Stability analysis of multiserver discrete-time queueing systems with renewal-type server interruptions

Morozov, Evsey; Fiems, Dieter; Bruneel, Herwig

doi:10.1016/j.peva.2011.07.002

Cited by 22 publications

(10 citation statements)

References 35 publications

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“…The model in [62] is a variant of this, where the service-time distribution is general and both the arrival process and the serverinterruption process are dependent on a common underlying finite-state Markov chain. Also relevant in this context is [63], where sufficient conditions for system stability, both in the single-server and the multi-server case, are established under renewal assumptions for arrival, service and interruption processes.…”

Section: Discrete-time Modelsmentioning

confidence: 99%

Analysis of a discrete-time single-server queue with an occasional extra server

Bruneel

Wittevrongel

2017

Performance Evaluation

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We consider a discrete-time queueing system having two distinct servers: one server, the "regular" server, is permanently available, while the second server, referred to as the "extra" server, is only allocated to the system intermittently. Apart from their availability, the two servers are identical, in the sense that the customers have deterministic service times equal to 1 fixed-length time slot each, regardless of the server that processes them. In this paper, we assume that the extra server is available during random "up-periods", whereas it is unavailable during random "down-periods". Up-periods and down-periods occur alternately on the time axis. The up-periods have geometrically distributed lengths (expressed in time slots), whereas the distribution of the lengths of the down-periods is general, at least in the first instance. Customers enter the system according to a general independent arrival process, i.e., the numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution.For this queueing model, we are able to derive closed-form expressions for the steadystate probability generating functions (pgfs) and the expected values of the numbers of customers in the system at various observation epochs, such as the start of an up-period, the start of a down-period and the beginning of an arbitrary time slot. At first sight, these formulas, however, appear to contain an infinite number of unknown constants. One major issue of the mathematical analysis turns out to be the determination of these constants. In the paper, we show that restricting the pgf of the down-periods to be a rational function of its argument, brings about the crucial simplification that the original infinite number of unknown constants appearing in the formulas can be expressed in terms of a finite number of independent unknowns. The latter can then be adequately determined based on the bounded nature of pgfs inside the complex unit disk, and an extensive use of properties of polynomials.Various special cases, both from the perspective of the arrival distribution and the down-period distribution, are discussed. The results are also illustrated by means of relevant numerical examples.Possible applications of this type of queueing model are numerous: the extra server could be the regular server of another similar queue, helping whenever an idle period occurs in its own queue; a geometric distribution for these idle times is then a very natural modelling assumption. A typical example would be the situation at the check-in counter at a gate in an airport: the regular server serves customers with a low-fare ticket, while the extra server gives priority to the business-class and first-class customers, but helps checking regular customers, whenever the priority line is empty. 1

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Section: Discrete-time Modelsmentioning

confidence: 99%

Analysis of a discrete-time single-server queue with an occasional extra server

Bruneel

Wittevrongel

2017

Performance Evaluation

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“…1 such that .k/ i that are distributed as B .k/ . We then obtain, for an appropriate x 0 , the following lower bound for the probability (28) [where we also use Equation (29)]:…”

Section: Proof Let At the Instant T Bmentioning

confidence: 99%

Maximal flow‐level stability of best‐rate schedulers in heterogeneous wireless systems

Jacko

Morozov

Potakhina

et al. 2015

Trans. Emerging Tel. Tech.

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We investigate flow‐level stability of schedulers in parallel‐service wireless systems, which is important for maximizing the base station's capacity to serve the heterogeneous flows that are within the base station's power range. We model such a system as a multi‐class queueing system with multiple preemptive servers, in which flows of different classes randomly arrive and depart once their flow is completed. The channel condition of a flow varies randomly over time because of fading and mobility. The evolution of the channel condition is assumed to be Markovian and class dependent. We focus on a general family of the best‐rate schedulers that, whenever possible, serve flows that are in the channel condition corresponding to the highest achievable class‐dependent transmission rate (i.e. the best rate). We prove under mild assumptions that any best‐rate scheduler achieves maximal stability, that is, stabilizes the system whenever possible, in all systems with generally distributed class‐dependent arrivals and flow sizes. Copyright © 2015 John Wiley & Sons, Ltd.

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“…Later on, Zhang et al [18] studied the noncooperative and cooperative strategies of customers in the context of a retrial queueing model where the server is subject to vacations, breakdowns and repairs. However, although more attention has been paid on the analysis of discrete time queueing models with unreliable server, such as Atencia and Moreno [1], Bruneel [2], Fiems et al [5][6], Moreno [9], Morozov et al [10], Vinck and Bruneel [14], Wang and Zhang [16][17], among others, it is found that there is no work concerning the equilibrium balking behavior in the discrete-time queues with server breakdowns and repairs.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium Balking Strategies in the Geo/Geo/1 Queues with Server Breakdowns and Repairs

Yang

Wang

Zhao

2014

Quality Technology & Quantitative Management

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______________________________________________________________________Abstract: This paper studies the equilibrium behavior of customers in the Geo/Geo/1 queue with server breakdowns and repairs. The reliability factor is introduced into the economic analysis and customers' behavior analysis of the discrete-time queues. The server is subject to breakdowns and repairs when providing service to customers. Upon arrival, the customers decide for themselves whether to enter the system or balk based on a natural reward-cost structure and available information on hand. The behavior of customers is investigated with various levels of information regarding three states of server and the system queue length. By deriving and solving a set of system difference equations, we obtain the stationary distribution of the system and the mean sojourn time of an arriving customer. Equilibrium strategies for the customers under different levels of information are derived and the behaviors of the customers under these strategies are investigated. Finally we illustrate the effect of several parameters on the equilibrium behavior via numerical experiments.

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Stability analysis of multiserver discrete-time queueing systems with renewal-type server interruptions

Cited by 22 publications

References 35 publications

Analysis of a discrete-time single-server queue with an occasional extra server

Analysis of a discrete-time single-server queue with an occasional extra server

Maximal flow‐level stability of best‐rate schedulers in heterogeneous wireless systems

Equilibrium Balking Strategies in the Geo/Geo/1 Queues with Server Breakdowns and Repairs

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