Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

Abstract

A discrete-time two-dimensional quasi-birth-and-death process (2d-QBD process), denoted by {Y n } = {(X 1,n , X 2,n , J n )}, is a two-dimensional skip-free random walk {(X 1,n , X 2,n )} on Z 2 + with a supplemental process {J n } on a finite set S 0 . The supplemental process {J n } is called a phase process. The 2d-QBD process {Y n } is a Markov chain in which the transition probabilities of the two-dimensional process {(X 1,n , X 2,n )} vary according to the state of the phase process {J n }. This modulation is assumed to be space homogeneous except for the boundaries of Z 2 + . Under certain conditions, the directional exact asymptotic formulae of the stationary distribution of the 2d-QBD process have been obtained in Ref. [7]. In this paper, we give an example of 2d-QBD process and proofs of some lemmas and propositions appeared in Ref. [7].

Kernel Method -- An Analytic Approach for Tail Asymptotics in Stationary Probabilities of 2-Dimensional Queueing Systems

“…, M } and transition diagram given in Figure 1, where A i,j and A (k) i,j are all nonnegative matrices of size M × M with M < ∞, and i,j A i,j and i,j A (k) i,j (k = 0, 1, 2) are stochastic. theorem (Theorem 2.1 in [70]) could not be confirmed to be 1 that would ensure that the 2d-QBD has the same types of tail asymptotics as that for the random walk in the quarter plane.…”

“…Examples of using large deviations approach can be found in Borovkov and Mogul'skii [6] and Lieshout and Mandjes [54],. A method based on geometric properties of the model, initiated by Miyazawa, is robust including Miyazawa [62,64,63,65], Miyazawa and Rolski [66], Ozawa [69], Ozawa and Kobayashi [70]. When for exact tail asymptotics, the asymptotic analysis and the Tauberian-like theorem components were integrated into this geometric method.…”

“…(1) Under a stability condition (see, for example [70]), let π m,n;k be the stationary probability vector. Then, the fundamental form (in matrix form) is given by:…”

“…The random walk in the quarter plane modulated by a Markov chain was also considered in [70], which was referred to as the discrete-time, two-dimensional quasi-birth-and-death process (2d-QBD). Instead of special examples, the authors of [70] considered a general setting for this type of Markovian model.…”

“…The random walk in the quarter plane modulated by a Markov chain was also considered in [70], which was referred to as the discrete-time, two-dimensional quasi-birth-and-death process (2d-QBD). Instead of special examples, the authors of [70] considered a general setting for this type of Markovian model. Their method can be considered as a combination of the geometric method, introduced by Miyazawa in [63], and the asymptotic analysis, an important component of the kernel method.…”