Nonstationary queues with Interrrupted Poisson arrivals and unreliable/repairable servers

A queueing model having a nonstationary Interrupted Poisson arrival process (IPP(t)),s time-dependent exponential unreliable/repairable servers and finite capacityc is introduced, and an approximation method for analysis of it is developed and tested. Approximations are developed for the time-dependent queue length moments and the system viewpoint waiting time distributions and moments. The approximation involves state-space partitioning and numerically integrating partial-moment differential equations (PMDEs). Surrogate distribution approximations (SDA's) are used to close the system of PMDEs. The approximations allow for analysis using only (s + 1)(s + 6) differential equations for the queue length moments rather than the 2(c + 1)(s +1) equations required by the classic method of numerically integrating the full set of Kolmogorov-forward equations. Effectively hours of cpu time are reduced to minutes for even modest capacity systems. Approximations for waiting time distributions and moments are developed.This research was partially funded by National Science Foundation grant ECS-8404409.     

Approximations for the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">GI</Emphasis> type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.      

Behaviour of queueing approximations based on sample moments

We investigate moment–based queueing approximations in the presence of sampling error. Let L be the steady–state mean number in the system for a GI/M/1 queue. We focus on the estimation of L under the assumption that only sample moments of the interarrival–time distribution are known. A simulation experiment is carried out for several interarrival–time distributions. For each case, sample moments from the interarrival–time distribution are matched to an approximating phase–type distribution and the corresponding estimate L is obtained. We show that the sampling error in the moments induces bias as well as variability in L. Based on our simulation experiment, we suggest matching only two moments when the sample coefficient of variation is low or when sample size is low; otherwise, matching three moments is preferable.     

Simple Approximations for the GI/G/c Queue-II: The Moments,the Inverse Distribution Function and the Loss Function of the Number in the System and of the Queue Delay

Based on results obtained in part I of this paper, approximations for the first four moments of the number in the system are developed and thence used to approximate the inverse distribution function (IDF) and the loss functions (LF), employing Shore's general approximations. Existing approximations for the first two moments of queueing time in a GI/G/l queue serve to approximate the IDF and the LF of queueing time in the corresponding GI/G/c queue. The accuracy attained is generally satisfactory, while a remarkable algebraic simplicity is preserved. A numerical example demonstrates the applicability of some of the new approximations to solve optimization problems.     

A graphical investigation of error bounds for moment-based queueing approximations

Many approximations of queueing performance measures are based on moment matching. Empirical and theoretical results show that although approximations based on two moments are often accurate, two-moment approximations can be arbitrarily bad and sometimes three-moment approximations are far better. In this paper, we investigate graphically error bounds for two- and three-moment approximations of three performance measures forGI/M/ · type models. Our graphical analysis provides insight into the adequacy of two- and three-moment approximations as a function of standardized moments of the interarrival-time distribution. We also discuss how the behavior of these approximations varies with other model parameters and with the performance measure being approximated.     

A Maximum Entropy Analysis of the G/G/1 Queue at Equilibrium

The principle of maximum entropy is used to analyse a G/G/1 queue at equilibrium when the constraints involve only the first two moments of the interarrival-time and service-time distributions. Robust recursive relations for the queue-length distribution are determined, and a probability density function analogue is characterized. Furthermore, connections with classical queueing theory and operational analysis are established, and an overall approximation, based on the concept of ‘global’ maximum entropy, is introduced. Numerical examples provide useful information on how critically system behaviour is affected by the distributional form of the interarrival and service times, and favourable comparisons are made with diffusion and other approximations. Comments on the implication of the work to the analysis of more general queueing systems are included.     

