Zero-automatic networks
type: Item
instance of: scholarly article
title: Zero-automatic networks
zbMATH Open document ID: 1198.90090
author: Thu-Ha Dao-Thi
author: Jean Mairesse
published in: Discrete Event Dynamic Systems
publication date: 2010-04-21T00:00:00Z
review text: One continues the study of zero-automatic queues introduced in [Adv. Appl. Probab. 39, No. 2, 429--461 (2007; Zbl 1121.60094)] which may be viewed as a synthesis of a queue and a random walk on some infinite group or monoid, or as a synthesis and wide generalization of the simple queue and the G-queue with positive and negative customers. One takes \(N\) zero-automatic queues corresponding to possibly different groups or monoids. For a non-empty subset of queues there is an external arrival process of customers. After completion of its service, a customer is either routed to another queue or leaves the network, two routing policies are assumed: Jackson-like and Kelly-like networks. Customers of different types from the set \(\Sigma \) may enter, the buffer content is given by the word \(\Sigma ^*\), the arrival and service rates are \(\lambda \) and \(\mu \), respectively. The set of possible buffer contents is of the form \(L=\{u_k\ldots u_1\in \Sigma ^*~|~u_i\in Next(u_{i+1})\}\), where \(Next(a)\subset \Sigma \) is the subset of types which do not interact with type \(a\). Let \((X_n)_n\) be the \(L\)-valued random variables corresponding to the successive buffer contents. Let \(|X_n|\) be the length of \(X_n\), set \(\hat{\gamma}=\lim_{n\to \infty}|X_n|/n\). The stability condition for the queue is \(\lambda \hat{\gamma}<\mu \). Under this condition a set of so-called twisted traffic equations is associated with the queue, they have a unique solution \((\rho ,r)\). Set \(q(a)=r(a)/r(Next(a))\) for all \(a\in \Sigma \). The stationary distribution of buffer content satisfies \[ \pi (u)=(1-\rho )\rho ^kq(u_k)\dots q(u_2)q(u_1). \] The term \((1-\rho )\rho ^k\) accounts for the buffer length, and the term \(q(u_k)\dots q(u_2)q(u_1)\) accounts for the precise buffer content. At network level there are two layers: the twisted traffic equations of the individual queues, and the network traffic equations. The stationary distribution has a ``double product form'': with …
Mathematics Subject Classification ID: 90B22
Mathematics Subject Classification ID: 60K25
zbMATH DE Number: 5696498
zbMATH Keywords: zero-automatic queue
zbMATH Keywords: product form
zbMATH Keywords: Jackson network
zbMATH Keywords: Kelly network
reviewed by: László Lakatos
MaRDI profile type: Publication
full work available at URL: s10626-008-0048-1
OpenAlex ID: W2007147072
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DOI: 10.1007/S10626-008-0048-1
description: scientific article; zbMATH DE number 5696498