Christ and Avi-Itzhak (2002) analyzed a queueing system with two competing servers who determine their service rates so as to optimize their individual utilities. The system is formulated as a two-person game; Christ and Avi-Itzhak proved the existence of a unique Nash equilibrium which is symmetric. In this paper, we explore globally optimal solutions. We prove that the unique Nash equilibrium is generally strictly inferior to a globally optimal solution and that optimal solutions are symmetric and require the servers to adopt service rates that are smaller than those occurring in equilibrium. Furthermore, given a symmetric globally optimal solution, we show how to impose linear penalties on the service rates so that the given optimal solution becomes a unique Nash equilibrium. When service rates are not observable, we show how the same effect is achieved by imposing linear penalties on a corresponding signal.

Autoregressive arrival models are described by a few parameters and provide a simple means to obtain analytical models for matching the first- and second-order statistics of measured data. We consider a discrete-time queueing system where the service time of a customer occupies one slot and the arrival process is governed by a discrete autoregressive process of order 1 (a DAR(1) process) which is characterized by an arbitrary stationary batch size distribution and a correlation coefficient. The tail behaviors of the queue length and the waiting time distributions are examined. In particular, it is shown that, unlike in the classical queueing models with Markovian arrival processes, the correlation in the DAR(1) model results in nongeometric tail behavior of the queue length (and the waiting time) if the stationary distribution of the DAR(1) process has infinite support. A complete characterization of the geometric tail behavior of the queue length (and the waiting time) is presented, showing the impact of the stationary distribution and the correlation coefficient when the stationary distribution of the DAR(1) process has finite support. It is also shown that the stationary distribution of the queue length (and the waiting time) has a tail of regular variation with index -β − 1, by finding an explicit expression for the tail asymptotics when the stationary distribution of the DAR(1) process has a tail of regular variation with index -β.

It is known that for decreasing hazard rate (DHR) service times the foreground-background discipline (FB) minimizes the mean delay in the M/G/1 queue among all work-conserving and nonanticipating service disciplines. It is believed that a similar result is valid for increasing mean residual lifetime (IMRL) service times. However, on the one hand, we point out a flaw in an earlier proof of the latter result and construct a counter-example that demonstrates that FB is not necessarily optimal within class IMRL. On the other hand, we prove that the mean delay for FB is smaller than that of the processor-sharing discipline within class IMRL, giving a weaker version of an earlier hypothesis.

Admission control can be employed to avoid congestion in queueing networks subject to overload. In distributed networks, the admission decisions are often based on imperfect measurements on the network state. In this paper, we study how the lack of complete state information affects the system performance, by considering a simple network model for distributed admission control. The stability region of the network is characterized and it is shown how feedback signaling makes the system very sensitive to its parameters.

We consider a Jackson network in which some of the nodes have an infinite supply of work: when all the customers queued at such a node have departed, the node will process a customer from this supply. Such nodes will be processing jobs all the time, so they will be fully utilized and experience a traffic intensity of 1. We calculate flow rates for such networks, obtain conditions for stability, and investigate the stationary distributions. Standard nodes in this network continue to have product-form distributions, while nodes with an infinite supply of work have geometric marginal distributions and Poisson inflows and outflows, but their joint distribution is not of product form.

The aim of this paper is to evaluate the performance of the optimal policy (the Gittins index policy) for open tax problems of the type considered by Klimov in the undiscounted limit. In this limit, the state-dependent part of the cost is linear in the state occupation numbers for the multi-armed bandit, but is quadratic for the tax problem. The discussion of the passage to the limit for the tax problem is believed to be largely new; the principal novelty is our evaluation of the matrix of the quadratic form. These results are confirmed by a dynamic programming analysis, which also suggests how the optimal policy should be modified when resources can be freely deployed only within workstations, rather than system-wide.

We derive several algorithms for the busy period distribution of the canonical Markovian fluid flow model. One of them is similar to the Latouche-Ramaswami algorithm for quasi-birth-death models and is shown to be quadratically convergent. These algorithms significantly increase the efficiency of the matrix-geometric procedures developed earlier by the authors for the transient and steady-state analyses of fluid flow models.

