JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 30, 448-470 (1970) Distribution of Time above a Threshold for Markov Processes* LAWRENCE D. STONE AND BARRY BELKIN Daniel H. Wagner, Associates, Paoli, Pennsylvania 19301 AND MARTIN AVERY SNYDER Bryn Mawr College, Bryn Mawr, Pennsylvania 19010 and Daniel H. Wagner, Associates, Paoli, Pennsylvania 19301 Submitted by K. J. Astrom Received: June 26, 1969 1. INTRODUCTION In this paper, we are primarily concerned with the problem of finding the distribution of the time a Markov process, {X, : t 3 0}, with stationary transition probabilities spends above a fixed level in a given time interval. We let H,(t) be the amount of time the process spends at or above the level 1 in the interval [0, t]. Then we wish to findF,(t, T) = P[H,(t) < 7 1 X0 = x]. We attack this problem by finding an equation whose unique solution is the double transform of F, . The time above a threshold problem arises quite naturally in the context of communication when signal strength is assumed to be a Markov process. Assuming that communication is possible only when the signal strength is above a threshold 1, F,(t, T) gives the distribution of the amount of time in [0, t] during which communication is possible. In general, knowledge of F, will be of interest whenever it is of importance to measure the cumulative time a stochastic process (representing possibly radiation, noise, or pressure) spends above a critical level. The time above a threshold problem has been considered in the context of the fluctuations of a random walk. Let N, be the number of positive partial sums among the first 1z partial sums in a random walk. It has been shown by Anderson [l] and Spitzer [2] that if the increments of the random walk have mean zero and finite variance, then N&z has a limiting arcsine distribution. * This work was partially supported by the Naval Analysis Programs, Office of Naval Research under Contract No. N00014-69-C-0278. 448