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24
Sojourn Time Asymptotics in the M/G/1 Processor Sharing Queue
 QUEUEING SYSTEMS
, 1998
"... We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index \Gamma , noninteger, iff the sojourn time distribution is regularly varying of index \Gamma . This result is derived from a new expression for the LaplaceStieltjes transform of the sojo ..."
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Cited by 56 (9 self)
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We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index \Gamma , noninteger, iff the sojourn time distribution is regularly varying of index \Gamma . This result is derived from a new expression for the LaplaceStieltjes transform of the sojourn time distribution. That expression also leads to other new properties for the sojourn time distribution. We show how the moments of the sojourn time can be calculated recursively and prove that the kth moment of the sojourn time is finite iff the kth moment of the service time is finite. In addition, we give a short proof of a heavy traffic theorem for the sojourn time distribution, prove a heavy traffic theorem for the moments of the sojourn time, and study the properties of the heavy traffic limiting sojourn time distribution when the service time distribution is regularly varying. Explicit formulas and multiterm expansions are provided for the case that the service time has a Pareto...
Heavytraffic limits for loss proportions in singleserver queues
 Queueing Syst
, 2004
"... Abstract. We establish heavytraffic stochasticprocess limits for the queuelength and overflow stochastic processes in the standard singleserver queue with finite waiting room (G/G/1/K). We show that, under regularity conditions, the content and overflow processes in related singleserver models ..."
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Abstract. We establish heavytraffic stochasticprocess limits for the queuelength and overflow stochastic processes in the standard singleserver queue with finite waiting room (G/G/1/K). We show that, under regularity conditions, the content and overflow processes in related singleserver models with finite waiting room, such as the finite dam, satisfy the same heavytraffic stochasticprocess limits. As a consequence, we obtain heavytraffic limits for the proportion of customers or input lost over an initial interval. Except for an interchange of the order of two limits, we thus obtain heavytraffic limits for the steadystate loss proportions. We justify the interchange of limits in M/GI/1/K and GI/M/1/K special cases of the standard GI/GI/1/K model by directly establishing local heavytraffic limits for the steadystate blocking probabilities.
Waiting Time Asymptotics in the Single Server Queue with Service in Random Order
, 2003
"... We consider the single server queue with service in random order. For a large class of heavytailed service time distributions, we determine the asymptotic behavior of the waiting time distribution. For the special case of Poisson arrivals and regularly varying service time distribution with inde ..."
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Cited by 8 (3 self)
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We consider the single server queue with service in random order. For a large class of heavytailed service time distributions, we determine the asymptotic behavior of the waiting time distribution. For the special case of Poisson arrivals and regularly varying service time distribution with index it is shown that the waiting time distribution is also regularly varying, with index 1 #, and the prefactor is determined explicitly. Another
Limits for cumulative input processes to queues
 Prob. Eng. Inf. Sci
, 1999
"... Abstract We establish functional central limit theorems (FCLTs) for a cumulative input process to a fluid queue from the superposition of independent onoff sources, where the on periods and off periods may have heavytailed probability distributions. Variants of these FCLTs hold for cumulative busy ..."
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Cited by 7 (1 self)
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Abstract We establish functional central limit theorems (FCLTs) for a cumulative input process to a fluid queue from the superposition of independent onoff sources, where the on periods and off periods may have heavytailed probability distributions. Variants of these FCLTs hold for cumulative busy time and idle time processes associated with standard queueing models. The heavytailed onperiod and offperiod distributions can cause the limit process to have discontinuous sample paths, e.g., to be a nonBrownian stable process or more general L'evy process, even though the converging processes have continuous sample paths. Consequently, we exploit the Skorohod M1 topology on the function space D of rightcontinuous functions with left limits. The limits here combined with the previously established continuity of the reflection map in the M1 topology implies both heavytraffic and nonheavytraffic FCLTs for buffercontent processes in stochastic fluid networks. Keywords: functional central limit theorems, invariance principles, heavytraffic limit theorems, stochastic fluid networks, cumulative input processes, cumulative busytime processes, heavytailed probability distributions, stable processes, L'evy processes, communication networks 1.
An overview of Brownian and nonBrownian FCLTs for the singleserver queue
 Queueing Systems
, 2000
"... We review functional central limit theorems (FCLTs) for the queuecontent process in a singleserver queue with finite waiting room and the firstcome firstserved service discipline. We emphasize alternatives to the familiar heavytraffic FCLTs with reflected Brownian motion (RBM) limit process tha ..."
