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For the stationary loss systems M/M/m/K and GI/M/m/K, we study two quantities: the number of lost customers during the time interval (0,t] (the first system only), and the number of lost customers among the first n customers to arrive (both systems). We derive explicit bounds for the total variation distances between the distributions of these quantities and compound Poisson–geometric distributions. The bounds are small in the light traffic case, i.e., when the loss of a customer is a rare event. To prove our results, we show that the studied quantities can be interpreted as accumulated rewards of stationary renewal reward processes, embedded into the queue length process or the process of queue lengths immediately before arrivals of new customers, and apply general results by Erhardsson on compound Poisson approximation for renewal reward processes. 相似文献
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Torkel Erhardsson 《Probability Theory and Related Fields》2000,117(2):145-161
We consider a stationary version of a renewal reward process, i.e., a renewal process where a random variable called a reward
is associated with each renewal. The rewards are nonnegative and I.I.D., but each reward may depend on the distance to the
next renewal. We give an explicit bound for the total variation distance between the distribution of the accumulated reward
over the interval (0,L] and a compound Poisson distribution. The bound depends in its simplest form only on the first two joint moments of T and Y (or I{Y > 0}), where T is the distance between successive renewals and Y is the reward. If T and Y are independent, and LE(Y) (or LP(Y > 0)) is bounded or Y binary valued, then the bound is O(E(Y)) as E(Y) → 0 (or O(P(Y > 0)) as P(Y > 0) → 0). To prove our result we generalize a Poisson approximation theorem for point processes by Barbour and Brown, derived
using Stein's method and Palm theory, to the case of compound Poisson approximation, and combine this theorem with suitable
couplings.
Received: 1 March 1999 / Revised version: 2 August 1999 /?Published online: 31 May 2000 相似文献
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