Synergetic effects in multichannel queuing systems with group arrivals of customers

In recent years, experts in a modelling of computer networks have shown great interest to an application of queuing systems with an infinite number of channels. This interest mainly is determined by a convenience of such systems calculations because of a queue absence and so a possibility for customers to choose channels freely. But real computer networks contain finite numbers of channels. So, it is necessary to establish an application of systems with infinite number of channels for their modelling. In this paper, we establish conditions, in which an aggregation of n oneserver systems into multiserver system for n ^да reduce to a queue disappearance. These conditions are based on limit theorems on a convergence of a number of busy channels in multiserver system to a number of busy channels in a system with the infinite number of channels and on a convergence of specially normed random process, described a number of customers arrived into a system up to a moment t to some limit the Wiener process. To obtain these results, we used the «moment» Kolmogorov-Chencov condition, an upper bound for a probability that a Gaussian process exceeds a high level and the Donsker-Prokhorov invariance principle. We take as initial a oneserver queuing system with a Poisson or deterministic input flow and group arrivals of customers including retrial queuing systems widely used in applications. In this paper, a model of n - server queuing system is considered, in which e _n (t) is a number of customers, arrived in the system up to the moment t inclusively, e _n (0) = 0, Me _n (t) = nm(t), where m(t) is a non decreasing function, q _n (t) is the number of busy servers at the moment t, q _n (0) = 0, xj is the service time of j -th customer, xj, j > 1, is the sequence of independent and identically distributed random variables with the distribution function F(t) (F (t) = 1 - F (t) ), which has a continuous and bounded by some positive number density. Assume that 0 < t ₁ < t ₂ < ... is a Poisson flow, N(t) = max(i: t _t < t), t > 0, is a Poisson process with the intensity a >0, r|j, ,. • • are independent and identically distributed random variables with integer and positive values, M| = f, Dr|j = f ₂ < да N (t) and S(t) = 2 Ik is the generalized Poisson process describing an arrival of | customers at the moment tk, k > 1. Then, input k=1 n flow into the n channel queuing system is defined by the equality e _n (t) = 2 Sk (t), where S ₁ (t),..., S _n (t) are independent copies of k=1 the process S(t) . A main result of this paper is the following statement. Theorem. If for some T >0the inequality J F(t)dm(t) <1 is hold, then PI sup q _n (t) = n I ^ 0, n ^ да. 0 V0

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