An approximate transient analysis of the M(t)/M/1 queue

An approach, based on recent work by Stern [56], is described for obtaining the approximate transient behavior of both the M/M/1 and M(t)/M/1 queues, where the notation M(t) indicates an exponential arrival process with time-varying parameter λ(t). The basic technique employs an M/M/1K approximation to the M/M/1 queue to obtain a spectral representation of the time-dependent behavior for which the eigen values and eigenvectors are real. Following a general survey of transient analysis which has already been accomplished, Stern's M/M/1/K approximation technique is examined to determine how best to select a value for K which will yield both accurate and computationally efficient results. It is then shown how the approximation technique can be extended to analyze the M(t)/M/1 queue where we assume that the M(t) arrival process can be approximated by a discretely time-varying Poisson process. An approximate expression for the departure process of the M/M/1 queue is also proposed which implies that, for an M(t)/M/1 queue whose arrival process is discretely time-varying, the departure process can be approximated as discretely time-varying too (albeit with a different time-varying parameter). In all cases, the techniques and approximations are examined by comparison with exact analytic results, simulation or alternative discrete-time approaches.