Minimizing age of information under arbitrary arrival model with arbitrary packet size

We consider a single source-destination pair, where information updates (in short, updates) arrive at the source at arbitrary time instants. For each update, its size, i.e. the service time required for complete transmission to the destination, is also arbitrary. At any time, the source may choose which update to transmit, while incurring transmission cost that is proportional to the duration of transmission. We consider the age of information (AoI) metric that quantifies the staleness of the update (information) at the destination. At any time, AoI is equal to the difference between the current time, and the arrival time of the latest update (at the source) that has been completely transmitted (to the destination). The goal is to find a causal (i.e. online) scheduling policy that minimizes the sum of the AoI and the transmission cost, where the possible decisions at any time are (i) whether to preempt the update under transmission upon arrival of a new update, and (ii) if no update is under transmission, then choose which update to transmit (among the available updates). In this paper, we propose a causal policy called SRPT+ that at each time, (i) preempts the update under transmission if a new update arrives with a smaller size (compared to the remaining size of the update under transmission), and (ii) if no update is under transmission, then from the set of available updates with size less than a threshold (which is a function of the transmission cost and the current AoI), begins to transmit the update for which the ratio of the reduction in AoI upon complete transmission (if not preempted in future) and the remaining size, is maximum. We characterize the performance of SRPT+ using a metric called the competitive ratio, i.e. the ratio of the cost of causal policy and the cost of an optimal offline policy (that knows the entire input in advance), maximized over all possible inputs. We show that the competitive ratio of SRPT+ is at most 5. In the special case when there is no transmission cost, we further show that the competitive ratio of SRPT+ is at most 3.