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

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Abstract
Let {(Xn,Jn)} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {Fij}, Fij(B) = ℙ[X1 ∈ B, J1 = j | J0 = i], B ∈ B(ℝ), i, j ∈ E. If Fij([x,∞))/(1-H(x)) → Wij ∈ [0,∞), as x → ∞, 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 Wij > 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[supn≥0Sn > x] → (−𝔼Xn)−1 ∫x∞ ℙ[Xn > u]du as x → ∞, where Sn = ∑1nXk, 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.

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