On the first meeting or crossing of two independent trajectories for some counting processes. (Q2574557)

The methodology to study the first meeting or crossing (f.m./c.) of counting processes with a given boundary was developed by the authors [Adv.\ Appl.\ Probab.\ 28, 853--876 (1996; Zbl 0857.60085)] and is used here to solve a related problem of the f.m./c.\ between two independent trajectories of different stochastic processes. Three cases are examined successively: the renewal process starting above or under the trajectory (an important distinction between these two situations was found in the above citied paper) of a compound Poisson process, a compound binomial process and a linear birth process with immigration. The trajectory of the renewal process is always viewed as a fixed boundary, otherwise each case is treated separately. The randomized versions of the Abel-Gontscharoff and Appell polynomials are important mathematical tools to treat the above and under f.m./c.\ cases, respectively. The approach is mainly analytical without any need of probabilistic arguments. The direct relevance of the compound Poison trajectory case to queueing and dam theories is discussed, whereas the relation of the compound binomial case to the ruin problem for some discrete-time risk model is only shortly mentioned.