  In this paper, we consider performing packet managements in discrete time
status updating system, focusing on determining the stationary AoI-distribution
of the system. Firstly, let the queue model be Ber/G/1/1, we obtain the
AoI-distribution by introducing a two-dimensional AoI-stochastic process and
solving its steady state, which describes the random evolutions of AoI and age
of packet in system simultaneously. In this case, actually we analyze a more
general queue called probabilistic preemption Ber/G/1/1, where the packet
service is allowed to be preempted with certain probabilities. As a special
case, stationary AoI-distribution for the system with Ber/Geo/1/1 queue is
obtained either. For the system having size 2, two specific queues are
considered, i.e., the Ber/Geo/1/2 and Ber/Geo/1/2* queues. The core idea to
find the stationary AoI-distribution is that the random transitions of
three-dimensional vector including AoI at the receiver, the packet age in
service, and the age of waiting packet can be fully described, such that a
three-dimensional AoI process is constituted. The stationary distribution of
three-dimensional process then gives the stationary AoI distribution as one of
its marginal distributions. For both cases, the explicit expressions of
AoI-distribution are derived, thus giving the complete description of the
steady state AoI for the system. For all the cases, since the steady state of a
larger-dimensional AoI process is solved, so that except the AoI-distribution,
we obtain more. For instance, the distributions of packet system time and
waiting time for size-two updating system, and the so-called violation
probabilities that AoI exceeds certain threshold.
