Robustness of Rootfinding in Single-Server Queueing Models

Chaudhry, M. L.; Harris, Carl M.; Marchal, William G.

doi:10.1287/ijoc.2.3.273

Cited by 55 publications

(29 citation statements)

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“…A proof for the existence of this root can be given and is similar to the case discussed in Theorem 3, Chaudhry eta/. 10 It should, however, be pointed out here that it is only the far tail that can be approximated very accurately by this single term z 1 • To obtain the tail probabilities, assume…”

Section: Approximations For Tail Probabilitiesmentioning

confidence: 89%

On ‘OR Enactment: The Theatrical Metaphor’

Wainwright¹

1993

Journal of the Operational Research Society

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this is done by finding roots of the denominator of the probability generating function of Wq and then resolving the generating function into partial fractions. Numerical examples are given showing the use of the required roots, even when there is a large number of them . The method discussed in this paper avoids spectrum factorization and uses both closed-and non-closed forms of interarrival-arid service-time distributions. Approximations for the tail probabilities in terms of one or three roots taken in ascending order of magnitude are also discussed . The exact computational results that can be obtained from the methods of this study should prove useful to both practitioners and queueing theorists dealing with bounds, inequalities, approximations, and simulation results.

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Section: Approximations For Tail Probabilitiesmentioning

confidence: 89%

On ‘OR Enactment: The Theatrical Metaphor’

Wainwright¹

1993

Journal of the Operational Research Society

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“…Therefore, the most general method relies on numerical techniques. Chaudhry et al [6] have developed a software program to solve root-finding problems in queueing theory numerically, which works in our experience for almost all distributions.…”

Section: A2 Numerical Determination Of the Rootsmentioning

confidence: 99%

Bounds and Approximations for the Fixed-Cycle Traffic-Light Queue

Broek

Leeuwaarden²,

Adan

et al. 2006

Transportation Science

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This paper deals with the fixed-cycle traffic-light (FCTL) queue, where vehicles arrive at an intersection controlled by a traffic light and form a queue. The traffic light alternates between green and red periods, and delayed vehicles are assumed to depart during the green period at equal time intervals. The key performance characteristic in the FCTL queue is the so-called mean overflow, defined as the mean queue length at the end of a green period. An exact solution for the mean overflow is available, but it has been considered to be of little practical value because it requires some numerical procedures. Therefore, most of the literature on the FCTL queue is about deriving approximations for the mean overflow. In deriving these approximations, most authors first approximate the FCTL queue by a bulk-service queue, approximate the mean overflow in the bulk-service queue, and use this as an approximation for the mean overflow in the FCTL queue. So far no quantitative comparison of both models has been given. We compare both models and assess the quality of the approximation for various settings of the parameter values. In this comparison and throughout the paper we do not restrict ourselves to Poisson arrivals, but consider a more general arrival process instead. We discuss the numerical issues that need to be resolved to calculate the exact expression for the mean overflow in both queues and show that clear computational schemes are available. Next, we present several bounds and approximations of the mean overflow that do not require numerical procedures. In particular, we derive a new approximation based on the heavy traffic limit and a scaling argument. We compare the new bounds and approximation with the existing ones. We elaborate on the impact of several parameters, like the length of the green and red period and the variance of the arrival distribution. Each of these parameters turns out to be crucial.

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“…In [9] it is shown that the condition that A is infinitely-divisible, or the somewhat weaker condition that A(z) has no zeros inside the unit circle, are sufficient for the roots of z s = A(z) on and within the unit circle to be distinct. However, examples exist of A(z) having zeros inside the unit circle and at the same time having distinct roots (see e.g.…”

Section: Fourier Series Representationmentioning

confidence: 99%

“…Example 3.5). It is therefore that in both [9] and [20] the urge of finding a necessary condition for distinctness is expressed. In this respect, we have the following result: Lemma 3.7.…”

Section: Fourier Series Representationmentioning

confidence: 99%

Analytic Computation Schemes for the Discrete-Time Bulk Service Queue

Janssen

Leeuwaarden²

2005

Queueing Syst

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In commonly used approaches for the discrete-time bulk service queue, the stationary queue length distribution follows from the roots inside or outside the unit circle of a characteristic equation. We present analytic representations of these roots in the form of sample values of periodic functions with analytically given Fourier series coefficients, making these approaches more transparent and explicit. The resulting computational scheme is easy to implement and numerically stable. We also discuss a method to determine the roots by applying successive substitutions to a fixed point equation. We outline under which conditions this method works, and compare these conditions with those needed for the Fourier series representation. Finally, we present a solution for the stationary queue length distribution that does not depend on roots. This solution is explicit and well-suited for determining tail probabilities up to a high accuracy, as demonstrated by some numerical examples.

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Robustness of Rootfinding in Single-Server Queueing Models

Cited by 55 publications

References 0 publications

On ‘OR Enactment: The Theatrical Metaphor’

On ‘OR Enactment: The Theatrical Metaphor’

Bounds and Approximations for the Fixed-Cycle Traffic-Light Queue

Analytic Computation Schemes for the Discrete-Time Bulk Service Queue

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