Speaker: Uri Yechiali, Tel Aviv University, Israel
Title: Explicit solutions for continuous-time QBD processes by using relations between matrix geometric analysis and the probability generating functions method
Time and Place: April 12, 2024 at 14h30 in Salle Lagrange Gris, Inria, Sophia-Antipolis
Abstract: Two main methods are used to solve continuous-time QBD processes: Matrix Geometric (MG) and Probability Generating Functions (PGFs). The MG method requires a numerical solution (via successive substitutions) of a matrix R satisfying the quadratic equation A0+RA1+ R2A2 = 0. The PGFs method involves a vector G(z) of unknown generating functions satisfying H(z) G(z)T= b(z)T where the vector b(z) includes 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 A0, A1 and A2; (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. Examples are presented showing that in many QBD problems the above three matrices are triangular.
