Queuing system evolution in a diffusion approximation scheme
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QUEUING SYSTEM EVOLUTION IN A DIFFUSION APPROXIMATION SCHEME A. V. Mamonova a and Iu. F. Grizab
UDC 519.21
An asymptotic diffusion approximation scheme is investigated as applied to the requirement evolution in semi-Markov queuing systems. In proving the diffusion approximation theorem, the compensating operator of the corresponding extended Markov process is used. This problem is solved with the help of a phase merging procedure. Keywords: queuing system of type [ SM | M | ¥ ] N , averaging and diffusion approximation scheme, semimarkov process, stationary phase merging, compensating operator, generating operator, singular perturbation problem.
A queuing system (QS) of [ SM | M |1| ¥ ] N -type is defined by a semimarkov process that specifies an incoming requirement flow arriving at the input of the system from the outside; the service time at each of N nodes of the system is exponentially distributed, no more than one requirement can be simultaneously served by each node of the system, and the queue of requirements is unbounded at each node of the system [1–3]. In [4], such a system is considered in an averaging scheme, and an averaging theorem is formulated and proved that makes it possible to obtain a limiting process r( t ), t ³ 0. To completely analyze the system, fluctuations of a random evolution should be investigated with respect to the averaged evolution [5]. In this work, the diffusion approximation theorem is formulated and proved for the case when r ( t ) = r = const and r is determined from the balance condition C( r ) = 0 [3]. The mentioned problem is solved by the introduction of a small parameter of a series e ® 0 (e > 0). The algorithm of stationary phase merging [1, 6–8] is used. In proving the theorem, an asymptotic decomposition of the compensating operator (CO) of an extended Markov renewal process is performed and a singular perturbation problem is solved for such a CO. 1. BASIC DENOTATIONS, PROBLEM STATEMENT, AND PRELIMINARY RESULTS A semimarkov process k( t ) , t ³ 0 , in a measurable phase space ( E , e ) is specified by a semimarkov kernel Q ( x, B , t ) = Gx ( t ) P ( x, B ) , x Î E, B Î e, t ³ 0 .
(1)
P ( x, B ) = P {k n + 1 Î B | k n = x}, x Î E, B Î e,
(2)
A stochastic kernel determines transition probabilities of an embedded Markov chain k n , n ³ 0 . A family of distribution functions Gx ( t ) = P {q n + 1 £ t | k n = x} = P {q x £ t}, x Î E,
(3)
determines sojourn times in states q x , x Î E. A counting process n ( t ) = max{n : t n £ t}, t ³ 0 ,
(4)
a
National University of State Tax Service of Ukraine, Irpen’, Ukraine, [email protected]. bFree International University of Moldova, Kishinev, Moldova. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 136–145, May–June 2009. Original article submitted August 19, 2008. 1060-0396/09/4503-0455
©
2009 Springer Science+Business Media, Inc.
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determines the number of renewal moments t n = t n + 1 + q n , n ³ 1, t 0 = 0 .
(5)
We write the expected service time in the form ¥
g ( x ): = Eq x = ò Gx ( t )dt ,
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