32.00
*0.9600E-01*0.1011
* 33.68
*0.3000E-02*
*
*
*
*
*
*
*
Facultade de Informatica. A Coruña. Junio 2005
194
APPROXIMATE ANALYTICAL SOLUTIONS
 Diffusion method: Packet communication network
*******************************************************************
* NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* LINIA 60 * 32.00
*0.8000E-01*0.8343E-01* 33.37
*0.2500E-02*
*
*
*
*
*
*
*
* LINIA 62 * 32.00
*0.2667E-01*0.2703E-01* 32.44
*0.8333E-03*
*
*
*
*
*
*
*
* LINIA 63 * 32.00
*0.8800E-01*0.9222E-01* 33.53
*0.2750E-02*
*******************************************************************
MEMORY USED:
19812 WORDS OF 4 BYTES
( 2.48 % OF TOTAL MEMORY)
103.0
100.5
135.6
103.2
134.5
100.3
102.8
103.8
100.8
99.14
101.0
101.2
121
122 /END/
Facultade de Informatica. A Coruña. Junio 2005
195
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods

In this family of methods we establish an iterative computation
of the result from a reasonable conjecture.

They have neither any theoretical justification of the iteration
convergence nor the coincidence of the limit of the iteration with
the theoretical result; however the experience shows that there
are no counter-examples in the normal domain of use of such
methods.
Facultade de Informatica. A Coruña. Junio 2005
196
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods

They are useful tools to study cases not covered by exact
methods, as, for instance:
o closed queuing networks with FIFO stations that have with nonexponentially distributed service time
o systems with customers seizing simultaneously more than one server
(for example, in a disk input-output there are simultaneous seizing of
the disk and the path between disks and memory).
o systems with customers affected of priorities

They are also useful to reduce the computing time in methods
derived from the BCMP theorem when they are applied to large
dimension systems.
Facultade de Informatica. A Coruña. Junio 2005
197
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system

This case is identical to the previous one with an important
difference:
o the disk path to memory is shared by several disks and should be taken
into account in the modelling process
o the scheduling policy of disks accesses in the control unit is SLTF
(Shortest Latency Time First).

As in this case the basic assumption of analytical models is not
fulfilled, we are obliged to build an iterative model starting with
the assumption that there will not be conflicts in the path; from
this assumption it is possible to compute the throughput in the
disks. With this data we re-compute the path conflict time due to
the lost rotations. This time is introduced in the disks service
time and we restart the iteration.
Facultade de Informatica. A Coruña. Junio 2005
198
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
1 /DECLARE/ QUEUE CPU,DISC(4),TERMINAL;
2
3
4
5
6
7
8
9
10 /STATION/
11
12
13
14
==> .4;
15
==> .4;
REAL
REAL
REAL
REAL
PROB1(4)=(2,1.5,1,0.5);
PROB2(4)=(1.5,2,3,3.5);
TR1,TR2;
SD(4),VP,VPN,LT,TF,VR=3600,TAU,UC,
SK(4)=(13.,12.,11.,10.);
REAL EPS;
CLASS C1,C2;
INTEGER I,N;
NAME=CPU;
SCHED=PS;
SERVICE(C1)=CST(8.52);
SERVICE(C2)=CST(12.);
TRANSIT(C1)=DISC,PROB1,TERMINAL,C1,0.6,TERMINAL,C2,0
TRANSIT(C2)=DISC,PROB2,TERMINAL,C1,0.6,TERMINAL,C2,0
Facultade de Informatica. A Coruña. Junio 2005
199
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
/STATION/ NAME=DISC;
TRANSIT=CPU;
/STATION/ NAME=DISC(1);
SERVICE=EXP(SD(1));
/STATION/ NAME=DISC(2);
SERVICE=EXP(SD(2));
/STATION/ NAME=DISC(3);
SERVICE=EXP(SD(3));
/STATION/ NAME=DISC(4);
SERVICE=EXP(SD(4));
/STATION/ NAME=TERMINAL;
TYPE=INFINITE;
INIT(C1)=N;
SERVICE(C1)=EXP(30000.);
SERVICE(C2)=EXP(60000.);
TRANSIT=CPU;
/CONTROL/ CLASS=ALL QUEUE;
Facultade de Informatica. A Coruña. Junio 2005
200
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
33 /EXEC/
34
35
36
37
38
39
40
41
42
43
44
==> TF;
45
46
BEGIN
TAU:=60.*1000./VR;
TF:=TAU/10.;
LT:=TAU/2.;
FOR N:=150 STEP 150 UNTIL 750 DO
BEGIN
PRINT("NOMBRE D’USUARIS =",N);
EPS:=1.;
VP:=0.;
WHILE EPS>=1.E-4 DO
BEGIN
FOR I:=1 STEP 1 UNTIL 4 DO SD(I):=SK(I)+LT+VP+
PRINT;
SOLVE;
Facultade de Informatica. A Coruña. Junio 2005
201
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
47
48
==>
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
UC:=0.;
FOR I:=1 STEP 1 UNTIL 4 DO UC := UC + TF * MTH
RUPUT(DISC(I));
VPN:=TAU*UC/(1.-UC);
EPS:=ABS(VP-VPN)/VPN;
VP:=VPN;
END;
TR1:=MCUSTNB(CPU,C1);
TR2:=MCUSTNB(CPU,C2);
FOR I:= 1 STEP 1 UNTIL 4 DO
BEGIN
TR1:=TR1+MCUSTNB(DISC(I),C1);
TR2:=TR2+MCUSTNB(DISC(I),C2);
END;
TR1:=TR1/MTHRUPUT(TERMINAL,C1);
TR2:=TR2/MTHRUPUT(TERMINAL,C2);;
PRINT("RESPONSE TIME OF CLASS C1 =",TR1);
PRINT("RESPONSE TIME OF CLASS C2 =",TR2);
END;
END;
Facultade de Informatica. A Coruña. Junio 2005
202
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
150
- MEAN VALUE ANALYSIS ("MVA") *******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
*(C1
)* 8.520
*0.1088
*0.1539
* 12.06
*0.1277E-01*
*(C2
)* 12.00
*0.1873
*0.2650
* 16.98
*0.1560E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.1468
*0.1719
* 26.92
*0.6384E-02*
*(C1
)* 23.00
*0.9790E-01*0.1146
* 26.92
*0.4257E-02*
*(C2
)* 23.00
*0.4894E-01*0.5729E-01* 26.92
*0.2128E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 149.0
*0.4200E+05*0.3547E-02*
*(C1
)*0.3000E+05*0.0000E+00* 63.84
*0.3000E+05*0.2128E-02*
*(C2
)*0.6000E+05*0.0000E+00* 85.12
*0.6000E+05*0.1419E-02*
*
*
*
*
*
*
*
*******************************************************************
Facultade de Informatica. A Coruña. Junio 2005
203
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
150
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.2960
*0.4188
* 14.76
*0.2837E-01*
*(C1
)* 8.520
*0.1088
*0.1539
* 12.05
*0.1277E-01*
*(C2
)* 12.00
*0.1872
*0.2649
* 16.98
*0.1560E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.72
*0.1514
*0.1782
* 27.91
*0.6383E-02*
*(C1
)* 23.72
*0.1009
*0.1188
* 27.91
*0.4256E-02*
*(C2
)* 23.72
*0.5046E-01*0.5938E-01* 27.91
*0.2127E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 148.9
*0.4200E+05*0.3546E-02*
*(C1
)*0.3000E+05*0.0000E+00* 63.83
*0.3000E+05*0.2128E-02*
*(C2
)*0.6000E+05*0.0000E+00* 85.11
*0.6000E+05*0.1418E-02*
*******************************************************************
Facultade de Informatica. A Coruña. Junio 2005
204
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
150
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.2960
*0.4188
* 14.76
*0.2837E-01*
*(C1
)* 8.520
*0.1088
*0.1539
* 12.05
*0.1277E-01*
*(C2
)* 12.00
*0.1872
*0.2649
* 16.98
*0.1560E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.72
*0.1514
*0.1782
* 27.91
*0.6383E-02*
*(C1
)* 23.72
*0.1009
*0.1188
* 27.91
*0.4256E-02*
*(C2
)* 23.72
*0.5046E-01*0.5938E-01* 27.91
*0.2127E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 148.9
*0.4200E+05*0.3546E-02*
*(C1
)*0.3000E+05*0.0000E+00* 63.83
*0.3000E+05*0.2128E-02*
*(C2
)*0.6000E+05*0.0000E+00* 85.11
*0.6000E+05*0.1418E-02*
*******************************************************************
RESPONSE TIME OF CLASS C1 =
204.6
RESPONSE TIME OF CLASS C2 =
439.5
Facultade de Informatica. A Coruña. Junio 2005
205
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
300
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.5905
* 1.426
* 25.19
*0.5659E-01*
*(C1
)* 8.520
*0.2170
*0.5239
* 20.57
*0.2547E-01*
*(C2
)* 12.00
*0.3735
*0.9017
* 28.97
*0.3112E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.2929
*0.4134
* 32.46
*0.1273E-01*
*(C1
)* 23.00
*0.1953
*0.2756
* 32.46
*0.8490E-02*
*(C2
)* 23.00
*0.9761E-01*0.1378
* 32.46
*0.4244E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 297.1
*0.4200E+05*0.7074E-02*
*(C1
)*0.3000E+05*0.0000E+00* 127.3
*0.3000E+05*0.4245E-02*
*(C2
)*0.6000E+05*0.0000E+00* 169.8
*0.6000E+05*0.2830E-02*