Higher order approximations for tandem queueing networks

In this paper a higher order approximation for single server queues and tandem queueing networks is proposed and studied. Different from the most popular two-moment based approximations in the literature, the higher order approximation uses the higher moments of the interarrival and service distributions in evaluating the performance measures for queueing networks. It is built upon the MacLaurin series analysis, a method that is recently developed to analyze single-node queues, along with the idea of decomposition using higher orders of the moments matched to a distribution. The approximation is computationally flexible in that it can use as many moments of the interarrival and service distributions as desired and produce the corresponding moments for the waiting and interdeparture times. Therefore it can also be used to study several interesting issues that arise in the study of queueing network approximations, such as the effects of higher moments and correlations. Numerical results for single server queues and tandem queueing networks show that this approximation is better than the two-moment based approximations in most cases.     

On the inapproximability of <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">K</Emphasis>: why two moments of job size distribution are not enough

The M/G/K queueing system is one of the oldest models for multiserver systems and has been the topic of performance papers for almost half a century. However, even now, only coarse approximations exist for its mean waiting time. All the closed-form (nonnumerical) approximations in the literature are based on (at most) the first two moments of the job size distribution. In this paper we prove that no approximation based on only the first two moments can be accurate for all job size distributions, and we provide a lower bound on the inapproximability ratio, which we refer to as “the gap.” This is the first such result in the literature to address “the gap.” The proof technique behind this result is novel as well and combines mean value analysis, sample path techniques, scheduling, regenerative arguments, and asymptotic estimates. Finally, our work provides insight into the effect of higher moments of the job size distribution on the mean waiting time.     

A numerically efficient method for the MAP/D/1/K queue via rational approximations

The Markovian Arrival Process (MAP), which contains the Markov Modulated Poisson Process (MMPP) and the Phase-Type (PH) renewal processes as special cases, is a convenient traffic model for use in the performance analysis of Asynchronous Transfer Mode (ATM) networks. In ATM networks, packets are of fixed length and the buffering memory in switching nodes is limited to a finite numberK of cells. These motivate us to study the MAP/D/1/K queue. We present an algorithm to compute the stationary virtual waiting time distribution for the MAP/D/1/K queue via rational approximations for the deterministic service time distribution in transform domain. These approximations include the well-known Erlang distributions and the Padé approximations that we propose. Using these approximations, the solution for the queueing system is shown to reduce to the solution of a linear differential equation with suitable boundary conditions. The proposed algorithm has a computational complexity independent of the queue storage capacityK. We show through numerical examples that, the idea of using Padé approximations for the MAP/D/1/K queue can yield very high accuracy with tractable computational load even in the case of large queue capacities.This work was done when the author was with the Bilkent University, Ankara, Turkey and the research was supported by TÜBITAK under Grant No. EEEAG-93.     

An interpolation approximation for the GI/G/1 queue based on multipoint Padé approximation

The performance evaluation of many complex manufacturing, communication and computer systems has been made possible by modeling them as queueing systems. Many approximations used in queueing theory have been drawn from the behavior of queues in light and heavy traffic conditions. In this paper, we propose a new approximation technique, which combines the light and heavy traffic characteristics. This interpolation approximation is based on the theory of multipoint Padé approximation which is applied at two points: light and heavy traffic. We show how this can be applied for estimating the waiting time moments of the ^GI/G/1 queue. The light traffic derivatives of any order can be evaluated using the MacLaurin series analysis procedure. The heavy traffic limits of the ^GI/G/1 queue are well known in the literature. Our technique generalizes the previously developed interpolation approximations and can be used to approximate any order of the waiting time moments. Through numerical examples, we show that the moments of the steady state waiting time can be estimated with extremely high accuracy under all ranges of traffic intensities using low orders of the approximant. We also present a framework for the development of simple analytical approximation formulas. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analyzing retrial queues by censoring

In this paper we analyze the M/M/c retrial queue using the censoring technique. This technique allows us to carry out an asymptotic analysis, which leads to interesting and useful asymptotic results. Based on the asymptotic analysis, we develop two methods for obtaining approximations to the stationary probabilities, from which other performance metrics can be obtained. We demonstrate that the two proposed approximations are good alternatives to existing approximation methods. We expect that the technique used here can be applied to other retrial queueing models.     

