On relationship between loss probability and utilization factor on real-time basis

It is known that for sources that output symbols according to Poisson process the loss probability coincides with the utilization factor of the channel for any code on the realtime basis. It is also known for continuous-time Markov chains they take the same value for any code. This coincidence means that minimizing loss probability gives us the maximum spare time to transmit another information. That is, we can enjoy two merits by optimizing one parameter. In this paper, we specify the source class where we have such a benefit. Concretely, as a super class including Poisson processes and continuous-time Markov chains, we define a class of sources that traverse a finite state space with arbitrary random stay time. For this source class, we show that the necessary and sufficient condition for the coincidence of values between the loss probability and the utilization factor is that for each state the stay time obeys an exponential distribution. Assuming that the stay time obeys an exponential distribution, we derive the minimum loss probability. Moreover, we derive the minimum loss probability in the case that a single code is common in all states.