Exit probabilities of markov modulated constrained random walks

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2018
Başoğlu Kabran, Fatma
Let X be the constrained random walk on Z2+ with increments (0, 0), (1, 0), (−1, 1), (0, −1) whose jump probabilities are determined by the state of a finite state Markov chain M. X represents the lengths of two queues of customers (or packets, tasks, etc.) waiting for service from two servers working in tandem; the arrival of customers occur with rate λ(Mk), service takes place at rates μ1(Mk), and μ2(Mk) where Mk denotes the current state of the Markov chain M. We assume that the average arrival rate is less than the average service rates, i.e., X is assumed stable. Stability implies that X moves in cycles that restart each time it hits the origin. Let τn be the first time X hits the line ∂An={x:x(1)+x(2)=n}, i.e., when the sum of the queue lengths equals n for the first time ; if the queues share a common buffer, τn represents the time of a buffer overflow and pn = P(x,m)(τn < τ0) is the probability that a given cycle ends with a buffer overflow, i.e., system failure. Let Y be the same random walk as X but only constrained on∂2 = {y ∈ Z×Z+ : y(2) = 0} and its jump probabilities for the first component reversed. Let B = {y ∈ Z2 : y(1) = y ( 2 ) } and let τ be the first time Y hits B . For x ∈ R 2+ , with x (1) + x (2)<1 define xn = ⌊nx⌋ and let m ∈ M denote the initial point of the Markov chain M. We show that P((n−xn(1),xn(2)),m)(τ < ∞) approximates P((xn(1),xn(2)),m)(τn < τ0) with exponentially vanishing relative error when x(1) > 0. We then construct a class of harmonic functions for (Y,M) and use their linear combinations to develop approximate formulas for P(y,m)(τ < ∞). The construction is based on points on a characteristic surface associated with Y defined through the eigenvalues of a matrix whose components depend on the transition matrix of the modulating chain and the jump probabilities of Y . We indicate possible applications of our results and approach in finance and insurance.