Vacation and Polling Models with Retrials

Boxma, OJ Onno; Resing, Jacques

doi:10.1007/978-3-319-10885-8_4

Cited by 9 publications

(9 citation statements)

References 19 publications

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“…We consider a single server cyclic polling system with retrials and so-called glue periods. This model was first introduced in [7] for a single station vacation model and a two-station model with switchover times. Further in [1] this was extended to an N -station model with switchover times.…”

Section: The Modelmentioning

confidence: 99%

“…A first attempt to study a polling model which combines retrials and glue periods is [7], which mainly focuses on the case of a single server and a single station, but also outlines how that analysis can be extended to the case of two stations. In [1] an N -station polling model with retrials and with constant glue periods is considered, for the case of gated service discipline at all stations.…”

Section: Introductionmentioning

confidence: 99%

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Performance analysis of polling systems with retrials and glue periods

Abidini¹,

Boxma²,

Kim³

et al. 2017

Queueing Syst

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We consider gated polling systems with two special features: (i) retrials, and (ii) glue or reservation periods. When a type-i customer arrives, or retries, during a glue period of station i, it will be served in the next visit period of the server to that station. Customers arriving at station i in any other period join the orbit of that station and will retry after an exponentially distributed time. Such polling systems can be used to study the performance of certain switches in optical communication systems.For the case of exponentially distributed glue periods, we present an algorithm to obtain the moments of the number of customers in each station. For generally distributed glue periods, we consider the distribution of the total workload in the system, using it to derive a pseudo conservation law which in its turn is used to obtain accurate approximations of the individual mean waiting times.We also consider the problem of choosing the lengths of the glue periods, under a constraint on the total glue period per cycle, so as to minimize a weighted sum of the mean waiting times.

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Section: The Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Performance analysis of polling systems with retrials and glue periods

Abidini¹,

Boxma²,

Kim³

et al. 2017

Queueing Syst

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“…Motivated by questions regarding the performance modelling and analysis of optical networks, the study of polling models with retrials and glue periods was initiated in Boxma and Resing [5]. In these models, just before the server arrives at a station there is some glue period.…”

Section: Introductionmentioning

confidence: 99%

“…Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. In [5], the joint queue length process is analysed both at embedded time points (beginnings of glue periods, visit periods and switch-over periods) and at arbitrary time points, for the model with two queues and deterministic glue periods. This analysis is later on extended in Abidini, Boxma and Resing [2] to the model with a general number of queues.…”

Section: Introductionmentioning

confidence: 99%

Heavy traffic analysis of a polling model with retrials and glue periods

2018

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We present a heavy traffic analysis of a single-server polling model, with the special features of retrials and glue periods. The combination of these features in a polling model typically occurs in certain optical networking models, and in models where customers have a reservation period just before their service period. Just before the server arrives at a station there is some deterministic glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. As this model defies a closed-form expression for the queue length distributions, our main focus is on their heavy-traffic asymptotics, both at embedded time points (beginnings of glue periods, visit periods and switch periods) and at arbitrary time points. We obtain closed-form expressions for the limiting scaled joint queue length distribution in heavy traffic and use these to accurately approximate the mean number of customers in the system under different loads.taking limits in known expressions for the Laplace-Stieltjes transform (LST) of the waiting-time distribution. Alternatively, Olsen and van der Mei [17] provide similar results, by studying the behaviour of the descendant set approach (a numerical computation method, cf. Konheim, Levy and Srinivasan [10]) in the heavy traffic limit. For the derivation of heavy traffic asymptotics for our model, however, we will use results from branching theory, mainly those presented in Quine [18]. Earlier, these results have resulted in heavy traffic asymptotics for conventional polling models, see van der Mei [15]. We will use the same method as presented in that paper, but for a different class of polling system that models the dynamics of optical networks. In addition, for some steps of the analysis, we will present new and straight forward proofs, while other steps require a different approach. Furthermore, we will derive asymptotics for the joint queue length process at arbitrary time points, as opposed to just the marginal processes as derived in [15]. Due to the additional intricacies of the model at hand, we will need to overcome many arising complex difficulties, as will become apparent later.The rest of the paper is organized as follows. In Section 2, we introduce some notation and present a theorem from [18] on multitype branching processes with immigration. In Section 3, we describe in detail the polling model with retrials and glue periods and recall from [2] how the joint queue length process at some embedded time points in this model is related to multitype branching processes with immigration. Next, we will derive heavy traffic results for our model. In Section 4, we consider the joint queue length process at the start of glue periods. In Section 5, we look at the joint queue length process at the start of visit and switch-over periods, while in Section 6, we consider the joint queue length process at ar...

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“…✩ This is an invited, considerably extended version ofBoxma and Resing (2014) [28]. The main additions are Sections 2.4 and 3.…”

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confidence: 99%

Analysis and optimization of vacation and polling models with retrials

Abidini

Boxma

Resing

2016

Performance Evaluation

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Vacation and Polling Models with Retrials

Cited by 9 publications

References 19 publications

Performance analysis of polling systems with retrials and glue periods

Performance analysis of polling systems with retrials and glue periods

Heavy traffic analysis of a polling model with retrials and glue periods

Analysis and optimization of vacation and polling models with retrials

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