Waiting Time Tail Research Articles

SET-VALUED PERFORMANCE APPROXIMATIONS FOR THE QUEUE GIVEN PARTIAL INFORMATION

In order to understand queueing performance given only partial information about the model, we propose determining intervals of likely values of performance measures given that limited information. We illustrate this approach for the mean steady-state waiting time in the $GI/GI/K$ queue. We start by specifying the first two moments of the interarrival-time and service-time distributions, and then consider additional information about these underlying distributions, in particular, a third moment and a Laplace transform value. As a theoretical basis, we apply extremal models yielding tight upper and lower bounds on the asymptotic decay rate of the steady-state waiting-time tail probability. We illustrate by constructing the theoretically justified intervals of values for the decay rate and the associated heuristically determined interval of values for the mean waiting times. Without extra information, the extremal models involve two-point distributions, which yield a wide range for the mean. Adding constraints on the third moment and a transform value produces three-point extremal distributions, which significantly reduce the range, producing practical levels of accuracy.

Extremal models for the [formula omitted] waiting-time tail-probability decay rate

We identify interarrival-time and service-time distributions that yield tight upper and lower bounds on the asymptotic decay rate of the steady-state waiting-time tail probability in the GI∕GI∕K queue, given the first two moments of those underlying distributions plus alternative additional information. We apply Tchebycheff systems after providing a concise review.

The Single Server Queue with Mixing Dependencies

We study a single server queue, where a certain type of dependence is introduced between the service times, or between the inter-arrival times, or both between the service times and the inter-arrival times. This dependence arises via mixing, i.e., a parameter pertaining to the distribution of the service times, or of the inter-arrival times, is itself considered to be a random variable. We give a duality result between such queueing models and the corresponding insurance risk models, for which the respective dependence structures have been studied before. For a number of examples we provide exact expressions for the waiting time distribution, and compare these to the ones for the standard M/M/1 queue. We also investigate the effect of dependence and derive first order asymptotics for some of the obtained waiting time tails. Finally, we discuss this dependence concept for the waiting time tail of the G/M/1 queue.

Note—Computing Time-Dependent Waiting Time Probabilities in M(t)/M/s(t) Queuing Systems

In this note we present algorithms that compute, exactly or approximately, time-dependent waiting time tail probabilities and the time-dependent expected waiting time in M(t)/M/s(t) queuing systems.

Approximating M/G/1 Waiting Time Tail Probabilities

We propose a new approximation formula for the waiting time tail probability of the M/G/1 queue with FIFO discipline and unlimited waiting space. The aim is to address the difficulty of obtaining good estimates when the tail probability has non-exponential asymptotics. We show that the waiting time tail probability can be expressed in terms of the waiting time tail probability of a notional M/G/1 queue with truncated service time distribution plus the tail probability of an extreme order statistic. The Cramér–Lundberg approximation is applied to approximate the tail probability of the notional queue. In essence, our technique extends the applicability of the Cramér–Lundberg approximation to cases where the standard Lundberg condition does not hold. We propose a simple moment-based technique for estimating the parameters of the approximation; numerical results demonstrate that our approximation can yield very good estimates over the whole range of the argument.

LOWER BOUNDS FOR LRD/GI/1 QUEUES WITH SUBEXPONENTIAL SERVICE TIMES

We investigate the tail distribution of the virtual waiting times in a LRD/GI/1 queue where the arrival process is long-range dependent (LRD) and the service times are independent and identically distributed (i.i.d.) random variables. We present two lower bounds on the stationary waiting time tail asymptotics, which illustrate the different dominating components that influence server performance under various conditions. In particular, we show that the tail distribution of the stationary waiting time is bounded below by that of the associated LRD/D/1 queues resulting from replacing all random service times by the mean. This shows the performance impact purely due to the long-range dependency of the arrival process. On the other hand, when the service times are subexponential, we show that the tail distribution of the stationary waiting time is bounded below by that of the corresponding D/GI/1 queue by replacing the dependent arrival process with its associated independent version. This shows the minimum performance impact due to the tail distribution of the service times. The above two lower bounds indicate that the performance of LRD/GI/1 queues will be dominated by the heavier tail of the corresponding LRD/D/1 and D/GI/1 queues. These features are further illustrated and quantified through examples and via numerous simulation experiments.

