Nonexponential Service Time Distributions Research Articles

Stationary queue length of a single-server queue with negative arrivals and nonexponential service time distributions

In this paper, a single-server queue with negative customers is considered. The arrival of a negative customer will remove one positive customer that is being served, if any is present. An alternative approach will be introduced to derive a set of equations which will be solved to obtain the stationary queue length distribution. We assume that the service time distribution tends to a constant asymptotic rate when time t goes to infinity. This assumption will allow for finding the stationary queue length of queueing systems with non-exponential service time distributions. Numerical examples for gamma distributed service time with fractional value of shape parameter will be presented in which the steady-state distribution of queue length with such service time distributions may not be easily computed by most of the existing analytical methods.

Decay of tails at equilibrium for FIFO join the shortest queue networks

In join the shortest queue networks, incoming jobs are assigned to the shortest queue from among a randomly chosen subset of $D$ queues, in a system of $N$ queues; after completion of service at its queue, a job leaves the network. We also assume that jobs arrive into the system according to a rate-$\alpha N$ Poisson process, $\alpha<1$, with rate-1 service at each queue. When the service at queues is exponentially distributed, it was shown in Vvedenskaya et al. [Probl. Inf. Transm. 32 (1996) 15-29] that the tail of the equilibrium queue size decays doubly exponentially in the limit as $N\rightarrow\infty$. This is a substantial improvement over the case D=1, where the queue size decays exponentially. The reasoning in [Probl. Inf. Transm. 32 (1996) 15-29] does not easily generalize to jobs with nonexponential service time distributions. A modularized program for treating general service time distributions was introduced in Bramson et al. [In Proc. ACM SIGMETRICS (2010) 275-286]. The program relies on an ansatz that asserts, in equilibrium, any fixed number of queues become independent of one another as $N\rightarrow\infty$. This ansatz was demonstrated in several settings in Bramson et al. [Queueing Syst. 71 (2012) 247-292], including for networks where the service discipline is FIFO and the service time distribution has a decreasing hazard rate. In this article, we investigate the limiting behavior, as $N\rightarrow \infty$, of the equilibrium at a queue when the service discipline is FIFO and the service time distribution has a power law with a given exponent $-\beta$, for $\beta>1$. We show under the above ansatz that, as $N\rightarrow\infty$, the tail of the equilibrium queue size exhibits a wide range of behavior depending on the relationship between $\beta$ and $D$. In particular, if $\beta>D/(D-1)$, the tail is doubly exponential and, if $\beta<D/(D-1)$, the tail has a power law. When $\beta=D/(D-1)$, the tail is exponentially distributed.

Asymptotic independence of queues under randomized load balancing

Randomized load balancing greatly improves the sharing of resources while being simple to implement. In one such model, jobs arrive according to a rate-?N Poisson process, with ?<1, in a system of N rate-1 exponential server queues. In Vvedenskaya et al. (Probl. Inf. Transm. 32:15---29, 1996), it was shown that when each arriving job is assigned to the shortest of D, D?2, randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as N??. This is a substantial improvement over the case D=1, where queue sizes decay exponentially. The reasoning in Vvedenskaya et al. (Probl. Inf. Transm. 32:15---29, 1996) does not easily generalize to jobs with nonexponential service time distributions. A modularized program for treating randomized load balancing problems with general service time distributions was introduced in Bramson et al. (Proc. ACM SIGMETRICS, pp. 275---286, 2010). The program relies on an ansatz that asserts that, for a randomized load balancing scheme in equilibrium, any fixed number of queues become independent of one another as N??. This allows computation of queue size distributions and other performance measures of interest. In this article, we demonstrate the ansatz in several settings. We consider the least loaded balancing problem, where an arriving job is assigned to the queue with the smallest workload. We also consider the more difficult problem, where an arriving job is assigned to the queue with the fewest jobs, and demonstrate the ansatz when the service discipline is FIFO and the service time distribution has a decreasing hazard rate. Last, we show the ansatz always holds for a sufficiently small arrival rate, as long as the service distribution has 2 moments.