*******************************************************************
Facultade de Informatica. A Coruña. Junio 2005
206
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
300
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.5902
* 1.424
* 25.17
*0.5657E-01*
*(C1
)* 8.520
*0.2169
*0.5233
* 20.56
*0.2546E-01*
*(C2
)* 12.00
*0.3733
*0.9007
* 28.95
*0.3111E-01*
*
*
*
*
*
*
*
* DISC
1 * 24.50
*0.3118
*0.4521
* 35.52
*0.1273E-01*
*(C1
)* 24.50
*0.2079
*0.3015
* 35.52
*0.8486E-02*
*(C2
)* 24.50
*0.1039
*0.1507
* 35.52
*0.4242E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 297.0
*0.4200E+05*0.7071E-02*
*(C1
)*0.3000E+05*0.0000E+00* 127.3
*0.3000E+05*0.4243E-02*
*(C2
)*0.6000E+05*0.0000E+00* 169.7
*0.6000E+05*0.2828E-02*
*******************************************************************
Facultade de Informatica. A Coruña. Junio 2005
207
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
300
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.5902
* 1.424
* 25.17
*0.5657E-01*
*(C1
)* 8.520
*0.2169
*0.5233
* 20.56
*0.2546E-01*
*(C2
)* 12.00
*0.3733
*0.9007
* 28.95
*0.3111E-01*
*
*
*
*
*
*
*
* DISC
1 * 24.50
*0.3118
*0.4521
* 35.52
*0.1273E-01*
*(C1
)* 24.50
*0.2079
*0.3014
* 35.52
*0.8486E-02*
*(C2
)* 24.50
*0.1039
*0.1507
* 35.52
*0.4242E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 297.0
*0.4200E+05*0.7071E-02*
*(C1
)*0.3000E+05*0.0000E+00* 127.3
*0.3000E+05*0.4243E-02*
*(C2
)*0.6000E+05*0.0000E+00* 169.7
*0.6000E+05*0.2828E-02*
*******************************************************************
RESPONSE TIME OF CLASS C1 =
289.4
RESPONSE TIME OF CLASS C2 =
632.8
Facultade de Informatica. A Coruña. Junio 2005
208
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
450
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.8763
* 6.438
* 76.66
*0.8399E-01*
*(C1
)* 8.520
*0.3220
* 2.366
* 62.60
*0.3780E-01*
*(C2
)* 12.00
*0.5543
* 4.072
* 88.16
*0.4619E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.4347
*0.7667
* 40.57
*0.1890E-01*
*(C1
)* 23.00
*0.2898
*0.5112
* 40.57
*0.1260E-01*
*(C2
)* 23.00
*0.1449
*0.2555
* 40.57
*0.6298E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 440.9
*0.4200E+05*0.1050E-01*
*(C1
)*0.3000E+05*0.0000E+00* 189.0
*0.3000E+05*0.6299E-02*
*(C2
)*0.6000E+05*0.0000E+00* 252.0
*0.6000E+05*0.4199E-02*
*******************************************************************
Facultade de Informatica. A Coruña. Junio 2005
209
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
450
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.8754
* 6.393
* 76.20
*0.8390E-01*
*(C1
)* 8.520
*0.3217
* 2.349
* 62.22
*0.3776E-01*
*(C2
)* 12.00
*0.5537
* 4.044
* 87.64
*0.4614E-01*
* DISC
1 * 25.33
*0.4781
*0.9129
* 48.36
*0.1888E-01*
*(C1
)* 25.33
*0.3188
*0.6086
* 48.36
*0.1259E-01*
*(C2
)* 25.33
*0.1594
*0.3043
* 48.36
*0.6292E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 440.5
*0.4200E+05*0.1049E-01*
*(C1
)*0.3000E+05*0.0000E+00* 188.8
*0.3000E+05*0.6293E-02*
*(C2
)*0.6000E+05*0.0000E+00* 251.7
*0.6000E+05*0.4195E-02*
*******************************************************************
Facultade de Informatica. A Coruña. Junio 2005
210
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
450
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
*0.8754
* 6.393
* 76.20
*0.8390E-01*
*(C1
)* 8.520
*0.3217
* 2.349
* 62.22
*0.3776E-01*
*(C2
)* 12.00
*0.5537
* 4.044
* 87.64
*0.4614E-01*
*
*
*
*
*
*
*
* DISC
1 * 25.32
*0.4781
*0.9127
* 48.35
*0.1888E-01*
*(C1
)* 25.32
*0.3187
*0.6085
* 48.35
*0.1259E-01*
*(C2
)* 25.32
*0.1593
*0.3042
* 48.35
*0.6292E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 440.5
*0.4200E+05*0.1049E-01*
*(C1
)*0.3000E+05*0.0000E+00* 188.8
*0.3000E+05*0.6293E-02*
*(C2
)*0.6000E+05*0.0000E+00* 251.7
*0.6000E+05*0.4195E-02*
*******************************************************************
RESPONSE TIME OF CLASS C1 =
594.4
RESPONSE TIME OF CLASS C2 =
1376.
Facultade de Informatica. A Coruña. Junio 2005
211
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
600
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
* 1.000
* 93.51
* 975.7
*0.9584E-01*
*(C1
)* 8.520
*0.3675
* 34.37
* 796.7
*0.4313E-01*
*(C2
)* 12.00
*0.6325
* 59.15
* 1122.
*0.5271E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.4960
*0.9841
* 45.63
*0.2157E-01*
*(C1
)* 23.00
*0.3307
*0.6561
* 45.63
*0.1438E-01*
*(C2
)* 23.00
*0.1653
*0.3280
* 45.63
*0.7187E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 503.2
*0.4200E+05*0.1198E-01*
*(C1
)*0.3000E+05*0.0000E+00* 215.7
*0.3000E+05*0.7188E-02*
*(C2
)*0.6000E+05*0.0000E+00* 287.5
*0.6000E+05*0.4792E-02*
*******************************************************************
Facultade de Informatica. A Coruña. Junio 2005
212
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
600
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
* 1.000
* 92.66
* 966.8
*0.9584E-01*
*(C1
)* 8.520
*0.3675
* 34.05
* 789.5
*0.4313E-01*
*(C2
)* 12.00
*0.6325
* 58.61
* 1112.
*0.5271E-01*
*
*
*
*
*
*
*
* DISC
1 * 25.71
*0.5544
* 1.244
* 57.69
*0.2157E-01*
*(C1
)* 25.71
*0.3696
*0.8295
* 57.69
*0.1438E-01*
*(C2
)* 25.71
*0.1848
*0.4147
* 57.69
*0.7187E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 503.2
*0.4200E+05*0.1198E-01*
*(C1
)*0.3000E+05*0.0000E+00* 215.7
*0.3000E+05*0.7188E-02*
*(C2
)*0.6000E+05*0.0000E+00* 287.5
*0.6000E+05*0.4792E-02*
*******************************************************************
RESPONSE TIME OF CLASS C1 =
4997.
RESPONSE TIME OF CLASS C2 = 0.1271E+05
Facultade de Informatica. A Coruña. Junio 2005
213
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
750
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
* 1.000
* 243.5
* 2541.
*0.9584E-01*
*(C1
)* 8.520
*0.3675
* 89.49
* 2075.
*0.4313E-01*
*(C2
)* 12.00
*0.6325
* 154.0
* 2922.
*0.5271E-01*
*
*
*
*
*
*
*
* DISC
1 * 23.00
*0.4960
*0.9841
* 45.64
*0.2157E-01*
*(C1
)* 23.00
*0.3307
*0.6561
* 45.64
*0.1438E-01*
*(C2
)* 23.00
*0.1653
*0.3280
* 45.64
*0.7187E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 503.2
*0.4200E+05*0.1198E-01*
*(C1
)*0.3000E+05*0.0000E+00* 215.7
*0.3000E+05*0.7188E-02*
*(C2
)*0.6000E+05*0.0000E+00* 287.5
*0.6000E+05*0.4792E-02*
*******************************************************************
Facultade de Informatica. A Coruña. Junio 2005
214
APPROXIMATE ANALYTICAL SOLUTIONS
 Iterative methods: Conversational system
NOMBRE D’USUARIS =
750
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * THRUPUT *
*******************************************************************
* CPU
* 10.43
* 1.000
* 242.7
* 2532.
*0.9584E-01*
*(C1
)* 8.520
*0.3675
* 89.18
* 2067.
*0.4313E-01*
*(C2
)* 12.00
*0.6325
* 153.5
* 2912.
*0.5271E-01*
*
*
*
*
*
*
*
* DISC
1 * 25.71
*0.5544
* 1.244
* 57.69
*0.2157E-01*
*(C1
)* 25.71
*0.3696
*0.8295
* 57.69
*0.1438E-01*
*(C2
)* 25.71
*0.1848
*0.4147
* 57.69
*0.7187E-02*
*
*
*
*
*
*
*
* TERMINAL *0.4200E+05*0.0000E+00* 503.2
*0.4200E+05*0.1198E-01*
*(C1
)*0.3000E+05*0.0000E+00* 215.7
*0.3000E+05*0.7188E-02*
*(C2
)*0.6000E+05*0.0000E+00* 287.5
*0.6000E+05*0.4792E-02*
*******************************************************************
RESPONSE TIME OF CLASS C1 = 0.1266E+05
RESPONSE TIME OF CLASS C2 = 0.3251E+05
Facultade de Informatica. A Coruña. Junio 2005
215
OUTLINE
 INTRODUCTION
 CONCEPT OF QUEUE
 CONCEPT OF QUEUEING NETWORK
 NUMERICAL TECHNIQUES
 EXACT ANALYTICAL SOLUTIONS
 APPROXIMATE ANALYTICAL
SOLUTIONS
 SIMULATION TECHNIQUES
Facultade de Informatica. A Coruña. Junio 2005
216
SIMULATION TECHNIQUES
 Computer based simulation implies writing a computer
program which depicts the system operations.
 Running
this program permits us to "imitate" the
system behaviour in a very short time. Thus, we are
able to "observe" the system and collect performance
statistics.
Facultade de Informatica. A Coruña. Junio 2005
217
SIMULATION TECHNIQUES
 Simulation advantage are:


