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].
We consider a general QBD process as defining a FIFO queue and obtain the stationary distribution of the sojourn time of a customer in that queue as a matrix exponential distribution, which is identical to a phase-type distribution under a certain condition. Since QBD processes include many queueing models where the arrival and service process are dependent, these results form a substantial generalization of analogous results reported in the literature for queues such as the PH/PH/c queue. We also discuss asymptotic properties of the sojourn time distribution through its matrix exponential form.
In order to analyze stability of a two-queue model, we consider a two-dimensional quasibirth-and-death process (2d-QBD process), denoted by {Y (t)} = {((L 1 (t), L 2 (t)), J(t))}. The two-dimensional process {(L 1 (t), L 2 (t))} on Z 2 + is called a level process, where the individual processes {L 1 (t)} and {L 2 (t)} are assumed to be skip free. The supplemental process {J(t)} is called a phase process and it takes values in a finite set. The 2d-QBD process is a CTMC, in which the transition rates of the level process vary according to the state of the phase process like an ordinary (one-dimensional) QBD process. In this paper, we first state the conditions ensuring a 2d-QBD process is positive recurrent or transient and then demonstrate that the efficiency of a two-queue model can be estimated by using the conditions we obtained.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.