Analysis of stochastic fluid queues driven by local-time processes

Abstract

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.

On the excursions of reflected local-time processes and stochastic fluid queues

“…In this paper we shall derive an expression for the joint law of three random variables: the duration of a busy period, the duration of an idle period, and the maximum of Q over a busy period. The result is formulated as Theorem 1 in Section 3: (16) therein is new and extends some of the results of [8]. The result is expressed in terms of the process , which is in turn a function of the underlying Markov process X.…”

“…It is worth recalling [8] that if we consider (3) as a fixed point equation for Q then the process defined by (2) is the unique stationary and ergodic solution of (3). A typical sample path of Q is depicted in Figure 2 in Section 2.…”

“…In this respect, the process Q is thought of as the workload in a stochastic fluid queue. Amongst other things in [8], [9], and [13], expressions are derived for the marginal distribution of Q and the Laplace transform of the duration of typical idle and busy periods. In this paper we shall derive an expression for the joint law of three random variables: the duration of a busy period, the duration of an idle period, and the maximum of Q over a busy period.…”

“…In this paper we consider a model for which special cases have been studied in [8], [9], and [11], consisting of a priority queueing system where the high priority class is a stochastic process denoted by X and the low priority class is a process denoted by Q. Our interest is in studying Q.…”

“…It turns out that the problem can be expressed in general terms via an underlying strong Markov process X and its local time L at 0, a process which is considered as an input to Q. For further motivation to the physical problem, we refer the reader to [8], [9], and [11]. In what follows, we express the problem in mathematical terms.…”