Two-buffer fluid models with multiple ON-OFF inputs and threshold assistance Guy Latouche Université Libre de Bruxelles Département d’Informatique 1050 Brussels, Belgium latouche@ulb.ac.be Giang T. Nguyen Université Libre de Bruxelles Département d’Informatique 1050 Brussels, Belgium giang.nguyen@ulb.ac.be Zbigniew Palmowski University of Wroclaw Mathematical Institute 50-384 Wroclaw, Poland zbigniew.palmowski@gmail.com ABSTRACT We consider a two-buffer fluid model with N ON-OFF in- puts and threshold assistance, which is an extension of the same model with N = 1 in [18]. While the rates of change of both buffers are piecewise constant and dependent on the underlying Markovian phase of the model, the rates of change for Buffer 2 are also dependent on the specific level of Buffer 1. This is because both buffers share a fixed out- put capacity, the precise proportion of which depends on Buffer 1. The generalization of the number of ON-OFF in- puts necessitates slight modifications in the original rules of output-capacity sharing from [18], and considerably compli- cates both the theoretical analysis and numerical computa- tion of various performance measures. Here, we give a short explanation on how to derive the marginal probability distribution of Buffer 1, and bounds for that of Buffer 2. In an upcoming paper, we describe the procedures in more details. Furthermore, restricting Buffer 1 to a finite size, we determine its marginal probability distri- bution in the specific case of N = 1, thus providing numer- ical comparisons to the corresponding results in [18] where Buffer 1 is assumed to be infinite. We also demonstrate how this imposed restriction effects the bounds of marginal probabilities for Buffer 2. Categories and Subject Descriptors I.6.5 [Simulation and Modelling]: Model development; G.3 [Probability and Statistics]: [queueing theory, stochas- tic processes] General Terms Performance, Measurement 1. INTRODUCTION Stochastic fluid models have a wide range of real-life ap- plications, such as industrial and computer engineering, ac- tuarial science, environmental modeling and telecommuni- cations. A Markov-modulated single-buffer fluid model is a two-dimensional Markov process {X(t),ϕ(t): t ∈ R + }, where X(t) is the continuous level of the buffer, and ϕ(t) is the discrete phase of the underlying irreducible Markov chain that governs the rates of change. A practical and well- studied case is piecewise constant rates: the fluid is assumed to have a constant rate ci when ϕ(t)= i, for i in a finite state space S . The traditional approach for obtaining per- formance measures of Markov-modulated single-buffer fluids with piecewise constant rates is to use spectral analysis (see, among others, [17, 3, 19, 23, 12]). Over the last two decades, matrix analytic methods have gained a lot of attention as an alternative and algorithmically effective approach for an- alyzing these standard fluids (see, for instance, [22, 21, 1, 2, 6, 10, 5, 7, 11, 8]). In this paper, we consider a two-buffer fluid model {X(t),Y (t), ϕ1(t),ϕ2(t): t ∈ R + }, where X(t) ≥ 0 and Y (t) ≥ 0 repre- sent the levels of Buffers 1 and 2, respectively. At a given time t ≥ 0, the rates of change of Buffer 1 depend only on the underlying Markovian phase ϕ1(t); however, the rates of change of Buffer 2 depend on both ϕ2(t) and X(t). This is because while each buffer receives its own input sources, both buffers share a fixed output capacity c, in proportion dependent on the level of Buffer 1. More specifically, Buffer j receives N ON-OFF input sources, each has exponentially distributed ON- and OFF- intervals at corresponding rates αj and βj , and continuously generates fluid at rate Rj dur- ing ON- intervals, for j =1, 2. When the fluid level X(t) of Buffer 1 is above a certain threshold x ∗ > 0, Buffer 1 is al- located the total shared output capacity c, leaving Buffer 2 without any; when 0 <X(t) <x ∗ , Buffer j has output capacity cj , c1 + c2 = c; and when X(t) = 0, Buffer 1 has output capacity min{iR1,c1}, and Buffer 2 c− min{iR1,c1}, where i is the number of inputs of Buffer 1 being on at the time t. This theoretical model is an extension of the same model with N = 1 in [18], where the reader can find a compre- hensive account of practical applications in communication networks. We note that when X(t) = 0, the rule for output- capacity allocation in our general N ON-OFF input model differs to that in the single ON-OFF input model in [18], which is to allocate the total capacity c to Buffer 2. The to- tality rule is logical for the single ON-OFF input: when there is only one ON-OFF input for each buffer, Buffer 1 is empty only when its input is off; in that case, Buffer 2 can receive the whole output capacity c, until the moment the input of Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SMCTOOLS 2011, May 16, Paris, France Copyright © 2011 ICST 978-1-936968-09-1 DOI 10.4108/icst.valuetools.2011.245848 456