ANALYTICALLY EXPLICIT RESULTS FOR THE GI/C-MSP/1/∞ QUEUEING SYSTEM USING ROOTS

Abstract

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at prearrival epochs of the GI/C-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution. We also provide the steady-state system-length distribution at an arbitrary epoch by using the classical argument based on Markov renewal theory. The sojourn-time distribution has also been investigated. The prearrival epoch probabilities have been obtained using the method of roots which is an alternative approach to the matrix-geometric method and the spectral method. Numerical aspects have been tested for a variety of arrival- and service-time distributions and a sample of numerical outputs is presented. The proposed method not only gives an alternative solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $$ GI ^{[X]}/C$$ G I [ X ] / C - $$ MSP /1/\infty $$ M S P / 1 / ∞

“…In this connection, readers are referred to Chaudhry et al (1987), Chaudhry et al (1990), Tijms (2003) and Chaudhry et al (2012Chaudhry et al ( , 2013, who have used the roots method. In particular, regarding the details about the use of Rouché's theorem in the analysis of characteristic equation, the readers are referred to Cohen and Down (1996), Adan et al (2006) etc.…”

“…The purpose of studying this queueing model using roots is that we obtain computationally simple and analytically closedform solution to the infinite-buffer GI [X ] /C-MSP/1 queue using roots method. It may be noted here that Chaudhry et al (2012) obtained analytically explicit expressions for steadystate probabilities for GI/C-MSP/1 queue using roots. Further, Samanta et al (2015) have carried out the discrete-time analysis of GI/D-MSP/1/∞ model using roots.…”

“…Further, Samanta et al (2015) have carried out the discrete-time analysis of GI/D-MSP/1/∞ model using roots. Although this paper is an extension of Chaudhry et al (2012) to the batch arrival of variable capacity, we have carried out waiting-time analysis as well, one root approximation, derivation of expected busy and idle periods and extensive numerical results. It may also be remarked here that the matrix-geometric method (MGM) uses an iterative procedure to get steady-state probabilities at the pre-arrival epochs for this infinite-buffer GI [X ] /C-MSP/1 queue.…”

Analytic and computational analysis of the discrete-time GI/D-MSP/1 queue using roots

“…where ρ is the traffic intensity, is equal to the probability that the system is empty" does not hold in [1], whereas this condition holds in the present paper, for details see Chaudhry et al [13] and related references therein. It may be remarked that the analysis of GI/C-MSP/1/∞ queue in continuous-time has been carried out by Chaudhry et al [13].…”

“…It may be remarked that the analysis of GI/C-MSP/1/∞ queue in continuous-time has been carried out by Chaudhry et al [13]. Furthermore, the modelling of discrete-time queues is more involved and quite different from the analysis used for the corresponding continuous-time queueing models so that several results obtained in this paper are new and analytic analysis differs at many places.…”