Queues

D.R. Cox, Walter Smith

• Publisher: Chapman and Hall/CRC
• Year: 2020
• Print ISBN: 9781138460317
• eBook ISBN: 9781000154450
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

This is a classic book on Queues. First published in 1961 it is clearly and concisely introduces the theory of queueing systems and is still just as relevant today. The monograph is aimed at both students and operational research workers concerned with the practical investigations of queueing, although almost every statistician will find its contents of interest.

CHAPTER V

More Specialized Topics

5.1 Introduction

In this chapter we deal briefly with a number of unrelated special topics, chosen partly to show some of the more complicated mathematical problems connected with queueing, and partly to illustrate a few problems of practical importance. The subjects dealt with are:

(a) the method of stages, a device for dealing in a simple way with problems involving a certain type of non-exponential distribution (section 5.2);

(b) a method of studying queueing problems by the use of integral equations (section 5.3);

(c) simulation and Monte Carlo methods (section 5.4);

(d) series of queues (section 5.5);

(e) the busy-period distribution (section 5.6).

5.2 The method of stages

We have seen repeatedly in earlier sections the great simplification introduced whenever a probability distribution connected with the queueing process is exponential. Thus if the distribution of service-time is exponential, with p.d.f. ae^−σx, there is a constant probability σδt + o(δt) that a service operation in progress at t is completed in (t, t + δt), quite independently of how long the operation has been in progress. If all the random variables determining the system are independently exponentially distributed, the method of section 2.2 can be used to obtain simultaneous linear equations for the equilibrium probabilities, and simultaneous linear differential equations with constant coefficients for the non-equilibrium distribution.

In his pioneer work on congestion in telephone systems, A. K. Erlang suggested an ingenious method by which the simple arguments used for exponential distributions can be applied also for certain non-exponential distributions. The general idea has been outlined in section 1.4 (i) (c). In the simplest form of the method, applied to the distribution of service-time, we suppose there to be k stages of service, each with a service-time having the exponential p.d.f. σe^−σx. The stages need have no physical meaning. When stage 1 of service is complete the customer passes immediately to stage 2, and so on, the whole servicing operation being complete when the k^th stage of service is complete. The total service-time is thus the sum of k independent identically distributed random variables with the p.d.f. σe^−σx. The Laplace transform of the exponential distribution is σ/(σ + s), so that the Laplace transform for the total service-time is σ^k/(σ +...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• I. Introduction
• II. Some simple queues with random arrivals
• III. More about simple queues
• IV. Machine Interference
• V. More specialized topics
• Appendixes
• Author Index
• Subject Index