Moment Generating Function of the AoI in a Two-Source System with Packet Management

We consider a status update system consisting of two independent sources and one server in which packets of each source are generated according to the Poisson process and packets are served according to an exponentially distributed service time. We derive the moment generating function (MGF) of the age of information (AoI) for each source in the system by using the stochastic hybrid systems (SHS) under two existing source-aware packet management policies which we term self-preemptive and non-preemptive policies. In the both policies, the system (i.e., the waiting queue and the server) can contain at most two packets, one packet of each source; when the server is busy and a new packet arrives, the possible packet of the same source in the waiting queue is replaced by the fresh packet. The main difference between the policies is that in the self-preemptive policy, the packet under service is replaced upon the arrival of a new packet from the same source, whereas in the non-preemptive policy, this new arriving packet is blocked and cleared. We use the derived MGF to find the first and second moments of the AoI and show the importance of higher moments.