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 A_n = M | M | n | 0 with intensity of input flow nk and intensities of service at all n servers, p = k/p,. Denote P_n (p) the stationary blocking probability in the system A_n at a given p. Let a_n, b_n, n > 1, be two real sequences. For n ^ o>. ci we assume that a_n ·n if limsup - < 1. Let us say a_n ~ b_n, if b_nnn. " 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 = A_l = ... = 'k_m 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 c_k 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 nn_k, 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-,...,п_т =щ-. Assume that the numbers -,...,- are rational and Cj Cj Cj Cj rewrite them as - = -,...,-= -, where pairs of positive integers (p₂,q₂),..., (p_m,q_m) consist of mutually prime numbers. ^C1 92 ^c\ Чш Then, for the numbers n₂,...,n_m to be integers, it requires that number is a multiple of q₂q_m. Therefore the number щ should be divided by the smallest common multiple L of the numbers q₂,-..,q_m. Thus, all possible values of the numbers n_i,...,n_m, satisfying the basic equality (1), look like these Ъ Чш

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