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Strong approximations of open queueing networks. (English) Zbl 0758.60099

There are considered queueing systems with one or two stations, each of which has an infinite capacity waiting room. Customers are served in order of arrival and after service they are randomly routed to either another station or out of the system entirely. For the case of one station let \(Q(t)\), \(0\leq t\leq T\), be the queue length process, where \(T\) is large in comparison with interarrival and service times. The process \(Q(t)\) is strongly approximated with Brownian motion or reflected Brownian motion. This is done by proving exponential inequalities for the sup distance between \(Q\) and the limiting process. As corollaries some results for the rate of convergence of the distribution of \(Q(T)\) and for \(E Q(T)\) are obtained. Similar results are next proved for networks with two stations. The method of the paper is based on J. Komlós, P. Major and G. Tusnády approximation [Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, 111-131 (1975; Zbl 0308.60029) and ibid. 34, 33-58 (1976; Zbl 0307.60045)].

MSC:

60K25 Queueing theory (aspects of probability theory)
60F17 Functional limit theorems; invariance principles
60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
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