Tail decay rates in double QBD processes and related reflected random walks. (English) Zbl 1213.60151

Miyazawa, Masakiyo

Math. Oper. Res. 34, No. 3, 547-575 (2009).

Summary: A double quasi-birth-and-death (QBD) process is the QBD process whose background process is a homogeneous birth-and-death process, which is a synonym of a skip-free random walk in the two-dimensional positive quadrant with homogeneous reflecting transitions at each boundary face. It is also a special case of a 0-partially homogeneous chain introduced by Borovkov and Mogul’skii [Sib. Math. J. 37, No. 4, 647–682 (1996); translation from Sib. Mat. Zh. 37, No. 4, 745–782 (1996; Zbl 0878.60023)]. Our main interest is in the tail decay behavior of the stationary distribution of the double QBD process in the coordinate directions and that of its marginal distributions. In particular, our problem is to get their rough and exact asymptotics from primitive modeling data. We first solve this problem using the matrix analytic method. We then revisit the problem for the 0-partially homogeneous chain, refining existing results. We exemplify the decay rates for Jackson networks and their modifications.

MSC:

60K25	Queueing theory (aspects of probability theory)
90B25	Reliability, availability, maintenance, inspection in operations research
60F10	Large deviations
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

quasi-birth-and-death process; partially homogeneous chain; two-dimensional queues; stationary distribution; rough decay rate; exact asymptotics; reflected random walk; Jackson network with server cooperation