The transient M/G/1/0 queue: Some bounds and approximations for light traffic with application to reliability (Q1908327)

Summary: We consider the transient analysis of the \(M/G/1/0\) queue, for which \(P_n (t)\) denotes the probability that there are no customers in the system at time \(t\), given that there are \(n\) \((n = 0,1)\) customers in the system at time 0. The analysis, which is based upon coupling theory, leads to simple bounds on \(P_n (t)\) for the \(M/G/1/0\) and \(M/PH/1/0\) queues and improved bounds for the special case \(M/E_r/1/0\). Numerical results are presented for various values of the mean arrival rate \(\lambda\) to demonstrate the increasing accuracy of approximations based upon the above bounds in light traffic, i.e., as \(\lambda \to 0\). An important area of application for the \(M/G/1/0\) queue is as a reliability model for a single repairable component. Since most practical reliability problems have \(\lambda\) values that are small relative to the mean service rate, the approximations are potentially useful in that context. A duality relation between the \(M/G/1/0\) and \(GI/M/1/0\) queues is also described.