Parallel and Tandem Fluid Networks with Dependent Levy Inputs

Kella, Offer

doi:10.1214/aoap/1177005358

Cited by 54 publications

(79 citation statements)

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“…In particular, if J(t) = (J 1 (t), 0) ′ , P = (p ij ), with p 12 = 1 and zeroes elsewhere, if one chooses β 1 = β 2 = 0 and ω 1 = 0 in Theorem 6.1, then one obtains Theorem 3.2 of Dȩbicki, Mandjes and van Uitert [6]. Additionally, if one chooses β 1 = β 2 = 0 and supposes that J 1 is a subordinator, we recover the results of Kella [20].…”

Section: Generalitiessupporting

confidence: 74%

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Quasi-Product Forms for Lévy-Driven Fluid Networks

2007

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We study stochastic tree fluid networks driven by a multidimensional Lévy process. We are interested in (the joint distribution of) the steady-state content in each of the buffers, the busy periods, and the idle periods. To investigate these fluid networks, we relate the above three quantities to fluctuations of the input Lévy process by solving a multidimensional Skorokhod reflection problem. This leads to the analysis of the distribution of the componentwise maximums, the corresponding epochs at which they are attained, and the beginning of the first last-passage excursion. Using the notion of splitting times, we are able to find their Laplace transforms. It turns out that, if the components of the Lévy process are 'ordered', the Laplace transform has a so-called quasi-product form.The theory is illustrated by working out special cases, such as tandem networks and priority queues.

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Section: Generalitiessupporting

confidence: 74%

“…Prompted by a series of papers by Kella and Whitt [20,22,25,26], there has been a considerable interest in multidimensional generalizations of the classical storage model with nondecreasing Lévy input and constant release rate [34,Ch. 4].…”

Section: Introductionmentioning

confidence: 99%

Quasi-Product Forms for Lévy-Driven Fluid Networks

2007

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“…Relation with two coupled queues in tandem There is also a relation between the coupled processor model with two queues described above and two tandem queues which are coupled. The relation is similar to the one between the systems without coupling, which was pointed out for Lévy input in Kella [5] (see also [1]). …”

Section: Conclusion and Final Remarkssupporting

confidence: 74%

“…The Lindley-type recursion for queue 2 is similar to (5). In the sequel, we will derive the recursion for queue 1.…”

Section: The K-dimensional Modelmentioning

confidence: 99%

A coupled processor model with simultaneous arrivals and ordered service requirements

Badila

Resing

2015

Queueing Syst

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We study a coupled processor model with simultaneous arrivals. Under the assumption of ordered input, the Laplace-Stieltjes transform of the joint stationary amount of work in the system can be explicitly calculated by relating it to the amount of work in a parallel queueing system without coupling. This relation is first exploited for the system with two coupled queues. The method is then extended to higher dimensions.

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“…[13] for such a case: the process I 2 of [13, Equation (2.8)] is closely related to our process Q, but in that paper, the process I 2 is not the per se object of study. (See also [11] for a tandem fluid network case and, of course, the rather rich literature of Brownian fluid networks; see, e.g. [8] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Analysis of stochastic fluid queues driven by local-time processes

Konstantopoulos¹,

Kyprianou

Sirvio

et al. 2008

Adv. Appl. Probab.

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We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a reflected Lévy process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is typically (but not necessarily) singular with respect to the Lebesgue measure, a situation which, in view of the nonsmooth or bursty nature of several types of Internet traffic, is nowadays quite realistic. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period, and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a Lévy process (a subordinator), hence making the theory of Lévy processes applicable. Another important ingredient in our approach is the use of Palm calculus for stationary random point processes and measures.

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Parallel and Tandem Fluid Networks with Dependent Levy Inputs

Cited by 54 publications

References 0 publications

Quasi-Product Forms for Lévy-Driven Fluid Networks

Quasi-Product Forms for Lévy-Driven Fluid Networks

A coupled processor model with simultaneous arrivals and ordered service requirements

Analysis of stochastic fluid queues driven by local-time processes

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