Some Markov queueing networks with absolute priorities
type: Item
instance of: scholarly article
title: Some Markov queueing networks with absolute priorities
zbMATH Open document ID: 0533.60095
author: M. Ya. Postan
published in: Problems of Information Transmission
publication date: 1983-01-01T00:00:00Z
review text: An open network of N service systems (vertices) is considered. There are \(n_ i\) service channels in each vertex. M classes of customers are served in each vertex. The service time in the i-th vertex is independent of the customer class and has an exponential distribution with the parameter \(\mu_ i\). The queue size in each vertex is unbounded. To each vertex M independent customer flows are coming. The intensity of the k-th flow into the i-th vertex is \(\lambda_{ik}\). The customers of each class after being served in the i-th vertex go to the j-th vertex with probability r(i,j) or leave the network with probability \(r(i,N+1)\). In each vertex there is the same system of priorities. Priority of the customer is constant. Let \(\xi_{ik}(t)\) denote the number of customers of class k in the vertex i at the moment t. In the paper explicit formulas for stationary expected values \(\xi_{ik}=\lim M \xi_{ik}(t)\) are given. For the loss function \(F=\sum^{N}_{i=1}\sum^{M}_{k=1}c_ kM_{ik}\) an optimal rule for designing priorities to the classes of customers is given. The analogous results are derived for so called closed networks. In this case, for each i \(r(i,N+1)=0\). It is also assumed that \(\lambda_{ik}=0\) for each i and k.
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60K20
Mathematics Subject Classification ID: 90B22
zbMATH DE Number: 3846624
zbMATH Keywords: open network of service systems
zbMATH Keywords: priorities
zbMATH Keywords: loss function
zbMATH Keywords: closed networks
reviewed by: Piotr Staniewski
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3846624