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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.

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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.

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Markovian service process (MSP) is a model similar to the Markovian arrival process (MAP), where arrivals are replaced with service completions. The MSP can represent various queueing models such as vacation models, N -policy models and exceptional service models. We analyze MAP/MSP/1 queues and obtain a new sort of matrix-type factorization of the vector generating function for the stationary queue length. The MAP/MSP/1 queue is a very tractable model since its behavior is represented as a quasi-birth-and-death process.

We deal with a two-queue model with mixed exhaustive and gated-type Klimited services. This model can achieve various degrees of priority in each queue by varying the value of K. Analyzing the model by using the delay cycle technique, we obtain the Laplace-Stieltjes transforms of the waiting time distributions for both cases with zero and non-zero walking times.