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Article Excerpt
We study stochastic tree fluid networks driven by a multidimensional Levy 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 Levy 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 Levy 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.
Key words: n-dimensional Levy process; splitting time; quasi-product form; multidimensional Skorokhod problem; Levy-driven fluid network; tree fluid network; buffer content; busy period; idle period
1. Introduction. Prompted by a series of papers by Kella and Whitt (Kella [20, 22], Kella and Whitt [25, 26]), there has been considerable interest in multidimensional generalizations of the classical storage model with nondecreasing Levy input and constant release rate (Prabhu [34, Ch. 4]). In the resulting networks, often called stochastic fluid networks, the input into the buffers is governed by a multidimensional Levy process. Recently, motivated by work of Harrison and Williams on diffusion approximations (Harrison and Williams [16, 17]), the presence of product forms has been investigated (Kella [21, 23], Konstantopoulos et al. [28], Piera et al. [33]). Recall that the stationary buffer-content vector has a product form if it has independent components, meaning that the distribution of this vector is a product of the marginal distributions.
The results in these papers show that, apart from trivial cases, the stationary buffer-content vector of stochastic fluid networks never has a product form. Despite this "negative" result, we show that it may still be possible to express the joint distribution of the buffer content in terms of the marginal distributions. This is most evident in the Laplace domain. For certain tandem queues, for instance, the Laplace transform is a product that cannot be "separated"; we then say that the buffer-content vector has a quasi-product form.
In the literature on stochastic fluid networks, there has been a focus on the stationary buffer-content vector W or one of its components. Here, we are also interested in the stationary distribution of vector of ages of the busy periods B and idle periods I. The age of a busy (or idle) period is the amount of time that the buffer content has been positive (or zero) without being zero (positive). Knowing these, it is also possible to find the distribution of the remaining length of the busy (or idle) period and the total length of these periods.
We are interested in W, B, and I for a class of Levy-driven fluid networks with a tree structure, which we therefore call tree fluid networks. Our analysis of these networks relies on a detailed study of a related multidimensional Skorokhod reflection problem (see, e.g., Robert [35]). Using its explicit solution, we relate the triplet of vectors (W, B, I) to the fluctuations of a multidimensional Levy process X. We also prove that the stationary distribution of the buffer-content vector is unique.
Since our analysis of fluid tree networks is based on fluctuations of the process X, this paper also contributes to fluctuation theory for multidimensional Levy processes. Supposing that each of the components of X drifts to -[infinity], we write [bar.X] for the (vector of) componentwise maximums of X, G for the corresponding epochs at which they are attained, and H for the beginning of the first last-passage excursion. Under a certain independence assumption, if the components of G are "ordered," we express the Laplace transform of ([bar.X], G) in terms of the transforms of the marginals ([[bar.X].sub.j], [G.sub.j]). Since [X.sub.j] is a real-valued Levy process, the Laplace transform of ([[bar.X].sub.j], [G.sub.j]) is known if [X.sub.j] has one-sided jumps; see, for instance, Bertoin [4, Thm. VII.4].
We also examine the distribution of H under the measure [P.sup.[down arrow].sub.k], which is the law of X given that the process [X.sub.k] stays nonpositive. There exists a vast body of literature on (one-dimensional) Levy processes conditioned to stay nonpositive (or nonnegative); see the recent paper by Chaumont and Doney [5] for references. Under the measure [P.sup.[down arrow].sub.k], we also find the transform of ([bar.X], G). As a special case, we establish the Laplace transform of the maximum of a Levy process conditioned to stay below a subordinator, such as a (deterministic) positive-drift process.
By exploiting the solution of the aforementioned Skorokhod problem, the results that we obtain for the process X can be cast immediately into the fluid-network setting. For instance, the knowledge of ([bar.X], G) allows us to derive the Laplace transform of the stationary distribution of (W, B) in a tandem network and a priority system if there are only positive jumps, allowing Brownian input at the "root" station. That is, we characterize the joint law of the buffer-content vector and the busy-period vector. With the [P.sup.[down arrow].sub.k]-distribution of H, we establish the transform of the idle-period vector I for a special tandem network. Our formulas generalize all explicit results for tandem fluid networks known to date (in the Laplace domain), such as those obtained by Kella [20] and more recently by Debicki et al. [6]. Most notably, quasi-products appear in our formulas, even for idle periods.
To derive our results, we make use of the notion of splitting times. These essentially allow us to reduce the problem to the one-dimensional case. For real-valued Markov processes, splitting times have been introduced by Jacobsen [18]. Splitting times decompose ("split") a sample path of a Markov process into two independent pieces. A full description of the process before and after the splitting time can be given. However, since the splitting time is not necessarily a stopping time, the law of the second piece may differ from the original law of the Markov process (refer to Millar [31, 32] for further details and to Kersting and Memisoglu [27] for a recent contribution).
