Continuous feedback fluid queues
Introduction
In the area of modern telecommunication systems fluid queues are often used as burst scale models for multiplexers, see e.g. [15, Ch. 17], [18]. In such models the content process of a fluid buffer changes at a rate determined by some autonomous modulating stochastic process . The net fluid input rate at time t is then given by at times when . Of particular importance are Markov-modulated fluid models in which the background process is a Markov process, see e.g. [2], [11].
More general models, known as feedback fluid queues, were introduced in [1], [16]. Here, the behavior of the fluid buffer content is determined by the background process as before, but in turn the behavior of the background process now depends on the current buffer level. Loosely speaking, the background process behaves as a continuous-time Markov chain, but its ‘generator matrix’ now depends on the current buffer level y. Due to this feedback, the background process is actually no longer a Markov process.
Feedback fluid queues can be useful for studying certain production systems and for modeling modern telecommunication networks in which the network and the sources interact. For instance, the interaction between one or two TCP sources (i.e., traffic sources that use the transmission control protocol, as currently deployed in the Internet) and some buffer in the network was analyzed in [8]. In [12], [13], feedback fluid queues were used to study feedback schemes in access networks. These models have the feature that the matrix is a piecewise constant function of the buffer level y. Moreover, the fluid rates may also depend on y as piecewise constant functions . In the current paper we analyze a more general feedback model in which both and the fluid rates depend continuously on y. Clearly this explains the title of our paper, although a second interpretation is also possible, namely that the background process experiences an influence from the buffer continuously over time (i.e. in each arbitrarily small interval of time).
Apart from the relation with Markov-modulated and feedback fluid queues, our model also has a relation with extensions of the classical storage process, with state-dependent output. In the classical storage model, the input is a compound Poisson process, and the release rate (i.e. the rate at which the buffer is depleted) is constant, see [14]. Early extensions of this storage process are considered in [5], [9], and references therein, in which the release rate is state-dependent; in fact it is a strictly positive piecewise continuous function of the current buffer content. Another paper worth mentioning in this context is [3], where not only the release rate but also the arrival rate of customers is state-dependent.
In all of the above models with state-dependent release rate, the input process is allowed to have jumps. Also in the context of fluid queues, where the input process is gradual, some models were studied where the input and/or output rates are state-dependent. The authors of [7] allow the net input rates, i.e., the difference between the input and the output rates, to be piecewise constant functions of the buffer content. In [10] the case is solved in which these rates are piecewise continuous functions. In fact these two models are special cases of the feedback models in [12] and the current paper, respectively, since here also the fluid rates depend on the buffer content, while unlike [7], [10] also the behavior of the background process itself is influenced by the buffer content. Finally we mention the closely related (but independent) work in [4] where a generator approach is used to analyze a similar feedback fluid queue as here. However, the analysis is restricted to a background process with two states. Another difference is that in our case the buffer size is finite, while in [4] the buffer size is infinite.
We did not yet explain in detail what we mean by stating that the background process behaves ‘as a Markov chain, with a level-dependent generator matrix ’. A precise description of our model is given in Section 2, where also some technical assumptions are mentioned. In Section 3 we derive the Kolmogorov forward equations for the joint Markov process . We do this by carefully following the infinitesimal approach, also employed in standard Markov-modulated fluid models. The technicalities needed for a rigorous proof are deferred to the Appendix. In the final Section 4 we present the set of differential equations and boundary conditions that describe the stationary distribution, as well as a closed form solution for the case in which the background process has two states.
Section snippets
Model and preliminaries
In this section we explain the fluid model of interest more precisely. We also state some assumptions on the functions involved and introduce some notation.
Derivation of the Kolmogorov forward equations
In this section we derive the Kolmogorov forward equations for the joint process . First we summarize the derivation from [10] for the situation in which Q, but not necessarily R, is constant as a function of the buffer content. This derivation is completely analogous to that of the standard case, in which R and Q are fixed. Then we focus on the case in which is a non-constant function of y. Although at first sight it may seem obvious that again a similar system of forward
Stationary behavior
In this section we present the differential equations for the stationary distribution and give the closed form solution for the case in which the source process has two states. Theorem 4.1 Assuming it exists, the stationary distribution for the process satisfies the following system of (ordinary) differential and algebraic equations:with boundary conditionsand normalization condition
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