Operator-Geometric Solutions for the M/G/k Queue and its Variants

One of the classical, but yet unsolved queueing systems is the M/G/k queue with Poisson input, general service time distribution and serving facilities. In the present paper, this queue is analyzed as a piecewise–deterministic Markov process. First, an iteration formula for the transient distributions is derived. This formula is a new result for piecewise–deterministic Markov processes, too, since it yields their transition probability kernel. In the main part, recent results on stability of piecewise–deterministic Markov processes are used in order to show that the computation of the stationary distribution for the M/G/k queue can be reduced to the determination of the stationary distribution for a birth and death process with kernel entries. This results in operator–geometric solutions for the M/G/k queue. For the M/G/k retrial queue, a similar analysis is presented. Here, closed formulae for the transient distribution are derived. These expressions hold for the class of piecewise–deterministic processes without jumps from a border set, too. Additionally, it is sketched how these results can be generalized for BMAP/G/k queues.