ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

12. Probability and Statistics

1991 MS Classification: 60 Shmygevskyy, Mykola, National Technical University, Kyiv, Ukraine On a generalization of Jackson’s law for network in random environment Queuing networks with input flows controlled by Markov chain η(t) are considered. It means that moments of call’s input coincide with ones of Markov chain jumps. The rate of service time is directly proportional to a number of calls in the node. If the call is serviced in k-th node, then it enters in m-th node with probability p_{km} and leaves the network with probability 1 − ∑ p_{km} = p_{kr+1}, P = ∥p_{km}∥_{1}^{r} is the switching matrix, r is the number of network’s nodes. Stationary distribution of queuing network mentioned above hasn’t a multiplicative form. Thus Jackson’s law is not true for this network ([1]). It is proposed a generalization of Jackson’s law when the uncorrelated property substitutes the independence one. For the networks a sufficient condition of noncorrelatedness is found. It is necessary that Laplace transformations expressed in terms of infinitesimal characteristics and ergodic distribution of η(t) degenerate to 0 at the spectrum of switching matrix. Reference: 1. E. A. Lebedev, and A. A. Chechelnitsky, Limit theorems for some class of the service networks, Frontiers in Pure and Applied Probability, vol. 2, part 6, (1994), 374–381.