The $$MMAP\left[ K \right]/PH\left[ K \right]/1$$ queues with a last-come-first-served preemptive service discipline

This paper studies two queueing systems with a Markov arrival process with marked arrivals and PH-distribution service times for each type of customer. Customers (regardless of their types) are served on a last-come-first-served preemptive resume and repeat basis, respectively. The focus is on the stationary distribution of queue strings in the system and busy periods. Efficient algorithms are developed for computing the stationary distribution of queue strings, the mean numbers of customers served in a busy period, and the mean length of a busy period. Comparison is conducted numerically between performance measures of queueing systems with preemptive resume and preemptive repeat service disciplines. A counter-intuitive observation is that for a class of service time distributions, the repeat discipline performs better than the resume one. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analyticity of single-server queues in light traffic

Recently, several methods have been proposed to approximate performance measures of queueing systems based on their light traffic derivatives, e.g., the MacLaurin expansion, the Padé approximation, and interpolation with heavy traffic limits. The key condition required in all these approximations is that the performance measures be analytic when the arrival rates equal to zero. In this paper, we study theGI/G/1 queue. We show that if the c.d.f. of the interarrival time can be expressed as a MacLaurin series over [0, ), then the mean steady-state system time of a job is indeed analytic when the arrival rate to the queue equals to zero. This condition is satisfied by phase-type distributions but not c.d.f.'s without support [0, ), such as uniform and shifted exponential distributions. In fact, we show through two examples that the analyticity does not hold for most commonly used distribution functions which do not satisfy this condition.     

Exact and approximate numerical solutions of steady-state distributions arising in the queueGI/G/1

In this paper we first obtain, in a unified way, closed-form analytic expressions in terms of roots of the so-called characteristic equation (c.e.), and then discuss the exact numerical solutions of steady-state distributions of (i) actual queueing time, (ii) virtual queueing time, (iii) actual idle time, and (iv) interdeparture time for the queueGI/R/1, whereR denotes the class of distributions whose Laplace-Stieltjes transforms (LSTs) are rational functions (ratios of a polynomial of degree at mostn to a polynomial of degreen). For the purpose of numerical discussions of idle- and interdeparture-time distributions, the interarrival-time distribution is also taken to belong to the classR. It is also shown that numerical computations of the idle-time distribution ofR/G/1 queues can be done even ifG is not taken asR. Throughout the discussions it is assumed that the queue discipline is first-come-first-served (FCFS). For the tail of the actual queueing-time distribution ofGI/R/1, approximations in terms of one or more roots of the c.e. are also discussed. If more than one root is used, they are taken in ascending order of magnitude. Numerical aspects have been tested for a variety of complex interarrival- and service-time distributions. The analysis is not restricted to generalized distributions with phases such as Coxian-n (C _n ), but also covers nonphase type distributions such as uniform (U) and deterministic (D). Some numerical results are also presented in the form of tables and figures. It is expected that the results obtained from the present study should prove to be useful not only to practitioners, but also to queueing theorists who would like to test the accuracies of inequalities, bounds or approximations.     

On an Approximation to the Mean Response Times of Priority Classes in a Stable G/G/c/PR Queue

Analytic approximations are proposed for the mean response-times of R(≥ 2) priority classes in a stable G/G/c/PR queue with general class interarrival and service time distributions and c(≥ 2) parallel servers under pre-emptive resume (PR) scheduling. The generalized exponential (GE) distributional model is used to represent general distributions with known first two moments per class. The analysis is based on the extension of known heuristic arguments and earlier results regarding the study of the stable GE/GE/c/FCFS (c ≥ 1, single class) and GE/G/1/PR queues. Numerical examples illustrate the accuracy of the proposed approximations in relation to simulations involving different interarrival and service time distributions per class. Moreover, GE-type performance bounds on the system response time per class are defined. Comments on the role of the new mean response time expressions towards the approximation of the joint and marginal queue length distributions of a stable G/G/c/PR queue are included.     

