Chapter 17 Queueing Theory 2015 Mc GrawHill Education

Chapter 17 Queueing Theory © 2015 Mc. Graw-Hill Education. All rights reserved.

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 efficiently © 2015 Mc. Graw-Hill Education. All rights reserved. 2

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 3

17. 2 Basic Structure of Queueing Models • 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 © 2015 Mc. Graw-Hill Education. All rights reserved. 4

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 5

Basic Structure of Queueing Models • 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 © 2015 Mc. Graw-Hill Education. All rights reserved. 6

Basic Structure of Queueing Models • 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 © 2015 Mc. Graw-Hill Education. All rights reserved. 7

Basic Structure of Queueing Models © 2015 Mc. Graw-Hill Education. All rights reserved. 8

Basic Structure of Queueing Models • 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) © 2015 Mc. Graw-Hill Education. All rights reserved. 9

Basic Structure of Queueing Models • 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 © 2015 Mc. Graw-Hill Education. All rights reserved. 10

Basic Structure of Queueing Models • © 2015 Mc. Graw-Hill Education. All rights reserved. 11

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 12

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 tractable © 2015 Mc. Graw-Hill Education. All rights reserved. 13

The Role of the Exponential Distribution • Key properties of the exponential distribution – f. T(t) is a strictly decreasing function of t © 2015 Mc. Graw-Hill Education. All rights reserved. 14

The Role of the Exponential Distribution • 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 © 2015 Mc. Graw-Hill Education. All rights reserved. 15

The Role of the Exponential Distribution • © 2015 Mc. Graw-Hill Education. All rights reserved. 16

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 17

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 18

The Birth-and-Death Process • 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 © 2015 Mc. Graw-Hill Education. All rights reserved. 19

The Birth-and-Death Process • Key measures of performance for the queueing system © 2015 Mc. Graw-Hill Education. All rights reserved. 20

17. 6 Queueing Models Based on the Birthand-Death Process • Models have a Poisson input and exponential service times • The M/M/s model © 2015 Mc. Graw-Hill Education. All rights reserved. 21

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) © 2015 Mc. Graw-Hill Education. All rights reserved. 22

Queueing Models Based on the Birth-and. Death Process • The finite calling population variation of the M/M/s model – Given on Pages 760 -762 of the text – See next slide for diagram © 2015 Mc. Graw-Hill Education. All rights reserved. 23

Queueing Models Based on the Birth-and. Death Process © 2015 Mc. Graw-Hill Education. All rights reserved. 24

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 25

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 26

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 27

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 28

Priority-Discipline Queueing Models • Results for the preemptive priorities model – Total expected waiting time in the system changes – For the single server case: © 2015 Mc. Graw-Hill Education. All rights reserved. 29

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 30

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 31

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 32

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 33

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? © 2015 Mc. Graw-Hill Education. All rights reserved. 34

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 © 2015 Mc. Graw-Hill Education. All rights reserved. 35