Modeling & Simulation- Lecture -4-
5/11/2014
Queuing Systems
Queuing theory establishes a powerful tool in modeling and performance
analysis of many complex systems, such as computer networks, telecommunication
systems, call centers, manufacturing systems and service systems. Many real systems
can be modeled as networks of queues, such as the waiting line in a bank, a bus stop,
waiting line in airport, material flow in factory, and signal flow in network. So, each
of these simple systems consists of three major components:
• Entities:
Something that changes the state of the system. In many cases, particularly those
involving service systems, the entity may be a person. In the customer service center,
the entities are the customers. Entities do not necessarily have to be people; they can
also be objects. Similarly, in the factory example, the entities are components waiting
to be machined; entities can be signals waiting to be received in communication
networks.
• Queues:
Queues are the simulation term for lines. Entities generally wait in a queue until
it is their turn to be processed.
• Resources:
Processer or server that serves the entities those are in the queue. Example
factory machines, computer processer, nodes in communication networks. The
relationships among these components are illustrated in figure (1).
Queue or Buffer
Arrival
Departure
Server
Entities being served
Figure (1) Basic Queuing system component
Queues can be categorized to simple queue, or it can be chained to make a network
of queues; thus the departure from one queue enters to the next queue. Queuing
network can be opened network or closed network. Figure (2) shows the opened
network, while figure (3) shows the closed network representation.
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Lecturer: Hawraa Sh.
Modeling & Simulation- Lecture -4-
Arrival
5/11/2014
Queues
Departure
Server
Server
Figure (2) Opened Queuing Network
Arrival
Queues
Server
Server
Departure
Figure (3) Closed Queuing Network
Basic Queuing System Concepts
Each queuing system is described by characters: A / B /C / X / Y / Z, these
characters called Kendall’s notation:
 The first characteristic (A); specifies the arrival distribution (or the time between
two successive arrivals). The following standard abbreviations are used:
M = Exponential, D = Deterministic, Ek = Erlang type k, GI = General
Independent.
 The second characteristic (B); specifies the service probability distribution.
M = Exponential, D = Deterministic, Ek = Erlang type k, GI = General
Independent.
 The third characteristic (C); specifies the number of parallel identical servers in the
systems, denoted by 1 of theirs only one server and by C if theirs multiple server.
 The fourth characteristic (X); describes the queue length or the number of entities
allowable to enter the system.
 The fifth characteristic (Y); specifies the queue strategy.
A queue strategy determines the discipline for ordering Entities in a queue. It defines
the order in which they are served and the way in which resources are divided between
customers. Here are some of these strategies:
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Lecturer: Hawraa Sh.
Modeling & Simulation- Lecture -4
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First In First Out (FIFO).
Last In First Out (LIFO).
Priority Queue.
The shortest is served first.
Random Service.
Round Robin.
 Calling source for entities asking for service (Z): it considers being finite if
these entities coming from finite population or it can be infinite of the
population they are come from is infinite.
Performance Measurements
One approach to Performance measurement is to obtain the data by observing
the events and activities on an existing system. Mean response time, the total service
time, the workflow, the numbers of completed or aborted service requests, the total
waiting time, the queue length, the number of transactions completed per unit time, the
ratio of blocked connection requests, and The rollback completion time are the most
important performance measures.
When analyzing a system with a single queue, the most common performance
measures are obtained as follows:
 Arrival distribution (λ): or the average time between arrivals (Interarrival times =
1⁄𝜆). This distribution considers being the first statistical pointer that helped to
determine the model type.
 Service rate (  ): number of departing entines per unit time. Service time is
average between departures (Interdeparture time = 1⁄µ).
 Server utilization (ρ): or Traffic intensity describes the fraction of time that the
server is busy, or the mean fraction of active servers, in the case of multiple
servers.
𝜌=
𝜆
µ
 Queue length ( L ) : is the number of Entities in the system including both the
entities waiting for the service and receiving it .
 Waiting time ( W ) : is the total time that the Entities spend in the system waiting to
be served.
 𝑃𝑖 : The probability of a given number of Entities i in the system.
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Lecturer: Hawraa Sh.
Modeling & Simulation- Lecture -4-
5/11/2014
Little's Formula
Little's formula is the most used formula in performance evaluations. It is
important because it can be used with most queuing systems and applies to both single
server systems and network of servers as well. It state that
N = λ ∗ T , L = λ ∗ W, Lq = λ ∗ Wq
Where N is the average number of entities in the system, λ is the arrival rate to
the system, T is the average time spent in the system, W is the average waiting time in
whole system, Wq is the average waiting time in the queue, L is the average number in
the system, and Lq is the average number in the queue .
