Asymptotical analysis of queueing system MMPP|M|N with feedback

Let us consider a queueing system MMPP|M|N with two types of feedback. Customer, entering the system in accordance with the MMPP flow, is sent to the server if at least one of the N servers is idle, otherwise it goes into the orbit, where it delays for a random time distributed exponentially with the parameter σ. Customer services during a random time exponentially distributed with the parameter μ; at the moment of service completion, customer may leave the system with probability r0; perform instantaneous feedback with probability r1 or perform delayed feedback with probability r2. The problem is to obtain the probability distribution of the process n(t) is the number of occupied servers in the system and process i(t) is the number of customers in the orbit. In Section 2, we write system of Kolmogorov differential equations for stationary probability distribution P k t k n t n i t i P k n i { ( ) , ( ) , ( ) } ( , , ) and a system of Kolmogorov differential equations is written in matrix form for a vector of partial characteristic functions. Section 3 contains an asymptotic analysis of the matrix equation for the vector of partial characteristic functions of the number of customers in the orbit. The main results of the research are the following theorems. Theorem 1. In the considered feedback system under asymptotic condition 0 asymptotic characteristic function of the normalized number it() of customers has the form lim { } , jw i t jw M e e where parameter κ is a positive root x of the scalar equation R B I e ( )( ) 0, x x - 0 where row-vector R { (0), (1),..., ( )} R R R N is probability distribution of number of occupied servers in the system, and is a solution of the system of equations { } ( ) ( ) 0 - - x R A B I I 0 1 , Re 1. Theorem 2. In the considered feedback system asymptotic characteristic function of centered and normalized number of customers in the orbit has the form 2 2 ( ) ( ( ) ) 2 0 lim { } ( ) , jw jw i t M e w e - where 2 ( ) , ( ) - - - 0 0 0 0 RI e g B I e RI e φ B I e row-vectors φ и g are solutions of the following systems of equations: φ A B I I R I I ( ( )) ( ), - - - 0 1 0 1 φe 0; ( ( )) ( ), - - - 0 1 1 g A B I I R I B ge 0. Section 4 presents the results of a numerical experiment, performing its accuracy and area of applicability of the proposed asymptotic method comparing with the results of simulation.

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