Queuing Networks
Jean-Yves Le Boudec
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Contents
1.
The Class of Multi-Class Product Form Networks
2. The Elements of a Product-Form Network
3. The Product-Form Theorem
4. Computational Aspects
5. What this tells us
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1. Networks of Queues are Important but
May Be Tough to Analyze
Queuing networks are frequently used models
The stability issue may, in general, be a hard one
Necessary condition for stability (Natural Condition)
server utilization < 1
at every queue
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Instability Examples
Poisson arrivals ; jobs go
through stations 1,2,1,2,1
then leave
A job arrives as type 1, then
becomes 2, then 3 etc
Exponential, independent
service times with mean mi
Priority scheduling
Station 1 : 5 > 3 >1
Station 2: 2 > 4
Q: What is the natural
stability condition ?
A: λ (m1 + m3 + m5 ) < 1
λ (m2 + m4) < 1
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λ=1
m1 = m3 = m4 = 0.1
m2 = m5 = 0.6
Utilization factors
Station 1: 0.8
Station 2: 0.7
Network is unstable !
If λ (m1 + … + m5 ) < 1
network is stable;
why?
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Bramson’s Example 1: A Simple FIFO Network
Poisson arrivals; jobs go through
stations A, B,B…,B, A then leave
Exponential, independent service
times
Steps 2 and last: mean is L
Other steps: mean is S
Q: What is the natural stability
condition ?
A: λ ( L + S ) < 1
λ ( (J-1)S + L ) < 1
Bramson showed: may be
unstable whereas natural stability
condition holds
Bramson’s Example 2
A FIFO Network with Arbitrarily Small Utilization Factor
S
S
L
S
S
L
S
S
L
S
Utilization factor at every station ≤ 4 λ S
Network is unstable for
S ≤ 0.01
L ≤ S8
m = floor(-2 (log L )/L)
S
L
m queues
2 types of
customers
λ = 0.5 each type
routing as
shown,
… = 7 visits
FIFO
Exponential
service times,
with mean as
shown
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Take Home Message
The natural stability condition is necessary but may not be sufficient
We will see a class of networks where this never happens
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2. Elements of a Product Form Network
Customers have a class attribute
Customers visit stations according to Markov Routing
External arrivals, if any, are Poisson
2 Stations
Class = step, J+3 classes
Can you reduce the number
of classes ?
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Chains
Customers can switch class, but remain in the same chain
ν
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Chains may be open or closed
Open chain = with Poisson arrivals. Customers must eventually leave
Closed chain: no arrival, no departure; number of customers is constant
Closed network has only closed chains
Open network has only open chains
Mixed network may have both
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3 Stations
4 classes
1 open chain
1 closed chain
ν
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Bramson’s Example 2
A FIFO Network with Arbitrarily Small Utilization Factor
S
S
L
S
S
L
S
S
L
S
S
L
2 Stations
Many classes
2 open chains
Network is open
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Visit Rates
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2 Stations
5 classes
1 chain
Network is open
Visit rates
θ11 = θ13 = θ15 = θ22 = θ24 = λ
θsc = 0 otherwise
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ν
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Constraints on Stations
Stations must belong to a restricted catalog of stations
We first see a few examples, then give the complete catalog
Two categories: Insensitive (= Kelly-Whittle) and MSCCC
Example of Category 1 (insensitive station): Global Processor Sharing
One server
Rate of server is shared equally among all customers present
Service requirements for customers of class c are drawn iid from a distribution
which depends on the class (and the station)
Example of Category 1 (insensitive station): Delay
Infinite number of servers
Service requirements for customers of class c are drawn iid from a distribution
which depends on the class (and the station)
No queuing, service time = service requirement = residence time
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Example of Category 2 (MSCCC station): FIFO with B servers
B servers
FIFO queueing
Service requirements for customers of class c are drawn iid from an exponential
distribution, independent of the class (but may depend on the station)
Example of Category 2 (MSCCC station): MSCCC with B servers
B servers
FIFO queueing with constraints
At most one customer of each class is allowed in service
Service requirements for customers of class c are drawn iid from an exponential
distribution, independent of the class (but may depend on the station)
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Say which network
satisfies the
hypotheses for
product form
B (FIFO, Exp)
A
C (Prio, Exp)
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A station of Category 1 is any station that
satisfies the Kelly-Whittle property
Examples: Global or per-class PS, Global or per-class LCFSPR, Delay
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Stations of Category 2 must have Exponential,
class independent service requirements
