Queueing models for the analysis of communication systems

Abstract

Queueing models can be used to model and analyze the performance of various subsystems in telecommunication networks; for instance, to estimate the packet loss and packet delay in network routers. Since time is usually synchronized, discrete-time models come natural. We start this paper with a review of suitable discrete-time queueing models for communication systems. We pay special attention to two important characteristics of communication systems. First, traffic usually arrives in bursts, making the classic modeling of the arrival streams by Poisson processes inadequate and requiring the use of more advanced correlated arrival models. Second, different applications have different quality-of-service requirements (packet loss, packet delay, jitter, etc.). Consequently, the common first-come-first-served (FCFS) scheduling is not satisfactory and more elaborate scheduling disciplines are required. Both properties make common memoryless queueing models (M/M/1-type models) inadequate. After the review, we therefore concentrate on a discrete-time queueing analysis with two traffic classes, heterogeneous train arrivals and a priority scheduling discipline, as an example analysis where both time correlation and heterogeneity in the arrival process as well as non-FCFS scheduling are taken into account. Focus is on delay performance measures, such as the mean delay experienced by both types of packets and probability tails of these delays

Rejoinder on: Queueing models for the analysis of communication systems

“…Indeed, this is often possible, although properly identifying the counterpart is not always trivial. Before pointing to some issues with the continuous-time equivalent of the model in [1], we detail how a limiting operation can be performed for obtaining the mean queue content of a continuous-time M/G/1 queue from that of the discrete-time M/G/1 queue. We start with the continuous-time model and discretize the inter-arrival times and service times with respect to some slot length ∆.…”

“…More interesting limits can be obtained by assuming that trains only produce a packet with some probability. However, such an extension of the discrete-time model voids the discrete-time analysis of [1], as discussed further in the next section of this rejoinder. Secondly, the server processes a single packet per slot, the obvious limit being immediate service of packets.…”

“…In this section, we describe some of the potential extensions to the model and analysis in [1] inquired about by the discussants in their comments about the paper. These extensions can be roughly classified into (i) refinements of the modeling assumptions and (ii) the calculation of additional performance measures.…”

“…As class M does not "see" lower-priority classes and as the scheduling within higher priority-classes has no impact on the delay of class M , the analysis of the multi-class system reduces to the analysis of a queueing system with two priority classes, as in the present. The argument above however yields a model with heterogeneous trains (trains with different distributions for their lengths) within the high-priority class, whereas the model in [1] assumes homogeneous trains. Consequently, a direct application of the results of the two-class model is not possible.…”

“…However, we suspect that the transition from homogeneous to heterogeneous trains (or to more than two priority classes) is not that hard. Without detailing such analysis here, we mention that heterogeneity requires the inclusion of more than two vectors of infinite dimension of the number of messages of which the n-th packet arrives during a random slot in, for instance, expression (10) for P T in [1]. Note that the inclusion of more than one traffic class in one priority class is an interesting generalization in its own right.…”