Time-dependent analysis of an $M/M/c$ preemptive priority system with two priority classes

TL;DR

This paper develops a method to analyze the time-dependent behavior of a two-class $M/M/c$ preemptive priority queue, providing explicit Laplace transforms of transition functions and stationary distributions.

Contribution

It introduces a recursive approach using Ramaswami's formula to derive explicit Laplace transforms for the system's transition functions and stationary distribution.

Findings

Explicit Laplace transforms for transition functions are derived.

Recursive computation method for complex states is developed.

Results provide the most explicit expressions for such systems to date.

Abstract

We analyze the time-dependent behavior of an $M/M/c$ priority queue having two customer classes, class-dependent service rates, and preemptive priority between classes. More particularly, we develop a method that determines the Laplace transforms of the transition functions when the system is initially empty. The Laplace transforms corresponding to states with at least $c$ high-priority customers are expressed explicitly in terms of the Laplace transforms corresponding to states with at most $c - 1$ high-priority customers. We then show how to compute the remaining Laplace transforms recursively, by making use of a variant of Ramaswami's formula from the theory of $M/G/1$-type Markov processes. While the primary focus of our work is on deriving Laplace transforms of transition functions, analogous results can be derived for the stationary distribution: these results seem to yield the most explicit expressions known to date.