Rod J. Fretwell

Closed Form Performance Distributions of a Discrete Time GIg/D/1/N Queue with Correlated Traffic

A novel approach is applied to the study of a queue with general correlated traffic, in that the only features of the traffic which are taken into account are the usual measures of its correlation: the traffic is modelled as a batch renewal process. The batch renewal process is a precise tool for investigation into effects of correlation because it is the least biased choice of process which is completely determined by the infinite sets of measures of the traffic’s correlation (e.g. indices of dispersion, covariances or correlation functions).

A cost-effective approximation for SRD traffic in arbitrary multi-buffered networks

Abstract The question When may short range dependence (SRD) traffic be modelled by uncorrelated traffic with negligible error? is addressed in terms of performance metrics such as stationary queue length distribution (QLD), marginal blocking probability (BP) and mean queue length (MQL) for generic queueing network models (QNMs) of some multi-buffered communication systems such as asynchronous transfer mode (ATM) switch architectures and other high-speed networks with output port queueing. To this end, a cost-effective queue-by-queue decomposition algorithm is devised for the approximate analysis of arbitrary open first-come-first-served (FCFS) discrete-time QNMs with SRD arrivals, deterministic (D) service times, departure first (DF) buffer management simultaneity policy and repetitive service blocking with random destination (RS-RD). The shifted generalised geometric (sGGeo) batch renewal process, which exhibits geometrically declining count and interval covariances, is adopted to model external SRD traffic. To facilitate tractability with a tolerable accuracy and, thus, the understanding of the quantitative behaviour of SRD traffic in the interior of the network, the sGGeo process is approximated by an ordinary generalised geometric (GGeo) process whose first two moments of the count distribution match the corresponding first two sGGeo moments. Consequently, the sGGeo/D/1/N and GGeo/D/1/N queues, in conjunction with GGeo-type flow approximation formulae, play the role of analytic building blocks within the proposed decomposition scheme. Typical numerical tests are presented to (i) determine empirical conditions at a single queue level for there to be negligible approximation error in some performance metrics as a result of using a GGeo process in place of the sGGeo process, (ii) illustrate at a queueing network level the credibility of the decomposition algorithm via simulation and (iii) investigate the effect of external sGGeo-type traffic with varying degree of correlation upon network performance.

Batch renewal process: exact model of traffic correlation

The batch renewal process can conform exactly to any given (bounded) indices of dispersion or (infinite) sets of correlation functions and is the least biased choice of all possible processes given only such measures of correlation. It contains no feature extraneous to correlation. In particular, it implies no assumptions about traffic burst structure.

Correlated traffic modelling: Batch renewal and Markov modulated processes

The usual measures of traffic correlation do not determine the traffic process completely. Consequently there may be a number of traffic processes which induce the same correlation but which differ in other features and so differ in impact on performance.

LRD and SRD traffic: review of results and open issues for the batch renewal process

The batch renewal process is the least-biased choice of process given only the measures of count correlation and interval correlation at all lags.This paper reviews the batch renewal process, both for LRD (long-range-dependent) traffic and for SRD (short-range-dependent) traffic in the discrete space-discrete time domain, and in the wider context of general traffic in that domain. It shows some applications of the batch renewal process in simple queues and in queueing network models. The paper concludes with open research problems and issues arising from the discussion.

Analysis of Discrete-Time Queues with Space and Service Priorities for Arbitrary Arrival Processes

Finite buffer queues with service and space priorities are of great importance towards effective congestion control and quality of service (QoS) protection in high speed networks. This paper presents closed-form analytical solutions, based on the principle of maximum entropy (ME), for the performance evaluation of single server finite capacity G/G/1/N censored queues with R (R>1) distinct priority classes under preemptive-resume (PR) or head-of-line (HoL) service disciplines. Buffer management for these queues employs either partial buffer sharing (PBS) or complete buffer sharing (CBS) schemes. A vector, N, representing a sequence of thresholds (N1, N2,... ,NR) for each priority class jobs, is used to implement these buffer management schemes in order to provide quality of service constraints. Closed-form expressions for various performance metrics have been derived

Performance of Weighted Fair Queuing System with Multi-class Jobs

This paper analyses the weighted fair queueing (WFQ) system subject to three classes of network applications. The arrival traffic flows follow Poisson processes. The service times is exponentially distributed. The system is modelled as a two-dimensional Markov chain and use matrix geometric technique to solve its stationary probabilities. The determination of the steady state probabilities can be used to compute the performance measures of the system, such as the mean queue length, the throughput and the mean response time. Numerical experiments corroborate the theoretical results are offered and make the developed model as an effective tool for examining the performance of WFQ systems.

Analysis of Correlated Traffic by Batch Renewal Process

The discovery of the batch renewal process (BRP) provides means to investigate the effect of traffic correlation independent of any other traffic characteristics. BRP is considered to be the least-biased choice traffic model, given only the measures of correlation, because the BRP is completely determined by the complete (infinitely denumerable) sets of count correlation and of interval correlation. A procedure is described in this paper on how to construct a unique BRP from given correlations. This procedure gives telecommunication engineers the means to how to generate the least-biased traffic model from the statistics of covariances alone.

General Solution of Queueing Model with Batch Renewal Arrival by Two Approaches

Queueing models with finite capacity and batch arrival or batch service processes are widely recognised as powerful tools for representing transportation networks, flexible manufacturing systems and predicting their performance. Although a considerable progress has been made in the field of analysis of a queue with single server and batch service in the literature, more work is needed to study more realistic type of queues with both batch arrival and service processes. This paper gives two general solutions to queues with continuous time batch renewal processes. Which are dual embedded Markov chains and supplementary variable technique respectively, and the two approaches are shown essentially equivalent.

Performance Evaluation of Weighted Fair Queuing System Using Matrix Geometric Method

This paper analyses a multiple class single server M/M/1/K queue with finite capacity under weighted fair queuing (WFQ) discipline. The Poisson process has been used to model the multiple classes of arrival streams. The service times have exponential distribution. We assume each class is assigned a virtual queue and incoming jobs enter the virtual queue related to their class and served in FIFO order.We model our system as a two dimensional Markov chain and use the matrix-geometric method to solve its stationary probabilities. This paper presents a matrix geometric solution to the M/M/1/K queue with finite buffer under (WFQ) service. In addition, the paper shows the state transition diagram of the Markov chain and presents the state balance equations, from which the stationary queue length distribution and other measures of interest can be obtained. Numerical experiments corroborating the theoretical results are also offered.