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On functionals of a marked Poisson process observed by a renewal process. (English) Zbl 0995.60082
Consider the batch Poisson arrival process to an empty service system where the server starts service only at inspection times when the number of arrived customers exceeds some level $$N$$. The sequence of inspection times is assumed to be a renewal sequence. The authors derive functional equations and formulas for the multidimensional transform of the vector of (termination time, termination index, queue length at termination time, queue length at the last inspection before termination). Here termination time is the inspection time when the server observes the first exceedance of $$N$$ after the last end of a busy period. Several special cases are considered and the paper is opened with similar formulas in a more general point process setting.

##### MSC:
 60K10 Applications of renewal theory (reliability, demand theory, etc.) 60K15 Markov renewal processes, semi-Markov processes 60K25 Queueing theory (aspects of probability theory)
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