Limit theorem for multichannel networks in heavy traffic
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LIMIT THEOREM FOR MULTICHANNEL NETWORKS IN HEAVY TRAFFIC A. V. Livinskayaa† and E. A. Lebedeva‡
UDC 519.21
Abstract. Multichannel stochastic networks are considered. Poisson arrivals at each node of the networks have time-dependent characteristics. For heavy traffic conditions, an approximate Gaussian process is set up and a functional limit theorem is proved. Keywords: multichannel stochastic network, heavy traffic, Gaussian approximation.
The basic mathematical model analyzed in the present paper is a queuing network of r servers, with Poisson arrivals n i ( t ) with the leading function L i ( t ), i = 1, 2,... , r (nondecreasing right-continuous function) at the ith server. Each of the r nodes operates as a multichannel stochastic system. Once a request arrives, its service starts immediately. The service time at the ith node is exponentially distributed with parameter m i , i = 1, 2,... , r . After the service is completed at the ith node the r
request arrives for service at the j th node with probability p ij and leaves the network with probability p ir + 1 = 1- å p ij , j =1
P = ||
r p ij || 1
is the network routing matrix. An additional node with number r + 1 is interpreted as network output.
According to the notation used in stochastic network theory, we will denote the model described above by [M t | M | ¥] r. Let Qi ( t ), i = 1, 2,... , r , be the number of requests at the i th node at time t . By the process of service of requests in an [ M t | M | ¥ ] r network we will mean an r-dimensional process Q¢ ( t ) = (Q1 ( t ),... , Qr ( t )) . The main purpose of the present study is to analyze the process Q ( t ), t ³ 0, under critical load. Following [1], let us introduce the space of functions D [ 0, T ] that are defined on the interval [ 0, T ] , take finite real values, have a left-hand limit at each point, and are right-continuous (left-continuous for t = T ). A sequence of functions U
x n ( t ) Î D [ 0, T ] is said to converge to x 0 ( t ) Î D [ 0, T ] in the uniform topology U (x n ( t ) Þ x 0 ( t )) if n ®¥
sup | x n ( t ) - x 0 ( t ) | ® 0 as n ® ¥ . Denote the space of continuous functions defined on the interval [ 0, T ] by C [ 0, T ] .
tÎ[ 0 ,T ]
The critical load is due to the following behavior of the parameters of the network. Condition 1. The arrivals depend on n (series number) so that on any finite interval [ 0, T ] U
n -1 L(in ) ( nt ) Þ L(i0 ) ( t ) Î C [ 0, T ], i = 1, 2,... , r . n ®¥
(1)
Let us consider two practically important cases where Condition 1 is satisfied. Thereby, we temporarily assume that t
the Poisson flow n i ( t ) is regular: L i ( t ) = ò l i ( u )du , where l i ( u ) is an instantaneous value of the parameter (see, for 0
example, [2, p. 100]). Such a flow can naturally be called a Poisson flow with variable parameter. a
Taras Shevchenko National University, Kyiv, Ukraine, † [email protected]; ‡ [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 6, November–December, 2012, pp. 106–113. Original article submitted January 10, 201
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