Compressor Design for Timely Status Updates over Erasure Channel

A compression problem for status updating system with an erasure channel is considered. The samples (updates) generated are fed into a variable-rate quantizer and encoded into binary strings. These encoded samples are sent to the receiver over an erasure channel where each sample has a probability $\epsilon_{0}$ of being erased. To improve the information freshness, we consider a zero-wait policy, where a new sample is generated right away after the transmission of the previous is completed, regardless of whether the transmission is successful or not. In other words, the transmitter immediately acquires a new sample rather than attempting to retransmit the previous one when a transmission failure occurs. Based upon this policy, we formulate an optimization problem to jointly design the quantizer and encoder to minimize the age of information (Aol) subject to a mean-squared error (MSE) distortion constraint of the samples. We prove that the uniform quantizer and the AoI-optimal code are asymptotically optimal in the sense that, as the distortion tends to zero, this solution attains the minimum average age asymptotically. Furthermore, we prove that the optimal average age is asymptotically linear with respect to the log MSE with a slope of $-\frac{3-\epsilon_{0}}{4\left(1-\epsilon_{0}\right)}$.