On Multiserver Retrial Queues: History, Okubo-Type Hypergeometric Systems and Matrix Continued-Fractions

Abstract

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/non-dominant" 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: 1) the irregular singularity at 0; 2) the dominant regular singularity, which yields asymptotic series via classic methods like the Frobenius vector expansion; and 3) the point 1, whose Taylor series coefficients are the factorial moments.

Proof of the conjecture on the stability of a multiserver retrial queue

“…A queueing system is called stable when the Markov process describing the dynamics of the queueing system is positive recurrent and unstable otherwise. In this paper we solve the conjecture made by Avram et al [6], on stability condition for the M/M/s retrial queue with Bernoulli acceptance, abandonment and feedback. More specifically, we proved that the Markov process describing this queueing system is positive recurrent if ρ ∞ < 1 and transient if ρ ∞ > 1, where ρ ∞ is the traffic load under the saturation condition of the orbit.…”

“…We consider the M/M/s retrial queue, given by Avram et al [6]. Primary customers arrive according to a Poisson process with rate λ.…”

“…Avram et al [6] made a conjecture (Remark 6.2 of [6]) about the stability condition for the above mentioned model. A queueing system is called stable when the Markov process describing the dynamics of the queueing system is positive recurrent and unstable otherwise.…”

“…Avram et al [6] considered a more general model than the classical multiserver retrial queue. If there is a free server when a primary customer arrives, then this customer has three possibilities: (i) get served, (ii) abandon, or (iii) join the orbit.…”