We derive explicit closed expressions for the moment generating functions of whole collections of quantities associated with the waiting time till the occurrence of composite events in either discrete or continuous-time models. The discrete-time models are independent, or Markov-dependent, binary trials and the events of interest are collections of successes with the property that each two consecutive successes are separated by no more than a fixed number of failures. The continuous-time models are renewal processes and the relevant events are clusters of points. We provide a unifying technology for treating both the discrete and continuous-time cases. This is based on first embedding the problems into similar ones for suitably selected Markov chains or Markov renewal processes, and second, applying tools from the exponential family technology.
|Journal||Journal of Applied Probability|
|Publication status||Published - 1999|