Optimal Source Codes for Timely Updates

A transmitter observing a sequence of independent and identically distributed random variables seeks to keep a receiver updated about its latest observations. The receiver need not be apprised about each symbol seen by the transmitter, but needs to output a symbol at each time instant $t$. If at time $t$ the receiver outputs the symbol seen by the transmitter at time $U(t)\leq t$, the age of information at the receiver at time $t$ is $t-U(t)$. We study the design of lossless source codes that enable transmission with minimum average age at the receiver. We show that the asymptotic minimum average age can be attained up to a constant gap by the Shannon codes for a tilted version of the original pmf generating the symbols, which can be computed easily by solving an optimization problem. Furthermore, we exhibit an example with alphabet $\X$ where Shannon codes for the original pmf incur an asymptotic average age of a factor $O(\sqrt{\log |\X|})$ more than that achieved by our codes. Underlying our prescription for optimal codes is a new variational formula for integer moments of random variables, which may be of independent interest. Also, we discuss possible extensions of our formulation to randomized schemes and to the erasure channel, and include a treatment of the related problem of source coding for minimum average queuing delay.