there is no theoretical limitation and it allows us to study
systems not able to be studied by means of analytical or
numerical techniques.
It is easy to learn and to apply.
 Simulation disadvantages are:


the effort (mainly time) to develop and to debug a
simulation program.
the execution time, that can be quite long to obtain results
with enough significance.
Facultade de Informatica. A Coruña. Junio 2005
218
SIMULATION TECHNIQUES
 The complexity of the simulation model depend on the
detail of the system behaviour representation.
 To build up simulation models it is convenient to use:

modelling languages as QNAP2, RESQ, PAWS, etc., that make
easier to build up queuing network models and that include
analytical, numerical and simulation procedures.

simulation languages, such as SIMSCRIPT, GPSS, SIMULA,
etc., that offer greater simulation capabilities.

high level programming language for building up very detailed
models.
Facultade de Informatica. A Coruña. Junio 2005
219
SIMULATION TECHNIQUES
 The intrinsic problems of any simulation are:

random numbers generation

simulated time management

extraction of statistical estimations of the simulated behaviour

evaluation of the confidence interval of the estimations
Facultade de Informatica. A Coruña. Junio 2005
220
SIMULATION TECHNIQUES
 Conversational system

Identical to the previously studied by analytical
techniques

Influence of the memory
multiprogramming factor
Facultade de Informatica. A Coruña. Junio 2005
management
by
a
221
SIMULATION TECHNIQUES
 Conversational system
Disk 1
Disk 2
CPU
Disk 3
Terminals
Memory
management
Facultade de Informatica. A Coruña. Junio 2005
Disk 4
222
SIMULATION TECHNIQUES
 Conversational system
1 /DECLARE/ QUEUE CPU,DISC(4),MEM,SMEM,TERMINAL,R(2);
2
REAL PROF1(4)=(2.,3.5,4.5,5.);
3
REAL PROF2(4)=(1.5,3.5,6.5,10.);
4
REAL TR1,TR2;
5
REAL D;
6
CLASS C1,C2;
7
INTEGER I,N,M;
8
REF CUSTOMER C;
Facultade de Informatica. A Coruña. Junio 2005
223
SIMULATION TECHNIQUES
 Conversational system
9 /STATION/ NAME=CPU;
10
SCHED=PS;
11
SERVICE(C1)=BEGIN
12
CST(8.52);
13
D := UNIFORM(0., 6.);
14
FOR I := 1 STEP 1 UNTIL 4 DO
15
IF D <= PROF1(I) THEN TRANSIT(DISC(I));
16
C:=R(1).FIRST;
17
WHILE C.FATHER <> CUSTOMER DO C:=C.NEXT;
18
TRANSIT(C,OUT);
19
V(SMEM);
20
IF D <= 5.6 THEN TRANSIT(TERMINAL,C1)
21
22
ELSE TRANSIT(TERMINAL,C2);
END;
Facultade de Informatica. A Coruña. Junio 2005
224
SIMULATION TECHNIQUES
 Conversational system
23
SERVICE(C2)=BEGIN
24
CST(12.);
25
D := UNIFORM(0., 11.);
26
FOR I := 1 STEP 1 UNTIL 4 DO
27
IF D <= PROF2(I) THEN TRANSIT(DISC(I));
28
C:=R(2).FIRST;
29
WHILE C.FATHER <> CUSTOMER DO C:=C.NEXT;
30
TRANSIT(C,OUT);
31
V(SMEM);
32
IF D <= 10.6 THEN TRANSIT(TERMINAL,C1)
33
ELSE TRANSIT(TERMINAL,C2);
34
END;
35 /STATION/ NAME=DISC;
36
TRANSIT=CPU;
37 /STATION/ NAME=DISC(1);
38
SERVICE=EXP(23.);
39 /STATION/ NAME=DISC(2);
40
SERVICE=EXP(22.);
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225
SIMULATION TECHNIQUES
 Conversational system
41 /STATION/ NAME=DISC(3);
42
SERVICE=EXP(21.);
43 /STATION/ NAME=DISC(4);
44
SERVICE=EXP(20.);
45 /STATION/ NAME=MEM;
46
SERVICE=BEGIN
47
P(SMEM);
48
TRANSIT(CPU);
49
END;
50 /STATION/ NAME=SMEM;
51
TYPE=SEMAPHORE,MULTIPLE(M);
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SIMULATION TECHNIQUES
 Conversational system
52 /STATION/ NAME=TERMINAL;
53
TYPE=INFINITE;
54
INIT(C1)=6*N/10;
55
INIT(C2)=4*N/10;
56
SERVICE(C1)=BEGIN
57
EXP(30000.);
58
TRANSIT(NEW(CUSTOMER),R(1),C1);
59
END;
60
SERVICE(C2)=BEGIN
61
EXP(60000.);
62
TRANSIT(NEW(CUSTOMER),R(2),C2);
63
END;
64
TRANSIT=MEM;
65 /CONTROL/ TMAX=5000000.;
66
CLASS=ALL QUEUE;
67
ACCURACY=ALL QUEUE,ALL CLASS;
68
OPTION=NRESULT;
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SIMULATION TECHNIQUES
 Conversational system
69 /EXEC/
BEGIN
70
FOR N:=150 STEP 150 UNTIL 750 DO
71
FOR M:=1 STEP 1 UNTIL 20 DO
72
BEGIN
73
PRINT(" ");
74
PRINT("NUMERO DE USUARIOS =",N);
75
PRINT("FACTOR DE MULTIPROGRAMACION =",M);
76
SIMUL;
77
PRINT("TIEMPO DE RESPUESTA DE LA CLASE C1
=",MRESPONS
==> E(R(1))," +/-"),CRESPONSE(R(1));
78
PRINT("TIEMPO DE RESPUESTA DE LA CLASE C2
=",MRESPONS
==> E(R(2))," +/-",CRESPONSE(R(2)));
79
PRINT("PRODUCTIVIDAD C1 =",MTHRUPUT(TERMINAL,C1));
80
PRINT("PRODUCTIVIDAD C2 =",MTHRUPUT(TERMINAL,C2));
81
PRINT("PRODUCTIVIDAD TOTAL =",MTHRUPUT(TERMINAL));
82
END;
83
END;
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SIMULATION TECHNIQUES
 Conversational system
NUMERO DE USUARIOS =
150
FACTOR DE MULTIPROGRAMACION =
TIEMPO DE RESPUESTA
TIEMPO DE RESPUESTA
PRODUCTIVIDAD C1 =
PRODUCTIVIDAD C2 =
PRODUCTIVIDAD TOTAL
DE LA CLASE C1 =
DE LA CLASE C2 =
0.2059E-02
0.1356E-02
= 0.3415E-02
NUMERO DE USUARIOS =
150
FACTOR DE MULTIPROGRAMACION =
TIEMPO DE RESPUESTA
TIEMPO DE RESPUESTA
PRODUCTIVIDAD C1 =
PRODUCTIVIDAD C2 =
PRODUCTIVIDAD TOTAL
1
1203.
1368.
+/+/-
132.0
145.4
292.9
499.6
+/+/-
14.44
23.42
2
DE LA CLASE C1 =
DE LA CLASE C2 =
0.2131E-02
0.1380E-02
= 0.3511E-02
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SIMULATION TECHNIQUES
 Communication network

Identical to the previously studied by analytical
techniques
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SIMULATION TECHNIQUES
 Communication network
1 /DECLARE/ QUEUE INTEGER J;
2
QUEUE GEN,LINIA(8,8),R(4,4);
3
INTEGER DIREC(8,4)=(0,2,2,2,
4
1,4,4,8,
5
4,0,4,4,
6
2,3,6,8,
7
6,6,0,6,
8
8,4,5,8,
9
8,8,8,0,
10
2,4,6,7);
11
12
REAL TRAB,TRAC,TRAD,TRBA,TRBC,TRBD,TRCA,TRCB,TRCD,TR
==> DA,TRDB,TRDC;
13
INTEGER OD(4,4) = ( 0, 60,140,240,
14
315,315,365,390,
15
470,590,590,630,
16
730,880,930,930);
17
INTEGER INIC(4) = (1, 3, 5, 7);
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SIMULATION TECHNIQUES
 Communication network
18
INTEGER I,K;
19
CUSTOMER INTEGER ORIG,DESTI;
20
REF CUSTOMER C;
21 /EXEC/
FOR I := 1 STEP 1 UNTIL 8 DO FOR K := 1 STEP 1 UNTIL
==> 8 DO
22
LINIA(I,K).J := K;
23 /STATION/ NAME = GEN;
24
TYPE = SOURCE;
25
SERVICE = BEGIN
26
EXP(60000./930.);
27
I:= RINT(1,930);
28
ORIG := 1;
29
DESTI := 1;
30
WHILE (I > OD(ORIG,DESTI)) DO
31
IF DESTI < 4 THEN DESTI := DESTI + 1
32
ELSE
33
BEGIN
34
ORIG := ORIG + 1;
35
DESTI := 1;
36
END;
37
C := NEW(CUSTOMER);
38
TRANSIT(C,R(ORIG,DESTI));
39
TRANSIT(LINIA(INIC(ORIG),DIREC(INIC(ORIG
==> ),DESTI)));
40
END;
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SIMULATION TECHNIQUES
 Communication network
42 /STATION/ NAME = LINIA;
43
SERVICE = BEGIN
44
EXP(256.*8./64.);
45
I:=DIREC(J,DESTI);
46
IF I=0 THEN
47
BEGIN
48
C := R(ORIG,DESTI).FIRST;
49
WHILE C <> SON DO C := C.NEXT;
