Mathematical Methods of Operations Research (1997) 45:325-345 Range Reliability in Random Walks PIERRE VALLOIS D6partement de Math6matiques, Universit6 de Nancy I, B.P. 239, 54506 Vandoeuvre les Nancy Cedex, France CHARLES S. TAPIERO ESSEC, B.P. 105, 95021 Cergy-Pontoise, France Abstract: This paper introduces a definition of reliability based on a process range. Thus, process failure is defined when the range of a process first reaches a given and unacceptable level. The Mean Time To Failure (MTTF) which is defined as the mean of the first time for a range to attain a given amplitude is then calculated for an asymmetric random walk process. The probability distribution of the range is then given and the process reliability over long periods of system operations are then calculated. Applications such as the control of wings movements, stock price and exchange rates volatility (defined in terms of reliability) are also used to motivate the usefulness of range processes in reliability studies. Finally, we point out that there is necessarily a relationship between the range reliability and the propensity of a series to become chaotic. Key Words: Stochastic Processes, Amplitude Process, Reliability. 1 Introduction Run length statistics have, ever since the 1940's been used in various areas (for example, see Mood 1940, Wald and Wolfowitz, 1940, Wolfowitz, 1943, Mosteller, 1941). Recently, they have been applied to radar astronomy, DNA sequencing and to many other fields. In particular in quality control and relia- bility engineering (see Sehwager 1983, Fou and Koutras, 1994). These studies have emphasized the control of a process trajectory however, rather than a process volatility which can be measured by its variance or by its range. The purpose of this paper is to consider an approach to process reliability based on run length distributions of the range when the underlying process is a random walk process, rather than a process location. For example, aging and learning (improving) processes as well as stock price or exchange rates processes can be modeled by such random walks. Traditionally, process malfunction is defined by undesirable performance expressed by location statistics. Process reliability is then defined by the propensity of the systems not to attain a set of given states over a time interval (0, t). Few reliability studies relate to a process variability, 1432-2994/97/45:3/325-345 $2.50 9 1997 Physica-Verlag, Heidelberg