# Scheduling to Minimize Age of Information with Multiple Sources

We consider a G/G/1 queueing system with a single server, where updates arrive from different sources stochastically with possibly different update inter-generation time distributions. The server can transmit/serve at most one update at any time, with potentially different transmission/service times for updates belonging to distinct sources. The age of information (AoI) of any source is a function of the time difference between the departure time of successive updates of that source. Each fully/partially transmitted update incurs a fixed (energy) cost, and the goal of the scheduler is to minimize the linear combination of the sum of the age of information across all sources and the total energy cost. We propose a simple non-preemptive randomized scheduling algorithm that randomly marks arriving updates from a source to be eligible for transmission with a fixed probability and discards them otherwise. Every time the server becomes free, it chooses a source for transmission randomly with another fixed probability and begins to transmit the most recently marked update of the chosen source. Both the respective probabilities are chosen by solving a convex program. The competitive ratio of the proposed algorithm (against a non-preemptive offline optimal algorithm) is shown to be 3 plus the maximum of the ratio of the variance and the mean of the inter-arrival time distribution of sources. For several common distributions such as exponential, uniform and Rayleigh, the competitive ratio is at most 4. For preemptive policies, a G/M/1 system is considered and a non-preemptive policy is shown to have competitive ratio (against a preemptive offline optimal algorithm) at most 5 plus the maximum of the ratio of the variance and the mean of the inter-arrival time distribution of sources.

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