The BMAP/G/1 queue: A tutorial

Abstract

Abstract. We present an overview of recent results related to the single server queue with general independent and identically distributed service times and a batch Markovian arrival process (BMAP). The BMAP encompasses a wide range of arrival processes and yet, mathematically, the BMAP/G/1 model is a relatively simple matrix generalization of the M/G/1 queue. Stationary and transient distributions for the queue length and waiting time distributions are presented. We discuss numerical algorithms for computing these quantities, which exploit both matrix analytic results and numerical transform inversion. Two-dimensional transform inversion is used for the transient results. The idea of a BMAP is to keep the tractability of the Poisson arrival process but significantly generalize it in ways that allow the inclusion of dependent interarrival times, non-exponential interarrival-time distributions, and correlated batch sizes. The BMAP includes as special cases both phase type renewal processes (which include the Erlang, Ek, and hyperexponential, Hk, renewal processes) and non-renewal processes such as the Markov modulated Poisson process

Response Time Distributions in Networks of Queues

“…The LST of F W (t), the actual waiting time distribution of an arbitrary customer in the queue, is given by [26] …”

Quasi-Birth-and-Death Processes with Rational Arrival Process Components

“…Thus Asmussen and Bladt's quite general definition proved to be analytically equivalent to the Markovian Arrival Process [10] (MAP) in terms of its representation in the form of two matrices C and D.…”

On the performance of circuit‐switched networks in the presence of correlated traffic

“…Let I ns = ns and R ns = ns − I ns denote the integral and fractional parts of ns, respectively. Given that the superposition and splitting of MMPPs are again an MMPP [14,21,23], the traffic arrival process at a network channel can be approximated by a new twostate MMPP c (with superscript c to denote the traffic at a network channel). The parameter matrices of the MMPP c can be selected to match those of the superposition of I ns MMPP s and one MMPP R ; the MMPP R with superscript R denotes the resulting traffic from the splitting of the MMPP s , with the splitting probability R ns .…”