TY - JOUR

T1 - Explicit solutions for continuous-Time qbd processes by using relations between matrix geometric analysis and the probability generating functions METHOD

AU - Hanukov, Gabi

AU - Yechiali, Uri

N1 - Publisher Copyright:
Copyright © Cambridge University Press 2020.

PY - 2021/7

Y1 - 2021/7

N2 - 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 A0 + RA1 + R2A2 = 0. PGFs involve a row vector of unknown generating functions satisfying where the row vector contains unknown boundary probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and 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; and (ii) the stability condition is readily extracted.

AB - 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 A0 + RA1 + R2A2 = 0. PGFs involve a row vector of unknown generating functions satisfying where the row vector contains unknown boundary probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and 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; and (ii) the stability condition is readily extracted.

KW - calculation of the rate matrix R

KW - continuous-Time QBD processes

KW - matrix geometric

KW - probability generating functions

UR - http://www.scopus.com/inward/record.url?scp=85077733498&partnerID=8YFLogxK

U2 - 10.1017/S0269964819000470

DO - 10.1017/S0269964819000470

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AN - SCOPUS:85077733498

SN - 0269-9648

VL - 35

SP - 565

EP - 580

JO - Probability in the Engineering and Informational Sciences

JF - Probability in the Engineering and Informational Sciences

IS - 3

ER -