Busy period distribution in state-dependent queues
type: Item
instance of: scholarly article
title: Busy period distribution in state-dependent queues
zbMATH Open document ID: 0661.60115
DOI: 10.1007/BF01158903
published in: Queueing Systems
publication date: 1987-01-01T00:00:00Z
review text: A state-dependent M/M/1 queueing system is considered. It is assumed that the arrival rate \(\lambda =\lambda (v(t))\) depends on the current work v(t) in the system and that the mean service time of a job arriving at time moment \(t^*\) is \(1/\mu (v(t^*))\). The arrival rate is fast while the mean service requests are supposed to be small. The density function \(p(x,t)=(\partial /\partial t)P_ r\{T<t| v(0)=x\}\) of residual length of a busy period is studied using singular perturbation techniques (here \(T=\inf \{t:v(t)=0\})\). Approximations to p(x,t) for two different cases are constructed: the first case has \(r(x)=\lambda (x)/\mu (x)<1\), \(x\in [0,\infty)\); the second case has \(r(x)<1\), \(x\in (0,\infty)\) and \(r(0)=1\).
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 90B22
zbMATH DE Number: 4080589
zbMATH Keywords: state-dependent
zbMATH Keywords: residual length of a busy period
zbMATH Keywords: singular perturbation techniques
author: Charles Knessl
author: B. J. Matkowsky
author: Zeev Schuss
author: Charles Tier
MaRDI profile type: Publication
cites work: Q4101255
cites work: Q3207312
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full work available at URL: https://doi.org/10.1007/bf01158903
OpenAlex ID: W2040294627
description: scientific article; zbMATH DE number 4080589