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.

References

1. Latouche, G. (2011). "Level-Independent Quasi-Birth-and-Death Processes". Wiley Encyclopedia of Operations Research and Management Science. doi:10.1002/9780470400531.eorms0461. ISBN 9780470400531.
2. Gautam, Natarajan (2012). Analysis of Queues: Methods and Applications. CRC Press. ISBN 9781439806586.
3. Latouche, G.; Pearce, C. E. M.; Taylor, P. G. (1998). "Invariant measures for quasi-birth-and-death processes". Communications in Statistics. Stochastic Models. 14: 443. doi:10.1080/15326349808807481.
4. Palugya, S. N.; Csorba, M. T. J. (2005). "Modeling Access Control Lists with Discrete-Time Quasi Birth-Death Processes". Computer and Information Sciences - ISCIS 2005. Lecture Notes in Computer Science. 3733. p. 234. doi:10.1007/11569596_26. ISBN 978-3-540-29414-6.
5. Asmussen, S. R. (2003). "Markov Additive Models". Applied Probability and Queues. Stochastic Modelling and Applied Probability. 51. pp. 302–339. doi:10.1007/0-387-21525-5_11. ISBN 978-0-387-00211-8.
6. Kroese, D. P.; Scheinhardt, W. R. W.; Taylor, P. G. (2004). "Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process". The Annals of Applied Probability. 14 (4): 2057. doi:10.1214/105051604000000477.
7. Motyer, A. J.; Taylor, P. G. (2006). "Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators". Advances in Applied Probability. 38 (2): 522. doi:10.1239/aap/1151337083.