On Global Near Optimality of Special Periodic Protocols for Fluid Polling Systems with Setups

Matveev, Alexey S.; Feoktistova, V.; Bolshakova, K.

doi:10.1007/s10957-016-0923-0

Cited by 7 publications

(4 citation statements)

References 22 publications

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“…By the same derivation for M as in the proof of Theorem 7, for A := A i A w(i,1) • • • A w(i,IL−1) , its spectral radius ̺(A) < 1. Since T is an affine operator mapping from R K + to R K + , it follows from Lemma 5.1 in Matveev et al (2016) that T is a contraction map in R K + , and possesses a unique fixed point (i.e., g i ), which in turn implies that (B.26) and (B.27) admit a unique solution (in R ILK + ). We can also verify that (q e,k (a i ), k ∈ K, i ∈ I L ) solves (B.26) and (B.27).…”

Section: B3 Characterization Of the First Momentmentioning

confidence: 95%

“…(An affine operator is positively invariant if it maps R K + into itself; see (Matveev et al 2016, p.10). ) By Lemma 5.1 in Matveev et al (2016), if ̺(A ′ ) < 1, where ̺(A ′ ) denotes the spectral radius of the matrix A ′ , then the positively invariant affine operator Γ ′ (q) is a contraction mapping in R K + . Hence, we next show that ̺(A ′ ) < 1.…”

Section: The Sb-pr Controlmentioning

confidence: 99%

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Asymptotic Optimality of the Binomial-Exhaustive Policy for Polling Systems with Large Switchover Times

Dong

Perry

2020

Preprint

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We study an optimal-control problem of polling systems with large switchover times, when a holding cost is incurred on the queues. In particular, we consider a stochastic network with a single server that switches between several buffers (queues) according to a pre-specified order, assuming that the switchover times between the queues are large relative to the processing times of individual jobs. Due to its complexity, computing an optimal control for such a system is prohibitive, and so we instead search for an asymptotically optimal control. To this end, we first solve an optimal control problem for a deterministic relaxation (namely, for a fluid model), that is represented as a hybrid dynamical system. We then "translate" the solution to that fluid problem to a binomial-exhaustive policy for the underlying stochastic system, and prove that this policy is asymptotically optimal in a large-switchover-time scaling regime, provided a certain uniform integrability (UI) condition holds. Finally, we demonstrate that the aforementioned UI condition holds in the following cases: (i) the holding cost has (at most) linear growth, and all service times have finite second moments; (ii) the holding cost grows at most at a polynomial rate (of any degree), and the service-time distributions possess finite moment generating functions. Our proofs that the UI condition holds in these two cases may be of an independent interest.

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Section: B3 Characterization Of the First Momentmentioning

confidence: 95%

Section: The Sb-pr Controlmentioning

confidence: 99%

Asymptotic Optimality of the Binomial-Exhaustive Policy for Polling Systems with Large Switchover Times

Dong

Perry

2020

Preprint

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“…Following the same derivation as for M in the proof of Theorem 4.1, we can show that for A := A i A w(i,1) • • • A w(i,I−1) , its spectral radius (A) < 1. Since T is an affine operator mapping from R K + to R K + , it follows from Lemma 5.1 in Matveev et al (2016) that T is a contraction map in R K + , and possesses a unique fixed point (i.e., g i ), which in turn implies that (5.3) and (5.4) admit a unique solution (in R IK + ). Therefore, it must be the case that (E [Q k (A i )] , k ∈ K, i ∈ I) is the unique solution to (5.1).…”

Section: Characterization Of the First Momentmentioning

confidence: 96%

Existence and Approximations of Moments for Polling Systems under the Binomial-Exhaustive Policy

Hu,

Dong,

Perry

2020

Preprint

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We establish sufficient conditions for the existence of moments of the steady-state queue in polling systems operating under the binomial-exhaustive policy (BEP). We assume that the server switches between the different buffers according to a pre-specified table, and that switchover times are incurred whenever the server moves from one buffer to the next. We further assume that customers arrive according to independent Poisson processes, and that the service and switchover times are independent random variables with general distributions. We then propose a simple scheme to approximate the moments, which is shown to be asymptotically exact as the switchover times increase without bound, and whose computation complexity does not grow with the order of the moment. Finally, we demonstrate that the proposed asymptotic approximations for the moments is related to the fluid limit under a large-switchover-time scaling; thus, similar approximations can be easily derived for other server-switching policies, by simply identifying the fluid limits under those controls. Numerical examples demonstrate the effectiveness of our approximations for the moments under BEP and other policies, and their improving accuracy as the switchover times increase.

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“…The fluid model of the polling system is presented by Matveev et al [113]. A queue is interpreted as the fluid level which decreases when the server serves the queue.…”

Section: Non-discrete Polling Systems and Polling Networkmentioning

confidence: 99%

Polling Systems and Their Application to Telecommunication Networks

Vishnevsky

2021

Mathematics

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The paper presents a review of papers on stochastic polling systems published in 2007–2020. Due to the applicability of stochastic polling models, the researchers face new and more complicated polling models. Stochastic polling models are effectively used for performance evaluation, design and optimization of telecommunication systems and networks, transport systems and road management systems, traffic, production systems and inventory management systems. In the review, we separately discuss the results for two-queue systems as a special case of polling systems. Then we discuss new and already known methods for polling system analysis including the mean value analysis and its application to systems with heavy load to approximate the performance characteristics. We also present the results concerning the specifics in polling models: a polling order, service disciplines, methods to queue or to group arriving customers, and a feedback in polling systems. The new direction in the polling system models is an investigation of how the customer service order within a queue affects the performance characteristics. The results on polling systems with correlated arrivals (MAP, BMAP, and the group Poisson arrivals simultaneously to all queues) are also considered. We briefly discuss the results on multi-server, non-discrete polling systems and application of polling models in various fields.

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On Global Near Optimality of Special Periodic Protocols for Fluid Polling Systems with Setups

Cited by 7 publications

References 22 publications

Asymptotic Optimality of the Binomial-Exhaustive Policy for Polling Systems with Large Switchover Times

Asymptotic Optimality of the Binomial-Exhaustive Policy for Polling Systems with Large Switchover Times

Existence and Approximations of Moments for Polling Systems under the Binomial-Exhaustive Policy

Polling Systems and Their Application to Telecommunication Networks

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