By G. Sh. Tsitsiashvili, M. A. Osipova

This monograph offers very important learn leads to the parts of queuing idea, danger concept, graph idea and reliability concept. The analysed stochastic community types are aggregated platforms of parts in random environments. to build and to examine a good number of varied stochastic community types it's attainable by way of an explanation of latest analytical effects and a building of calculation algorithms in addition to of the appliance of bulky conventional ideas this kind of positive method is in a previous exact research of an algebraic version part and results in an visual appeal of recent unique stochastic community types, algorithms and alertness to computing device technology and knowledge technologies.Accuracy and asymptotic formulation, extra calculation algorithms were built because of an advent of regulate parameters into analysed types, a discount of multi-dimensional difficulties to at least one dimensional difficulties, a comparative research, a photograph interpretation of community types, an research of recent types features, a decision of exact distributions periods or rules of subsystems aggregation, proves of recent statements.

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Systems with a competition of customers In this section a multiserver queueing system with a customers competition for a minimal waiting time is considered. This consideration is compared with the previous section consideration and it is shown that the servers competition decreases significantly the tail of the waiting time limit distribution in a comparison with the customers competition. Consider a queueing system D|GI|m|∞ with the input flow 0 = t1 ≤ t2 = t1 + 1 ≤ t3 = t2 + 1, . . and a group arrival.

7) in the system B : for t → ∞ ∞ m pk−1 (1 − p)km . lim sup P (un > t) ∼ lim inf P (un > t) ∼ G (t) n→∞ n→∞ k=1 (i) Remark 6. 6 they are dependent. 3) is obtained for an arbitrary recurrent input flow. 5. 1. Suppose that 0 < b ≤ 1 then there are positive α, m0 so that for 0 < t < 1/m0 the inequality G(t) ≥ αtb is true and so G(t) ≤ exp(−αtb ). Put m > m0 and denote ∞ Jm = m mG (t)dt = Um + Vm , 0 1/m Um = ∞ m mG (t)dt, Vm = m mG (t)dt. 0 1/m Then 1/m 1/m m exp(−αmtb )dt = Um ≤ 0 = m exp(−((αm)1/bt)b ) 0 m (αm)1/b (αm)1/b /m exp(−v b )dv ∼ 0 = O(m m−1 Vm ≤ mG 1−1/b m (αm)1/b dt(αm)1/b = (αm)1/b ∞ exp(−v b )dv = 0 ), m → ∞, ∞ (1/m) ∞ G(t)dt ≤ m exp(−α(m − 1)m−b ) 1/m G(t)dt = 0 = o(m1−1/b), m → ∞.

Property 2. If F (x) ∈ S∗ then FI (x) ∈ S where F I (x)= Property 3. Suppose that Q(x)= − ln F (x), Q (x) = q(x). Each condition is sufficient for the inclusion F (x) ∈ S∗ : 1) lim sup xq(x) < ∞, x→∞ 2) lim q(x) = 0, lim xq(x) = ∞, q(x) ∈ R(−δ), δ ∈ (0, 1]. x→∞ x→∞ The property 3 leads to the following statement. 1. Each condition is sufficient for the inclusion F (x)∈S∗ : a) ∃ a < −1, l(x) ∈ L1 so that F (x) = l(x)xa, b) ∃ a ∈ (−1, 0), l(x) ∈ L1 so that q(x) = l(x)xa. 1. f. 1 then 1 − F (x) ∈ S∗.

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