We consider the asymptotic behaviour of the stationary tail probabilities in the discrete-time GI/G/1-type queue with countable background state space. These probabilities are presented in matrix form with respect to the background state space, and shown to be the solution of a Markov renewal equation. Using this fact, we consider their decay rates. Applying the Markov renewal theorem, it is shown that certain reasonable conditions lead to the geometric decay of the tail probabilities as the level goes to infinity. We exemplify this result using a discrete-time priority queue with a single server and two types of customer.

We consider an M/GI/1 queue with two types of customers, positive and negative, which cancel each other out. The server provides service to either a positive customer or a negative customer. In such a system, the queue length can be either positive or negative and an arrival either joins the queue, if it is of the same sign, or instantaneously removes a customer of the opposite sign at the end of the queue or in service. This study is a generalization of Gelenbe's original concept of a queue with negative customers, where only positive customers need services and negative customers arriving at an empty system are lost or need no service. In this paper, we derive the transient and the stationary probability distributions for the major performance measures in terms of generating functions and Laplace transforms. It has been shown that the previous results for the system with negative arrivals of zero service time are special cases of our model. In addition, we obtain the stationary waiting time distribution of this system in terms of a Laplace transform.

We derive results that show the impact of aggregation in a queueing network. Our model consists of a two-stage queueing system where the first (upstream) queue serves many flows, of which a certain subset arrive at the second (downstream) queue. The downstream queue experiences arbitrary interfering traffic. In this setup, we prove that, as the number of flows being aggregated in the upstream queue increases, the overflow probability of the downstream queue converges uniformly in the buffer level to the overflow probability of a single queueing system obtained by simply removing the upstream queue in the original two-stage queueing system. We also provide the speed of convergence and show that it is at least exponentially fast. We then extend our results to non-i.i.d. traffic arrivals.

The issue of ‘fairness’ is raised frequently in the context of evaluating queueing policies, notably in relation to telecommunications and computer systems where it may be of no lesser importance than the conventional measures of performance. Comparisons of the fairness of various systems and policies are often awkward due to lack of generally accepted definitions and measures for this important property. The purpose of this work is to propose possible fairness measures enabling us to quantitatively measure and compare the level of fairness associated with G/G/R queueing systems. We define and discuss order (of service) fairness and use an axiomatic approach for developing a measure for it in the G/D/1 case. The measure obtained for the G/D/1 system is then generalized and applied to the G/G/R class of systems. A practical implication of this work is that, for a wide class of service disciplines, the variance of the waiting time can be used as a yardstick for comparing fairness levels.

This paper studies recurrence properties of autoregressive (AR) processes with ‘super-heavy-tailed’ innovations. Specifically, we study the case where the innovations are distributed, roughly speaking, as log-Pareto random variables (i.e. the tail decay is essentially a logarithm raised to some power). We show that these processes exhibit interesting and somewhat surprising behaviour. In particular, we show that AR(1) processes, with the usual root assumption that is necessary for stability, can exhibit null-recurrent as well as transient dynamics when the innovations follow a log-Cauchy-type distribution. In this regime, the recurrence classification of the process depends, somewhat surprisingly, on the value of the constant pre-multiplier of this distribution. More generally, for log-Pareto innovations, we provide a positive-recurrence/null-recurrence/transience classification of the corresponding AR processes.

This paper studies a discrete-time single-server queue with two independent inputs and service interruptions. One of the inputs to the queue is an independent and identically distributed process. The other is a much more general process and it is not required to be Markov nor is it required to be stationary. The service interruption process is also general and it is not required to be Markov or to be stationary. This paper shows that a stochastic decomposition property for the virtual waiting-time process holds in the discrete-time single-server queue with service interruptions. To the best of the author's knowledge, no stochastic decomposition results for virtual waiting-time processes in non-work-conserving queues, such as queues with service interruptions, have been obtained before and only work-conserving queues have been studied in the literature.

