Abstract

We address the transient analysis of networks of queues with exponential service times. Such networks can easily have such a huge state space that their exact transient analysis is unfeasible. In this paper we propose an approximate transient analysis technique based on decomposing the queues of the network using a compact and approximate representation of the departure process of each queue. Namely, we apply time-inhomogeneous Markov arrival processes (IMAP) to describe the stream of clients leaving the queues. By doing so, the overall approximate model of the network is a time-inhomogeneous continuous time Markov chain (ICTMC) with significantly less number of states than there are in the original Markov chain. The proposed construction of the output IMAP of a queue is based on its transient state probabilities. We illustrate the approach first on a single M/M/1 queue and analyze the goodness of fitting of the departure process by numerical examples. Then we extend the approach to networks of queues and evaluate the precision of the resulting technique on several simple numerical examples by comparing the exact and the approximate transient probabilities of the queues.

Highlights

  • Queuing networks (QN) are employed in many application areas such as computer and telecommunications networks, manufacturing systems, transport, logistics or even systems biology

  • The technique relies on approximating the departure process of each queue by an inhomogeneous Markov arrival processes (IMAP)

  • The departure process approximation we applied is state-based, i.e., the states of an output IMAP are related to the states of the corresponding queue

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Summary

INTRODUCTION

Queuing networks (QN) are employed in many application areas such as computer and telecommunications networks, manufacturing systems, transport, logistics or even systems biology. Even if the state space is large, special characteristics can be exploited to carry out an exact analysis Such situations are limited in practice to networks of infinite server queues [4, 14]. The method we propose in this paper is based on decomposing the queues of the network and representing the departure process of each queue by a time-inhomogeneous Markov arrival process (IMAP). Markov arrival processes (MAP) were introduced in [17] and several steady state solution techniques based on matrix analytic methods have been proposed to study single queues [12] or networks of queues in an approximate manner [9,10,11].

CONSIDERED QN CLASS
TIME-INHOMOGENEOUS MAPS
APPROXIMATE ANALYSIS OF QNS BY IMAP BASED DECOMPOSITION
NUMERICAL ILLUSTRATION
Tandem queue
A more complex model
CONCLUSIONS
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