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18
Waitingtime tail probabilities in queues with longtail servicetime distributions
 QUEUEING SYSTEMS
, 1994
"... We consider the standard GI/G/1 queue with unlimited waiting room and the firstin firstout service discipline. We investigate the steadystate waitingtime tail probabilities P(W> x) when the servicetime distribution has a longtail distribution, i.e., when the servicetime distribution fails ..."
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Cited by 73 (23 self)
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We consider the standard GI/G/1 queue with unlimited waiting room and the firstin firstout service discipline. We investigate the steadystate waitingtime tail probabilities P(W> x) when the servicetime distribution has a longtail distribution, i.e., when the servicetime distribution fails to have a finite moment generating function. We have developed algorithms for computing the waitingtime distribution by Laplace transform inversion when the Laplace transforms of the interarrivaltime and servicetime distributions are known. One algorithm, exploiting Pollaczek’s classical contourintegral representation of the Laplace transform, does not require that either of these transforms be rational. To facilitate such calculations, we introduce a convenient twoparameter family of longtail 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 Paretolike tails, i.e., are of order x − r for r> 1. We use this family of longtail distributions to investigate the quality of approximations based on asymptotics for P(W> x) as x → ∞. We show that the asymptotic approximations with these longtail servicetime distributions can be remarkably inaccurate for typical x values of interest. We also derive multiterm asymptotic expansions for the waitingtime 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 waitingtime tail probabilities in this case. When working with servicetime data, we suggest using empirical Laplace transforms.
Realtime delay estimation based on delay history
, 2007
"... 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 realtime delay estimators based on recent customer delay experience. The main estimators considered are: (i) the delay of the ..."
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Cited by 14 (4 self)
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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 realtime 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 delayhistory 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 multiserver 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 manyserver and classical heavytraffic limiting regimes.
On the inapproximability of M/G/K: Why two moments of job size distribution are not enough
"... The M/G/K queueing system is one of the oldest model 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 closedform (nonnumerical) approximations in the literature a ..."
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Cited by 8 (2 self)
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The M/G/K queueing system is one of the oldest model 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 closedform (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.
The effect of higher moments of job size distribution on the performance of an . . .
, 2008
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Finding the optimal quantum size: Sensitivity analysis of the M/G/1 roundrobin queue
"... 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 roundrobin fashion. The processor sharing (PS) policy is an idealization of the quantumbased roundrobin scheduling in the limit where the quantum size becomes ..."
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Cited by 3 (0 self)
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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 roundrobin fashion. The processor sharing (PS) policy is an idealization of the quantumbased roundrobin 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 roundrobin 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.
Recent Asymptotic Results in the Probabilistic Analysis of Schedule Makespans
, 1995
"... 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 p ..."
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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 flowshop 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 online greedy scheduling policy, the resolution of a longstanding o...
On the catalyzing effect of randomness on the perflow throughput in wireless networks. Technical report. Available from http://net.tlabs.tuberlin.de/∼florin/lib/randnet.pdf and also from arXiv.org
, 2013
"... Abstract—This paper investigates the throughput capacity of a flow crossing a multihop wireless network, whose geometry is characterized by general randomness laws including Uniform, Poisson, HeavyTailed distributions for both the nodes ’ densities and the number of hops. The key contribution is t ..."
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Abstract—This paper investigates the throughput capacity of a flow crossing a multihop wireless network, whose geometry is characterized by general randomness laws including Uniform, Poisson, HeavyTailed distributions for both the nodes ’ densities and the number of hops. The key contribution is to demonstrate how the perflow throughput depends on the distribution of 1) the number of nodes Nj inside hops ’ interference sets, 2) the number of hops K, and 3) the degree of spatial correlations. The randomness in both Nj ’s and K is advantageous, i.e., it can yield larger scalings (as large as Θ(n)) than in nonrandom settings. An interesting consequence is that the perflow capacity can exhibit the opposite behavior to the network capacity, which was shown to suffer from a logarithmic decrease in the presence of randomness. In turn, spatial correlations along the endtoend path are detrimental by a logarithmic term. I.
Stochastic Models and Analysis for Resource Management in Server Farms
"... 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 trialanderror becomes ..."
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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 trialanderror 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 multiserver 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 FirstComeFirstServed (FCFS) scheduling discipline,
On MarkovKrein Characterization of Mean Sojourn Time in Queueing Systems
, 2011
"... We present a new analytical tool for three queueing systems which have defied exact analysis so far: (i) the classical M/G/k multiserver system, (ii) queueing systems with fluctuating arrival and service rates, and (iii) the M/G/1 roundrobin queue. We argue that rather than looking for exact expre ..."
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Cited by 1 (1 self)
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We present a new analytical tool for three queueing systems which have defied exact analysis so far: (i) the classical M/G/k multiserver system, (ii) queueing systems with fluctuating arrival and service rates, and (iii) the M/G/1 roundrobin 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 lighttraffic asymptote and given first n ( = 2, 3) moments of the job size distribution, analogous to the classical MarkovKrein 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 hyperexponential 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
On Multiplexing Flows: Does it Hurt or Not?
"... Abstract—This paper analyzes queueing behavior subject to multiplexing a stochastic process M(n) of flows, and not a constant as conventionally assumed. By first considering the case when M(n) is iid, it is shown that flows ’ multiplexing ‘hurts’ the queue size (i.e., the queue size increases in dis ..."
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Abstract—This paper analyzes queueing behavior subject to multiplexing a stochastic process M(n) of flows, and not a constant as conventionally assumed. By first considering the case when M(n) is iid, it is shown that flows ’ multiplexing ‘hurts’ the queue size (i.e., the queue size increases in distribution). The simplicity of the iid case enables the quantification of the ‘best’ and ‘worst ’ distributions of M(n), i.e., minimizing/maximizing the queue size. The more general, and also realistic, case when M(n) is Markovmodulated reveals an interesting behavior: flows ’ multiplexing ‘hurts ’ but only when the multiplexed flows are sufficiently long. An important caveat raised by such observations is that the conventional approximation of M(n) by a constant can be very misleading for queueing analysis. I.