An alternative method for computing system-length distributions of BMAP/R/1 and BMAP/D/1 queues using roots

Abstract

In this paper, we present closed-form expressions for system-length distributions in terms of roots outside the unit disk of the characteristic equation of the BMAP/R/1 queue, where arrival process is batch Markovian arrival process (BMAP) and R represents a class of distributions having rational Laplace–Stieltjes transform. The unknown boundary vector has been evaluated using the roots (inside and on the unit disk) of the characteristic equation. Several numerical results are presented for a variety of arrival and service-time distributions including phase-type (PH) and matrix-exponential (ME) which cover a wide variety of distributions that arise in applications. Using an approximated representation of ME distribution, results for BMAP/D/1 queue are also presented. We compared our result with the results obtained using classical matrix analytic method as well as cyclic reduction algorithm. It is shown that the computation-time of the proposed method does not depend upon the parameters like traffic intensity and correlation co-efficient. The method is analytically quite simple and easy to implement.