Synergetic effect in network with hyper exponential distributions of service times

In this paper, a problem of construction of queuing network with multiserver nodes and vanishing queues with large numbers of servers is solved. In queing systems such synergetic effects are analyzed suffieciently detailed. But in queuing networks an analysis of these effects are more complicated because of mixing of outgoing flows. This problem is closely connected with intensively developing researchs of networks with infinite numbers of servers in which there are no queues. An interest to these networks is arised by a simplicity of probability distributions in the networks so by manifold applications like claud computing and systmes of public service. The networks with service times which have hyperexponential distributions are analyzed. That is service times distributions are probability mixtures of finite numbers of exponential distributions. Any network node is divided into few M | M | n | ю systems with special numbers of servers. As a result initial queuing network is transformed into Jackson type network. So, to analyze synergetic effect it is enough to consider separate multiserver systems of M | M | n | ю type for which this effects may be analyzed sufficiently simple. In such a way calculational and constructive mehtods are combuined. Main element of this construction is an algorithm of a transformation of queuing network with hyperexponential distributions of service times into network with exponentialy distributed service times. Main property of this transformation is a retention of servers numbers in different nodes. This algorithm is based on ergodicity condition essentially. Then product theorem for the Jackson type queuing network is used. This theorem allows to calculate limit distribution of customers numbers in the network by limit distribution of customers numbers in different nodes. So, to obtain synergetic effect in the network it is possible to use synergetic effect in the nodes. Considered model of queing network may be spread onto networks with completely monotone distributions of service times. Here complete monotone distribution is distribution with probability density f(t), satisfying the inequalities (-1) d f() > o, t > O, k > 1. _dt Well known results allow to approximate any completely monotone distribution F(t) by hyperexponential distribution G(t), satisfying for any e > O the inequality sup_t>_O | F(t) - G(t) |< s. In queing theory completely monotone distributions are the Weibull, Pareto, and other subexponential distributions. To approximate probability of queue existence in networks with completely monotone distribution it is possible using stability theorems. A possibility to consider synergetic effects in separate nodes of the network in the regime of heavy traffic is discussed also.

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