Tail Behaviour of the Area Under the Queue Length Process of the Single Server Queue with Regularly Varying Service Times (original) (raw)
On the integral of the workload process of the single server queue
Journal of Applied Probability
This paper is devoted to a study of the integral of the workload process of the single server queue, in particular during one busy period. Firstly, we find asymptotics of the area 𝒜 swept under the workload process W(t) during the busy period when the service time distribution has a regularly varying tail. We also investigate the case of a light-tailed service time distribution. Secondly, we consider the problem of obtaining an explicit expression for the distribution of 𝒜. In the general GI/G/1 case, we use a sequential approximation to find the Laplace—Stieltjes transform of 𝒜. In the M/M/1 case, this transform is obtained explicitly in terms of Whittaker functions. Thirdly, we consider moments of 𝒜 in the GI/G/1 queue. Finally, we show asymptotic normality of .
Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times
Stochastic Processes and their Applications, 2004
This paper considers a stable GI/GI/1 queue with subexponential service time distribution. Under natural assumptions we derive the tail behaviour of the busy period of this queue. We extend the results known for the regular variation case under minimal conditions. Our method of proof is based on a large deviations result for subexponential distributions.
Tail Asymptotics for the Busy Period in the GI/G/1 Queue
Mathematics of Operations Research, 2001
We characterise the tail behaviour of the busy period distribution in the GI=G=1 queue under the assumption that the tail of the service time distribution is of intermediate regular variation. This extends a result of De Meyer and Teugels 16] who treated the M/G/1 queue with a regularly varying service time distribution. Our method of proof is, opposed to the one in 16], probabilistic and reveals an insightful relationship between the busy period and the cycle maximum.
Waiting Time Asymptotics in the Single Server Queue with Service in Random Order
Queueing Systems, 2004
We consider the single server queue with service in random order. For a large class of heavy-tailed 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 −ν, it is shown that the waiting time distribution is also regularly varying, with index 1 − ν, and the pre-factor is determined explicitly.
Queueing Systems, 2013
We study the asymptotic behavior of the tail probabilities of the waiting time and the busy period for the M/G/1/K queues with subexponential service times under three different service disciplines: FCFS, LCFS, and ROS. Under the FCFS discipline, the result on the waiting time is proved for the more general G I /G/1/K queue with subexponential service times and lighter interarrival times. Using the wellknown Laplace-Stieltjes transform (LST) expressions for the probability distribution of the busy period of the M/G/1/K queue, we decompose the busy period into a sum of a random number of independent random variables. The result is used to obtain the tail asymptotics for the waiting time distributions under the LCFS and ROS disciplines.
The M/G/1 queue with heavy-tailed service time distribution
IEEE Journal on Selected Areas in Communications, 1998
In modern teletraffic applications of queueing theory, service time distributions B(t) B(t) B(t) with a heavy tail occur, i.e., 10B(t) Ct 0 B(t) Ct 0 B(t) Ct 0 for t ! 1 t ! 1 t ! 1 with >
LOGARITHMIC ASYMPTOTICS FOR STEADY-STATE TAIL PROBABILITIES IN A SINGLE-SERVER QUEUE
1993
We consider the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, but without any explicit independence conditions on the interarrival and service times. We find conditions for the steady-state waiting-time distribution to have small-tail asymptotics of the form x-1 logP(W＞x) → θ*as x → ∞for θ*＞0. We require only stationarity of the basic sequence of
Heavy-traffic asymptotics for the single-server queue with random order of service
Operations Research Letters, 2005
We consider the waiting time distribution of the Gl/Gl /1 queue where customers are served in random order; inter-arrival and service times may have finite or infinite variance. Our main result shows that the waiting time in heavy traffic can be written as a product of two random variables. Our proof is based on the intuitively appealing fact that in heavy traffic, the queue length stays constant during the sojourn time of a customer. For the special finite variance case, our result settles a conjecture of . 2000 Mathematics Subject Classification: 60K25.
Waiting-time tail probabilities in queues with long-tail service-time distributions
Queueing Systems, 1994
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 twoparameter 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.
