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

Let {(Xn, Jn )} be a stationary Markov-modulated random walk on × E (E is finite), defined by its probability transition matrix measure F ={ F ij }, F ij (B) = [X1 ∈ B, J1 = j | J0 = i], B ∈ ( ), i, j ∈ E .I fF ij ([x, ∞))/(1 − H (x)) → W ij ∈ [0, ∞) ,a sx →∞ , 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 Xn 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 [supn≥0 Sn > x ]→ (− Xn ) −1 ∞ x [Xn > u] du as x →∞ ,w hereSn = n Xk , S0 = 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.