Conditions for stability and instability of retrial queueing systems with general retrial times

Abstract

We study the stability of single server retrial queues under general distribution for retrial times and stationary ergodic service times, for three main retrial policies studied in the literature: classical linear, constant and control policies. The approach used is the renovating events approach to obtain sufficient stability conditions by strong coupling convergence of the process modeling the dynamics of the system to a unique stationary ergodic regime. We also obtain instability conditions by convergence in distribution to improper limiting sequences.

Connect Time Limits and Performance Measures in a Dial-Up Modem Pool System

“…The research on retrial queues is fairly extensive and extremely rigorous in mathematical analysis. A complete literature review of the field is beyond the scope of this study, and would, to a great extent, duplicate the work in previous surveys (see Amador & Artalejo, 2009;Artalejo & Gómez-Corral, 2008;Falin, 1990;Kernane, 2008;and Yang & Templeton, 1987).…”

Extended generator and associated martingales for M/G/1 retrial queue with classical retrial policy and general retrial times

“…In Kernane [17], the stability condition was determined for an M/G/1 retrial queue with general retrial times and classical retrial policy by assuming also that the process of service times is stationary and ergodic. In the references [2,20,21] , only some approximations and simulations are presented for models with phase-type retrial time.…”

“…The first attempt to generalize the retrial time distribution was done by Kapyrin [16] for a classical retrial policy, but Falin [10] has shown later that his approach was incorrect. In Kernane [17], the stability condition was determined for an M/G/1 retrial queue with general retrial times and classical retrial policy by assuming also that the process of service times is stationary and ergodic. In the references [2,20,21] , only some approximations and simulations are presented for models with phase-type retrial time.…”