Researching of Retrial Queueing system MMPP|GI|1 by using asymptotic analysis method on heavy load condition

In the article we research the queueing system with the orbit (retrial queueing system) with MMPP input flow, which diagonal matrix of the arrival rates associated with each state is pA and generator matrix of Markov chain n(t) is q, service time of a customer is distributed by general independent law B(x). The task is to obtain probability distribution for the number of calls in the orbit. Let following notation: i(t) - stochastic process describing the number of calls in the orbit, n(t) - Markov chain controlling ММРР input flow, z(t) - remaining service time and k(t) defines a state of service. The stochastic process with variable component number {1, n(t), i(t), z(t)}, {0, n(t), i(t)} of the system states in time is Markov, thus the Kolmogorov system of differential equation was written for obtaining probability distribution {P(0,n,i,t); P(1,n,i,z,t)} the RQ-system states. The system was considered at stationary state in matrix form. The transition to the characteristic functions in the equations system was made, then the asymptotic analysis method on heavy load condition was applied. After mathematical transformations the asymptotic characteristic function of the calls number in the orbit was obtained as: h(u) = (1-- u-) , so it is characteristic function of gamma- P(1 -P) 1 I Ь j-1 1 distribution with parameters R = — I VIE +—\ I , а = 1 +--R , where Ь, b are first and second Ь У 2Ь ) Ьст moments of service distribution law, vector V is a solution of the system VQ = R(bI -1). Also b in the paper some numerical results of asymptotic distribution parameters researching are presented.

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