Ladder heights and the Markov-modulated M/G/1 queue
This paper is concerned with the Markov-modulated M/G/1 queue with both the arrival rate and the distribution of service time depending on the actual state of the Markov environmental process. For analysing the distribution of the steady-state waiting time of this queueing system, a random walk setting is considered which leads to a Wiener-Hopf formulation of this problem. In particular, a matrix generalization of the notion of the ascending ladder height distribution is introduced. In distinction from the classical approach, the Wiener-Hopf problem is formulated in a continuous-time setting. Here, the main idea is to study the intensity matrix of a certain Markov jump process which arises through a time-reversal representation of a fundamental occupation measure. The general results are applied to compute characteristics like the steady-state probability of emptiness as well as expectation and second moment of the steady-state waiting time. It is shown that the waiting time distribution is phase-type if the service time distributions have this property. The present paper is an interesting completion of earlier research on this queueing system, where other approaches have been used such as the complex plane method of \textit{G. J. K. Regterschot} and \textit{J. H. A. de Smit} [Math. Oper. Res. 11, 465-483 (1986; Zbl 0619.60093)] or the computational approach of \textit{M. F. Neuts} [Structured stochastic matrices of the M/G/1 type and their applications (1989; Zbl 0695.60088)].
scientific article; zbMATH DE number 4215114
scientific article