The G/M/1 Queueing System

Abstract

Consider a first-come first-served single-server queue. Assume that service requirements are independent and follow an identical exponential distribution with a parameter μ. This assumption implies that during a (not necessarily continuous) period of time of length t, the number of customers possibly served by the server has a Poisson distribution with parameter μt. The reason we use the term “possibly” is that in practice what can happen is that during that period or part of it, the system might be empty and hence, although service is ready to be provided, there is no one there to enjoy it. One more thing to observe here is that if one stops the clock when the server is idle, then the departure process under the new clock is a Poisson process. As always, the interarrival times have some continuous distribution with a density function denoted here by g(t) for t ≥ 0 and that they are independent. In other words, the arrivals form a renewal process. Finally, we assume independence between the arrival and the service processes.