A multi-class queuing network analysis methodology for improving hospital emergency department performance

This paper derives an open queuing network model of an emergency department (ED) design intended to increase the capacity of an ED to treat patients. The methodology captures hospital-specific differences in patient acuity mix, arrival patterns and volumes, and efficiencies of processes in a single common computational model. A spreadsheet implementation of the resulting queuing equations is used by managers, in real time, to size ED areas using waiting time and overflow probability as quality of service targets. Non-homogeneous arrival patterns, non-exponential service time distributions, and multiple patient types are all incorporated. The methodology has been applied to a fleet of hospitals for validation. Results from one of them are used to demonstrate the methodology. Scope and purpose Population growth, closure of emergency departments (EDs) nearby, or seasonal peak variation can cause ED patient wait times to increase dramatically. Prolonged waits induce patients to leave the ED before receiving treatment, creating a public health problem. The purpose of this paper is to introduce a new paradigm of ED care that reduces “walk-aways”, and increases ED access, through an operational research method that customizes to any hospital through the use of hospital-specific data elements.

Variability Functions for Parametric-Decomposition Approximations of Queueing Networks

We propose an enhancement to the parametric-decomposition method for calculating approximate steady-state performance measures of open queueing networks with non-Poisson arrival processes and nonexponential service-time distributions. Instead of using a variability parameter ca2 for each arrival process, we suggest using a variability function ca2ρ, 0 &lt; ρ &lt; 1, for each arrival process i.e, the variability parameter should be regarded as a function of the traffic intensity ρ of a queue to which the arrival process might go. Variability functions provide a convenient representation of different levels of variability in different time scales for arrival processes that are not nearly renewal processes. Variability functions enable the approximations to account for long-range effects in queueing networks that cannot be addressed by variability parameters. For example, the variability functions provide a way to address the heavy-traffic bottleneck phenomenon, in which exceptional variability (either high or low) in the input has little impact in a series of queues with low-to-moderate traffic intensities, and then has a big impact when it reaches a later queue with a relatively high traffic intensity. The variability functions also enable the approximations to characterize irregular periodic deterministic external arrival processes in a reasonable way; i.e., if there are no batches, then ca2ρ should be 0 for ρ near 0 or 1, but ca2ρ can assume arbitrarily large values for appropriate intermediate ρ. We present a full network algorithm with variability functions, showing that the idea is implementable. We also show how simulations of single queues can be effectively exploited to determine variability functions for difficult external arrival processes.

Perturbation analysis of closed queueing networks with general service time distributions

Perturbation analysis of closed queuing networks with nonexponential service time distributions is studied. Perturbation analysis formulas using realization probabilities are extended to these networks. A perturbation generation function, which generalizes the perturbation generation rule, is defined. equations for realization probability and formulas for sensitivity of the system throughput with respect to service time distribution parameters are presented. The formulas provide an analytical method of calculating throughput sensitivity and an explanation of the application of perturbation analysis algorithms for networks with general service time distributions. The author focuses on the extension of concepts and intuitive explanations of the formulas rather than on mathematical derivations. >

Nonexponential networks of queues

We will propose a new, iterative method for approximately analyzing closed networks of queues with nonexponential service time distributions and FCFS scheduling. Our method is based on the Principle of Maximum Entropy and produces results which, first, are consistent with the fundamental Work Rate Theorem and, second, are exact for separable networks of queues. Considering accuracy and execution time characteristics, our method offers a viable alternative to Marie's homogeneous approximation method.

Performance of the Queueing Network Analyzer

This paper describes the performance of the Queueing Network Analyzer (QNA), a software package developed at Bell Laboratories to calculate approximate congestion measures for networks of queues. QNA is compared with simulations and other approximations of several open networks of single-server queues. This paper illustrates how to apply QNA and indicates the quality that can be expected from the approximations. The examples here demonstrate the importance of the variability parameters used in QNA to describe non-Poisson arrival processes and nonexponential service-time distributions. For these examples, QNA performs much better than the standard Markovian algorithm, which does not use variability parameters. The accuracy of the QNA results (e.g., the expected delays) in these examples is satisfactory for engineering purposes.

Routine Analysis of Periodic Queues

Abstract Many queues are characterized by predictable periodic fluctuations as well as by random fluctuations. Queues that arise at copiers or duplicators usually involve such predictable fluctuations in work-load arrival. This note describes a series of approximations that have been combined into a simple inexpensive computer program. This program is being routinely used by the Xerox Corporation to approximate the average turnaround time at nonstationary single-server queues with nonexponential service time distributions.