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 ∈ B, J
1 = j | J
0 = i], B ∈
B
(ℝ), i, j ∈ E. If F
ij
([x,∞))/(1-H(x)) → W
ij
∈ [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 𝔼X
n
< 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≥0
S
n
> x] → (−𝔼X
n
)−1 ∫
x
∞ ℙ[X
n
> u]du as x → ∞, where S
n
= ∑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.
ASYMPTOTIC BEHAVIORS OF WAITING TIME DISTRIBUTION FUNCTION (WTDF) ψ(t) AND ASYMPTOTIC SOLUTIONS OF CONTINUOUS-TIME RANDOM WALK (CTRW) PROBLEMS
