A multiserver retrial queue: regenerative stability analysis |
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Authors: | Evsey Morozov |
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Institution: | (1) Institute of Applied Mathematical Research, Karelian Research Centre RAS and Petrozavodsk University, Petrozavodsk, Russia |
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Abstract: | We consider a multiserver retrial GI/G/m queue with renewal input of primary customers, interarrival time τ with rate
, service time S, and exponential retrial times of customers blocked in the orbit. In the model, an arriving primary customer enters the system
and gets a service immediately if there is an empty server, otherwise (if all m servers are busy) he joins the orbit and attempts to enter the system after an exponentially distributed time. Exploiting
the regenerative structure of the (non-Markovian) stochastic process representing the total number of customers in the system
(in service and in orbit), we determine stability conditions of the system and some of its variations. More precisely, we
consider a discrete-time process embedded at the input instants and prove that if
and
, then the regeneration period is aperiodic with a finite mean. Consequently, this queue has a stationary distribution under
the same conditions as a standard multiserver queue GI/G/m with infinite buffer. To establish this result, we apply a renewal technique and a characterization of the limiting behavior
of the forward renewal time in the (renewal) process of regenerations. The key step in the proof is to show that the service
discipline is asymptotically work-conserving as the orbit size increases. Included are extensions of this stability analysis
to continuous-time processes, a retrial system with impatient customers, a system with a general retrial rate, and a system
with finite buffer for waiting primary customers. We also consider the regenerative structure of a multi-dimensional Markov
process describing the system.
This work is supported by Russian Foundation for Basic Research under grants 04-07-90115 and 07-07-00088. |
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Keywords: | Multiserver retrial queue General service time Renewal input Stability analysis Regenerations Impatient customers Finite buffer |
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