New Look at Finite Single Server Queue with Poisson Input and Semi-Markov Service Times

TL;DR

This paper develops new analytical formulas for the finite single server queue with Poisson arrivals and semi-Markov service times, providing insights into queue size distributions in both stationary and transient states.

Contribution

It introduces novel analytical formulas for the $M/SM/1/b$ queue, expanding the mathematical understanding of queues with semi-Markov service times.

Findings

Derived new formulas for queue size distribution

Analyzed both stationary and transient states

Established similarity with $BMAP/G/1/b$ systems

Abstract

The mathematics of the finite single server queue with Poisson input and semi-Markov service times($M/SM/1/b$) is similar to that used for $BMAP/G/1/b$ systems. This observation results in new analytical formulas for a queue size in the $M/SM/1/b$ system. Both stationary and the transient solutions are considered