The problem of optimal real-time transmission of a Markov source under constraints on the expected number of transmissions is considered, both for the discounted and long term average cases. This setup is motivated by applications where transmission is sporadic and the cost of switching on the radio and transmitting is significantly more important than the size of the transmitted data packet. For this model, we characterize the distortion-transmission function, i.e., the minimum expected distortion that can be achieved when the expected number of transmissions is less than or equal to a particular value. In particular, we show that the distortion-transmission function is a piecewise linear, convex, and decreasing function. We also give an explicit characterization of each vertex of the piecewise linear function.

To prove the results, the optimization problem is cast as a decentralized constrained stochastic control problem. We first consider the Lagrange relaxation of the constrained problem and identify the structure of optimal transmission and estimation strategies. In particular, we show that the optimal transmission is of a threshold type. Using these structural results, we obtain dynamic programs for the Lagrange relaxations. We identify the performance of an arbitrary threshold-type transmission strategy and use the idea of calibration from multi-armed bandits to determine the optimal transmission strategy for the Lagrange relaxation. Finally, we show that the optimal strategy for the constrained setup is a randomized strategy that randomizes between two deterministic strategies that differ only at one state. By evaluating the performance of these strategies, we determine the shape of the distortion-transmission function. These results are illustrated using an example of transmitting a birth-death Markov source.