Transient Analysis of the M/M/k/N/N Queue using a Continuous Time Homogeneous Markov System with Finite State Size Capacity

Abstract

In this article, the M/M/k/N/N queue is modeled as a continuous-time homogeneous Markov system with finite state size capacity (HMS/cs). In order to examine the behavior of the queue a continuous-time homogeneous Markov system (HMS) constituted of two states is used. The first state of this HMS corresponds to the source and the second one to the state with the servers. The second state has a finite capacity which corresponds to the number of servers. The members of the system which can not enter the second state, due to its finite capacity, enter the buffer state which represents the system's queue. In order to examine the variability of the state sizes formulae for their factorial and mixed factorial moments are derived in matrix form. As a consequence, the pmf of each state size can be evaluated for any t ∈ ℝ+. The theoretical results are illustrated by a numerical example.