A retrial system with two input streams and two orbit queues

Two independent Poisson streams of jobs flow into a single-server service system having a limited common buffer that can hold at most one job. If a type-i job (i=1,2) finds the server busy, it is blocked and routed to a separate type-i retrial (orbit) queue that attempts to re-dispatch its jobs at its specific Poisson rate. This creates a system with three dependent queues. Such a queueing system serves as a model for two competing job streams in a carrier sensing multiple access system. We study the queueing system using multi-dimensional probability generating functions, and derive its necessary and sufficient stability conditions while solving a boundary value problem. Various performance measures are calculated and numerical results are presented.

💡 Research Summary

The paper investigates a single‑server queueing system with a limited common buffer of size one that is fed by two independent Poisson arrival streams. When a type‑i job (i = 1, 2) arrives while the server is busy, it is blocked and routed to a dedicated type‑i retrial (orbit) queue. Each orbit queue has infinite capacity and attempts to resend its jobs to the server according to its own Poisson retrial rate ν_i, independent of the other orbit. Consequently the model consists of three interacting queues: the primary buffer, orbit 1 and orbit 2.

The authors formulate the stochastic dynamics as a three‑dimensional Markov process and introduce the multivariate probability generating function (PGF)

 G(z₀, z₁, z₂) = E