On an Adaptive-Quasi-Deterministic Transmission Policy Queueing Model [1]

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 caf´e 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 caf´e; 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 caf´e 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.