The idea to use splitting times in the context of stochastic networks is novel. The known results to date are obtained with Ito's formula (Konstantopoulos et al. [28]), a closely related martingale (Kella and Whitt [26]), or differential equations (Piera et al. [33]). Intuitively, these approaches all exploit a certain harmonicity. However, the results of Kyprianou and Palmowski [29] already indicate a relation between these approaches and splitting. Splitting has the advantage that it is insightful and that proofs are short. Moreover, it can also be used for studying more complicated systems (Dicker and Mandjes [8]).
This paper is essentially divided into two parts. In the first part, consisting of [section][section] 2-4, we analyze the fluctuations of an n-dimensional Levy processes X. The notion of splitting times is formalized in [section] 2. These splitting times are first used to study the distribution of ([bar.X], G) in [section] 3, and then to analyze the distribution of H under [P.sup.[down arrow].sub.k] in [section] 4. The second part of this paper deals with fluid networks. Section 5 ties these networks to fluctuations of X, so that the theory of the first part can be applied in [section] 6. Finally, in Appendix A, we derive some results for compound Poisson processes with negative drift. They are used in [section] 4.
2. Splitting times. This paper relies on the application of splitting times to a multidimensional LEvy process. After splitting times have been introduced, we study splitting at the maximum ([section] 2.1) and splitting at a last-passage excursion ([section] 2.2).
Throughout, let X = ([X.sub.1], ..., [X.sub.n])' be an n-dimensional Levy process, that is, a cadlag process with stationary, independent increments such that X(0) = [member of] [P.sup.n]. Without loss of generality, as in Bertoin [4], we work with the canonical measurable space ([OMEGA], F) = (D([0, [infinity]), [R.sup.d] [union] {[partial derivative]}), B), where B is the Borel [sigma]-field generated by the Skorokhod topology, and [partial derivative] is an isolated point that serves as a cemetery state. In particular, X is the coordinate process. Unless otherwise stated, "almost surely" refers to P. All vectors are column vectors.
The following assumption is used extensively throughout this paper:
D [X.sub.k](t) [right arrow] - [infinity] almost surely, for every k.
We emphasize that a dependence between components is allowed. In the sequel, [[bar.X].sub.k](t) (or [[X.sub.].sub.k](t)) is shorthand for [sup.sub.s[less than or equal to]t] [[X.bar].sub.k](s)). Due to D, [[bar.X].sub.k] := [[bar.X].sub.k] ([infinity]) is well-defined and almost surely finite for every k. Furthermore, we write [bar.X] = ([[bar.X].sub.1], ..., [[bar.X].sub.n])
The following two definitions are key to further analysis. The second definition is closely related to the first, but somewhat more care is needed on a technical level. Intuitively, for the purposes of this paper, there is no need to distinguish the two definitions.
DEFINITION 2.1. We say that a random time T is a splitting time for X under P if the two processes {X(t): [less than or equal to] t [less or equal to T]} and {X(T + t) - X(T): t [greater than or equal to] 0} are independent under P. We say that T is a splitting time from the left for X under [P if the two processes {X(t): [less than or equal to] t < T} and {X(T + t) - X(T-): t [greater than or equal to] 0} are independent under P.
Note that if X is a Levy process under P with respect to some filtration F that includes the natural filtration, any F-stopping time [tau] is a splitting time for X under P. In fact, the Levy assumption implies that {X([tau] + t) - X([tau]): t [greater than or equal to] 0} is not only independent of {X(t): [less than or equal to] t [less than or equal to] [tau]}, but also that it has the same distribution as {x(t): t [greater than or equal to] 0).
We need some notions related to the initial behavior of X. For k = 1, ..., n, set [[bar.R].sub.k] = inf{t > 0: [X.sub.k] (t) = [[bar.X].sub.k] (t)}. Since {[[bar.X].sub.k] (t) - [X.sub.k] (t): t [greater than or equal to] 0} is a Markov process under P with respect to the filtration generated by X (see Proposition VI.1 of Bertoin [4]), the Blumenthal zero-one law shows that either [[bar.R].sub.k] > almost surely (0 is then called irregular for {[[bar.X].sub.k](t) - [X.sub.k](t): t [greater than or equal to] 0}) or [[bar.R].sub.k] = almost surely (0 is then called regular for {[[bar.X].sub.k] (t) - [X.sub.k] (t): t [greater than or equal to] 0}). We also set [[R.bar].sub.k] = inf{t > 0: [X.sub.k](t) = [X.sub.k](t)}, and define regularity of for {[X.sub.k] (t) - [[X.bar].sub.k](t): t [greater than or equal to] 0} similarly as for {[[bar.X].sub.k] (t) - [X.sub.k] (t): t [greater than or equal to] 0}. If [[bar.R].sub.k] = almost surely, we introduce
[[bar.S].sub.k] = [[bar.S].sup.x.sub.k] := inf{t > 0: [X.sub.k] (t) [not equal to] [[bar.X].sub.k] (t)}.
Again, either [[bar.S].sub.k] = almost surely (0 is then called an instantaneous point for {[X.sub.k](t) - [X.sub.k](t): t > 0}) or [[bar.S].sub.k] > almost surely (0 is then called a holding point for {[[bar.X].sub.k] (t) - [X.sub.k](t): t [greater than or equal to]...
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