ON MULTISERVER RETRIAL QUEUES: HISTORY, OKUBO-TYPE HYPERGEOMETRIC SYSTEMS AND MATRIX CONTINUED-FRACTIONS

F. Avram (), D. Matei () and Y. Q. Zhao ()
Additional contact information
F. Avram: Département de Mathématiques, Avenue de l'Université Pau, France
D. Matei: Institute of Mathematics "Simion Stoilow" of the Romanian Academy, Bucharest, P.O. Box 1-764, RO-014700, Romania
Y. Q. Zhao: School of Mathematics and Statistics, Carleton University Ottawa, Canada

Asia-Pacific Journal of Operational Research (APJOR), 2014, vol. 31, issue 02, 1-28

Abstract: In this paper, we study two families of QBD processes with linear rates: (a) the multiserver retrial queue and its easier relative; and (b) the multiserver M/M/∞ Markov modulated queue. The linear rates imply that the stationary probabilities satisfy a recurrence with linear coefficients; as known from previous work, they yield a"minimal/nondominant" solution of this recurrence, which may be computed numerically by matrix continued-fraction methods. Furthermore, the generating function of the stationary probabilities satisfies a linear differential system with polynomial coefficients, which calls for the venerable but still developing theory of holonomic (or D-finite) linear differential systems. We provide a differential system for our generating function that unifies problems (a) and (b), and we also include some additional features and observe that in at least one particular case we get a special "Okubo-type hypergeometric system", a family that recently spurred considerable interest.The differential system should allow further study of the Taylor coefficients of the expansion of the generating function at three points of interest: (i) the irregular singularity at 0; (ii) the dominant regular singularity, which yields asymptotic series via classic methods like the Frobenius vector expansion; and (iii) the point 1, whose Taylor series coefficients are the factorial moments.

Keywords: Retrial queue; stationary probabilities; Okubo hypergeometric system; minimal solution (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (1)

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DOI: 10.1142/S0217595914400016

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