Stability Analysis of GI/G/c/K Retrial Queue with Constant Retrial Rate

TL;DR

This paper analyzes the stability of a general GI/G/c/K retrial queue with constant retrial rate, providing minimal stability conditions applicable to various special cases and real-world systems.

Contribution

It establishes the first minimal sufficient stability conditions for a very general GI/G/c/K retrial queue with constant retrial rate, covering new particular cases like deterministic service and Erlang models.

Findings

Derived minimal stability conditions with probabilistic interpretation

Unified framework for various queueing models including new particular cases

Applicable to systems like telephone exchanges and TCP transfers

Abstract

We consider a GI/G/c/K-type retrial queueing system with constant retrial rate. The system consists of a primary queue and an orbit queue. The primary queue has $c$ identical servers and can accommodate the maximal number of $K$ jobs. If a newly arriving job finds the full primary queue, it joins the orbit. The original primary jobs arrive to the system according to a renewal process. The jobs have general i.i.d. service times. A job in front of the orbit queue retries to enter the primary queue after an exponentially distributed time independent of the orbit queue length. Telephone exchange systems, Medium Access Protocols and short TCP transfers are just some applications of the proposed queueing system. For this system we establish minimal sufficient stability conditions. Our model is very general. In addition, to the known particular cases (e.g., M/G/1/1 or M/M/c/c systems), the proposed model covers as particular cases the deterministic service model and the Erlang model with constant retrial rate. The latter particular cases have not been considered in the past. The obtained stability conditions have clear probabilistic interpretation.