On first passage time for waiting processes

During the last 40 years waiting processes were considered in the literature in various fields of probability theory and statistics. They were investigated by D. V. Lindley [8], D. G. Kendall [3] and A. A. Borovkov [1] in queueing theory, by E. S. Page [10] and R. A. Khan [4], [5] within the cumulative sums algorithm, by N. Prabhu [11] in the theory of storage control and others. The time of the passage of the process across an a priori given level N became the critical value in various contexts in some papers. In the present paper we deduce the asymptotics of the expectation of the first passage time as N --> infinity. Although in the discrete case the result of the first part of the paper follows from the report of V. I. Lotov [9], it can also be obtained in another way bg extending the proof given by V. A. Labkovskii [7]. In the second part of the paper a definition is given of a waiting process with continuous time induced by a stochastically continuous random process with independent increments. A result similar to the one mentioned above appears to be valid in this case, too.