On an Adaptive-Quasi-Deterministic Transmission Policy Queueing Model

We analyze, further and deeper, a recently proposed technique for addressing the Massive Access Problem (MAP), an issue in telecommunications which arises when too many devices transmit packets to a gateway in quick succession. This technique, the Adaptive-Quasi-Deterministic Transmission Policy (AQDTP) is a special case of “traffic shaping” which involves delaying some packets at the points of origin to alleviate congestion at the routers. One nice feature of AQDTP is that it loses no packets and allows an infinite buffer. In this work, to clarify the approach in a general queueing theory framework, and to move beyond the original telecommunications application, we frame these potential delays as time spent at a café by customers before proceeding to a service facility. We first present some sample-path results that significantly refine and expand upon what was shown in previous work, and then present further results under a general stationary ergodic stochastic framework. In the sample-path realm, we give conditions that ensure AQDTP will not change the total delay and sojourn time of any customer as compared to what that customer would have experienced if there was no café; but we also prove that AQDTP can never reduce the total delay. The difference is that, under AQDTP, some of that delay is spent at the café instead of in the queue/line at the service facility. In a stochastic framework, our focus is on stability and constructing proper stationary versions of the model. Under i.i.d. assumptions we dig deeper by proving Harris recurrence of an underlying two-dimensional Markov process, and explicitly find positive recurrent regeneration points.