On a formula of Harald Cramér

Abstract

Abstract In 1930 H. Cramer determined the probability that the total claim never exceeds the risk reserve in the interval (0, ∞) in an insurance risk process in which the initial risk reserve is x, clients pay premium with a constant unit rate, claims occur in accordance with a homogeneous Poisson process, and the risk sums are independent and identically distributed positive random variables. In 1930 F. Pollaczek and in 1932 A. Y. Khintchine determined the limit of the probability that the waiting time is ⋜x for a single server queue with Poisson input and general service times. In 1952 the author determined the limit of the probability that the virtual waiting time is ⋜x for a single server queue with Poisson input and general service times. Surprisingly, in all three cases we have the same limit. In this paper this curious coincidence is explained, and the aforementioned results are extended to the cases where claims occur in accordance with a recurrent process in the risk process and customers arrive ...