Quasi-birth-and-death Processes Research Articles

QBD Processes Associated with Jacobi–Koornwinder Bivariate Polynomials and Urn Models

We study a family of quasi-birth-and-death (QBD) processes associated with the so-called first family of Jacobi–Koornwinder bivariate polynomials. These polynomials are orthogonal on a bounded region typically known as the swallow tail. We will explicitly compute the coefficients of the three-term recurrence relations generated by these QBD polynomials and study the conditions under we can produce families of discrete-time QBD processes. Finally, we show an urn model associated with one special case of these QBD processes.

Quasi-birth-and-death processes and multivariate orthogonal polynomials

The aim of this paper is to study some models of quasi-birth-and-death (QBD) processes arising from the theory of bivariate orthogonal polynomials. First we will see how to perform the spectral analysis in the general setting as well as to obtain results about recurrence and the invariant measure of these processes in terms of the spectral measure supported on some domain Ω⊂Rd. Afterwards, we will apply our results to several examples of bivariate orthogonal polynomials, namely product orthogonal polynomials, orthogonal polynomials on a parabolic domain and orthogonal polynomials on the triangle. We will focus on linear combinations of the Jacobi matrices generated by these polynomials and produce families of either continuous or discrete-time QBD processes. Finally, we show some urn models associated with these QBD processes.

Tail Decay Rates in Double QBD Processes and Related Reflected Random Walks

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 homogenous chain introduced by Borovkov and Mogul'skii. 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 homogenous chain, refining existing results. We exemplify the decay rates for Jackson networks and their modifications.

Busy period analysis of finite QBD processes

We present the number of customers served and the length of a busy period for finite quasi birth and death (QBD) processes where either one or both of the arrival or service processes can be serially correlated or interdependent. Special cases include the G/G/1/K, M/G/1/K, and G/M/1/K queues. The resulting algorithms are linear algebraic in nature and are easily implemented. The solutions allow studies on how the moments and correlations in the arrival and service processes affect the busy period. This includes the probability of serving exactly n customers during a busy period and the moments of the length of the busy period for different system (queue) sizes. We present an example of a QBD process where arrival and service processes are strongly dependent.

Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators

We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators

We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

Quasi-Birth-and-Death Processes with an Explicit Rate Matrix

We determine the equilibrium distribution for a class of quasi-birth-and-death (QBD) processes using the matrix-geometric method, which requires the determination of the rate matrix R. In contrast to most QBD processes, the class under consideration allows for an explicit description of R, by exploiting its probabilistic interpretation. We show that the problem of finding each element of R reduces to counting lattice paths in the transition diagram. For resolving this counting problem, we present an extended version of the classical Ballot theorem. We also present various examples of queueing models that fit into the class.

Block-Structured Fluid Queues Driven by QBD Processes

In this paper, a unified approach for studying block-structured fluid models is proposed by means of the RG-factorization. When the stochastic environment (or background) is assumed to be a quasi-birth-and death (QBD) process, with either infinitely many levels or finitely many levels, the Laplace transform for the stationary probability distribution of the buffer content is expressed in terms of the R-measure. At the same time, the Laplace-Stieltjes transforms for both the conditional distribution and the conditional mean of a first passage time in such a fluid queue are derived by the same approach.

The group inverse of finite homogeneous QBD processes

Generalized inverses of I−P, where P is a stochastic matrix, play an important role in the theory of Markov chains. In particular, the group inverse (I−P)# has a probabilistic interpretation and is well suited for algorithmic implementation. We determine (I−P)# for finite homogeneous quasi-birth-and-death (QBD) processes by exploiting both the structure of the process and the probabilistic properties of the group inverse.

Truncation and augmentation of level-independent QBD processes

In the study of quasi-birth-and-death (QBD) processes, the first passage probabilities from states in level one to the boundary level zero are of fundamental importance. These probabilities are organized into a matrix, usually denoted by G. The matrix G is the minimal nonnegative solution of a matrix quadratic equation. If the QBD process is recurrent, then G is stochastic. Otherwise, G is sub-stochastic and the matrix equation has a second solution G sto, which is stochastic. In this paper, we give a physical interpretation of G sto in terms of sequences of truncated and augmented QBD processes. As part of the proof of our main result, we derive expressions for the first passage probabilities that a QBD process will hit level k before level zero and vice versa, which are of interest in their own right. The paper concludes with a discussion of the stability of a recursion naturally associated with the matrix equation which defines G and G sto. In particular, we show that G is a stable equilibrium point of the recursion while G sto is an unstable equilibrium point if it is different from G.

Solution of finite QBD processes

We provide an explicit matrix analytic solution for finite quasi birth and death (QBD) processes, directly expressed in terms of process parameters. We show that this solution has the same asymptotic complexity of previously proposed non-explicit solutions and is more general than some of them. Moreover, it can be easily extended to the case of generalized QBD processes.

Solution of finite QBD processes

We provide an explicit matrix analytic solution for finite quasi birth and death (QBD) processes, directly expressed in terms of process parameters. We show that this solution has the same asymptotic complexity of previously proposed non-explicit solutions and is more general than some of them. Moreover, it can be easily extended to the case of generalized QBD processes.

Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes

Quasi-Birth-and-Death (QBD) processes have been analysed in detail by many authors. This paper considers the calculation of equilibrium distributions in level dependent QBD processes which are an extension of the classical QBD process. In addition to the general case, we consider a number of special cases of level dependent QBD processes, presenting algorithms for computing the equilibrium distribution