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Cited by 7 (1 self)
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We review functional central limit theorems (FCLTs) for the queuecontent process in a singleserver queue with finite waiting room and the firstcome firstserved service discipline. We emphasize alternatives to the familiar heavytraffic FCLTs with reflected Brownian motion (RBM) limit process that arise with heavytailed probability distributions and strong dependence. Just as for the familiar convergence to RBM, the alternative FCLTs are obtained by applying the continuous mapping theorem with the reflection map to previously established FCLTs for partial sums. We consider a discretetime model and first assume that the cumulative netinput process has stationary and independent increments, with jumps up allowed to have infinite variance or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable process, whose steadystate distribution can be calculated by numerically inverting its Laplace transform. For a sequence of models, the queue need not be in heavy traffic, and the limit can be a general reflected Lévy process. When the Lévy process representing the net input has no negative jumps, the steadystate distribution of the reflected Lévy process again can be calculated by numerically inverting its Laplace transform. We also establish FCLTs for the queuecontent process when the input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence. Then the limiting input process is a Gaussian process. When the limiting netinput process is also a Gaussian process and there is unlimited waiting room, the steadystate distribution of the limiting reflected Gaussian process can be conveniently approximated.
CONVERGENCE OF THE ALLTIME SUPREMUM OF A LÉVY PROCESS IN THE HEAVYTRAFFIC REGIME
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HEAVYTRAFFIC ANALYSIS OF THE MAXIMUM OF AN ASYMPTOTICALLY STABLE RANDOM WALK
, 902
"... Abstract. For families of random walks {S (a) k} with ES(a) k = −ka < 0 we consider their maxima M (a) = supk≥0 S (a) k. We investigate the asymptotic behaviour of M (a) as a → 0 for asymptotically stable random walks. This problem appeared first in the 1960’s in the analysis of a singleserver ..."
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Cited by 2 (0 self)
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Abstract. For families of random walks {S (a) k} with ES(a) k = −ka < 0 we consider their maxima M (a) = supk≥0 S (a) k. We investigate the asymptotic behaviour of M (a) as a → 0 for asymptotically stable random walks. This problem appeared first in the 1960’s in the analysis of a singleserver queue when the traffic load tends to 1 and since then is referred to as the heavytraffic approximation problem. Kingman and Prokhorov suggested two different approaches which were later followed by many authors. We give two elementary proofs of our main result, using each of these approaches. It turns out that the main technical difficulties in both proofs are rather similar and may be resolved via a generalisation of the Kolmogorov inequality to the case of an infinite variance. Such a generalisation is also obtained in this note. Assume that {Xi} ∞ i=1 is a sequence of i.i.d. random variables with a zero expectation: EX1 = 0. Define a random walk S0 = 0, Sk = k∑ Xi for k ≥ 1. i=1 Along with the random walk {Sk}, for each a> 0 define a random walk {S (a)} via Now we can define
Heavytraffic limit theorems for the heavytailed GI/G/1 queue
, 1997
"... The classic GI=G=1 queueing model of which the tail of the service time and/or the interarrival time distribution behaves as t \Gammav S(t) for t !1, 1 ! v ! 2 and S(t) a slowly varying function at infinity, is investigated for the case that the traffic load a approaches one. Heavytraffic limit t ..."
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Cited by 1 (1 self)
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The classic GI=G=1 queueing model of which the tail of the service time and/or the interarrival time distribution behaves as t \Gammav S(t) for t !1, 1 ! v ! 2 and S(t) a slowly varying function at infinity, is investigated for the case that the traffic load a approaches one. Heavytraffic limit theorems are derived for the case that these tails have a similar behaviour at infinity as well as for the case that one of these tails is heavier than the other one. These theorems state that the contracted waiting time \Delta(a)w, with w the actual waiting time for the stable GI=G=1 queue and \Delta(a) the contraction coefficient, converges in distribution for a " 1. Here \Delta(a) is that root of the contraction equation which approaches zero from above for a " 1. The structure of this contraction equation is determined by the character of the two tails. The LaplaceStieltjes transforms of the limiting distributions are derived. For nonsimilar tails the limiting distributions are explicit...