Polling systems in heavy traffic: higher moments of the delay

Mei, R.D. van der

doi:10.1016/s1388-3437(97)80032-7

Cited by 15 publications

(24 citation statements)

References 8 publications

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“…, N), and hence,Mŵ =ŵ, which shows thatŵ is indeed a right eigenvector ofM. Similar arguments can be used to show thatv is a left eigenvector ofM (along the lines discussed in the Appendix of [40]). The details are omitted for compactness of the presentation, and are left as an exercise to the reader.…”

Section: Lemmasupporting

confidence: 54%

Towards a unifying theory on branching-type polling systems in heavy traffic

Mei

2007

Queueing Syst

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For a broad class of polling models the evolution of the system at specific embedded polling instants is known to constitute a multi-type branching process (MTBP) with immigration. In this paper it is shown that for this class of polling models the vector that describes the state of the system at these polling instants, say X = (X 1 , . . . , X M ), satisfies the following heavy-traffic behavior (under mild assumptions):where γ is a known M-dimensional vector, (α, μ) has a gamma-distribution with known parameters α and μ, and where ρ is the load of the system. This general and powerful result is shown to lead to exact-and in many cases even closed-form-expressions for the Laplace-Stieltjes Transform (LST) of the complete asymptotic queue-length and waiting-time distributions for a broad class of branchingtype polling models that includes many well-studied polling models policies as special cases. The results generalize and unify many known results on the waiting times in polling systems in heavy traffic, and moreover, lead to new exact results for classical polling models that have not been Part of this research has been funded by the Dutch BSIK/BRICKS project.observed before. To demonstrate the usefulness of the results, we derive closed-form expressions for the LST of the waiting-time distributions for models with cyclic globallygated polling regimes, and for cyclic polling models with general branching-type service policies. As a by-product, our results lead to a number of asymptotic insensitivity properties, providing new fundamental insights in the behavior of polling models.

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Section: Lemmasupporting

confidence: 54%

Towards a unifying theory on branching-type polling systems in heavy traffic

Mei

2007

Queueing Syst

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“…The result can be obtained along the lines similar to the derivation of the results for the one-stage polling models in [14,15].…”

Section: Resultsmentioning

confidence: 81%

Polling Models with Two-Stage Gated Service: Fairness Versus Efficiency

Mei

Resing

2007

Lecture Notes in Computer Science

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Abstract.We consider an asymmetric cyclic polling system with general service-time and switch-over time distributions with so-called twostage gated service at each queue, an interleaving scheme that aims to enforce fairness among the different customer classes. For this model, we (1) obtain a pseudo-conservation law, (2) describe how the mean delay at each of the queues can be obtained recursively via the so-called Descendant Set Approach, and (3) present a closed-form expression for the expected delay at each of the queues when the load tends to unity (under proper heavy-traffic scalings), which is the main result of this paper. The results are strikingly simple and provide new insights into the behavior of two-stage polling systems, including several insensitivity properties of the asymptotic expected delay with respect to the system parameters. Moreover, the results provide insight in the delay-performance of two-stage gated polling compared to the classical one-stage gated service policies. The results show that the two-stage gated service policy indeed leads to better fairness compared to one-stage gated service, at the expense of a decrease in efficiency. Finally, the results also suggest simple and fast approximations for the expected delay in stable polling systems. Numerical experiments demonstrate that the approximations are highly accurate for moderately and heavily loaded systems.

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“…Solve a(x) = y for x to find a −1 (y) = ln(1 − y/ρ i )/(−b i ). Now substituting this in Equation (27) it follows after some simplification that the generalized trapezoidal distribution U * i coincides with the density function of U * i for ROS given in (23). This means that for the case of exponential service-time distributions, the sojourn-time distributions for ROS and PS coincide.…”

Section: Exponential Service Timesmentioning

confidence: 98%

“…It is well known that if a gamma random variable has parameters α and µ then its length-biased version has parameters α + 1 and µ. The following result gives an expression for the higher moments of the waiting times in heavy traffic (proven in [27,28]): …”

Section: First-come-first-servedmentioning

confidence: 99%

The impact of scheduling policies on the waiting-time distributions in polling systems

Bekker¹,

Vis²,

Dorsman

et al. 2014

Queueing Syst

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We consider polling models consisting of a single server that visits the queues in a cyclic order. In the vast majority of papers that have appeared on polling models, it is assumed that at each of the individual queues the customers are served on a First-Come-First-Served (FCFS) basis. In this paper we study polling models where the local scheduling policy is not FCFS, but instead, is varied as Last-ComeFirst-Served (LCFS), Random Order of Service (ROS), Processor Sharing (PS) and Shortest-Job-First (SJF). The service policies are assumed to be either gated or globally gated. The main result of the paper is the derivation of asymptotic closed-form expressions for the Laplace-Stieltjes transform (LST) of the scaled waiting-time and sojourn-time distributions under heavy-traffic assumptions. For FCFS service the asymptotic sojourn-time distribution is known to be of the form U Γ, where U and Γ are uniformly and gamma distributed with known parameters. In this paper we show that the asymptotic sojourntime distribution (1) for LCFS is also of the form U Γ, (2) for ROS is of the formŨ Γ whereŨ has a trapezoidal distribution, and (3) for PS and SJF is of the formŨ * Γ whereŨ * has a generalized trapezoidal distribution. These results are rather intriguing and lead to new fundamental insight in the impact of the local scheduling policy on the performance of polling models. As a by-product the heavy-traffic results suggest simple closed-form approximations for the complete waiting-time and sojourn-time distributions for stable systems with arbitrary load values. The accuracy of the approximations is evaluated by simulations.

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Polling systems in heavy traffic: higher moments of the delay

Cited by 15 publications

References 8 publications

Towards a unifying theory on branching-type polling systems in heavy traffic

Towards a unifying theory on branching-type polling systems in heavy traffic

Polling Models with Two-Stage Gated Service: Fairness Versus Efficiency

The impact of scheduling policies on the waiting-time distributions in polling systems

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