Polling Systems With Simultaneous Batch Arrivals

Mei, R.D. van der

doi:10.1081/stm-100002274

Cited by 12 publications

(25 citation statements)

References 22 publications

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“…In a series of papers, van der Mei and co-authors explore the use of the Descendant Set Approach (DSA) [17] to derive exact expressions the waiting-time distributions in models with mixtures of exhaustive and gated service and cyclic [39] or periodic [26] server routing. Following a similar approach, van der Mei also derives the exact asymptotics waiting-time distribution in cyclic queueing models with simultaneous batch arrivals [42]. Kroese [18] studies continuous polling systems in heavy traffic with unit Poisson arrivals on a ring and shows that the steady-state number of customers at each queue has approximately a gamma-distribution.…”

Section: Introductionmentioning

confidence: 97%

“…Initiated by the pioneering work of Coffman et al [12,13], the analysis of the heavy-traffic behavior of polling models has gained a lot of interest over the past decade. This has led to the derivation of asymptotic expressions for key performance metrics, such as the moments and distributions of the waiting times and the queue lengths, for a variety of model variants, including for example models with mixtures of exhaustive and gated service policies with cyclic server routing [39], periodic server routing [26,27], simultaneous batch arrivals [42], continuous polling [18], amongst others. In this context, a remarkable observation is that in the heavy-traffic behavior of polling models a central role is played by the gamma-distribution, which occurs in the analysis of these different model variants as the limiting distribution of the (scaled) cycle times and the marginal queue-lengths at polling instants.…”

Section: Introductionmentioning

confidence: 99%

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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: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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

Mei

2007

Queueing Syst

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“…A remarkable observation is that HT limits for models with more than two queues have only been rigorously proven for models driven by Poisson -or compound-Poisson [25] -processes, models that do satisfy the MTBP-structure (see references above), whereas HT limits for models with renewal arrivals, which generally violate the MTBP structure, have only been obtained on the basis of conjectures [6,7,15]. For this reason, in this paper we consider perhaps one of the simplest polling models that does violate the MTBP structure, and propose a new method to derive rigorous proofs for HT asymptotics.…”

Section: Introductionmentioning

confidence: 99%

“…[16] for details). By exploring the branching structure of the model, Van der Mei and coauthors [14,21,22,23,24,25] explore the recursive relations of the Descendant Set Approach 2 (DSA) [9] to derive closed-form expressions for the asymptotic delay distribution in HT for polling systems with an MTBP-structure in a general parameter setting, both for cyclic and periodic server routing. Kudoh et al [11] use the classical buffer-occupancy technique, which is based on an expression for the probability generating function of the joint queue-length distribution at successive polling instants, to derive explicit expressions for the second moment of the delay in fully symmetric systems with gated or exhaustive service at each queue for models with two, three and four queues.…”

Section: Introductionmentioning

confidence: 99%

Polling models with renewal arrivals: A new method to derive heavy-traffic asymptotics

Mei

Winands

2007

Performance Evaluation

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We consider asymmetric cyclic polling systems with an arbitrary number of queues, general service-time distributions, zero switch-over times, gated service at each queue, and with general renewal arrival processes at each of the queues. For this classical model, we propose a new method to derive closed-form expressions for the expected delay at each of the queues when the load tends to 1, under proper heavy-traffic (HT) scalings. In the literature on polling models, rigorous proofs of HT limits have only been obtained for polling models with Poisson-type arrival processes, whereas for renewal arrivals HT limits are based on conjectures [6,7,15]. Therefore, the main contribution of this paper lies in the fact that we propose a new method to rigorously prove HT limits for a class of non-Poisson-type arrivals. The results are remarkably simple and provide new fundamental insight and reveal explicitly how the expected delay at each of the queues depends on the system parameters, and in particular on the interarrival-time distributions at each of the queues. The results also suggest simple approximations for the expected delay in stable polling systems. Numerical results show that the approximations are highly accurate when the system load is roughly 90% or more.

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“…In polling systems near saturation, waitingtimes approach a gamma distribution and higher moment effects vanish under heavy traffic [24].…”

Section: Port 1 Analysismentioning

confidence: 99%