In this paper we present closed-form expressions for the distribution of the virtual (actual) queueing time for the BMAP/R/1 and BMAP/D/1 queues, where `R' represents a class of distributions having rational Laplace‒Stieltjes transforms. The closed-form analysis is based on the roots of the underlying characteristic equation. Numerical aspects have been tested for a variety of arrival and service-time distributions and results are matched with those obtained using the matrix-analytic method (MAM). Further, a comparative study of computation time of the proposed method with the MAM has been carried out. Finally, we also present closed-form expressions for the distribution of the virtual (actual) system time. The proposed method is analytically quite simple and easy to implement.

We derive the MacLaurin series for the moments of the system time and the delay with respect to the parameters in the service time or interarrival time distributions in the GI/G/1 queue. The coefficients in these series are expressed in terms of the derivatives of the interarrival time density function evaluated at zero and the moments of the service time distribution, which can be easily calculated through a simple recursive procedure. The light traffic derivatives can be obtained from these series. For the M/G/1 queue, we are able to recover the formulas for the moments of the system time and the delay, including the Pollaczek–Khinchin mean-value formula.