Coding under a fidelity criteria of a class of sources which emit randomly occurring messages is investigated. This class of sources models information carrying processes entering into communication networks. Messages emitted by computer terminals, teletypes, vocoders, and other such devices serve as actual examples. For this class of sources the rate distortion function is derived, and source coding and converse source coding theorems are proven. Employing these theorems, an operational definition of the rate distortion function in terms of message queueing delay, and transmission delay is presented. This definition relates the rate distortion function with the message network delay which is an important measure of performance of a communication network. Also, for such a class of sources, which emit randomly occurring messages, an adaptive data compression scheme is investigated. This scheme utilizes observations of the network congestion to determine the amount of compression a message receives, with the object of minimizing the message delay for a given distortion level.