A matrix-analytic solution for the DBMAP/PH/1 priority queue

Priority queueing models have been commonly used in telecommunication systems. The development of analytically tractable models to determine their performance is vitally important. The discrete time batch Markovian arrival process (DBMAP) has been widely used to model the source behavior of data traffic, while phase-type (PH) distribution has been extensively applied to model the service time. This paper focuses on the computation of the DBMAP/PH/1 queueing system with priorities, in which the arrival process is considered to be a DBMAP with two priority levels and the service time obeys a discrete PH distribution. Such a queueing model has potential in performance evaluation of computer networks such as video transmission over wireless networks and priority scheduling in ATM or TDMA networks. Based on matrix-analytic methods, we develop computation algorithms for obtaining the stationary distribution of the system numbers and further deriving the key performance indices of the DBMAP/PH/1 priority queue. AMS subject classifications: 60K25 · 90B22 · 68M20 The work was supported in part by grants from RGC under the contracts HKUST6104/04E, HKUST6275/04E and HKUST6165/05E, a grant from NSFC/RGC under the contract N_HKUST605/02, a grant from NSF China under the contract 60429202.     

Two-Class Priority Queueing System with State-Dependent Arrivals

In this paper, we present a performance analysis of a 2-dimensional preemptive priority queueing system with state-dependent arrivals. Using a Markovian formulation we first compute the steady state distribution for the queue length of both classes. Then, waiting times and busy periods are characterized through (i) first and second moments and (ii) the approximation of their cumulative distribution functions (cdf) and Laplace–Stieltjes transforms (LST). We derive these approximations connecting bounds in the Laplace domain with bounds on the original time domain. We also, study the behavior of the inter-departure time for each class. Finally, we conclude the paper with a set of computational experiments testing our results.     

Mean queue size in a queue with discrete autoregressive arrivals of order <Emphasis Type="Italic">p</Emphasis>

We consider a discrete time single server queueing system where the arrival process is governed by a discrete autoregressive process of order p (DAR(p)), and the service time of a customer is one slot. For this queueing system, we give an expression for the mean queue size, which yields upper and lower bounds for the mean queue size. Further we propose two approximation methods for the mean queue size. One is based on the matrix analytic method and the other is based on simulation. We show, by illustrations, that the proposed approximations are very accurate and computationally efficient.     

Approximations for multi-server queues: System interpolations

This paper provides a unifying method of generating and/or evaluating approximations for the principal congestion measures in aGI/G/s queueing system. The main focus is on the mean waiting time, but approximations are also developed for the queue-length distribution, the waiting-time distribution and the delay probability for the Poisson arrival case. The approximations have closed forms that combine analytical solutions of simpler systems, and hence they are referred to as system-interpolation approximations or, simply, system interpolations. The method in this paper is consistent with and generalizes system interpolations previously presented for the mean waiting time in theGI/G/s queue.     

Exact and approximate numerical solutions to steady-state single-server queues:M/G/1 — a unified approach

This paper presents a unified approach for the numerical solutions of anM/G/1 queue. On the assumption that the service-time distribution has a rational Laplace-Stieltjes transform (LST), explicit closed-form expressions have been obtained for moments, distributions of system length and waiting time (in queue) in terms of the roots of associated characteristic equations (c.e.'s). Approximate analyses for the tails of the distributions based on one or more roots are also discussed. Numerical aspects have been tested for a variety of complex service-time distributions including but not restricted to only mixed generalized Erlang and generalized hyperexponential. A sample of numerical computations is also included. It is hoped that the results obtained would prove to be beneficial to both practitioners and theorists dealing with bounds, inequalities, approximations, and other aspects.