Queueing systems with long-range dependent input process and subexponential service times

We analyze the asymptotic tail distribution of stationary waiting times and stationary virtual waiting times in a single-server queue with long-range dependent arrival process and subexponential service times. We investigate the joint impact of the long range dependency of the arrival process and of the tail distribution of the service times. We consider two traffic models that have been widely used to characterize the long-range dependence structure, namely, the M/G/ 8 input model and the Fractional Gaussian Noise (FGN) model. We focus on the response times of the customers in a First-Come First-Serve (FCFS) queueing system, although the results carry through to the backlog distribution of the system with any arbitrary queueing discipline. When the arrival process is driven by an M/G/ 8 input model we show that if the residual service time tail distribution F e is lighter than the residual session duration G e , then the stationary waiting time is dominated by the long-range dependence structure, which is determined by the residual session duration G e . If the residual service time distribution F e is heavier than the residual session duration G e , then the tail distribution of the stationary waiting time is dominated by that of the residual service time. When the arrival process is modeled by an FGN, we show that the waiting time tail distribution is asymptotically equal to the tail distribution of the residual service time if the latter is asymptotically heavier than Weibull distribution with shape parameter 2-2H, where H is the Hurst parameter of the FGN. If, however, this residual service time is asymptotically lighter than Weibull distribution with shape parameter 2-2H, then the waiting time tail distribution is dominated by the dependence structure of the arrival process so that it is asymptotically equal to Weibull distribution with shape parameter 2-2H.

An approximation for waiting time tail probabilities in multiclass systems

This article proposes a simple exponential approximation for waiting time tail probabilities in multiclass systems. The approximation is evaluated by making comparisons with simulation results, which show that the proposed approximation is adequate.

Asymptotics for M/G/1 low-priority waiting-time tail probabilities

We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptive-resume disciplines. We show that the low-priority steady-state waiting-time can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waiting-time distribution. We exploit this structures to determine the asymptotic behavior of the tail probabilities. Unlike the FIFO case, there is routinely a region of the parameters such that the tail probabilities have non-exponential asymptotics. This phenomenon even occurs when both service-time distributions are exponential. When non-exponential asymptotics holds, the asymptotic form tends to be determined by the non-exponential asymptotics for the high-priority busy-period distribution. We obtain asymptotic expansions for the low-priority waiting-time distribution by obtaining an asymptotic expansion for the busy-period transform from Kendall’s functional equation. We identify the boundary between the exponential and non-exponential asymptotic regions. For the special cases of an exponential high-priority service-time distribution and of common general service-time distributions, we obtain convenient explicit forms for the low-priority waiting-time transform. We also establish asymptotic results for cases with long-tail service-time distributions. As with FIFO, the exponential asymptotics tend to provide excellent approximations, while the non-exponential asymptotics do not, but the asymptotic relations indicate the general form. In all cases, exact results can be obtained by numerically inverting the waiting-time transform.

Waiting-time tail probabilities in queues with long-tail service-time distributions

We consider the standardGI/G/1 queue with unlimited waiting room and the first-in first-out service discipline. We investigate the steady-state waiting-time tail probabilitiesP(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 contourintegral 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 orderx −r forr>1. We use this family of long-tail distributions to investigate the quality of approximations based on asymptotics forP(W>x) asx→∞. We show that the asymptotic approximations with these long-tail service-time distributions can be remarkably inaccurate for typicalx values of interest. We also derive multi-term asymptotic expansions for the waiting-time tail probabilities in theM/G/1 queue. Even three terms of this expansion can be remarkably inaccurate for typicalx 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.