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3. The Product Form Theorem
Stationary distrib of numbers of customers has product form
Each term depends only on the station
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Stability
Stability depends only on every station in isolation being stable
When service rates are constant, this is the natural condition
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Product form and independence
In an open network
Product form => independence of stations in stationary regime
No longer true in a closed or mixed network
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Phase-Type Distributions
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Phase Type Distributions
Product form theorem requires service times to be
Either exponential (category 2 stations) i.e MSCCC including FIFO)
Or Phase type (category 1 stations)
Phase type distributions can approximate any distribution (for the topology
of weak convergence)
Stationary Distribution depends only on mean service time
(Insensitivity of category 1)
Therefore, it is reasonable to assume that the product form theorem applies
if we replace a phase type distribution by any distribution (even heavy
tailed)
Was done formally in some cases [8]
Take home message:
Stations of category 1 may have any service time distribution, class dependent
Stations of category 2 must have exponential distrib, class independent
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4. Computational Aspects
Station Function
The station function, used in the Product Form theorem, is the stationary
distribution of the station in isolation
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Different Stations may have same station
equivalent service rate
FIFO single server, global PS and global LCFSPR with class independent
mean service time have same station functions
Check this
Therefore they have the same equivalent service rate and have the same
effect in a network as long as we are interested in the distribution of
numbers of customers
Hence mean response times are the same
But distributions of response times may differ
Compare PS to FIFO
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Algorithms for Mixed Networks
Open networks: all stations are independent; solve one station in isolation
Mixed Networks: suppress open chains (suppression theorem)
Closed networks: the problem is computing the normalizing constant;
Many methods exist, optimized for different types of very large networks
Convolution algorithms: fairly general, applies to tricky cases (MSCCC),
requires storing normalizing constant (large)
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Throughput Theorem
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Example
N = nb customers at Gate
K = total population
Product Form theorem:
μ
ν
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Algorithms for Mixed Networks
Open networks: all stations are independent; solve one station in isolation
Mixed Networks: suppress open chains (suppression theorem)
Closed networks: the problem is computing the normalizing constant;
Many methods exist, optimized for different types of very large networks
Convolution algorithms: fairly general, applies to tricky cases (MSCCC),
requires storing normalizing constant (large)
Mean Value Analysis does not require computing the normalizing constant,
but does not apply (yet ?) to all cases
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The Arrival Theorem and Mean Value
Analysis (MVA) version 1
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The Arrival Theorem and Mean Value
Analysis (MVA) version 1
MVA version 1 uses the arrival theorem in a closed network where all
stations are
FIFO or Delay
or equivalent
Based on 3 equations and iteration on population:
Mean response time for a class c customer at a FIFO station (arrival theorem):
Little’s formula:
Total number of customers gives :
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MVA Version 2
Applies to more general networks;
Uses the decomposition and complement network theorems
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is equivalent to:
where the service rate μ*(n4) is the throughput of
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5. What this tells us
A rich class of networks with interesting properties
Simple stability conditions
Disciplines such as PS are insensitive to anything except mean service times
Classes can be anything; this is a very rich modelling paradigm
Only average visit rates matter
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Example: Model of Internet
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Product form holds if network can be modelled by a Kelly –Whittle station
Requires that rate allocated to class c flows has the form
Statistics of network depend only on traffic intensities
Flow durations and think times may be anything and may be correlated
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Requires that rate allocated to class c flows has the form
Such an allocation is called « balanced fair » and is the only one with
insensitivity property
Is numerically closed to proportional fairness (TCP)
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Questions
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