The Basic M/G/1 Queueing System

Abstract

In the previous chapters we made extensive use of the memoryless properties of the exponential distribution to study the dynamics of the M/M/1 and other queueing systems, as well as the service-time distributions and interarrival time distributions, which were exponentially distributed. Due to the memoryless property of the exponential distribution, the evolution of such systems from any point in time forward is independent of past history. Thus, the memoryless property allowed us to specify the state of the system at an arbitrary point in time and to write equations describing the system dynamics conveniently. If the service system has service times that are drawn from a general distribution , then the memoryless property is lost, and it is then necessary to choose observation times carefully in order that the state of the system at the observation times can be easily specified. That is, if we choose the observation times carefully, we may be able to specify the state of the system conveniently, and further, we may succeed in having the evolution of the process from that point forward be independent of past history. Suppose, for example, that we choose our observation times as those instants in time when a customer has just completed service. At those points in time, both the arrival process, which is memoryless, and the service process, which is not necessarily memoryless, start over again. Thus, in order to determine the future evolution of the system , it is necessary to know only the number of customers left in the system immediately following customer departures. Define to be the number of customers left in the system by the departing customer. Then, according to our previous observations, the process is Markovian. We call an embedded Markov chain, which we introduced in Chapter 2. We say that we have "embedded a Markov chain at the points of customer departure." Following our notation of the previous chapters, we denote the number of customers in the system,