Discrete Packet Management: Analysis of Age of Information of Discrete Time Status Updating Systems

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.