Bounds for a discrete-time multi-server queue with an application to cable networks

Denteneer, T.J.J.; Leeuwaarden, Johan S. H. van; Resing, Jacques

doi:10.1016/s1388-3437(03)80209-3

Cited by 8 publications

(7 citation statements)

References 2 publications

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“…The quantity α n can be interpreted as the number of slots in frame n that are used for handling requests, and the sum in (1) equals the total number of new data packets for which transmission is granted. For more background on this application, we refer to [2,3].…”

Section: Motivationmentioning

confidence: 99%

“…, it is readily seen that the probability transition matrix is an M/G/1-type stochastic matrix, see [10]. Hence, the ergodicity condition is the usual condition stating that the average drift should be negative, which in this case reduces to inequality (3). For an alternative proof of (3), based on Foster's criterion, the reader is referred to [2].…”

Section: Lemma 1 the Markov Chainmentioning

confidence: 99%

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The acquisition queue

2007

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We propose a new queueing model named the acquisition queue. It differs from conventional queueing models in that the server not only serves customers, but also performs acquisition of new customers. The server has to divide its energy between both activities. The number of newly acquired customers is uncertain, and the effect of the server's acquisition efforts can only be seen after some fixed time period δ (delay).The acquisition queue constitutes a (δ + 1)-dimensional Markov chain. The limiting queue length distribution is derived in terms of its probability generating function, and an exact expression for the mean queue length is given. For large values of δ the numerical procedures needed for calculating the mean queue length become computationally cumbersome. We therefore complement the exact expression with a fluid approximation.One of the key features of the acquisition queue is that the server performs more acquisition when the queue is small. Together with the delay, this causes the queue length process to show a strongly cyclic behavior. We propose and investigate several ways of planning the acquisition efforts. In particular, we propose an acquisition scheme that is designed

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Section: Motivationmentioning

confidence: 99%

Section: Lemma 1 the Markov Chainmentioning

confidence: 99%

The acquisition queue

2007

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“…as shown in Denteneer et al [5]. Using the same notation as for the fixed boundary model, the behavior of the backlog throughout a frame follows from…”

Section: Backlog In Flexible Boundary Modelmentioning

confidence: 99%

“…In Denteneer et al [5] bounds on the mean queue length and mean packet delay have been derived for the fixed boundary model. In Sala et al [11] the flexible boundary model has been investigated through simulation, which shows that inducing arrival slots at the beginning of each frame might positively influence the system's performance.…”

Section: Introductionmentioning

confidence: 99%

A discrete-time queueing model with periodically scheduled arrival and departure slots

Leeuwaarden¹,

Denteneer

Resing

2006

Performance Evaluation

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We consider a time-slotted queueing model where each time slot can either be an arrival slot, in which new packets arrive, or a departure slot, in which packets are transmitted and hence depart from the queue. The slot scheduling strategy we consider describes periodically, and for a fixed number of time slots, which slots are arrival and departure slots. We consider a static and a dynamic strategy. For both strategies, we obtain expressions for the probability generating function of the steady-state queue length and the packet delay. The model is motivated by cable-access networks, which are often regulated by a request-grant procedure in which actual data transmission is preceded by a reservation procedure. Time slots can then either be used for reservation or for data transmission.

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“…We consider a discrete-time queue (in queueing theory language a multi-server queue, see [6]), defined by the recursion X n+1 = max{X n − s, 0} + A n , n ∈ Z .…”

Section: Introductionmentioning

confidence: 99%

A Discrete Queue, Fourier Sampling on Szegö Curves and Spitzer Formulas

Janssen

Leeuwaarden²

2005

Int. J. Wavelets Multiresolut Inf. Process.

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We consider a discrete-time multi-server queue for which the moments of the stationary queue length can be expressed in terms of series over the zeros in the closed unit disk of a queue-specific characteristic function. In many important cases these zeros can be considered to be located on a queue-specific curve, called generalized Szegö curve. By adopting a special parametrization of these Szegö curves, the relevant zeros occur as equidistant samples of a 2π-periodic function whose Fourier coefficients can be determined analytically. Thus the series occurring in the expressions for the moments can be written as Fourier aliasing series with terms given in analytic form. This gives rise to formulas for e.g. the mean and variance of the queue length that are reminiscent of Spitzer's identity for the moment generating function of the steady-state waiting time for a G/G/1 queue. Indeed, by considering the queue under investigation as a G/G/1 queue, the new formulas for the mean and variance also follow from Spitzer's identity. The approach in this paper can also be used to compute the probability distribution function of the queue length in analytic form.

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Bounds for a discrete-time multi-server queue with an application to cable networks

Cited by 8 publications

References 2 publications

The acquisition queue

The acquisition queue

A discrete-time queueing model with periodically scheduled arrival and departure slots

A Discrete Queue, Fourier Sampling on Szegö Curves and Spitzer Formulas

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