Queues with Dropping Functions and General Arrival Processes

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.

Burst ratio in a single-server queue

“…Now we can easily finish the proof. Namely, combining (1) with (28), (23) and (27) finishes the proof of Theorem 1.…”

The Single-Server Queue with the Dropping Function and Infinite Buffer

“…There is, however, a simpler way to compute the loss ratio. It is based on the following formula (see, e.g., [16]):…”

“…The analysis based on the queueing theory research on queues with the dropping function was initiated with approximate solutions given in [10,11] and followed by exact results [12][13][14][15][16][17][18][19], obtained for various traffic and service assumptions. In particular, in [10], an approximate analysis 2 Mathematical Problems in Engineering of the model with batch Poisson arrivals and linear dropping function was presented.…”

“…In [15], the model with multiple service stations was studied. In [16,17], systems with autocorrelated arrivals were considered, with and without batch arrivals, respectively. Finally, [18,19] deal with a generalization of the model to the case with continuous packet volumes, continuous buffer capacity, and an acceptance function (rather than dropping function), which enqueues an arriving packet with probability that depends on the remaining buffer capacity.…”

“…In all the recent studies, the dropping function does not have a fixed shape; any form of this function can be used. As shown in, e.g., [16], this allows us to control precisely several performance characteristics of interest. Therefore, the dropping function is not further specified in this paper; it can assume any form.…”