50
TRANSIT(C,OUT);
51
TRANSIT (OUT);
52
END;
53
TRANSIT (LINIA(J,I));
54
END;
55
56 /CONTROL/ TMAX= 300000;
57
ACCURACY= ALL QUEUE;
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SIMULATION TECHNIQUES
 Communication network
59 /EXEC/
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
BEGIN
PRINT;
SIMUL;
TRAB:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+
MRESPONSE(LINIA(4,3));
TRAC:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,4))+
MRESPONSE(LINIA(4,6))+MRESPONSE(LINIA(6,5));
TRAD:=MRESPONSE(LINIA(1,2))+MRESPONSE(LINIA(2,8))+
MRESPONSE(LINIA(8,7));
TRBA:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,2))+
MRESPONSE(LINIA(2,1));
TRBC:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,6))+
MRESPONSE(LINIA(6,5));
TRBD:=MRESPONSE(LINIA(3,4))+MRESPONSE(LINIA(4,8))+
MRESPONSE(LINIA(8,7));
TRCA:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+
MRESPONSE(LINIA(8,2))+MRESPONSE(LINIA(2,1));
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SIMULATION TECHNIQUES
 Communication network
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
TRCD:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,8))+
MRESPONSE(LINIA(8,7));
TRCB:=MRESPONSE(LINIA(5,6))+MRESPONSE(LINIA(6,4))+
MRESPONSE(LINIA(4,3));
TRDA:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,2))+
MRESPONSE(LINIA(2,1));
TRDB:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,4))+
MRESPONSE(LINIA(4,3));
TRDC:=MRESPONSE(LINIA(7,8))+MRESPONSE(LINIA(8,6))+
MRESPONSE(LINIA(6,5));
PRINT(TRAB,TRAC,TRAD);
PRINT(TRBA,TRBC,TRBD);
PRINT(TRCA,TRCB,TRCD);
PRINT(TRDA,TRDB,TRDC);
END;
Facultade de Informatica. A Coruña. Junio 2005
235
SIMULATION TECHNIQUES
 Communication network
***SIMULATION WITH SPECTRAL METHOD ***
.. TIME =
300000.00 , NB SAMPLES =
512 , CONF. LEVEL = 0.95
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
*
*
*
*
*
*
*
* GEN
* 64.03
* 1.000
* 1.000
* 64.03
*
4684*
*
+/* 1.829
*0.0000E+00*0.0000E+00* 1.829
*
*
*
*
*
*
*
*
*
* LINIA 2 * 30.88
*0.1246
*0.1477
* 36.61
*
1210*
*
+/* 2.017
*0.1055E-01*0.1607E-01* 3.188
*
*
*
*
*
*
*
*
*
* LINIA 9 * 31.34
*0.1347
*0.1531
* 35.63
*
1289*
*
+/* 1.999
*0.1170E-01*0.1384E-01* 2.597
*
*
*
*
*
*
*
*
*
* LINIA 12 * 31.07
*0.7519E-01*0.8277E-01* 34.20
*
726*
*
+/* 2.504
*0.7059E-02*0.1367E-01* 3.444
*
*
*
*
*
*
*
*
*
Facultade de Informatica. A Coruña. Junio 2005
236
SIMULATION TECHNIQUES
 Communication network
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
* LINIA 16 * 34.91
*0.5632E-01*0.6102E-01* 37.82
*
484*
*
+/* 4.215
*0.9778E-02*0.1091E-01* 5.125
*
*
*
*
*
*
*
*
*
* LINIA 20 * 32.59
*0.7801E-01*0.8174E-01* 34.15
*
718*
*
+/* 2.378
*0.7146E-02*0.8014E-02* 2.579
*
*
*
*
*
*
*
*
*
* LINIA 26 * 34.57
*0.4171E-01*0.4353E-01* 36.08
*
362*
*
+/* 3.707
*0.6404E-02*0.6778E-02* 4.122
*
*
*
*
*
*
*
*
*
* LINIA 27 * 32.92
*0.1857
*0.2277
* 40.37
*
1691*
*
+/* 1.851
*0.1625E-01*0.2106E-01* 3.039
*
*
*
*
*
*
*
*
*
* LINIA 30 * 35.30
*0.7448E-01*0.8044E-01* 38.12
*
633*
*
+/* 3.819
*0.9264E-02*0.1115E-01* 4.429
*
*
Facultade de Informatica. A Coruña. Junio 2005
237
SIMULATION TECHNIQUES
 Communication network
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
* LINIA 32 * 28.69
*0.1243E-01*0.1243E-01* 28.69
*
130*
*
+/* 4.742
*0.2918E-02*0.2918E-02* 4.742
*
*
*
*
*
*
*
*
*
* LINIA 38 * 30.73
*0.1345
*0.1547
* 35.35
*
1313*
*
+/* 1.734
*0.1065E-01*0.1591E-01* 2.745
*
*
*
*
*
*
*
*
*
* LINIA 44 * 32.35
*0.6922E-01*0.7420E-01* 34.68
*
642*
*
+/* 2.659
*0.8011E-02*0.9049E-02* 3.228
*
*
*
*
*
*
*
*
*
* LINIA 45 * 31.92
*0.9321E-01*0.1024
* 35.07
*
876*
*
+/* 2.950
*0.8071E-02*0.1099E-01* 3.485
*
*
*
*
*
*
*
*
*
* LINIA 48 * 29.55
*0.6609E-01*0.6892E-01* 30.81
*
671*
*
+/* 2.910
*0.6452E-02*0.6856E-02* 2.968
*
*
*
*
*
*
*
*
*
Facultade de Informatica. A Coruña. Junio 2005
238
SIMULATION TECHNIQUES
 Communication network
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
* LINIA 56 * 32.19
*0.1548
*0.1929
* 40.11
*
1443*
*
+/* 2.196
*0.1281E-01*0.2053E-01* 3.851
*
*
*
*
*
*
*
*
*
* LINIA 58 * 32.63
*0.1008
*0.1104
* 35.71
*
927*
*
+/* 2.415
*0.9855E-02*0.1273E-01* 3.771
*
*
*
*
*
*
*
*
*
* LINIA 60 * 33.53
*0.8171E-01*0.8924E-01* 36.62
*
731*
*
+/* 2.046
*0.8194E-02*0.9643E-02* 3.040
*
*
*
*
*
*
*
*
*
* LINIA 62 * 35.31
*0.2860E-01*0.2887E-01* 35.64
*
243*
*
+/* 5.058
*0.5594E-02*0.5755E-02* 5.120
*
*
*
*
*
*
*
*
*
* LINIA 63 * 34.11
*0.9402E-01*0.1019
* 36.97
*
827*
*
+/* 2.188
*0.8583E-02*0.9742E-02* 2.344
*
*
Facultade de Informatica. A Coruña. Junio 2005
239
SIMULATION TECHNIQUES
 Communication network
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
* R
2 *0.0000E+00*0.0000E+00*0.1180
* 110.9
*
319*
*
+/*0.0000E+00*0.0000E+00*0.3193E-01* 9.902
*
*
*
*
*
*
*
*
*
* R
3 *0.0000E+00*0.0000E+00*0.1920
* 141.5
*
407*
*
+/*0.0000E+00*0.0000E+00*0.2514E-01* 14.12
*
*
*
*
*
*
*
*
*
* R
4 *0.0000E+00*0.0000E+00*0.1836
* 113.8
*
484*
*
+/*0.0000E+00*0.0000E+00*0.2303E-01* 8.691
*
*
*
*
*
*
*
*
*
* R
5 *0.0000E+00*0.0000E+00*0.1280
* 106.1
*
362*
*
+/*0.0000E+00*0.0000E+00*0.1609E-01* 7.103
*
*
*
*
*
*
*
*
*
* R
7 *0.0000E+00*0.0000E+00*0.8297E-01* 110.1
*
226*
*
+/*0.0000E+00*0.0000E+00*0.1289E-01* 9.790
*
*
Facultade de Informatica. A Coruña. Junio 2005
240
SIMULATION TECHNIQUES
 Communication network
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
* R
8 *0.0000E+00*0.0000E+00*0.4081E-01* 94.18
*
130*
*
+/*0.0000E+00*0.0000E+00*0.7638E-02* 9.646
*
*
*
*
*
*
*
*
*
* R
9 *0.0000E+00*0.0000E+00*0.2082
* 136.4
*
458*
*
+/*0.0000E+00*0.0000E+00*0.2374E-01* 6.981
*
*
*
*
*
*
*
*
*
* R
10 *0.0000E+00*0.0000E+00*0.2370
* 110.8
*
642*
*
+/*0.0000E+00*0.0000E+00*0.2296E-01* 6.225
*
*
*
*
*
*
*
*
*
* R
12 *0.0000E+00*0.0000E+00*0.7255E-01* 102.2
*
213*
*
+/*0.0000E+00*0.0000E+00*0.7541E-02* 7.903
*
*
*
*
*
*
*
*
*
* R
13 *0.0000E+00*0.0000E+00*0.1787
* 114.3
*
469*
*
+/*0.0000E+00*0.0000E+00*0.2146E-01* 8.399
*
*
Facultade de Informatica. A Coruña. Junio 2005
241
SIMULATION TECHNIQUES
 Communication network
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
* R
14 *0.0000E+00*0.0000E+00*0.2829
* 116.2
*
730*
*
+/*0.0000E+00*0.0000E+00*0.3170E-01* 6.388
*
*
*
*
*
*
*
*
*
* R
15 *0.0000E+00*0.0000E+00*0.8914E-01* 110.0
*
243*
*
+/*0.0000E+00*0.0000E+00*0.1678E-01* 9.066
*
*
*
*
*
*
*
*
*
*******************************************************************
... END OF SIMULATION ...
111.2
105.9
137.5
111.4
144.0
107.3
110.4
117.1
111.4
99.80
103.1
110.8
Facultade de Informatica. A Coruña. Junio 2005
242
SIMULATION TECHNIQUES
Token ring network

