Queueing Networks

WD04.1 Exact Aggregation & Decomposition Results for Product Form Queueing

• Richard J. Boucherie; University of Twente, Amsterdam, , The Netherlands; r.j.boucherie@math.utwente.nl

Aggregation and decomposition results for queueing networks allow the replacement of a part of the network by an equivalent station. Such results have proven to be valuable for numerical solutions of queueing networks and also provide insight into the nature of product form equilibrium distributions. We provide an overview of the evolution of exact aggregation and decomposition results starting with elementary results for Jackson-type networks and ending with networks of generalized quasi-reversible nodes linked via state-dependent routing mechanisms.

WD04.2 Traffic Flows & Product Form Solutions in Stochastic Transfer Networks

• Masakiyo Miyazawa; Science University of Tokyo, Yamazaki 2641, Noda, Chiba, 278-8150 , Japan; miyazawa@is.noda.sut.ac.jp,, http://queue3.is.noda.sut.ac.jp/
• H. Takada; , Noda, , Japan;

We focus on product form and related tractable stationary distributions in a general class of stochastic networks with finite number of nodes such that their network states are changed through signal transfers as well as internal transitions. We propose an abstract model and consider the stationary distribution for the network state. We introduce conditional traffic rates for arrivals and departures and based on which, we study the conditions under which the network has product form or other form of tractable solutions.

WD04.3 Product-Form Bounds & Approximations for the Stationary Distribution of a Queueing Network with Batch Services & Customer Coalescence

• Antonis Economou; University of Athens, Athens, , Greece; aeconom@internet.gr

We consider a queueing network with batch services and customer coalescence, whose stationary distribution is not tractable. In particular, the stations of this network are not quasi-reversible. Under a certain limited modification of the transition rate matrix, it is shown that every station becomes quasi-reversible; hence, the modified network has a product-form stationary distribution. We determine this distribution and investigate whether it can be used as an approximation or a bound for the stationary distribution of the original network. We prove that under a certain condition, the modified network provides a lower bound for the original network. Extensive numerical studies show that the modified network provides quite good approximations for the performance of the original network even when the above condition fails to hold.

WD04.4 State-Dependent Coupling of Non-Quasireversible Nodes

• Xiuli Chao; North Carolina State University, Dept. of IE, Raleigh, NC 27695; xchao@eos.ncsu.edu
• William Henderson; University of Adelaide, Dept. of Applied Math., Teletraffic Research Ctr., Adelaide, SA 5005 , Australia;
• Peter Taylor; University of Adelaide, Dept. of Applied Math, Adelaide, 5005 , Australia; ptaylor@maths.adelaide.edu.au

We extend results for networks of queues with state-independent transitions to networks with state-dependent transitions. We begin with a network in which the transition rates governing the stochastic behavior of the individual nodes in isolation depend only on the state of local information, i.e., the state of the node. However, the routing probabilities, which couple the nodes into a network, are state-dependent. Assuming that the network has a stationary distribution, we construct another network with state-dependent transition rates at each node and obtain its stationary distribution. The result extends the recent results of Miyazawa & Chao (1998), Henderson & Taylor (1999) and Miyazawa & Takada (1999), who obtain similar results for networks of quasi-reversible queues in which the stationary distribution has a 'product form'.