Marked Markovian Arrivals in a Tandem G-Network with Blocking

Abstract

Queueing networks with blocking have broad applications in computer modelling and manufacturing. The present paper focusses on the MMAP[2]/M/1/ ∞ →·/M/1/K + 1 G-queue with blocking. This network consists of a sequence of two single-server stations with an infinite queue allowed before the first server and an intermediate queue of finite capacity K ≥ 0 allowed between servers. This restriction results in the blocking of the first server whenever a unit having completed its service in Station 1 cannot enter into Station 2 due to K + 1 units are accommodated into Station 2. There are two types of arrivals, called units and signals, which are modelled by a single Markovian arrival process with marked transitions. Each unit is served at Stations 1 and 2 in that order, and then it exits of the network. A signal induces the last unit in queue or in service, if there is one, to leave the network instantly, and it has no effect otherwise. Our purpose is to study the influence of the dependence between units and signals on the performance evaluation of the continuous-time Markov chain describing the state of the network at arbitrary times, which constitutes a quasi-birth-and-death process. We present tractable formulas for a variety of probabilistic descriptors, with special emphasis on the distribution of inter-departure times.