8 nodes

Uniform traffic
Facultade de Informatica. A Coruña. Junio 2005
243
SIMULATION TECHNIQUES
Token ring network
1 /DECLARE/ QUEUE INTEGER N, M;
2
QUEUE NUS(8),ESP(8),S,R;
3
INTEGER I;
4
REAL TARR;
5
CUSTOMER REAL TSERV;
6
CUSTOMER INTEGER ORIGEN,DESTI;
7
REF CUSTOMER C,D;
8
FLAG SEM;
9
CLASS TOK,MIS;
10
Facultade de Informatica. A Coruña. Junio 2005
244
SIMULATION TECHNIQUES
Token ring network
11 /STATION/ NAME = S;
12
TYPE = SOURCE;
13
SERVICE = BEGIN
14
EXP(TARR);
15
ORIGEN := RINT(1,8);
16
DESTI := RINT(1,7);
17
IF DESTI >= ORIGEN THEN DESTI := DESTI + 1;
18
TSERV := EXP(800.); & microsegons
19
C := NEW(CUSTOMER);
20
TRANSIT(C,R,MIS);
21
TRANSIT(ESP(ORIGEN),MIS);
22
END;
23
Facultade de Informatica. A Coruña. Junio 2005
245
SIMULATION TECHNIQUES
Token ring network
24 /STATION/ NAME = NUS;
25
TYPE = MULTIPLE(2);
26
SERVICE(TOK) = BEGIN
27
WHILE ESP(N).NB > 0 DO
28
BEGIN
29
D := ESP(N).FIRST;
30
CST(D.TSERV);
31
TRANSIT(D,NUS(M));
32
UNSET(SEM);
33
WAIT(SEM);
34
END;
35
CST(20); & microsegons
36
TRANSIT(NUS(M));
37
END;
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246
SIMULATION TECHNIQUES
Token ring network
38
SERVICE(MIS) = BEGIN
39
IF N = ORIGEN THEN
40
BEGIN
41
SET(SEM);
42
TRANSIT(OUT);
43
END;
44
IF N = DESTI THEN
45
BEGIN
46
C := R.FIRST;
47
WHILE SON <> C DO C := C.NEXT;
48
TRANSIT(C,OUT);
49
END;
50
CST(1.6); & microsegons
51
TRANSIT(NUS(M));
52
END;
53
54 /STATION/ NAME = NUS(1);
55
INIT(TOK) = 1;
Facultade de Informatica. A Coruña. Junio 2005
247
SIMULATION TECHNIQUES
Token ring network
57 /CONTROL/ TMAX = 10000000; CLASS = ALL QUEUE;
58
ACCURACY = ALL QUEUE, ALL CLASS;
59
60 /EXEC/
BEGIN
61
FOR I := 1 STEP 1 UNTIL 8 DO
62
BEGIN
63
NUS(I).N := I;
64
ESP(I).N := I;
65
NUS(I).M := I + 1;
66
END;
67
NUS(8).M := 1;
68
FOR TARR := 100000, 10000, 1000, 800 DO
69
BEGIN
70
PRINT;
71
PRINT ("TEMPS ENTRE ARRIBADES ",TARR,"MICROSEG");
72
SIMUL;
73
END;
74
END;
Facultade de Informatica. A Coruña. Junio 2005
248
SIMULATION TECHNIQUES
Token ring network
TEMPS ENTRE ARRIBADES
0.1000E+06MICROSEG
***SIMULATION WITH SPECTRAL METHOD ***
... TIME = 10000000.00 , NB SAMPLES =
512 , CONF. LEVEL = 0.95
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
* NUS
1 * 20.11
*0.6248E-01*0.1250
* 20.11
*
62141*
*
+/*0.1347
*0.3563E-03*0.7126E-03*0.1347
*
*
*(TOK
)* 20.14
*0.6247E-01*0.1249
* 20.14
*
62038*
*
+/*0.1349
*0.3569E-03*0.7137E-03*0.1349
*
*
*(MIS
)* 1.398
*0.7199E-05*0.1440E-04* 1.398
*
103*
*
+/*-1.000
*0.1365E-05*0.2730E-05*-1.000
*
*
Facultade de Informatica. A Coruña. Junio 2005
249
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/-
2 * 20.07
*0.6237E-01*0.1247
* 20.07
*
*0.8790E-01*0.3143E-03*0.6286E-03*0.8790E-01*
)* 20.10
*0.6236E-01*0.1247
* 20.10
*
*0.8881E-01*0.3151E-03*0.6302E-03*0.8881E-01*
)* 1.382
*0.7119E-05*0.1424E-04* 1.382
*
*-1.000
*0.1691E-05*0.3381E-05*-1.000
*
3 * 20.09
*0.6242E-01*0.1248
* 20.09
*
*0.7185E-01*0.2305E-03*0.4610E-03*0.7185E-01*
)* 20.12
*0.6241E-01*0.1248
* 20.12
*
*0.7276E-01*0.2315E-03*0.4629E-03*0.7276E-01*
)* 1.351
*0.6959E-05*0.1392E-04* 1.351
*
*-1.000
*0.1862E-05*0.3724E-05*-1.000
*
4 * 20.08
*0.6239E-01*0.1248
* 20.08
*
*0.1148
*0.3504E-03*0.7009E-03*0.1148
*
)* 20.11
*0.6238E-01*0.1248
* 20.11
*
*0.1140
*0.3514E-03*0.7028E-03*0.1140
*
)* 1.522
*0.7839E-05*0.1568E-04* 1.522
*
*-1.000
*0.1646E-05*0.3292E-05*-1.000
*
Facultade de Informatica. A Coruña. Junio 2005
62140*
*
62037*
*
103*
*
62140*
*
62037*
*
103*
*
62140*
*
62037*
*
103*
*
250
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/-
5 * 20.19
*0.6274E-01*0.1255
* 20.19
*
*0.1325
*0.3989E-03*0.7978E-03*0.1325
*
)* 20.22
*0.6273E-01*0.1255
* 20.22
*
*0.1470
*0.3994E-03*0.7988E-03*0.1470
*
)* 1.382
*0.7119E-05*0.1424E-04* 1.382
*
*-1.000
*0.1337E-05*0.2674E-05*-1.000
*
6 * 20.11
*0.6249E-01*0.1250
* 20.11
*
*0.1433
*0.4475E-03*0.8951E-03*0.1433
*
)* 20.14
*0.6248E-01*0.1250
* 20.14
*
*0.1429
*0.4480E-03*0.8960E-03*0.1429
*
)* 1.398
*0.7199E-05*0.1440E-04* 1.398
*
*-1.000
*0.1526E-05*0.3053E-05*-1.000
*
7 * 20.11
*0.6249E-01*0.1250
* 20.11
*
*0.1094
*0.3093E-03*0.6186E-03*0.1094
*
)* 20.15
*0.6249E-01*0.1250
* 20.15
*
*0.9936E-01*0.3094E-03*0.6187E-03*0.9936E-01*
)* 1.398
*0.7199E-05*0.1440E-04* 1.398
*
*-1.000
*0.1483E-05*0.2966E-05*-1.000
*
Facultade de Informatica. A Coruña. Junio 2005
62140*
*
62037*
*
103*
*
62140*
*
62037*
*
103*
*
62140*
*
62037*
*
103*
*
251
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/-
8 * 20.17
*0.6268E-01*0.1254
* 20.17
*0.1637
*0.4324E-03*0.8647E-03*0.1637
)* 20.20
*0.6267E-01*0.1253
* 20.20
*0.1642
*0.4327E-03*0.8654E-03*0.1642
)* 1.367
*0.7039E-05*0.1408E-04* 1.367
*-1.000
*0.1334E-05*0.2667E-05*-1.000
1 *0.0000E+00*0.0000E+00*0.9553E-03* 734.8
*-1.000
*0.0000E+00*0.8521E-03*-1.000
)*0.0000E+00*0.0000E+00*0.9553E-03* 734.8
*-1.000
*0.0000E+00*0.8521E-03*-1.000
2 *0.0000E+00*0.0000E+00*0.7404E-03* 528.8
*-1.000
*0.0000E+00*0.6695E-03*-1.000
)*0.0000E+00*0.0000E+00*0.7404E-03* 528.8
*-1.000
*0.0000E+00*0.6695E-03*-1.000
3 *0.0000E+00*0.0000E+00*0.8659E-03* 541.2
*-1.000
*0.0000E+00*0.4874E-03*-1.000
)*0.0000E+00*0.0000E+00*0.8659E-03* 541.2
*-1.000
*0.0000E+00*0.4874E-03*-1.000
Facultade de Informatica. A Coruña. Junio 2005
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
62140*
*
62037*
*
103*
*
13*
*
13*
*
14*
*
14*
*
16*
*
16*
*
252
SIMULATION TECHNIQUES
Token ring network
* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/-
4 *0.0000E+00*0.0000E+00*0.7169E-03* 1434.
*-1.000
*0.0000E+00*0.6916E-03*-1.000
)*0.0000E+00*0.0000E+00*0.7169E-03* 1434.
*-1.000
*0.0000E+00*0.6916E-03*-1.000
5 *0.0000E+00*0.0000E+00*0.1484E-02* 1060.
*-1.000
*0.0000E+00*0.9771E-03*-1.000
)*0.0000E+00*0.0000E+00*0.1484E-02* 1060.
*-1.000
*0.0000E+00*0.9771E-03*-1.000
6 *0.0000E+00*0.0000E+00*0.9811E-03* 754.7
*-1.000
*0.0000E+00*0.9433E-03*-1.000
)*0.0000E+00*0.0000E+00*0.9811E-03* 754.7
*-1.000
*0.0000E+00*0.9433E-03*-1.000
7 *0.0000E+00*0.0000E+00*0.9871E-03* 759.3
*-1.000
*0.0000E+00*0.6366E-03*-1.000
)*0.0000E+00*0.0000E+00*0.9871E-03* 759.3
*-1.000
*0.0000E+00*0.6366E-03*-1.000
Facultade de Informatica. A Coruña. Junio 2005
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
5*
*
5*
*
14*
*
14*
*
13*
*
13*
*
13*
*
13*
*
253
SIMULATION TECHNIQUES
Token ring network
* ESP
8 *0.0000E+00*0.0000E+00*0.1368E-02* 912.0
*
15*
*
+/*-1.000
*0.0000E+00*0.1041E-02*-1.000
*
*
*(MIS
)*0.0000E+00*0.0000E+00*0.1368E-02* 912.0
*
15*
*
+/*-1.000
*0.0000E+00*0.1041E-02*-1.000
*
*
* S
*0.9609E+05* 1.000
* 1.000
*0.9609E+05*
103*
*
+/*-1.000
*0.0000E+00*0.0000E+00*-1.000
*
*
*
*
*
*
*
*
*
* R
*0.0000E+00*0.0000E+00*0.8148E-02* 791.0
*
103*
*
+/*-1.000
*0.0000E+00*0.2265E-02*-1.000
*
*
*(MIS
)*0.0000E+00*0.0000E+00*0.8148E-02* 791.0
*
103*
*
+/*-1.000
*0.0000E+00*0.2265E-02*-1.000
*
*
*
*
*
*
*
*
*
*******************************************************************
... END OF SIMULATION ...
Facultade de Informatica. A Coruña. Junio 2005
254
SIMULATION TECHNIQUES
Token ring network
TEMPS ENTRE ARRIBADES
0.1000E+05MICROSEG
***SIMULATION WITH SPECTRAL METHOD ***
... TIME = 10000000.00 , NB SAMPLES =
512 , CONF. LEVEL = 0.95
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
* NUS
1 * 21.31
*0.6245E-01*0.1249
* 21.31
*
58612*
*
+/*0.4077
*0.1228E-02*0.2456E-02*0.4077
*
*
*(TOK
)* 21.66
*0.6238E-01*0.1248
* 21.66
*
57608*
*
+/*0.4775
*0.1230E-02*0.2460E-02*0.4775
*
*
*(MIS
)* 1.402