On the exact asymptotics of the busy period in GI/G/1 queues
Advances in Applied Probability, 2006
In this paper we study the busy period in GI/G/1 work-conserving queues. We give the exact asymptotics of the tail distribution of the busy period under the light tail assumptions. We also study the workload process in the M/G/1 system conditioned to stay positive.
Scale functions of Lévy processes and busy periods of finite-capacity M/GI/1 queues
Journal of Applied Probability, 2004
In this paper we use the exit time theory for Lévy processes to derive new closed-form results for the busy period distribution of finite-capacity fluid M/G/1 queues. Based on this result, we then obtain the busy period distribution for finite-capacity queues with on–off inputs when the off times are exponentially distributed.
In this paper, a discrete-time single server queueing system with infinite buffer size and geometrically distributed arrivals is considered. We derive the functional equations and analyze the distribution of the number of customers served during a busy period for geometrically distributed service time as well as for deterministic service time. We also show that in the limiting case the results obtained in this paper are consistent with the corresponding continuous-time counterparts by Medhi [1].
On a Random Sum Formula for the Busy Period of the M/G/Infinity Queue With Applications
2001
A random sum formula is derived for the forward recurrence time associated with the busy period length of the MjGj1 queue. This result is then used to घiङ provide a necessary and suaecient condition for the subexponentiality of this forward recurrence time, and घiiङ establish a stochastic comparison in the convex increasing घvariabilityङ ordering between the busy periods in two MjGj1 queues with service times comparable in the convex increasing ordering.
Scale functions of L�vy processes and busy periods of finite-capacity M/GI/1 queues
Journal of Applied Probability, 2004
In this paper we use the exit time theory for Lévy processes to derive new closed form results for the busy period distribution of finite capacity fluid M/G/1 queues. Based on this result we then obtain the busy period distribution for finite capacity queues with on-off inputs when the off times are exponentially distributed.
Queue length asymptotics for the multiple-server queue with heavy-tailed Weibull service times
Queueing Systems, 2019
We study the occurrence of large queue lengths in the GI / GI / d queue with heavy-tailed Weibull-type service times. Our analysis hinges on a recently developed sample path large-deviations principle for Lévy processes and random walks, following a continuous mapping approach. Also, we identify and solve a key variational problem which provides physical insight into the way a large queue length occurs. In contrast to the regularly varying case, we observe several subtle features such as a non-trivial trade-off between the number of big jobs and their sizes and a surprising asymmetric structure in asymptotic job sizes leading to congestion.
On queues with service and interarrival times depending on waiting times
2007
We consider an extension of the standard G/G/1 queue, described by the equation W D = max{0, B − A + Y W }, where P[Y = 1] = p and P[Y = −1] = 1 − p. For p = 1 this model reduces to the classical Lindley equation for the waiting time in the G/G/1 queue, whereas for p = 0 it describes the waiting time of the server in an alternating service model.
On the transition from heavy traffic to heavy tails for the M/G/1 queue: the regularly varying
2011
Extended proof of Lemma 3.2. This is an extended proof of Lemma 3.2 from Olvera-Cravioto et al. (2009) that includes the case when α = 3. The value α = 3 constitutes the boundary between infinite and finite variance, and results about the asymptotic behavior of P (S n > x) usually imply additional technical subtleties. For this reason most authors have ignored this specific value of α.
On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case
The Annals of Applied Probability, 2011
Extended proof of Lemma 3.2. This is an extended proof of Lemma 3.2 from Olvera-Cravioto et al. (2009) that includes the case when α = 3. The value α = 3 constitutes the boundary between infinite and finite variance, and results about the asymptotic behavior of P (S n > x) usually imply additional technical subtleties. For this reason most authors have ignored this specific value of α.
The busy period for the M0/G/1/m system with service time dependent of the queue length
Journal of Applied Mathematics and Computational Mechanics, 2013
We consider the M θ /G/1/m system wherein the service time depends on the queue length and it is determined at the beginning of customer service. Using an approach based on the potential method proposed by V. Korolyuk, the Laplace transforms for the distribution of the number of customers in the system on the busy period and for the distribution function of the busy period are found.