2020/01/03 by Gabi Hanukov, Uri Yechiali
Business, Management and Accounting · Mathematics · Social Sciences · #Advanced Queuing Theory Analysis #Random Matrices and Applications #Transportation Planning and Optimization
paper · doi:10.1017/s0269964819000470
openalex publication_date 2020/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25
Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equation A 0 + RA 1 + R 2 A 2 = 0. PGFs involve a row vector G(z) of unknown generating functions satisfying H(z)G(z)^\textrmT = b(z)^\textrmT, where the row vector b(z) contains unknown “boundary” probabilities calculated as functions of roots of the matrix H ( z ). We show that: (a) H ( z ) and b(z) can be explicitly expressed in terms of the triple A 0 , A 1 , and A 2 ; (b) when each matrix of the triple is lower (or upper) triangular, then (i) R can be explicitly expressed in terms of roots of det [H(z)] ; and (ii) the stability condition is readily extracted.