Распределение ресурсов в многоканальной системе массового обслуживания с блокировкой на основе синергетических эффектов

Строится оценка скорости сходимости к нулю вероятности отказа в многоканальной системе обслуживания, моделирующей телекоммуникационную сеть, при стремлении к бесконечности количества серверов и нагрузки. С ее помощью решается задача разделения ресурсов между различными пользователями телекоммуникационной сети.

The allocation of resources in multichannel loss queuing system based on synergistic effects

In this paper, we consider n - server loss system under the assumption that the intensity of the input flow is proportional to n. We investigate the convergence of the blocking probability in this system to zero at n ^ o>. A similar problem arises in the design of modern data transmission systems. A specific of suggested asymptotic results is that we did not obtain accuracy formulas or solutions of optimization problems for the transmission systems. Consider queuing system An = M | M | n | 0 with intensity of input flow nk and intensities of service at all n servers, p = k/p,. Denote Pn (p) the stationary blocking probability in the system An at a given p. Let an, bn, n > 1, be two real sequences. For n ^ o>. ci we assume that an ·n if limsup - < 1. Let us say an ~ bn, if bnnn. " К IT Теорема 1. The following limit ratio is true: P„(1) ~. -, n V 7Ш Теорема 2. At p < 1 following relations are valid f n ln2p 1 IT f n ln2p p - exp|--- J-J- ±Pn (p^ exp| -v 2 ]\тт\8 ~ - ^ 2 Suppose that we have m independent Poisson flows of customers with intensities 'k = Al =... = 'km and parallel servers with the intensity of service at each of them equal to We assume that the service of the k-th flow customer is realized on ck servers, 1 < k < m. We shoud like to distribute the servers between the flows so that the blocking probabilities P(k)(1) for each of the flow k = l,...,m are about the same. Let the number of servers in the k-th subsystem be nnk, from Theorem 1 we obtain that the basic equations p -11 IT pr lnp J\ Пи'У p - 1 «1 _ _ »m (1) Распределение ресурсов в многоканальной системе массового обслуживания с блокировкой are fulfilled. We rewrite these equations in the form и2 =щ-,...,пт =щ-. Assume that the numbers -,...,- are rational and Cj Cj Cj Cj rewrite them as - = -,...,-= -, where pairs of positive integers (p2,q2),..., (pm,qm) consist of mutually prime numbers. C1 92 c\ Чш Then, for the numbers n2,...,nm to be integers, it requires that number is a multiple of q2qm. Therefore the number щ should be divided by the smallest common multiple L of the numbers q2,-..,qm. Thus, all possible values of the numbers ni,...,nm, satisfying the basic equality (1), look like these Ъ Чш

Tsitsiashvili G.S.1, 2 , Osipova M.A.1, 2 , Samouylov K.E. 3 , Gaidamaka Y.V. 3
Редакция журнала "Вестник ТГУ. УВТиИ"
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  • 1 Far East Federal University
  • 2 Institute for Applied Mathematics, Far Eastern Branch of Russian Academy Sciences
  • 3 Russian Friendship University of Peoples
многоканальная система массового обслуживания с отказами; телекоммуникационная сеть; модели телетрафика; multiserver queuing system with blocking; telecommunication network; models of teletrack
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