When to Preempt in a Status Update System?

We consider a time slotted status update system with an error-free preemptive queue. The goal of the sampler-scheduler pair is to minimize the age of information at the monitor by sampling and transmitting the freshly sampled update packets to the monitor. The sampler-scheduler pair also has a choice to preempt an old update packet from the server and transmit a new update packet to the server. We formulate this problem as a Markov decision process and find the optimal sampling policy. We show that it is optimal for the sampler-scheduler pair to sample a new packet immediately upon the reception of an update packet at the monitor. We also show that the optimal choice for the scheduler is to preempt an update packet in the server, if the age of that packet crosses a fixed threshold. Finally, we find the optimal preemption threshold when the range of the service time of the server is finite, otherwise we find the $\epsilon$-optimal preemption threshold.