  Bufferless and single-buffer queueing systems have recently been shown to be
effective in coping with escalated Age of Information (AoI) figures arising in
single-source status update systems with large buffers and FCFS scheduling. In
this paper, for the single-source scenario, we propose a numerical algorithm
for obtaining the exact distributions of both the AoI and the peak AoI (PAoI)
in (i) the bufferless $PH/PH/1/1/P(p)$ queue with probabilistic preemption with
preemption probability $p$, $0 \leq p \leq 1,$ and (ii) the single buffer
$M/PH/1/2/R(r)$ queue with probabilistic replacement of the packet in the queue
by the new arrival with replacement probability $r$, $0 \leq r \leq 1$. The
proposed exact models are based on the well-established theory of Markov Fluid
Queues (MFQ) and the numerical algorithms are matrix-analytical and they rely
on numerically stable and efficient vector-matrix operations. Moreover, the
obtained exact distributions are in matrix exponential form, making it amenable
to calculate the tail probabilities and the associated moments
straightforwardly. Firstly, we validate the accuracy of the proposed method
with simulations, and for sume sub-cases, with existing closed-form results. We
then comparatively study the AoI performance of the queueing systems of
interest under varying traffic parameters.
