Quasi-birth–death process

In queueing models, a discipline within the mathematical theory of probability, the quasi-birth–death process describes a generalisation of the birth–death process. As with the birth-death process it moves up and down between levels one at a time, but the time between these transitions can have a more complicated distribution encoded in the blocks.

Discrete time

The stochastic matrix describing the Markov chain has block structure

$P=\begin{pmatrix} A_1^\ast & A_2^\ast \\ A_0^\ast & A_1 & A_2 \\ & A_0 & A_1 & A_2 \\ && A_0 & A_1 & A_2 \\ &&& \ddots & \ddots & \ddots \end{pmatrix}$

where each of A₀, A₁ and A₂ are matrices and A*₀, A*₁ and A*₂ are irregular matrices for the first and second levels.

Continuous time

The transition rate matrix for a quasi-birth-death process has a tridiagonal block structure

$Q=\begin{pmatrix} B_{00} & B_{01} \\ B_{10} & A_1 & A_2 \\ & A_0 & A_1 & A_2 \\ && A_0 & A_1 & A_2 \\ &&& A_0 & A_1 & A_2 \\ &&&& \ddots & \ddots & \ddots \end{pmatrix}$

where each of B₀₀, B₀₁, B₁₀, A₀, A₁ and A₂ are matrices. The process can be viewed as a two dimensional chain where the block structure are called levels and the intra-block structure phases. When describing the process by both level and phase it is a continuous-time Markov chain, but when considering levels only it is a semi-Markov process (as transition times are then not exponentially distributed).

Usually the blocks have finitely many phases, but models like the Jackson network can be considered as quasi-birth-death processes with infinitely (but countably) many phases.

Stationary distribution

The stationary distribution of a quasi-birth-death process can be computed using the matrix geometric method.