TAILIEUCHUNG - Gordon and Newell queueing networks and copulas

In this paper we have found an analytical formula for a copula that connects the numbers Ni of customers in the nodes of a Gordon and Newell queueing network. We have considered two cases: The first one is the case of the network with 2 nodes, and the second one is the case of the network with at least 3 nodes. | Yugoslav Journal of Operations Research Vol 19 (2009), Number 1, 101-112 DOI: GORDON AND NEWELL QUEUEING NETWORKS AND COPULAS Daniel CIUIU Faculty of Civil, Industrial and Agricultural Buildings Technical University of Civil Engineering, Bucharest, Romania. Romanian Institute for Economic Forecasting, Bucharest, Romania. dciuiu@ Received: December 2007 / Accepted: May 2009 Abstract: In this paper we have found an analytical formula for a copula that connects the numbers Ni of customers in the nodes of a Gordon and Newell queueing network. We have considered two cases: the first one is the case of the network with 2 nodes, and the second one is the case of the network with at least 3 nodes. The analytical formula for the second case has been found for the most general case (none of the constants from a list is equal to a given value), and the other particular cases have been obtained by limit. Keywords: Gordon and newell queueing networks, copula. 1. INTRODUCTION A Jackson queueing network (see [7,4]) is an open queueing network with k nodes where the arrivals from outside network at the node i is exp(λi ) , the service at the node i is exp( μi ) , and after it finishes its service at the node i , a customer goes to the node j with the probability Pi j , or leaves the network with the probability Pi 0 . We know (see [7,4]) that the arrivals from inside or outside network at the node i are independent with the distribution exp(Λ i ) , where Λi is the solution of the system k ∑P j =1 ji ⋅ Λ j + λi = Λ i , i = 1, k (1) A Gordon and Newell queueing network (see [5]) is a closed queueing network with k nodes and N customers. The service time in the node i has the distribution 102 D. Ciuiu / Gordon and Newell Queueing Networks and Copulas exp( μi ) , and after the service in this node the customer goes to the node j with the probability Pi j . We have noticed that the matrix P as above is the transition matrix of an ergodic Markov

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