On Relationship between Ergodic Sojourn Time and Ergodic Residual Exit Time for Semi-Markov Processes

Abstract. For Markov chains in continuous time Keilson(1979) has shown that the relationship between the ergodic exit time TE and the ergodic sojourn time TV is identical to the relationship between the residual lifetime and the underlying lifetime at ergodicity in renewal theory. Keilson’s result relies upon the memory-less property of exponential distributions, and it would not hold true, in general, for semi-Markov processes. The purpose of this paper is to introduce a new per-formance measure called the ergodic residual exit time TW so as to prove that the relationship between TW and TV for semi-Markov processes is identical to the relationship between TE and TV for Markov chains in continuous time