We consider the standard GI/G/1 queue with unlimited waiting room and the first-in first-out service discipline. We investigate the steady-state waiting-time tail probabilities P(W> x) when the service-time distribution has a long-tail distribution, i.e., when the service-time distribution fails to have a finite moment generating function. We have developed algorithms for computing the waiting-time distribution by Laplace transform inversion when the Laplace transforms of the interarrival-time and service-time distributions are known. One algorithm, exploiting Pollaczek’s classical contour-integral representation of the Laplace transform, does not require that either of these transforms be rational. To facilitate such calculations, we introduce a convenient two-parameter family of long-tail distributions on the positive half line with explicit Laplace transforms. This family is a Pareto mixture of exponential (PME) distributions. These PME distributions have monotone densities and Pareto-like tails, i.e., are of order x − r for r> 1. We use this family of long-tail distributions to investigate the quality of approximations based on asymptotics for P(W> x) as x → ∞. We show that the asymptotic approximations with these long-tail service-time distributions can be remarkably inaccurate for typical x values of interest. We also derive multi-term asymptotic expansions for the waiting-time tail probabilities in the M/G/1 queue. Even three terms of this expansion can be remarkably inaccurate for typical x values of interest. Thus, we evidently must rely on numerical algorithms for determining the waiting-time tail probabilities in this case. When working with service-time data, we suggest using empirical Laplace transforms.

Motivated by interest in making delay announcements to arriving customers who must wait in call centers and related service systems, we study the performance of alternative real-time delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the last customer to enter service (LES), (ii) the delay experienced so far by the customer at the head of the line (HOL), and (iii) the delay experienced by the customer to have arrived most recently among those who have already completed service (RCS). We compare these delay-history estimators to the estimator based on the queue length (QL), which requires knowledge of the mean interval between successive service completions in addition to the queue length. We characterize performance by the mean squared error (MSE). We do analysis and conduct simulations for the standard GI/M/s multi-server queueing model, emphasizing the case of large s. We obtain analytical results for the conditional distribution of the delay given the observed HOL delay. An approximation to its mean value serves as a refined estimator. For all three candidate delay estimators, the MSE relative to the square of the mean is asymptotically negligible in the many-server and classical heavy-traffic limiting regimes.

The M/G/K queueing system is one of the oldest model for multi-server 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 (non-numerical) 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. 1

Makespan scheduling problems are in the mainstream of operations research, industrial engineering, and computer science. A basic multiprocessor version requires that n tasks be scheduled on m identical processors so as to minimize the makespan, i.e., the latest task finishing time. In the standard probability model considered here, the task durations are i.i.d. random variables with a distribution F , and the objective is to estimate the distribution of the makespan as a function of m, n, and F . This paper surveys probabilistic results for the multiprocessor scheduling problem and an important variant known as the permutation flow-shop problem. Several of the results are new; the others have appeared in the last few years. Because of the difficulty of exact analysis, the results take the form of limits as n ! 1 or as both m ! 1 and n ! 1 with m ! n. Some highlights of the survey are: a new asymptotic analysis of the on-line greedy scheduling policy, the resolution of a longstanding o...

We consider the round robin (RR) scheduling policy where the server processes each job in its buffer for at most a fixed quantum, q, in a round-robin fashion. The processor sharing (PS) policy is an idealization of the quantum-based round-robin scheduling in the limit where the quantum size becomes infinitesimal, and has been the subject of many papers. It is well known that the mean response time in an M/G/1/P S queue depends on the job size distribution via only its mean. However, almost no explicit results are available for the round-robin policy. For example, how does the variability of job sizes affect the mean response time in an M/G/1/RR queue? How does one choose the optimal quantum size in the presence of switching overheads? In this paper we present some preliminary answers to these fundamental questions. 1.

effect of higher moments of job size distribution on

We present a new analytical tool for three queueing systems which have defied exact analysis so far: (i) the classical M/G/k multi-server system, (ii) queueing systems with fluctuating arrival and service rates, and (iii) the M/G/1 round-robin queue. We argue that rather than looking for exact expressions for the mean response time as a function of the job size distribution, a more fruitful approach is to find distributions which minimize or maximize the mean response time given the first n moments of the job size distribution. We prove that for the M/G/k system in light-traffic asymptote and given first n ( = 2, 3) moments of the job size distribution, analogous to the classical Markov-Krein Theorem, these ‘extremal ’ distributions are given by the principal representations of the moment sequence. Furthermore, if we restrict the distributions to lie in the class of Completely Monotone (CM) distributions, then for all the three queueing systems, for any n, the extremal distributions under the appropriate “light traffic ” asymptotics are hyper-exponential distributions with finite number of phases. We conjecture that the property of extremality should be invariant to the system load, and thus our light traffic results should hold for general load as well, and propose potential strategies for a unified approach

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Server farms are popular architectures for computing infrastructures such as supercomputing centers, data centers and web server farms. As server farms become larger and their workloads more complex, designing efficient policies for managing the resources in server farms via trial-and-error becomes intractable. It is hard to predict the exact effect of various parameters on performance. Stochastic modeling and analysis techniques allow us to understand the performance of such complex systems and to guide design of policies to optimize the performance. However, most existing models of server farms are motivated by telephone networks, inventory management systems, and call centers. Modeling assumptions which hold for these problem domains are not accurate for computing server farms. There are numerous gaps between traditional models of multi-server systems and how today’s server farms operate. To cite a few: (i) Unlike call durations, supercomputing jobs and file sizes have high variance in service requirements and this critically affects the optimality and performance of scheduling policies. (ii) Most existing analysis of server farms focuses on the First-Come-First-Served (FCFS) scheduling discipline,