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By Hanson T.E., Johnson W.O.

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12). 3 is proved. Remark 2. 3 defines a case which is important for queueing networks and when a maximum of an ability to handle customers reaches for equal load coefficients in different network nodes. 12) solution is assumed in [36] for a deterministic vector optimization problem. 4. 12) has − → the single solution M (k) . For the route matrix Θ with positive elements (besides of − → − → θ00 ) define the vector M = M (Θ) = = (M1 , . 15) (1, M1, M2, . . , Mm) = (1, M1, M2, . . , Mm)Θ − → and put Θ(M) = {Θ : M (Θ) ∈ M}.

Divide a time halfaxis t ≥ 0 into cycles which consist of an idle interval and a service time interval. Each cycle length coincides with a sum of two independent random variables: first of them has exponential distribution wuith the parameter λ and second of them coincides with a service time. The mean cycle length equals 1/λ + b1 where b1 is the mean service time. From the integral renewal theorem [7, chapter 9, § 4] obtain that the stationary output flow intensity satisfies the formula I(λ) = 1 + b1 λ −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∗. 2. Systems with a competition of servers In this section a multiserver queueing system with a competition of servers for a customer is considered. This system is compared with the classical GI|GI|m|∞ multiserver systems by abilities to handle customers and by tails of waiting time limit distribution. The tails calculation and comparison are based on the Embrechts– Veraverbeke formula.

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