Analysis of queueing systems with randomly changing population states

Abstract

AbstractThe queueing system, in which the property of the population is changing randomly, frequently is observed in various problems, and it is important to know its property. However, the analysis of the characteristics of such a system in general is difficult. From such a viewpoint, this paper considers the simplest example of the M/M/1 system, where the inter‐arrival‐time distribution or service‐time distribution changes following a two‐stage Markov chain. A strict solution is derived for the behavior of the system. Using the state‐transition rate diagram of the system, the system equation for the steady‐state is derived. Then the z‐transform of the steady‐state probability distribution and the Laplace transform of the waiting time distribution are obtained. Their inverse transforms and the mean behaviors such as the mean number of waiting customers and mean waiting time are derived. Then the fundamental properties of the system are examined. For example, when the systems have the same mean load, the mean waiting time is longer if the load difference between the two states is large, or if the mean sojourn time at each state is long. These properties have been noticed by simulation, but they are verified quantitatively by analysis. However, in this paper the major aim is only to present an accurate method of analysis, and a detailed examination of the system is left for another paper.