Subexponential asymptotics of a Markov-modulated random walk with queueing applications

Let {(X-n, J(n))} be a stationary Markov-modulated random walk on R x E (E is finite), defined by its probability transition matrix measure F = {F-ij}, F-ij(B) = P[X-1 is an element of B, J(1) = j \ J(0) = i], B is an element of B(R), i, j is an element of E. if F-ij([x, infinity))/(1 - H(x)) --> W-ij is an element of [0, infinity), as x --> infinity, for some long-tailed distribution function H, then the ascending ladder heights matrix distribution G(+)(x) (right Wiener-Hopf factor) has long-tailed asymptotics. if EXn < 0, at least one W-ij > 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by P[sup(n greater than or equal to 0) S-n > x] --> (-EXn)(-1) integral(x)(infinity) P[X-n > u] du as x --> infinity, where S-n = Sigma(1)(n) X-k, S-0 = 0. Two general queueing applications of this result are given. First, if the same asymptotic conditions are imposed on a Markov-modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI/GI/1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.