Approximation of the exit probability of a stable Markov modulated constrained random walk

Abstract

Let X be the constrained random walk on Z+2 having increments (1, 0), (-1,1), (0,-1) with jump probabilities ?(Mk) , ?1(Mk) , and ?2(Mk) where M is an irreducible aperiodic finite state Markov chain. The process X represents the lengths of two tandem queues with arrival rate ?(Mk) , and service rates ?1(Mk) , and ?2(Mk) ; the process M represents the random environment within which the system operates. We assume that the average arrival rate with respect to the stationary measure of M is less than the average service rates, i.e., X is assumed stable. Let ?n be the first time when the sum of the components of X equals n for the first time. Let Y be the random walk on Z× Z+ having increments (-1,0), (1, 1), (0,-1) with probabilities ?(Mk) , ?1(Mk) , and ?2(Mk). Supposing that the queues share a joint buffer of size n, pn=P(xn,m)(?n<?0) is the probability that this buffer overflows during a busy cycle of the system. To the best of our knowledge, the only methods currently available for the approximation of pn are classical large deviations analysis giving the exponential decay rate of pn and rare event simulation. Let ? be the first time the components of Y are equal. For x?R+2, x(1) + x(2) < 1 , x(1) > 0 , and xn= ? nx? , we show that P(n-xn(1),xn(2),m)(?<?) approximates P(xn,m)(?n<?0) with exponentially vanishing relative error as n? ?. For the analysis we define a characteristic matrix in terms of the jump probabilities of (X, M). The 0-level set of the characteristic polynomial of this matrix defines the characteristic surface; conjugate points on this surface and the associated eigenvectors of the characteristic matrix are used to define (sub/super) harmonic functions which play a fundamental role both in our analysis and the computation/approximation of P(y,m)(?< ?). © 2020, Springer Science+Business Media, LLC, part of Springer Nature.