Tail behaviour of the area under a random process, with applications to queueing systems, insurance and percolations

Rafa Kulik 0 1 Zbigniew Palmowski 0 1 0 Z. Palmowski ( ) Mathematical Institute, Wrocaw University , Pl. Grunwaldzki 2/4, 50-384 Wrocaw, Poland 1 R. Kulik Department of Mathematics and Statistics, University of Ottawa , 585 King Edward Av., K1N 6N5 Ottawa, ON, Canada The areas under the workload process and under the queueing process in a single-server queue over the busy period have many applications not only in queueing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases. In the past two decades an enormous amount of work on asymptotics for queueing systems has been done. The tail behaviour of steady-state queueing process {Q(t ), t 0}, the workload {W (t ), t 0} or the busy period in standard systems has been well understood in both light- and heavy-tailed case. Surprisingly, however, very little is known on tail behaviour of integral functionals of the form - If (T ) := f X(u) du, where {X(t ), t 0} is a stochastic process (typically, X = Q or X = W ), f is a deterministic function and T is either or deterministic (finite or infinite). Such integrals appear naturally in analysis of ATM. The reader is referred to references given in [9] and [21]. Recently, in [2], the authors connected mean bit rate in time varying M/M/1 queue with moments of the integral 0 Q(u) du in a corresponding standard M/M/1 system. However, applications of such integrals go beyond queueing systems. Let {S(t ), t 0} be a standard risk process. Integrals 0T 1{S(u)<0}S(u) du, where T is deterministic (i.e. integrated negative part of the risk process), are suggested in [8] as possible risk measures. Further extensions are given in a multivariate setting. Furthermore, as in [16], integrals 0 Q(u) du in Geo/Geo/1 queues and (as a limit) in M/M/1 systems have particular interpretation in compact percolations. Last but not least, if X is Lvy process, integrals 0 exp(X(u)) du have applications in financial mathematics; see [24]. Other applications are coming from actuarial science, where very often regulated processes are considered and integral functionals from a regulation random mechanism are investigated. 2 Subexponential asymptotics Consider a stable GI/GI/1 queue. Denote by {T , Ti , i 0} and {S, Si , i 0} two stationary i.i.d. and mutually independent sequences of interarrival and service times, respectively. Let T = 1/E[T ], S = 1/E[S] and = T /S . Let {Q(t ), t 0} be a stationary queueing process and = inf{t 0 : Q(t ) = 0} the corresponding busy period. We shall assume that the distribution F of service time S is subexponential (denoted as F S). The distribution G is subexponential when Conjecture 2.1 If F S, then Fig. 1 A typical behaviour of a heavy-tailed queueing process Similarly, for the workload process the heuristic is as follows. The most likely way for the area to be large is that one early big service time occurs and apart from this, everything in the cycle develops normally. Using LLN and ignoring random fluctuations, this leads to the conclusion that the workload goes to zero with negative rate (1 ). Thus the area exceeds level x iff the area of the triangle with the sides (1 ) and is greater than x, hence when which suggests the following conjecture. Conjecture 2.2 If F S, then W (u) du > x Statements (2) and (3) were proven in [21] and [9], respectively, under regularly varying assumption of the service time, that is F (x) = xL(x), where > 1, and L is slowly varying at infinity. Furthermore, in [21] one needs additionally that = 0 with > 0. The tail behaviour of the busy period can be identified in terms of F for a large subclass of S: see [5, 15] and [28]. In particular, using (5) in the regularly varying case, together with (2) and (3), yields exact asymptotics for area under queueing process and workload, respectively (see [21] and [9]). 3 Light-tailed asymptotics As in Sect. 2, we consider a stable GI/GI/1 queue. Here, we assume that the service time S is light-tailed, that is there exists > 0 such that E[exp( S)] < . Under the above assumptions for the queueing process, we have the following open problem: Open Problem 3.1 Find exact asymptotics of Q(u) du > x . for some constants and C. We suggest the above asymptotics believing that it should be the same as for the M/M/1 queue, which was found in [13] under two conjectures on p. 391 and is in the following form: In the proof the authors used the Laplace transform method. We are not aware of any probabilistic proof of this result, and we do not know if Conjectures 1 and 2 in [13] hold true. Unfortunately, we have not managed to produce any heuristic for this result either. The idea of the piecewise linear most likely trajectory seems to produce the wrong expression. In particular, define the new probability measure P by where Fn (...truncated)