*0.7039E-04*0.1408E-03* 1.402
*
1004*
*
+/*0.3061E-01*0.5425E-05*0.1085E-04*0.3061E-01*
*
Facultade de Informatica. A Coruña. Junio 2005
255
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/-
2 * 21.70
*0.6360E-01*0.1272
* 21.70
*
*0.5427
*0.1433E-02*0.2866E-02*0.5427
*
)* 22.06
*0.6353E-01*0.1271
* 22.06
*
*0.5847
*0.1434E-02*0.2868E-02*0.5847
*
)* 1.381
*0.6935E-04*0.1387E-03* 1.381
*
*0.4022E-01*0.4872E-05*0.9745E-05*0.4022E-01*
3 * 21.54
*0.6314E-01*0.1263
* 21.54
*
*0.5847
*0.1705E-02*0.3410E-02*0.5847
*
)* 21.90
*0.6307E-01*0.1261
* 21.90
*
*0.5530
*0.1706E-02*0.3412E-02*0.5530
*
)* 1.393
*0.6991E-04*0.1398E-03* 1.393
*
*0.3717E-01*0.4980E-05*0.9959E-05*0.3717E-01*
4 * 21.30
*0.6241E-01*0.1248
* 21.30
*
*0.4710
*0.1174E-02*0.2348E-02*0.4710
*
)* 21.64
*0.6234E-01*0.1247
* 21.64
*
*0.4629
*0.1175E-02*0.2349E-02*0.4629
*
)* 1.399
*0.7023E-04*0.1405E-03* 1.399
*
*0.3573E-01*0.4258E-05*0.8516E-05*0.3573E-01*
Facultade de Informatica. A Coruña. Junio 2005
58612*
*
57608*
*
1004*
*
58612*
*
57608*
*
1004*
*
58611*
*
57607*
*
1004*
*
256
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/-
5 * 21.02
*0.6161E-01*0.1232
* 21.02
*
*0.4159
*0.9359E-03*0.1872E-02*0.4159
*
)* 21.37
*0.6154E-01*0.1231
* 21.37
*
*0.4241
*0.9370E-03*0.1874E-02*0.4241
*
)* 1.415
*0.7103E-04*0.1421E-03* 1.415
*
*0.3856E-01*0.7813E-05*0.1563E-04*0.3856E-01*
6 * 21.06
*0.6171E-01*0.1234
* 21.06
*
*0.4502
*0.1248E-02*0.2497E-02*0.4502
*
)* 21.40
*0.6164E-01*0.1233
* 21.40
*
*0.4786
*0.1249E-02*0.2499E-02*0.4786
*
)* 1.413
*0.7095E-04*0.1419E-03* 1.413
*
*0.3104E-01*0.4489E-05*0.8978E-05*0.3104E-01*
7 * 21.39
*0.6269E-01*0.1254
* 21.39
*
*0.4933
*0.1266E-02*0.2532E-02*0.4933
*
)* 21.74
*0.6262E-01*0.1252
* 21.74
*
*0.3923
*0.1268E-02*0.2537E-02*0.3923
*
)* 1.410
*0.7079E-04*0.1416E-03* 1.410
*
*0.3545E-01*0.7663E-05*0.1533E-04*0.3545E-01*
Facultade de Informatica. A Coruña. Junio 2005
58611*
*
57607*
*
1004*
*
58611*
*
57607*
*
1004*
*
58611*
*
57607*
*
1004*
*
257
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/-
8 * 21.48
*0.6294E-01*0.1259
* 21.48
*
*0.6782
*0.1235E-02*0.2471E-02*0.6782
*
)* 21.83
*0.6287E-01*0.1257
* 21.83
*
*0.7549
*0.1235E-02*0.2470E-02*0.7549
*
)* 1.385
*0.6951E-04*0.1390E-03* 1.385
*
*0.3685E-01*0.4594E-05*0.9188E-05*0.3685E-01*
1 *0.0000E+00*0.0000E+00*0.1149E-01* 926.7
*
*-1.000
*0.0000E+00*0.3546E-02*-1.000
*
)*0.0000E+00*0.0000E+00*0.1149E-01* 926.7
*
*-1.000
*0.0000E+00*0.3546E-02*-1.000
*
2 *0.0000E+00*0.0000E+00*0.1344E-01* 980.8
*
*0.0000E+00*0.0000E+00*0.3167E-02* 125.4
*
)*0.0000E+00*0.0000E+00*0.1344E-01* 980.8
*
*0.0000E+00*0.0000E+00*0.3167E-02* 125.4
*
3 *0.0000E+00*0.0000E+00*0.1317E-01* 1013.
*
*0.0000E+00*0.0000E+00*0.3675E-02* 180.5
*
)*0.0000E+00*0.0000E+00*0.1317E-01* 1013.
*
*0.0000E+00*0.0000E+00*0.3675E-02* 180.5
*
Facultade de Informatica. A Coruña. Junio 2005
58611*
*
57607*
*
1004*
*
124*
*
124*
*
137*
*
137*
*
130*
*
130*
*
258
SIMULATION TECHNIQUES
Token ring network
* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/-
4 *0.0000E+00*0.0000E+00*0.1155E-01* 916.4
*-1.000
*0.0000E+00*0.3341E-02*-1.000
)*0.0000E+00*0.0000E+00*0.1155E-01* 916.4
*-1.000
*0.0000E+00*0.3341E-02*-1.000
5 *0.0000E+00*0.0000E+00*0.1060E-01* 914.1
*-1.000
*0.0000E+00*0.3776E-02*-1.000
)*0.0000E+00*0.0000E+00*0.1060E-01* 914.1
*-1.000
*0.0000E+00*0.3776E-02*-1.000
6 *0.0000E+00*0.0000E+00*0.9766E-02* 834.7
*-1.000
*0.0000E+00*0.2652E-02*-1.000
)*0.0000E+00*0.0000E+00*0.9766E-02* 834.7
*-1.000
*0.0000E+00*0.2652E-02*-1.000
7 *0.0000E+00*0.0000E+00*0.1208E-01* 1015.
*-1.000
*0.0000E+00*0.3174E-02*-1.000
)*0.0000E+00*0.0000E+00*0.1208E-01* 1015.
*-1.000
*0.0000E+00*0.3174E-02*-1.000
Facultade de Informatica. A Coruña. Junio 2005
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
126*
*
126*
*
116*
*
116*
*
117*
*
117*
*
119*
*
119*
*
259
SIMULATION TECHNIQUES
Token ring network
* ESP
8 *0.0000E+00*0.0000E+00*0.1295E-01* 959.5
*
135*
*
+/*0.0000E+00*0.0000E+00*0.3163E-02* 200.6
*
*
*(MIS
)*0.0000E+00*0.0000E+00*0.1295E-01* 959.5
*
135*
*
+/*0.0000E+00*0.0000E+00*0.3163E-02* 200.6
*
*
* S
* 9958.
* 1.000
* 1.000
* 9958.
*
1004*
*
+/* 670.0
*0.0000E+00*0.0000E+00* 670.0
*
*
*
*
*
*
*
*
*
* R
*0.0000E+00*0.0000E+00*0.9552E-01* 951.4
*
1004*
*
+/*0.0000E+00*0.0000E+00*0.9458E-02* 75.77
*
*
*(MIS
)*0.0000E+00*0.0000E+00*0.9552E-01* 951.4
*
1004*
*
+/*0.0000E+00*0.0000E+00*0.9458E-02* 75.77
*
*
*
*
*
*
*
*
*
*******************************************************************
... END OF SIMULATION ...
Facultade de Informatica. A Coruña. Junio 2005
260
SIMULATION TECHNIQUES
Token ring network
TEMPS ENTRE ARRIBADES
1000.
MICROSEG
***SIMULATION WITH SPECTRAL METHOD ***
... TIME = 10000000.00 , NB SAMPLES =
512 , CONF. LEVEL = 0.95
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
*
*
*
*
*
*
*
* NUS
1 * 61.63
*0.6358E-01*0.1272
* 61.63
*
20631*
*
+/* 6.193
*0.3718E-02*0.7435E-02* 6.193
*
*
*(TOK
)* 119.1
*0.6288E-01*0.1258
* 119.1
*
10562*
*
+/* 20.65
*0.3716E-02*0.7433E-02* 20.65
*
*
*(MIS
)* 1.390
*0.6999E-03*0.1400E-02* 1.390
*
10069*
*
+/*0.9350E-02*0.1776E-04*0.3552E-04*0.9350E-02*
*
Facultade de Informatica. A Coruña. Junio 2005
261
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/-
2 * 63.39
*0.6539E-01*0.1308
* 63.39
*
* 4.534
*0.3894E-02*0.7787E-02* 4.534
*
)* 122.5
*0.6468E-01*0.1294
* 122.5
*
* 16.47
*0.3904E-02*0.7809E-02* 16.47
*
)* 1.396
*0.7026E-03*0.1405E-02* 1.396
*
*0.1102E-01*0.2228E-04*0.4456E-04*0.1102E-01*
3 * 62.98
*0.6497E-01*0.1299
* 62.98
*
* 5.727
*0.3575E-02*0.7150E-02* 5.727
*
)* 121.7
*0.6427E-01*0.1285
* 121.7
*
* 29.96
*0.3579E-02*0.7158E-02* 29.96
*
)* 1.389
*0.6994E-03*0.1399E-02* 1.389
*
*0.9545E-02*0.1971E-04*0.3942E-04*0.9545E-02*
4 * 60.30
*0.6220E-01*0.1244
* 60.30
*
* 5.706
*0.4437E-02*0.8874E-02* 5.706
*
)* 116.4
*0.6149E-01*0.1230
* 116.4
*
* 19.09
*0.4432E-02*0.8865E-02* 19.09
*
)* 1.409
*0.7095E-03*0.1419E-02* 1.409
*
*0.1115E-01*0.2008E-04*0.4017E-04*0.1115E-01*
Facultade de Informatica. A Coruña. Junio 2005
20631*
*
10562*
*
10069*
*
20631*
*
10562*
*
10069*
*
20631*
*
10562*
*
10069*
*
262
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/-
5 * 60.82
*0.6274E-01*0.1255
* 60.82
*
* 7.048
*0.4138E-02*0.8275E-02* 7.048
*
)* 117.5
*0.6203E-01*0.1241
* 117.5
*
* 19.66
*0.4143E-02*0.8285E-02* 19.66
*
)* 1.404
*0.7069E-03*0.1414E-02* 1.404
*
*0.1153E-01*0.1863E-04*0.3726E-04*0.1153E-01*
6 * 58.71
*0.6056E-01*0.1211
* 58.71
*
* 6.512
*0.4101E-02*0.8203E-02* 6.512
*
)* 113.3
*0.5985E-01*0.1197
* 113.3
*
* 30.61
*0.4105E-02*0.8210E-02* 30.61
*
)* 1.411
*0.7102E-03*0.1420E-02* 1.411
*
*0.1168E-01*0.2026E-04*0.4051E-04*0.1168E-01*
7 * 57.14
*0.5913E-01*0.1183
* 57.14
*
* 6.698
*0.3162E-02*0.6325E-02* 6.698
*
)* 110.3
*0.5842E-01*0.1168
* 110.3
*
* 18.34
*0.3164E-02*0.6328E-02* 18.34
*
)* 1.409
*0.7096E-03*0.1419E-02* 1.409
*
*0.9951E-02*0.2002E-04*0.4005E-04*0.9951E-02*
Facultade de Informatica. A Coruña. Junio 2005
20631*
*
10562*
*
10069*
*
20631*
*
10562*
*
10069*
*
20630*
*
10561*
*
10069*
*
263
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/-
8 * 65.03
*0.6708E-01*0.1342
* 65.03
*
* 6.675
*0.3661E-02*0.7321E-02* 6.675
*
)* 125.7
*0.6638E-01*0.1328
* 125.7
*
* 32.18
*0.3660E-02*0.7320E-02* 32.18
*
)* 1.390
*0.6998E-03*0.1400E-02* 1.390
*
*0.1438E-01*0.1897E-04*0.3795E-04*0.1438E-01*
1 *0.0000E+00*0.0000E+00*0.7598
* 5756.
*
*0.0000E+00*0.0000E+00*0.1753
