Queues with service times and interarrival times depending linearly and randomly upon waiting times
type: Item
instance of: scholarly article
title: Queues with service times and interarrival times depending linearly and randomly upon waiting times
zbMATH Open document ID: 0707.60090
DOI: 10.1007/BF02411482
author: Ward Whitt
published in: Queueing Systems
publication date: 1990-01-01T00:00:00Z
review text: This paper studies the stochastic recursion \(W_{n+1}=[C_ nW_ n+X_ n]^+\) where the case \(C_ n\) corresponds to the classical Lindley recursion for the GI/G/1 queue. The sequence \(\{(C_ n,X_ n)\}\) may in part be stationary and ergodic rather than i.i.d. Stability criteria are derived and are found to depend crucially on the distribution of \(C_ 0\), in particular \(P(C_ 0>0)\) and E log \(C_ 0\). For example, if \(P(C_ 0>0)=1\), then \(W_ n\) converges in distribution when E log \(C_ 0<0\) whereas \(W_ n/C_ 0...C_{n-1}\) has a limit when E log \(C_ 0>0\) (this type of behaviour as well as the model is close to branching processes in a random environment). The paper also contains stochastic comparisons of systems with different parameters, a normal approximation for the steady-state limit W when E \(X_ 0>0\), and an application to the study of the problem of scheduling interarrival times.
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 90B22
zbMATH DE Number: 4161839
zbMATH Keywords: stationary
zbMATH Keywords: ergodic
zbMATH Keywords: Stability criteria
zbMATH Keywords: branching processes in a random environment
zbMATH Keywords: steady-state limit
zbMATH Keywords: problem of scheduling interarrival times
reviewed by: Søren Asmussen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4161839