Abstract
In a queueing system with the dropping function the arriving customer can be denied service (dropped) with the probability that is a function of the queue length at the time of arrival of this customer. The potential applicability of such mechanism is very wide due to the fact that by choosing the shape of this function one can easily manipulate several performance characteristics of the queueing system. In this paper we carry out analysis of the queueing system with the dropping function and a very general model of arrival process—the model which includes batch arrivals and the interarrival time autocorrelation, and allows for fitting the actual shape of the interarrival time distribution and its moments. For such a system we obtain formulas for the distribution of the queue length and the overall customer loss ratio. The analytical results are accompanied with numerical examples computed for several dropping functions.
Highlights
Consider a simple queueing system with a stream of arriving customers, the queue of customers waiting for service and one service station performing the service that takes random time
: The k-lag autocorrelation of batch interarrival times is CorrðkÞ 1⁄4 p DÀ0 1CðCkÀ1 À 1 pÞDÀ0 1C 1=Var; with C 1⁄4 ÀDÀ0 1ðD À D0Þ; where p is the stationary vector for intensity matrix C−I, i.e. it fulfills pðC À IÞ 1⁄4 1⁄20; . . . ; 0; p 1 1⁄4 1: The counting function of the BMAP process is defined as Pi;jðn; tÞ 1⁄4 PðNðtÞ 1⁄4 n; JðtÞ 1⁄4 jjNð0Þ 1⁄4 0; Jð0Þ 1⁄4 iÞ; where P denotes probability
We presented an analysis of the queueing system with the dropping function of arbitrary type, arbitrary distribution of the service time, and a very general customer arrival process, which allows for modeling of autocorrelation, batch arrivals and arbitrary shape of the interarrival time distribution
Summary
Consider a simple queueing system with a stream of arriving customers, the queue of customers waiting for service and one service station (server) performing the service that takes random time. Approach (b) cannot be used in most everyday-life applications of queueing systems This is due to the fact, that the operator of the queue has usually no means to reduce quickly the customer arrival rate (think of a bank, for instance). Besides networking, the dropping functions have great potential for applications in many other systems involving queueing of customers (or jobs, tasks etc.) This is connected with their powerful control capabilities. In [21], an approximate analysis of the queue with batch Poisson arrivals, linear dropping function and exponential service times was carried out. For these reasons, the previous analytical results on queues with dropping functions are of little use in networking and in other applications with sophisticated, autocorrelated traffic of customers.
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