The basic queueing system considered in this paper is the M/G/1 processor-sharing queue with or without impatience and with finite or infinite capacity. Under some mild assumptions, a criterion for the validity of the reduced-service-rate approximation is established when service times are heavy tailed. This result is applied to various models based on M/G/1 processor-sharing queues.

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite- and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrix-exponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrix-exponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.

Service is often provided in contexts where tasks or customers are impatient or perishable in that they have natural lifetimes of availability for useful service. Moreover, these lifetimes are usually unknown to the service provider. The question of how service might best be allocated to the currently waiting tasks or customers in such a context has been neglected and we propose three simple models. For each model, an index heuristic is developed and is assessed numerically. In all cases studied the heuristic comes close to optimality.

We consider the dynamic scheduling of a multiclass queueing system with two servers, one dedicated (server 1) and one flexible (server 2), with no arrivals. Server 1 is dedicated to processing type-1 jobs while server 2 is primarily responsible for processing type-2 jobs but can also aid server 1 with its work. We address when it is optimal for server 2 to aid server 1 with type-1 jobs rather than process type-2 jobs. The objective is to minimize the total holding costs incurred until all jobs in the system are processed and leave the system. We show that the optimal policy can exhibit one of three possible structures: (i) an exhaustive policy for type-2 jobs, (ii) a nonincreasing switching curve in the number of type-1 jobs and (iii) a nondecreasing switching curve in the number of type-1 jobs. We characterize the necessary and sufficient conditions under which each policy will be optimal. We also explore the use of the optimal policy for the problem with no arrivals as a heuristic for the problem with dynamic arrivals.

Stratified and simple random sampling (or testing) are two common methods used to investigate the number or proportion of items in a population with a particular attribute. Although it is known that cost factors and information about the strata in the population are often crucial in deciding whether to use stratified or simple random sampling in a given situation, the stochastic precedence ordering for random variables can also provide the basis for an interesting criteria under which these methods may be compared. It may be particularly relevant when we are trying to find as many special items as possible in a population (for example individuals with a disease in a country). Properties of this total stochastic order on the class of random variables are discussed, and necessary and sufficient conditions are established which allow the comparison of the number of items of interest found in stratified random sampling with the number found in simple random sampling in the stochastic precedence order. These conditions are compared with other results established on stratified and simple random sampling (testing) using different stochastic-order-type criteria, and applications are given for the comparison of sums of Bernoulli random variables and binomial distributions.

We exploit the structural properties of the BMAP/D/k system to carry out its algorithmic analysis. Specifically, we use these properties to develop algorithms for studying the distributions of waiting times in discrete time and the busy period. One of the structural properties used results from considering the system as having customers assigned in a cyclic order—which does not change the waiting-time distribution—and then studying only one arbitrary server. The busy period is defined as the busy period of an arbitrary single server based on this cyclic assignment of customers to servers. Finally, we study the marginal distribution of the joint queue length and phase of customer arrival. The structural property used for studying the queue length is based on the observation of the system every interval that is the length of one customer service time.

We consider a queue fed by a mixture of light-tailed and heavy-tailed traffic. The two traffic flows are served in accordance with the generalized processor sharing (GPS) discipline. GPS-based scheduling algorithms, such as weighted fair queueing (WFQ), have emerged as an important mechanism for achieving service differentiation in integrated networks. We derive the asymptotic workload behaviour of the light-tailed traffic flow under the assumption that its GPS weight is larger than its traffic intensity. The GPS mechanism ensures that the workload is bounded above by that in an isolated system with the light-tailed flow served in isolation at a constant rate equal to its GPS weight. We show that the workload distribution is, in fact, asymptotically equivalent to that in the isolated system, multiplied by a certain prefactor, which accounts for the interaction with the heavy-tailed flow. Specifically, the prefactor represents the probability that the heavy-tailed flow is backlogged long enough for the light-tailed flow to reach overflow. The results provide crucial qualitative insight in the typical overflow scenario.