Busy period analysis of the level dependent \(PH/PH/1/K\) queue
type: Item
instance of: scholarly article
title: Busy period analysis of the level dependent \(PH/PH/1/K\) queue
zbMATH Open document ID: 1222.60069
author: Ahmad Al Hanbali
published in: Queueing Systems
publication date: 2011-06-17T00:00:00Z
review text: The paper studies the transient behaviour of a state-dependent \(PH/PH/1/K\) queueing system. The author derives a closed form for the joint transform of the length of a busy period, the number of customers served during a busy period and the number of losses during a busy period. Earlier results on the analysis of the transient behaviour of traditional queueing systems (such as \(M/M/1/N\), \(M/GI/1/N\), \(GI/M/1/N\), etc.) and results on the joint transform of characteristics during a busy period have been obtained by \textit{L. Takács} [Introduction to the theory of queues. University Texts in the Mathematical Sciences. New York: Oxford University Press (1962; Zbl 0106.33502)], \textit{J. W. Cohen} [The single server queue. North-Holland Series in Applied Mathematics and Mechanics, Vol. 8. Amsterdam etc.: North-Holland Publishing Company (1982; Zbl 0481.60003)] and \textit{S. I. Rosenlund} [Adv. Appl. Probab. 8, 195--208 (1976; Zbl 0332.60070); Nav. Res. Logist. Q. 25, 107--119 (1978; Zbl 0378.60091)]. In this paper, the author distinguishes between two different types of losses: overflow losses that are due to an overflow of the buffer and losses due to an admission controller. Relatively simpler expressions for the joint transforms of the particular queueing systems \(M/PH/1/K\), \(M/PH/1/K\) (under a threshold policy) and \(PH/M/1/K\) are obtained.
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60J05
Mathematics Subject Classification ID: 68M20
Mathematics Subject Classification ID: 90B22
zbMATH DE Number: 5909322
zbMATH Keywords: \(PH/PH/1/K\) queue
zbMATH Keywords: phase-type distributions
zbMATH Keywords: level-dependent queues
zbMATH Keywords: busy period
zbMATH Keywords: transient analysis
zbMATH Keywords: absorbing Markov chains
zbMATH Keywords: matrix analytical approach
reviewed by: Vyacheslav Abramov
MaRDI profile type: Publication
full work available at URL: s11134-011-9213-6
OpenAlex ID: W2097592086
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DOI: 10.1007/S11134-011-9213-6
description: scientific article; zbMATH DE number 5909322