Asymptotic behaviour of a stopping time related to cumulative sum procedures and single-server queues

Abstract

For a sequence ξ 1 , ξ 2 , · ·· of i.i.d. random variables let X 0 = 0 and X k = max(X k –1 + ξ k , 0) for k = 1, 2, ···. Let . These stopping times are used in Page's (1954) one-sided cusum procedures and are also important in queueing theory. Various asymptotic properties of N x are derived.