Queueing situations arise in all aspects of work and life and are typified by the ‘queueing for service’. The theory of queueing gives a basis for understanding the various aspects of the problems and enables a quantitative assessment to be made. Therefore the theory enables these ‘service situations’ to be more effectively designed and operated.
The Poisson distribution and the negative exponential distribution are the two basic distributions in queueing theory. Their theory and general fields of application can be studied in general statistical theory texts and this book will deal primarily with their application to queueing theory.
The theory of this system and also the other systems described in later chapters can be found in most textbooks on queueing theory and a list of references is given at the end of this book.
In this chapter the M/M/I/N system is covered — the single-channel queueing system given in chapter 3 but with a constraint that the maximum number in the system cannot exceed N. Since the arrival rate has to be random and constant, this special case is generated when the number of potential customers is infinite, but they only join the system when n < N; when n = N, the customers arriving for service go elsewhere, that is, a non-captive system.
There are a large number of practical problems that are systems with either the arrival rate and/or service rate dependent on the number in the system. For example, consider a maintenance gang responsible for five large machines each of which breaks down randomly, at the average rate of once per week. Clearly the total arrival rate depends on the number already broken down (number in the system); again problems of telephone switchboards, car hire, incentives to service personnel, some stock control models are similar typical systems.
The general formulae for these systems were obtained by Pollaczeh-Khintchine, while Fry obtained a solution to systems with constant service times, that is, M/D/I/8 systems.
