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Queuing Models

Dr. Mahmoud Alrefaei

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Introduction

• Each one of us has spent a great deal of time waiting in lines.

• One example in the Cafeteria.

• Other examples of queues are– Printer queue– Customers in front of a cashier – Calls waiting for answer by a technical support

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What makes up a queue?• The System: A collection of objects under study.

– It is important to define the system boundaries.

• The Entities: The people, organisms, or objects that enter the system requiring some kind of service.

• The Servers: The people, organisms, or machines that perform the service required.

• The Queue: An accumulation of entities that have entered the system but have not been served.

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Types of Queues

Single Stage

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Multiple Stage - Manufacturing Plant

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Multi-channel Single Stage - Bank

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Parallel Single Stage - Supermarket

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Customer Discrimination – Bus station

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Converging Arrivals - Walk-in and Drive-thru

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Queue Discipline

• First Come First Served - FCFS– Most customer queues.

• Last Come First Served - LCFS– Packages, Elevator.

• Served in Random Order - SIRO– Entering Buses

• Priority Service– Multi-processing on a computer.– Emergency room.

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What factors affect system performance

• The Arrivals Process.– The time between any two successive arrivals – Does this depend on the number of people in the

system?– Finite populations.

• The Service Process.– The time taken to perform the service.– Does this depend on the number of people in the

system?• The number of servers operating in system.

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Measuring System Performance

• The total time an “entity” spends in the system (Denoted by W)

• The time an “entity spends in the queue.

(Denoted by Wq)

• The number of “entities” in the system.(Denoted by L)

• The number of “entities” in the queue.

(Denoted by Lq)

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Measuring System Performance

• The percentage of time the servers are busy (Utilization time)

• These quantities are variable over time.

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Poisson Process• Let {N(t), t>0} be the number of customers

arrive until the time t

• {N(t), is said to be a Poisson Process having rate , for >0, if– N(0) = 0– N(t+s) – N(t) Does not depend on the

previous history– N(t+s) – N(t) is independent of t.

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Poisson Process

The number of events in any interval of length t is Poisson distributed with mean t. That is for all s, t > 0 and n=0,1,2,...

.!

)()()( t

n

en

tnsNstNP

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Inter-Arrival Times

• What are inter-arrival times?

• Ti is the time between the (i-1)-st and the i-th events.

• Poisson Process can be used for modeling arrival process

0

1

2

3

0 2 4 6 8 10 12

Time (t)

A(t)

T1

T2

T3

S1 S2 S3

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Birth-Death Processes• A birth-death process is used to model

populations of entities in a system• The state of the system at time t is the number

of entities in the system at that time, often denoted by N(t).

• Births and deaths occur at a constant rate (like the Poisson process model)

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Birth-Death Processes

Pi,j(t) is defined to be the probability that there are j entities in the system at time t, given that there were i entities in the system at time 0.

0 1 2 j-1 j j+1

0 1 1j j

1 2j 1j

... ...

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Birth-Death Processes• The state of the system must be a non-

negative integer• Law 1

– A birth increases the state from j to j+1– The variable j is called the birth rate for

state j– A birth occurs between times t and t + t with

probability jt + o(t)

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Birth-Death Processes• Law 2

– A death decreases the state from j to j-1

– The variable j is called the death rate for state j (note that 0=0)

– A death occurs between times t and t + t with probability jt + o(t )

• Law 3– Births and Deaths are independent

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Birth-Death Processes• Can more than one event happen between

t and t + t?

• Why must 0=0?

• Knowledge of j and j completely specifies a Birth-Death process.

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Birth-Death Processes• Birth-Death Processes can be used to

model most M/M/... queuing systems.– An arrival is considered a “birth”.– A service completion is considered a “death”.

• Let Pi,j(t) be the probability N(t+s)–N(s)=j given that N(s)=i (or N(t)=j given N(0)=i

• It turns out that for many queuing systems, Pi,j(t) will approach a limit j as t gets larger.

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Birth-Death Processes• This limit will be independent of the

initial state i.j is called the steady state or

equilibrium probability of state j. j can be thought of as the probability that

at some instant in the future there are j entities in the system.

j can also be thought of as the fraction of time that there are j entities in the system.

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Exponential Distribution• The exponential distribution is

characterized by

1)(][

1)(F(t)

0

t

0

dtttfTE

edsetTP ts

222

0

22

1][][)(

)(][

TETETVar

dttftTE

te f(t)

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Exponential Distribution

• What is P(T>t) for an exponential distribution with parameter ? e- t

• P(T>t+s|T>s) is the probability of waiting a further time t after having already waited to time s.

• What is P(T>t+s|T>s) for an exponential distribution with parameter ? e- (t+s) / e- s = e- t

• Answer: P(T>t+s|T>s) = P(T>t) This is called the memoryless property

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An Example of Memoryless

• Suppose that the amount of time one spends in a bank is exponentially distributed with mean ten minutes.– What is ?– What is the probability that a customer will spend

more a quarter of an hour in the bank?• You have been waiting for ten minutes already. Now

what is the probability that you will spend more than a quarter of an hour in the bank?– What has a lack of memory, you or the

distribution?

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Birth-Death Processes

• Consider a M/M/1 queuing system.– Inter-arrival times are exponential with rate .

– Service times are exponential with rate .

• Suppose there are j entities in the system at time t.

What is the probability of an arrival in the interval (t,t + t]? Hint: use Taylor series expansion on F(t + t)-F(t) = 1 - e- t

= t + o(t)

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Birth-Death Processes• So the arrivals and service completions of

a M/M/1 queue are a birth-death process with the following rate diagram

0 1 2 j-1 j j+1

... ...

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Birth-Death Processes (Balance equations)

• It can be shown by induction that

• These are called balance equations• Therefore:

jjjjjjjj 1111

021

110

j

jj

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Birth-Death Processes• If we define the constants

• then

• and we know that

j

jjc

21

110

0 jj c

11

000

jj

jj c

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Birth-Death Processes

• So

• This means that the infinite sum of the cj’s must converge.

• If this sum is infinite then no steady-state distribution can exist.

1jj

0

c1

1π