2015 mcgraw-hill education. all rights reserved. chapter 17 queueing theory

© 2015 McGraw-Hill Education. All rights reserved.


Frederick S. Hillier Gerald J. Lieberman

Chapter 17

Queueing Theory


Introduction

• Queues (waiting lines) are part of everyday life, and an inefficient use of time

• Other types of inefficiencies– Machines waiting to be repaired– Ships waiting to be unloaded– Airplanes waiting to take off or land

• Queueing models– Determine how to operate a queueing system

most efficiently2


17.1 Prototype Example

• Emergency room at County Hospital is experiencing an increase in the number of visits– Patients are at peak usage hours often have

to wait– One doctor is on duty at all times– Proposal: add another doctor– Hospital’s management engineer is assigned

to study the proposal• Will use queueing theory models

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4

• Basic queueing process– Customers requiring service are generated

over time by an input source– Customers enter a queueing system and join

a queue if service not immediately available– Queue discipline rule is used to select a

member of the queue for service– Service is performed by the service

mechanism– Customer leaves the queueing system

17.2 Basic Structure of Queueing Models


Basic Structure of Queueing Models

• Calling population– Population from which arrivals come– Size may be assumed to be infinite or finite

• Calculations are far easier for infinite case

• Statistical pattern by which customers are generated over time must be specified– Common assumption: Poisson process

• Interarrival time– Time between consecutive arrivals

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• Balking– Customer refuses to enter queue if it is too

long

• Queue is characterized by the number of members it can contain– Can be infinite or finite

• Infinite is the standard assumption for most models

• Queue discipline examples– First-come-first-served, random, or other

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• Service mechanism– Parallel service channels are called servers

• Service time (holding time)– Time for service to be completed– Exponential distribution is frequently assumed

in practice

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• Model notation example– M/M/s

• First letter refers to distribution of interarrival times• Second letter indicates distribution of service times• Third letter indicates number of servers• M: exponential distribution• D: degenerate distribution• Ek: Erlang distribution

• G: general distribution (any arbitrary distribution allowed)

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• Transient condition of a queue– Condition when a queue has recently begun

operation

• Steady-state condition of a queue– Independent of initial state and elapsed time

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• Little’s formula– In a steady state queueing process:

• Where L is expected number of customers, is the mean arrival rate, and W is the waiting time

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17.3 Examples of Real Queueing Systems

• Classes of queueing systems– Commercial service systems

• Example: barbershop

– Transportation service systems• Example: cars waiting at a tollbooth

– Internal service systems• Customers are internal to the organization

– Social service systems• Example: judicial system

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17.4 The Role of the Exponential Distribution

• Operating characteristics of queueing systems determined by:– Probability distribution of interarrival times– Probability distribution of service times

• Negative values cannot occur in the probability distributions

• Exponential distribution– Meets goals of realistic, reasonable, simple,

and mathematically tractable13


The Role of the Exponential Distribution

• Key properties of the exponential distribution– fT(t) is a strictly decreasing function of t

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• Key properties of the exponential distribution – Lack of memory

• Probability distribution of remaining time until event is always the same

– The minimum of several independent exponential random variables has an exponential distribution

– A relationship exists with the Poisson distribution

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• Key properties of the exponential distribution – For all positive values of t:

• For small

– Unaffected by aggregation or disaggregation• Relevant primarily for verifying that the input

process is Poisson

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17.5 The Birth-and-Death Process

• Birth– Arrival of a new customer into the queueing

system

• Death– Departure of a served customer

• Birth-and-death process– Describes how the number of customers in

the queueing system changes as t increases

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The Birth-and-Death Process

• Individual births and deaths occur randomly– Lack of memory is characteristic of a Markov

chain

• Arrows in the diagram indicate possible transitions in the state of the system

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• Analysis is very difficult if the system is in a transient condition– Straightforward if a steady state condition

exists

• For any state of the system:– Mean entering rate equals mean leaving rate

• Called the balance equation for state n

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• Key measures of performance for the queueing system

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17.6 Queueing Models Based on the Birth-and-Death Process

• Models have a Poisson input and exponential service times

• The M/M/s model

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Queueing Models Based on the Birth-and-Death Process

• The M/M/s model as applied to the County Hospital example– See Pages 755-757 in the text

• The finite queue variation of the M/M/s model– Called the M/M/s/K model– Queue capacity is equal to (K − s)

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• The finite calling population variation of the M/M/s model– Given on Pages 760-762 of the text– See next slide for diagram

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17.7 Queueing Models Involving Nonexponential Distributions

• Poisson distribution does not apply when arrivals or service times are carefully scheduled or regulated– Mathematical analysis much more difficult

• Summary of models available for nonexponential service times– The M/G/1 model– The M/D/s model– The M/Ek/s model

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Queueing Models Involving Nonexponential Distributions

• Summary of models available for nonexponential input distributions– The GI/M/s model– The D/M/s model– The Ek/M/s model

• Other models deal with:– Hyperexponential distributions– Phase-type distributions

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17.8 Priority-Discipline Queueing Models

• Queue discipline based on a priority system– Assumes N priority classes exist– Poisson input process and exponential

service times are assumed for each priority class

• Nonpreemptive priorities– Customer being served cannot be ejected

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Priority-Discipline Queueing Models

• Preemptive properties– Lowest priority customer is ejected back into

the queue• Whenever higher priority customer enters

queueing system

• Results for the nonpreemptive priorities model– Little’s formula still applies– See Pages 771-772 in the text

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Priority-Discipline Queueing Models

• Results for the preemptive priorities model– Total expected waiting time in the system

changes– For the single server case:

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17.9 Queueing Networks

• Only a single service facility has been considered so far– Some problems have multiple service

facilities, or a queueing network

• Two basic kinds of networks– Infinite queues in series– Jackson networks


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Queueing Networks

• Equivalence property– Assume that a service facility with s servers

and an infinite queue has Poisson input with parameter λ and the same exponential service time distribution with parameter μ for each server (the M/M/s model) where s μ > λ

• Steady state output of this service facility is also a Poisson process with parameter λ


17.10 The Application of Queueing Theory

• Queueing system design involves the selection of:– Number of servers at a service facility– Efficiency of the servers– Number of service facilities– Amount of waiting space in the queue– Any priorities for different categories of

customers

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The Application of Queueing Theory

• Primary considerations in decision making– Cost of service capacity provided by the

queueing system– Consequences of making customers wait in

the queueing system

• Approaches– Establish how much waiting time is

acceptable– Determine the cost of waiting

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The Application of Queueing Theory

• Other issues– Waiting cost may not be proportional to

amount of waiting• Might be a nonlinear function

– Is it better to have a single fast server or multiple slower servers?

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17.11 Conclusions

• Queueing theory provides a basis for modeling queueing systems– Goal is to achieve an appropriate balance

between cost of service and cost of waiting

• The exponential distribution plays a fundamental role in queueing theory

• Priority-discipline queueing models– Appropriate when some categories of

customers given priority over others

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2015 mcgraw-hill education. all rights reserved. chapter 17 queueing theory

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