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Article Excerpt
We consider a Levy process that is reflected at and at K > 0. The reflected process is obtained by adding the difference between the local time at and the local time at K to the sum of the feeding Levy process and an initial condition. We define the loss rate to be the expectation of the local time at K at time 1 under stationary conditions. The main result of the paper is the identification of the loss rate in terms of the stationary measure of the reflected process and the characteristic triplet of the Levy process. We also derive asymptotics of the loss rate as K [right arrow] when the drift of the feeding process is negative and the Levy measure is light tailed. Finally, we extend the results for L6vy processes to hold for Markov-modulated Levy processes.
Key words: Levy process; reflection; Skorokhod problem; local time; loss rate; light tail; martingale; Lundberg equation; Markov-modulated Levy process; Cramer-Lundberg approximation; asymptotics
1. Introduction. In this paper, we investigate a Levy process {[X.sub.t]} reflected at and at K > 0. We construct the reflected process {[V.sup.K.sub.t]} as
[V.sup.K.sub.t] = [V.sup.K.sub.0] + [X.sub.t] + [L.sup.0.sub.t] - [L.sup.K.sub.t], (1)
where [L.sup.0.sub.t] and V.sup.K.sub.t are the local times at the respective boundaries, given as the solutions of a Skorokhod problem (see [section] 2 for details). Among possible applications, we mention finite capacity dam models, buffer systems and queueing systems; see, e.g., Asmussen [2], Bekker and Zwart [5], Cooper et al. [11], Jelenkovic [12], Moran [18], and Stadje [24]; and various telecommunication models, see, e.g., Jelenkovic [12], Kim and Shroff [15], and Zwart [26].
A first quantity of interest is of course the stationary distribution [[pi].sub.K]. There are various more-or-less independent studies around; see, in particular, Borovkov [9], Cooper et al. [11], Lindley [17], Siegmund [23], and Stadje [24]. The simplest representation appears to be that of Lindley [17] and of Siegmund [23], stating that
[[bar.[pi]].sub.K](y) = [[pi].sub.K]([y, K]) = P([X.sub.[tau][y-K, y]) [greater than or equal to] y), [less than or equal to] y [less than or equal to] K,
where [tau][u, v) = inf{t [greater than or equal to] 0: [X.sub.t] [not member of] [u, v)}, u [less than or equal to] [less than or equal to] v, and this is the one we will use. A short self-contained derivation is given in Asmussen [2, pp. 393-394]; see also Asmussen [1], Asmussen and Sigman [3], Ryan and Sigman [21], and Asmussen [2, IX.4 and XIV.3]. In this paper we are concerned with the loss rate [l.sup.K], defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (2)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] refers to the stationary situation. (In the sequel we will most often avoid this notation and write E[V.sup.K.sup.t], E[L.sup.K.sub.t], etc. This should cause no confusion because [V.sup.K.sub.t] is always assumed to be stationary.) This can be interpreted as the overflow rate in a dam model and as the bit loss rate in (say) a finite data buffer. Due to this importance for applications, there is much literature studying the loss rate or similar quantities, e.g., Bekker and Zwart [5], Jelenkovic [12], Kim and Shroff [15], Moran [18], and Zwart [26].
The main result of this paper, Theorem 3.1, is an identification of the loss rate lK in terms of known characteristics of {[X.sub.t]}--more precisely, the Levy triplet (see below) and [[pi].sub.K]. It is worth noting three related problems with an easy solution:
(i) Discrete-time two-sided reflected random walks given by the recursion
[V.sup.K.sub.n+1] = min[K, max(0, [V.sup.K.sub.n] + [Y.sub.n)],
with [Y.sub.1], [Y.sub.2], ..., i.i.d. Here,
[l.sup.k] = [[integral].sup.K.sub.0] E[[x + [Y.sub.1] - K].sup.+] [[pi].sub.K](dx).
(ii) The one-sided (at 0) reflected Levy process [V.sup.[infinity].sub.t] = [V.sup.[infinity].sub.0] + [X.sub.t] + [L.sup.0.sub.t]. Here, clearly in stationarity [L.sup.0.sub.t] has to balance the drift of [X.sub.t]. This simple conservation law immediately gives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
(iii) Certain Levy processes of a special structure, more precisely, such that one of [L.sup.0,c.sub.t], [L.sup.K,c.sub.t] (the continuous parts) vanishes. Here combinations of the arguments for the two previous cases easily yield an expression for [l.sup.k], as will be explained later.
However, the problem of identifying the loss rate [l.sup.k] appears nontrivial for a completely general Levy process with two-sided reflection. Our approach is to use martingale optional stopping to show that [l.sup.k] satisfies a certain equation. An additional nontrivial step of the analysis is the reduction of the resulting solution to a satisfying form, which essentially involves integrals with respect to [[pi].sub.K] and the Levy measure v as well as terms relating to the drift and a possible Brownian component. Note that the dependence on [[pi].sub.K] may appear disappointing at first sight, because [[pi].sub.K] itself is a function of the generating triplet. Even for the discrete-time case, we see, however, no way to get beyond this dependence, as is possible for one-sided reflection.
In [section] 4 we proceed by obtaining asymptotics for the loss rate [l.sup.k] as K [right arrow] [infinity], assuming negative drift so that [l.sup.k] [right arrow] 0, and light tails, which implies that the equation [kappa]([alpha]) = has a root [gamma] > (we exclude the cases where the sample paths of the process are decreasing, in which [l.sup.k] = 0), where [kappa] is the Levy exponent of {[X.sub.t]}; see [section] 2. Our result states that (we use the customary notation a(x) ~ b(x), x [right arrow] [infinity] to denote a(x)/b(x) [right arrow] 1, x [right arrow] [infinity]) [l.sup.k] ~ [De.sup.-yK], K [right arrow] [infinity], where the expression for D is in terms of quantities relating to Wiener-Hopf factorization and fluctuation theory for {[X.sub.t]}. This part of the paper can be seen as a continuous-time analogue of Pihlsgard [ 19] and as a strengthening of the logarithmic asymptotics of Kim and Shroff [15] to sharp asymptotics.
In [section] 5, we generalize the results of [section] 3 and [section] 4 to hold for a Markov-modulated Levy process.
Some important notation used in the paper: If t [??] [f.sub.t] is a function, we let [f.sub.t-] = [lim.sub.s[up arrow]t] [f.sub.s] and [f.sub.t] = [lim.sub.s[down arrow]t] [f.sub.s], provided that the limits exist. Also, we let [DELTA][f.sub.t] = [f.sub.t] - [f.sub.t-]. For a random variable (r.v.) Y and a measurable set A we sometimes use the notation (Y; A) to denote the r.v. Y(*)I(x [member of] A).
2. Preliminaries. In this section we give a brief background on Levy processes, included for easy reference. Standard references are Bertoin [7] and Sato [22]. We start with a probability space ([OMEGA], F, {[F.sub.t]}, P), where the filtration {[F.sub.t]} satisfies the usual conditions, i.e., it is augmented and right continuous. By a Levy process {[X.sub.t]} (with respect to {[F.sub.t]}) we understand a real-valued process with [X.sub.0] = and stationary independent increments, which is continuous in probability, i.e., [X.sub.t+s],...
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