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

Let X be the constrained random walk on $${\mathbb {Z}}_+^2$$ Z + 2 having increments (1, 0), $$(-\,1,1)$$ ( - 1 , 1 ) , $$(0,-\,1)$$ ( 0 , - 1 ) with jump probabilities $$\lambda (M_k)$$ λ ( M k ) , $$\mu _1(M_k)$$ μ 1 ( M k ) , and $$\mu _2(M_k)$$ μ 2 ( M k ) where M is an irreducible aperiodic finite state Markov chain. The process X represents the lengths of two tandem queues with arrival rate $$\lambda (M_k)$$ λ ( M k ) , and service rates $$\mu _1(M_k)$$ μ 1 ( M k ) , and $$\mu _2(M_k)$$ μ 2 ( M k ) ; 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 $$\tau _n$$ τ 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 $${{\mathbb {Z}}} \times {{\mathbb {Z}}}_+$$ Z × Z + having increments $$(-\,1,0)$$ ( - 1 , 0 ) , (1, 1), $$(0,-\,1)$$ ( 0 , - 1 ) with probabilities $$\lambda (M_k)$$ λ ( M k ) , $$\mu _1(M_k)$$ μ 1 ( M k ) , and $$\mu _2(M_k)$$ μ 2 ( M k ) . Supposing that the queues share a joint buffer of size n, $$p_n =P_{(x_n,m)}(\tau _n 0$$ x ( 1 ) > 0 , and $$x_n = \lfloor nx \rfloor $$ x n = ⌊ n x ⌋ , we show that $$P_{(n-x_n(1),x_n(2),m)}( \tau