Wiener-Hopf Factorizations for a Multidimensional Markov Additive Process and their Applications to Reflected Processes

Abstract

We extend the framework of Neuts' matrix analytic approach to a reflected process generated by a discrete time multidimensional Markov additive process. This Markov additive process has a general background state space and a real vector valued additive component, and generates a multidimensional reflected process. Our major interest is to derive a closed form formula for the stationary distribution of this reflected process. To this end, we introduce a real valued level, and derive new versions of the Wiener-Hopf factorization for the Markov additive process with the multidimensional additive component. In particular, it is represented by moment generating functions, and we consider the domain for it to be valid.Our framework is general enough to include multi-server queues and/or queueing networks as well as non-linear time series which are currently popular in financial and actuarial mathematics. Our results yield structural results for such models. As an illustration, we apply our results to extend existing results on the tail behavior of reflected processes.A major theme of this work is to connect recent work on matrix analytic methods to classical probabilistic studies on Markov additive processes. Indeed, using purely probabilistic methods such as censoring, duality, level crossing and time-reversal (which are known in the matrix analytic methods community but date back to Arjas & Speed [2] and Pitman [29]), we extend and unify existing results in both areas.

Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps

“…Hence, using the notation ν (1) n of Section 2, we can verify all the conditions of Theorem 4.1 of [28] since {A (k) n } is 1-arithmetic. Thus, we obtain (29).…”

“…The present version is valid as long as A (1) * (t) exists. This fact was formally proved in [29], but has often been ignored in the literature (see, e.g. [28]), even though it is crucial to the tail asymptotic problem.…”

“…By the existence of the stationary distribution, E ν U (σ U ) is finite. Then, as discussed in Section 2 of [29], it follows from censoring on the set U that…”

“…then we obviously have (28) by Lemma 7. Otherwise, (28) follows from (29) and (30) or their δ-arithmetic versions for each positive integer δ. Thus, it remains to prove (29) and (30).…”

A unified approach for large queue asymptotics in a heterogeneous multiserver queue

“…Recall that ν k specifies the conditional distribution of R(t 0,0 −) ≡ R(0−) under the stationary distribution of the embedded process {X(t 0,n −); n ∈ Z} given that L(0−) + 1 = L(0−) = k. Then, the stationary tail probability of L is given by the so-called cycle formula (see, e.g. Corollary 2.1 of [24]), i.e.…”