Reliability and Risk Controlled by the Simultaneous Presence of Random Events on a Time Interval

Abstract

The paper treats the important problem related to risk controlled by the simultaneous presence of critical events, randomly appearing on a time interval and shows that the expected time fraction of simultaneously present events does not depend on the distribution of events durations. In addition, the paper shows that the probability of simultaneous presence of critical events is practically insensitive to the distribution of the events durations. These counter-intuitive results provide the powerful opportunity to evaluate the risk of overlapping of random events through the mean duration times of the events only, without requiring the distributions of the events durations or their variance. A closed-form expression for the expected fraction of unsatisfied demand for random demands following a homogeneous Poisson process in a time interval is introduced for the first time. In addition, a closed-form expression related to the expected time fraction of unsatisfied demand, for a fixed number of consumers initiating random demands with a specified probability, is also introduced for the first time. The concepts stochastic separation of random events based on the probability of overlapping and the average overlapped fraction are also introduced. Methods for providing stochastic separation and optimal stochastic separation achieving balance between risk and cost of risk reduction are presented.

Domain-independent approach to risk reduction

“…The method of stochastic separation requires determining the expected time fraction of overlapping of a particular order, for risk-critical random events on a time interval. This can be done by using the analytical methods presented in (Todinov, 2017).…”

“…The demand is randomly located over a time interval of length L. It has been shown (Todinov, 2017) that the expected time fraction of overlapping of random demands (simultaneous random demands) of order k = 0, 1, 2,...,n is given by the binomial expansion of the expression [ and so on. Clearly,…”

“…It measures the expected fraction of unsatisfied demand for any number m 01 mn    of sources servicing the random demands. Stochastic separation can be done by reducing the probability  with which the random demands are initiated, by reducing the number of random demands n and by increasing the number of units m servicing the random demands (Todinov, 2017). The optimal number of units m servicing the random demands can be determined, which provides the optimal balance between acceptable risk level and cost.…”