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.^{[1][2]: 118} 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^[3]

${\displaystyle P={\begin{pmatrix}A_{1}^{\ast }&A_{2}^{\ast }\\A_{0}^{\ast }&A_{1}&A_{2}\\&A_{0}&A_{1}&A_{2}\\&&A_{0}&A_{1}&A_{2}\\&&&\ddots &\ddots &\ddots \end{pmatrix}}}$

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.^[4]

Continuous time

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

${\displaystyle Q={\begin{pmatrix}B_{00}&B_{01}\\B_{10}&A_{1}&A_{2}\\&A_{0}&A_{1}&A_{2}\\&&A_{0}&A_{1}&A_{2}\\&&&A_{0}&A_{1}&A_{2}\\&&&&\ddots &\ddots &\ddots \end{pmatrix}}}$

where each of B₀₀, B₀₁, B₁₀, A₀, A₁ and A₂ are matrices.^[5] The process can be viewed as a two dimensional chain where the block structure are called levels and the intra-block structure phases.^[6] 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.^[6][7]

Stationary distribution

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