Decomposing the queue length distribution of processor-sharing models into queue lengths of permanent customer queues

Abstract

We obtain a decomposition result for the steady state queue length distribution in egalitarian processor-sharing (PS) models. In particular, for multi-class egalitarian PS queues, we show that the marginal queue length distribution for each class equals the queue length distribution of an equivalent single class PS model with a random number of permanent customers. Similarly, the mean sojourn time (conditioned on the initial service requirement) for each class can be obtained by conditioning on the number of permanent customers. The decomposition result implies linear relations between the marginal queue length probabilities, which also hold for other PS models such as the egalitarian PS models with state-dependent system capacity that only depends on the total number of customers in the system. Based on the exact decomposition result for egalitarian PS queues, we propose a similar decomposition for discriminatory processor-sharing (DPS) models, and numerically show that the approximation is accurate for moderate differences in service weights.

A survey on discriminatory processor sharing

“…In [13] the authors develop an approximate but more tractable decomposition result for DPS which turns out to be quite accurate for moderate differences in service weights.…”

“…From equations (13) and (14), the authors of [49] conclude that by giving larger weight to class 1 (g 1 > g 2 ), the number of class-1 jobs can be reduced, whereas the number of class-2 jobs will not change significantly. In contrast, if class 2 were given higher preference (g 2 > g 1 ), the number of class-2 jobs will not be reduced, while the number of class-1 jobs would drastically increase.…”

Insensitive bounds for the moments of the sojourn time distribution in the M/G/1 processor-sharing queue

“…The probabilities P (N 1 = i) and P (N 2 = j) are in fact the marginal queue length probabilities of a two-class egalitarian processor-sharing queue (with two types of customer classes); see e.g. [4]. It is not difficult to show that…”

Analysis of a Polling System Modeling QoS Differentiation in WLANs

“…[9], [16]. The extension to non-persistent sources is provided in [3], [14], where a flow level model is introduced that is analysed using a Processor Sharing queueing model. Comparison with discrete event simulation shows that indeed the MAC layer can be adequately modeled via the Processor Sharing mechanism.…”

“…Although the flow level modeling of [3], [14], [17], [18], [19] captures the resource sharing behaviour of the MAC layer of 802.11 protocols, the essential behaviour at the packet level is not captured. At that level a flow consists of a series of packets that are transmitted one by one, where transmissions of different flows are intertwined.…”