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 -