* 1254.
*
)*0.0000E+00*0.0000E+00*0.7598
* 5756.
*
*0.0000E+00*0.0000E+00*0.1753
* 1254.
*
2 *0.0000E+00*0.0000E+00*0.7163
* 5564.
*
*0.0000E+00*0.0000E+00*0.1420
* 1172.
*
)*0.0000E+00*0.0000E+00*0.7163
* 5564.
*
*0.0000E+00*0.0000E+00*0.1420
* 1172.
*
3 *0.0000E+00*0.0000E+00*0.7218
* 5436.
*
*0.0000E+00*0.0000E+00*0.1340
* 1028.
*
)*0.0000E+00*0.0000E+00*0.7218
* 5436.
*
*0.0000E+00*0.0000E+00*0.1340
* 1028.
*
Facultade de Informatica. A Coruña. Junio 2005
20630*
*
10561*
*
10069*
*
1319*
*
1319*
*
1285*
*
1285*
*
1325*
*
1325*
*
264
SIMULATION TECHNIQUES
Token ring network
* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/-
4 *0.0000E+00*0.0000E+00*0.6485
*0.0000E+00*0.0000E+00*0.1442
)*0.0000E+00*0.0000E+00*0.6485
*0.0000E+00*0.0000E+00*0.1442
5 *0.0000E+00*0.0000E+00*0.7021
*0.0000E+00*0.0000E+00*0.1777
)*0.0000E+00*0.0000E+00*0.7021
*0.0000E+00*0.0000E+00*0.1777
6 *0.0000E+00*0.0000E+00*0.6706
*0.0000E+00*0.0000E+00*0.1593
)*0.0000E+00*0.0000E+00*0.6706
*0.0000E+00*0.0000E+00*0.1593
7 *0.0000E+00*0.0000E+00*0.6491
*0.0000E+00*0.0000E+00*0.1391
)*0.0000E+00*0.0000E+00*0.6491
*0.0000E+00*0.0000E+00*0.1391
Facultade de Informatica. A Coruña. Junio 2005
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
5402.
1177.
5402.
1177.
5699.
1054.
5699.
1054.
5635.
1203.
5635.
1203.
5413.
1111.
5413.
1111.
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
1199*
*
1199*
*
1232*
*
1232*
*
1190*
*
1190*
*
1198*
*
1198*
*
265
SIMULATION TECHNIQUES
Token ring network
* ESP
8 *0.0000E+00*0.0000E+00*0.6883
* 5200.
*
1321*
*
+/*0.0000E+00*0.0000E+00*0.1291
* 907.1
*
*
*(MIS
)*0.0000E+00*0.0000E+00*0.6883
* 5200.
*
1321*
*
+/*0.0000E+00*0.0000E+00*0.1291
* 907.1
*
*
* S
* 992.0
* 1.000
* 1.000
* 992.0
*
10079*
*
+/* 23.59
*0.0000E+00*0.0000E+00* 23.59
*
*
*
*
*
*
*
*
*
* R
*0.0000E+00*0.0000E+00* 5.561
* 5517.
*
10069*
*
+/*0.0000E+00*0.0000E+00* 1.161
* 1063.
*
*
*(MIS
)*0.0000E+00*0.0000E+00* 5.561
* 5517.
*
10069*
*
+/*0.0000E+00*0.0000E+00* 1.161
* 1063.
*
*
*
*
*
*
*
*
*
*******************************************************************
... END OF SIMULATION ...
Facultade de Informatica. A Coruña. Junio 2005
266
SIMULATION TECHNIQUES
Token ring network
TEMPS ENTRE ARRIBADES
800.0
MICROSEG
***SIMULATION WITH SPECTRAL METHOD ***
... TIME = 10000000.00 , NB SAMPLES =
512 , CONF. LEVEL = 0.95
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
* NUS
1 * 109.5
*0.6738E-01*0.1348
* 109.5
*
12306*
*
+/* 6.905
*0.3586E-02*0.7173E-02* 6.905
*
*
*(TOK
)*0.2511E+05*0.6653E-01*0.1331
*0.2511E+05*
53*
*
+/*-1.000
*0.3590E-02*0.7179E-02*-1.000
*
*
*(MIS
)* 1.388
*0.8504E-03*0.1701E-02* 1.388
*
12253*
*
+/*0.6208E-02*0.9743E-05*0.1949E-04*0.6208E-02*
*
Facultade de Informatica. A Coruña. Junio 2005
267
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/-
2 * 101.4
*0.6239E-01*0.1248
* 101.4
*
* 5.621
*0.2796E-02*0.5592E-02* 5.621
*
)*0.2322E+05*0.6154E-01*0.1231
*0.2322E+05*
*-1.000
*0.2801E-02*0.5602E-02*-1.000
*
)* 1.401
*0.8583E-03*0.1717E-02* 1.401
*
*0.6746E-02*0.1243E-04*0.2486E-04*0.6746E-02*
3 * 96.70
*0.5950E-01*0.1190
* 96.70
*
* 5.425
*0.3074E-02*0.6148E-02* 5.425
*
)*0.2213E+05*0.5864E-01*0.1173
*0.2213E+05*
*-1.000
*0.3074E-02*0.6148E-02*-1.000
*
)* 1.409
*0.8635E-03*0.1727E-02* 1.409
*
*0.6752E-02*0.1281E-04*0.2563E-04*0.6752E-02*
4 * 102.6
*0.6311E-01*0.1262
* 102.6
*
* 5.116
*0.2576E-02*0.5152E-02* 5.116
*
)*0.2349E+05*0.6225E-01*0.1245
*0.2349E+05*
*-1.000
*0.2576E-02*0.5152E-02*-1.000
*
)* 1.407
*0.8622E-03*0.1724E-02* 1.407
*
*0.6784E-02*0.1131E-04*0.2261E-04*0.6784E-02*
Facultade de Informatica. A Coruña. Junio 2005
12306*
*
53*
*
12253*
*
12306*
*
53*
*
12253*
*
12306*
*
53*
*
12253*
*
268
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* NUS
*
+/*(TOK
*
+/*(MIS
*
+/-
5 * 107.5
*0.6615E-01*0.1323
* 107.5
*
* 3.395
*0.1742E-02*0.3485E-02* 3.395
*
)*0.2464E+05*0.6529E-01*0.1306
*0.2464E+05*
*-1.000
*0.1747E-02*0.3494E-02*-1.000
*
)* 1.399
*0.8573E-03*0.1715E-02* 1.399
*
*0.5035E-02*0.1366E-04*0.2733E-04*0.5035E-02*
6 * 97.64
*0.6008E-01*0.1202
* 97.64
*
* 3.719
*0.3193E-02*0.6385E-02* 3.719
*
)*0.2235E+05*0.5922E-01*0.1184
*0.2235E+05*
*-1.000
*0.3198E-02*0.6396E-02*-1.000
*
)* 1.402
*0.8588E-03*0.1718E-02* 1.402
*
*0.6413E-02*0.1194E-04*0.2389E-04*0.6413E-02*
7 * 104.8
*0.6473E-01*0.1295
* 104.8
*
* 4.550
*0.3379E-02*0.6758E-02* 4.550
*
)*0.2446E+05*0.6387E-01*0.1277
*0.2446E+05*
*-1.000
*0.3384E-02*0.6769E-02*-1.000
*
)* 1.396
*0.8553E-03*0.1711E-02* 1.396
*
*0.4878E-02*0.1412E-04*0.2824E-04*0.4878E-02*
Facultade de Informatica. A Coruña. Junio 2005
12306*
*
53*
*
12253*
*
12306*
*
53*
*
12253*
*
12305*
*
52*
*
12253*
*
269
SIMULATION TECHNIQUES
Token ring network
* NUS
*
+/*(TOK
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/-
8 * 103.2
*0.6352E-01*0.1270
* 103.2
*
* 4.519
*0.2117E-02*0.4234E-02* 4.519
*
)*0.2410E+05*0.6266E-01*0.1253
*0.2410E+05*
*-1.000
*0.2115E-02*0.4230E-02*-1.000
*
)* 1.396
*0.8551E-03*0.1710E-02* 1.396
*
*0.5278E-02*0.1169E-04*0.2338E-04*0.5278E-02*
1 *0.0000E+00*0.0000E+00* 20.89
*0.1285E+06*
*0.0000E+00*0.0000E+00* 2.330
*0.1118E+05*
)*0.0000E+00*0.0000E+00* 20.89
*0.1285E+06*
*0.0000E+00*0.0000E+00* 2.330
*0.1118E+05*
2 *0.0000E+00*0.0000E+00* 19.79
*0.1297E+06*
*0.0000E+00*0.0000E+00* 2.213
*0.1044E+05*
)*0.0000E+00*0.0000E+00* 19.79
*0.1297E+06*
*0.0000E+00*0.0000E+00* 2.213
*0.1044E+05*
3 *0.0000E+00*0.0000E+00* 18.60
*0.1274E+06*
*0.0000E+00*0.0000E+00* 1.609
*0.1332E+05*
)*0.0000E+00*0.0000E+00* 18.60
*0.1274E+06*
*0.0000E+00*0.0000E+00* 1.609
*0.1332E+05*
Facultade de Informatica. A Coruña. Junio 2005
12305*
*
52*
*
12253*
*
1622*
*
1622*
*
1523*
*
1523*
*
1458*
*
1458*
*
270
SIMULATION TECHNIQUES
Token ring network
* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/* ESP
*
+/*(MIS
*
+/-
4 *0.0000E+00*0.0000E+00*
*0.0000E+00*0.0000E+00*
)*0.0000E+00*0.0000E+00*
*0.0000E+00*0.0000E+00*
5 *0.0000E+00*0.0000E+00*
*0.0000E+00*0.0000E+00*
)*0.0000E+00*0.0000E+00*
*0.0000E+00*0.0000E+00*
6 *0.0000E+00*0.0000E+00*
*0.0000E+00*0.0000E+00*
)*0.0000E+00*0.0000E+00*
*0.0000E+00*0.0000E+00*
7 *0.0000E+00*0.0000E+00*
*0.0000E+00*0.0000E+00*
)*0.0000E+00*0.0000E+00*
*0.0000E+00*0.0000E+00*
18.17
1.572
18.17
1.572
19.04
2.000
19.04
2.000
19.32
2.245
19.32
2.245
20.31
1.996
20.31
1.996
Facultade de Informatica. A Coruña. Junio 2005
*0.1232E+06*
*0.1280E+05*
*0.1232E+06*
*0.1280E+05*
*0.1240E+06*
*0.1414E+05*
*0.1240E+06*
*0.1414E+05*
*0.1273E+06*
*0.1378E+05*
*0.1273E+06*
*0.1378E+05*
*0.1297E+06*
*0.1299E+05*
*0.1297E+06*
*0.1299E+05*
1474*
*
1474*
*
1535*
*
1535*
*
1517*
*
1517*
*
1561*
*
1561*
*
271
SIMULATION TECHNIQUES
Token ring network
* ESP
8 *0.0000E+00*0.0000E+00* 19.63
*0.1249E+06*
1563*
*
+/*0.0000E+00*0.0000E+00* 1.951
*0.1021E+05*
*
*(MIS
)*0.0000E+00*0.0000E+00* 19.63
*0.1249E+06*
1563*
*
+/*0.0000E+00*0.0000E+00* 1.951
*0.1021E+05*
*
* S
* 811.1
* 1.000
* 1.000
* 811.1
*
12328*
*
+/* 14.66
*0.0000E+00*0.0000E+00* 14.66
*
*
*
*
*
*
*
*
*
* R
*0.0000E+00*0.0000E+00* 155.7
*0.1269E+06*
12253*
*
+/*0.0000E+00*0.0000E+00* 24.78
*0.2250E+05*
*
*(MIS
)*0.0000E+00*0.0000E+00* 155.7
*0.1269E+06*
12253*
*
+/*0.0000E+00*0.0000E+00* 24.78
*0.2250E+05*
*
*
*
*
*
*
*
*
*******************************************************************
... END OF SIMULATION ...
Facultade de Informatica. A Coruña. Junio 2005
272
SIMULATION TECHNIQUES
Ethernet network

