We define the family of transient Markov arrival processes (transient MAPs) which combine features of transient (or terminating) renewal processes and of the well-known MAPs: transient MAPs are point processes on the line, controlled by a finite Markov chain, which almost surely comprise a finite number of points. We analyze their basic properties.
"Transient Markov arrival processes." Ann. Appl. Probab. 13 (2) 628 - 640, May 2003. https://doi.org/10.1214/aoap/1050689597