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Research Papers

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

[+] Author and Article Information
M. T. Todinov

School of Engineering, Computing
and Mathematics,
Oxford Brookes University,
Oxford OX33 1HX, UK
e-mail: mtodinov@brookes.ac.uk

Manuscript received August 31, 2016; final manuscript received August 1, 2017; published online October 3, 2017. Assoc. Editor: Nasim Uddin.

ASME J. Risk Uncertainty Part B 4(2), 021003 (Oct 03, 2017) (11 pages) Paper No: RISK-16-1111; doi: 10.1115/1.4037519 History: Received August 31, 2016; Revised August 01, 2017

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.

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Figures

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Fig. 1

Probability of unsatisfied random demand as a function of the variance of the log-normal distribution for the demand times

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Fig. 2

Probability of unsatisfied random demand as a function of the spread of the uniform distribution modeling the demand times

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Fig. 3

Probability of unsatisfied random demand as a function of the variance of the normal distributions composing the mixture

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Fig. 4

Probability of unsatisfied random demand as a function of the half-length of the rectangular distributions composing the mixture

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Fig. 6

Overlapping of different order k for random demands on a time interval

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Fig. 5

Probability of unsatisfied random demand as a function of the number density of random demands, for a different number of sources

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Fig. 7

Expected fraction of unsatisfied random demand as a function of the number of consumers n, for different number of sources m servicing the demands

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Fig. 8

The expected fraction of unsatisfied demand as a function of the number of sources, at different demand time fractions ψ=d/L

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Fig. 9

(a) X-configuration and (b) Y-configuration for a fixed number of random events on a time interval

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