8 nodes

Uniform traffic
Facultade de Informatica. A Coruña. Junio 2005
273
SIMULATION TECHNIQUES
Ethernet network
1 /DECLARE/ QUEUE S, EST(8), BUS, R, SF;
2
CUSTOMER INTEGER I;
3
CUSTOMER REAL TSERV, TESP;
4
REAL TARR, SERV, ESP, TEMPS, TPROP, TEMPSC;
5
REF CUSTOMER C,D;
6
LABEL L;
7
FLAG FL, BL;
8
INTEGER CONF = 0;
9
Facultade de Informatica. A Coruña. Junio 2005
274
SIMULATION TECHNIQUES
Ethernet network
1 /DECLARE/ QUEUE S, EST(8), BUS, R, SF;
2
CUSTOMER INTEGER I;
3
CUSTOMER REAL TSERV, TESP;
4
REAL TARR, SERV, ESP, TEMPS, TPROP, TEMPSC;
5
REF CUSTOMER C,D;
6
LABEL L;
7
FLAG FL, BL;
8
INTEGER CONF = 0;
9
Facultade de Informatica. A Coruña. Junio 2005
275
SIMULATION TECHNIQUES
Ethernet network
10 /STATION/ NAME = S;
11
TYPE = SOURCE;
12
SERVICE = BEGIN
13
EXP(TARR);
14
I := RINT(1,8);
15
TSERV := EXP(SERV);
16
IF TSERV < 3.*TPROP THEN TSERV := 3.*TPROP;
17
C := NEW(CUSTOMER);
18
TRANSIT(C,R);
19
TRANSIT(EST(I));
20
END;
21
Facultade de Informatica. A Coruña. Junio 2005
276
SIMULATION TECHNIQUES
Ethernet network
22 /STATION/ NAME = EST;
23
SERVICE = BEGIN
24
L: IF BUS.NB = 0 THEN
25
BEGIN
26
C := NEW(CUSTOMER);
27
C.TSERV := TSERV;
28
UNSET(FL);
29
UNSET(BL);
30
TEMPS := TIME + TPROP;
31
TRANSIT(C, BUS);
32
WAIT(FL);
Facultade de Informatica. A Coruña. Junio 2005
277
SIMULATION TECHNIQUES
Ethernet network
33
34
35
36
37
38
39
40
41
42
43
44
IF CONF = 0 THEN
BEGIN
SET(BL);
TRANSIT(OUT);
END;
SET(BL);
CONF := 0;
TESP := EXP(ESP);
IF TESP<2.*TPROP THEN TESP := 2. * TPROP;
CST(TESP);
GOTO L;
END;
Facultade de Informatica. A Coruña. Junio 2005
278
SIMULATION TECHNIQUES
Ethernet network
45
46
47
48
49
50
51
52
53
54
55
56
57
IF (TIME < TEMPS) AND (CONF = 0) THEN
BEGIN
CONF := 1;
D := BUS.FIRST;
TEMPSC := TEMPS - TIME;
CST(TEMPSC);
TRANSIT(D, OUT);
SET(FL);
TESP := EXP(ESP);
IF TESP<2.*TPROP THEN TESP := 2. * TPROP;
CST(TESP);
GOTO L;
END;
Facultade de Informatica. A Coruña. Junio 2005
279
SIMULATION TECHNIQUES
Ethernet network
58
59
60
61
62
63
64
65
66
67
68
IF (TIME < TEMPS) AND (CONF = 1) THEN
BEGIN
TESP := EXP(ESP);
IF TESP<2.*TPROP THEN TESP := 2. * TPROP;
CST(TESP);
GOTO L;
END;
WAIT(BL);
GOTO L;
END;
Facultade de Informatica. A Coruña. Junio 2005
280
SIMULATION TECHNIQUES
Ethernet network
69 /STATION/ NAME = BUS;
70
SERVICE = BEGIN
71
CST(TSERV + 2.*TPROP);
72
C := R.FIRST;
73
WHILE C.FATHER <> FATHER DO C := C.NEXT;
74
TRANSIT(C, OUT);
75
SET(FL);
76
TRANSIT(OUT);
77
END;
78
Facultade de Informatica. A Coruña. Junio 2005
281
SIMULATION TECHNIQUES
Ethernet network
79 /STATION/ NAME = SF;
80
INIT = 1;
81
SERVICE = BEGIN
82
SET(FL);
83
SET(BL);
84
TRANSIT(OUT);
85
END;
86
87 /CONTROL/ TMAX = 100000.; ACCURACY = ALL QUEUE;
88
89 /EXEC/ BEGIN
90
TPROP := 0.01;
91
SERV := 0.8;
92
ESP := 0.1;
93
TEMPS := -TPROP;
94
FOR TARR := 5, 2.5, 1.25 DO SIMUL;
95
END;
Facultade de Informatica. A Coruña. Junio 2005
282
SIMULATION TECHNIQUES
Ethernet network
***SIMULATION WITH SPECTRAL METHOD ***
... TIME =
100000.00 , NB SAMPLES =
512 , CONF. LEVEL = 0.95
*******************************************************************
*
NAME
*
SERVICE * BUSY PCT *
CUST NB * RESPONSE *
SERV NB *
*******************************************************************
* S
*
* 5.073
+/-
* EST
*
+/-
* EST
*
+/-
* EST
*
+/-
* EST
*
+/-
* EST
*
+/-
*
19713*
*0.7027E-01*0.0000E+00*0.0000E+00*0.7027E-01*
*
1 *0.9595
* 1.000
* 1.000
* 5.073
*
2491*
*0.4293E-01*0.1102E-02*0.1180E-02*0.5266E-01*
*
2 *0.9714
*0.2390E-01*0.2459E-01*0.9873
*
2433*
*0.3115E-01*0.1343E-02*0.1491E-02*0.3891E-01*
*
3 *0.9993
*0.2364E-01*0.2425E-01*0.9966
*
2536*
*0.4504E-01*0.1101E-02*0.1159E-02*0.4904E-01*
*
4 *0.9660
*0.2534E-01*0.2609E-01* 1.029
*
2433*
*0.4483E-01*0.1456E-02*0.1518E-02*0.7335E-01*
*
5 *0.9709
*0.2350E-01*0.2407E-01*0.9892
*0.2321E-01*0.2356E-01*0.9856
*
2390*
*0.4695E-01*0.1439E-02*0.1497E-02*0.4730E-01*
*
Facultade de Informatica. A Coruña. Junio 2005
283
SIMULATION TECHNIQUES
Ethernet network
* EST
6 *0.9677
*0.2312E-01*0.2366E-01*0.9905
*
2389*
*
+/*0.3450E-01*0.9612E-03*0.1328E-02*0.4168E-01*
*
* EST
7 *0.9455
*0.2308E-01*0.2367E-01*0.9696
*
2441*
*
+/*0.3795E-01*0.1200E-02*0.1322E-02*0.3643E-01*
*
* EST
8 *0.9635
*0.2505E-01*0.2576E-01*0.9906
*
2600*
*
+/*0.3988E-01*0.1589E-02*0.1727E-02*0.4196E-01*
*
* BUS
*0.7994
*0.1624
*0.1624
*0.7994
*
20314*
*
+/*0.9631E-02*0.2830E-02*0.2830E-02*0.9631E-02*
*
* R
*0.0000E+00*0.0000E+00*0.1956
*0.9924
*
19713*
*
+/*0.0000E+00*0.0000E+00*0.5210E-02*0.2011E-01*
*
*******************************************************************
... END OF SIMULATION ...
Facultade de Informatica. A Coruña. Junio 2005
284
SIMULATION TECHNIQUES
Ethernet network
***SIMULATION WITH SPECTRAL METHOD ***
... TIME =
100000.00 , NB SAMPLES =
512 , CONF. LEVEL = 0.95
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
* S
* 2.523
* 1.000
* 1.000
* 2.523
*
39639*
*
+/*0.3238E-01*0.0000E+00*0.0000E+00*0.3238E-01*
*
* EST
1 * 1.190
*0.6058E-01*0.6443E-01* 1.266
*
5091*
*
+/*0.3643E-01*0.1890E-02*0.2066E-02*0.4254E-01*
*
* EST
2 * 1.189
*0.5951E-01*0.6372E-01* 1.274
*
5003*
*
+/*0.3580E-01*0.4234E-02*0.3192E-02*0.4344E-01*
*
* EST
3 * 1.174
*0.5754E-01*0.6086E-01* 1.242
*
4901*
*
+/*0.4252E-01*0.2450E-02*0.2627E-02*0.4427E-01*
*
* EST
4 * 1.191
*0.5883E-01*0.6312E-01* 1.278
*
4937*
*
+/*0.3555E-01*0.2305E-02*0.2675E-02*0.4592E-01*
*
* EST
5 * 1.198
*0.5927E-01*0.6323E-01* 1.278
*
4949*
*
+/*0.4266E-01*0.2633E-02*0.2816E-02*0.4674E-01*
*
Facultade de Informatica. A Coruña. Junio 2005
285
SIMULATION TECHNIQUES
Ethernet network
* EST
6 * 1.185
*0.5835E-01*0.6266E-01* 1.272
*
4926*
*
+/*0.3425E-01*0.2123E-02*0.2498E-02*0.4209E-01*
*
* EST
7 * 1.181
*0.5895E-01*0.6315E-01* 1.265
*
4993*
*
+/*0.3740E-01*0.2661E-02*0.3207E-02*0.5317E-01*
*
* EST
8 * 1.182
*0.5717E-01*0.6104E-01* 1.262
*
4837*
*
+/*0.3262E-01*0.2484E-02*0.3251E-02*0.4343E-01*
*
* BUS
*0.7520
*0.3316
*0.3316
*0.7520
*
44091*
*
+/*0.1085E-01*0.4436E-02*0.4436E-02*0.1085E-01*
*
* R
*0.0000E+00*0.0000E+00*0.5022
* 1.267
*
39637*
*
+/*0.0000E+00*0.0000E+00*0.1750E-01*0.2458E-01*
*
*******************************************************************
... END OF SIMULATION ...
Facultade de Informatica. A Coruña. Junio 2005
286
SIMULATION TECHNIQUES
Ethernet network
***SIMULATION WITH SPECTRAL METHOD ***
... TIME =
100000.00 , NB SAMPLES =
512 , CONF. LEVEL = 0.95
*******************************************************************
*
NAME
* SERVICE * BUSY PCT * CUST NB * RESPONSE * SERV NB *
*******************************************************************
* S
* 1.257
* 1.000
* 1.000
* 1.257
*
79568*
*
+/*0.9795E-02*0.0000E+00*0.0000E+00*0.9795E-02*
*
* EST
1 * 2.045
*0.2052
*0.2709
* 2.699
*
10034*
*
+/*0.5773E-01*0.8730E-02*0.1612E-01*0.1282
*
*
* EST
2 * 2.020
*0.2039
*0.2667
* 2.642
*
10096*
*
+/*0.7101E-01*0.8998E-02*0.1768E-01*0.1390
*
*
* EST
3 * 2.011
*0.1991
*0.2592
* 2.617
*
9904*
*
+/*0.6745E-01*0.8805E-02*0.1469E-01*0.1115
*
*
* EST
4 * 2.028
*0.2000
*0.2602
* 2.637
*
9864*
*
+/*0.5786E-01*0.6761E-02*0.1179E-01*0.1073
*
*
* EST
5 * 2.001
*0.2017
*0.2622
* 2.601
*
10079*
*
+/*0.6415E-01*0.6065E-02*0.1510E-01*0.1624
*
*
Facultade de Informatica. A Coruña. Junio 2005
287
SIMULATION TECHNIQUES
Ethernet network
* EST
6 * 2.035
*0.1993
*0.2610
* 2.664
*
9797*
*
+/*0.7039E-01*0.8648E-02*0.1688E-01*0.1333
*
*
* EST
7 * 2.021
*0.2007
*0.2594
* 2.611
*
9934*
*
+/*0.6259E-01*0.7496E-02*0.1742E-01*0.2369
*
*
* EST
8 * 2.014
*0.1985
*0.2571
* 2.608
*
9858*
*
+/*0.7488E-01*0.7814E-02*0.1403E-01*0.1413
*
*
* BUS
*0.5523
*0.6622
*0.6622
*0.5523
*
119899*
*
+/*0.5897E-02*0.7369E-02*0.7369E-02*0.5897E-02*
*
* R
*0.0000E+00*0.0000E+00* 2.096
* 2.635
*
79566*
*
+/*0.0000E+00*0.0000E+00*0.8279E-01*0.1023
*
*
*******************************************************************
... END OF SIMULATION ...
Facultade de Informatica. A Coruña. Junio 2005
288