Markov Chains
The most powerful analytic techniques for evaluating complex system performance
are based on the theory of Markov chains. A random process is called a Markov
Process if, conditional on the current state of the process, its future is independent of its
past. A Markov chain is a mathematical model for stochastic systems whose states,
discrete or continuous, are governed by transition probability. Transitions means the
change of the state of the system, and the probabilities associated with various statechanges are called transition probabilities. The collection of the states and transition
probabilities forms a Markov chain.
Many queuing models are in fact represents Markov processes. There are a
large number of real engineering, industrial, physical, biological, economics, and
social phenomena that can be described and analyzed as Markov chains.
The Birth-Death process
The Birth-Death process is a Continuous Time Markov Chains (CTMC) with
states {0, 1, 2…n-1} for which transition from state i may go only to state i-1 or i+1.
CTMC do not have discrete time or index. The state changes could occur at any
instant of time.
Queuing systems suppose that the entity arrival and departure operations to and
from the system basis on birth-death process. In queuing theory the term birth means
entity arrival to queuing line, and the term death means the departing entity after
getting the service.
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Lecturer: Hawraa Sh.
Modeling & Simulation- Lecture -4-
5/11/2014
M / M / 1 Queue
The M / M / 1 queue is the default representation of Kendall notation of queuing
systems, it contain from single server uses the FIFO service disciplines, and one
waiting line of infinite size. Figure (4) shows the M / M / 1 queue. And the state
diagram of this model explained in figure (5).
L
𝐿𝑞
λ
Server
µ
𝑊𝑞
W
Figure (4) The M / M / 1 queue
The relationship between the performance measurements of M / M / 1 is :
∴ W = Wq +
1
…………….((1))
µ
∴ Lq = λ ∙ Wq
…………….((2))
∴L=λ∙W
…………….((3))
This relationship defined with little's formula.
λ
λ
1
0
µ
λ
n-1
2
µ
λ
µ
λ
n+1
11
n
µ
µ
1
Figure (5) the state diagram of birth-death process for M / M /1
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Lecturer: Hawraa Sh.
Modeling & Simulation- Lecture -4-
5/11/2014
M / M / C Queue
The number of parallel servers is ( C ) where ( C >1 ) and the entities use the
FIFO service disciplines. In this model the arrival rate λn = λ , while the service rate
µn can be calculate as follows :
1 If (𝑛 ≥ 𝑐), the number of entities in the system is more than or equal to the
number of servers; this means all servers is busy; then the average of served
entities is (cµ ) .
2 If (𝑛 < 𝑐), the number of entities in the system is less than the number of
servers; this means that some servers are idle; then the average of served entities
is (nµ ).
Also, we can describe these points as follows:
nµ , 0 < 𝑛 < 𝑐
µn = {
cµ , 𝑐 ≤ 𝑛
λ
λ
λ
0
1
µ
λ
c-1
2
2µ
λ
c
cµ
cµ
Figure (6) the state diagram of birth-death process for M / M /C
We have two types of M / M / C , the first one consist of one waiting queue and all
the parallel servers share this queue, while in the other one each server have its own
waiting queue, the two types can be explain in a and b of figure (7).
µ
µ
Arrival
Server
1
Server
2
λ
Departure
µ
Server
n
(a) M / M / C model with one waiting queue.
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Lecturer: Hawraa Sh.
Modeling & Simulation- Lecture -4-
5/11/2014
λ/n
µ
λ/n
µ
Server
1
Server
2
Arrival
λ/n
Departure
µ
Server
n
(b) M / M /C model with one queue for each server.
Figure (7) M / M /C model
Tandem Queue
This model consist of (j) sequence stages, where (j>1). And each stage include
number of parallel server (i) where (i≥1). If entity completes its service at any server
or if new entity asking for a service this situation makes change in the state of system.
Figure (8) explain the model,
λ
µ𝟏
µ𝟏
Figure (8) Tandem queue model
Tandem Queue with Feedback
From the previous results, it become so clear for us that the system with two
sequence server is working as two independent M / M / 1 queues . And for applying
this relationship to tandem queue with feedback, we take the model in (a), the state
diagram in figure (9) as an example:
λ
P1
µ𝟎
C
µ𝟏
I
P2
Figure (9) Tandem queue with feedback
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Lecturer: Hawraa Sh.
Modeling & Simulation- Lecture -4-
5/11/2014
The arrival rate for the first server is λ0 and for the second server is λ1 and so the
departure in the steady state its done by λ0 , λ1 .
Entities arriving to the first server (C) from the outside with rate λ or from the second
server (I, feedback) with rate λ1 , thus
λ0 = λ + λ1
………….((1))
When entity arriving to the system, it enter server (C) and after complete its service
enter to server (I), then return to server (C) and finally depart from the system. We
conclude that the entity get service two times at server(C) and one time at server (I).
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Lecturer: Hawraa Sh.