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Simplification of inclusion-exclusion on intersections of unions with application to network systems reliability

机译：简化联合交叉处的包含/排除及其在网络系统可靠性中的应用

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摘要

Reliability of safety-critical systerris is a paramount issue in system engineering because in most practical situations the reliability of a non series-parallel network system has to be calculated. Some methods for calculating reliability use the probability principle of inclusion exclusion. When dealing with complex networks, this leads to very long mathematical expressions which are usually computationally very expensive to calculate. In this paper, we provide a new expression to simplify the probability principle of inclusion exclusion formula for intersections of unions which appear when calculating reliability on non series-parallel network systems. This new expression exploits the presence of many repeated events and has many fewer terms, which significantly reduces the computational cost. We also show that the general form of the probability principle of inclusion exclusion formula has a double exponential complexity, whereas the simplified form has only an exponential complexity with a linear exponent. Finally, we compare its computational efficiency against the sum of disjoint products method KDH88 for a simple artificial example and for a door management system, which is a safety-critical system in aircraft engineering. (C) 2018 Elsevier Ltd. All rights reserved.
机译：安全至关重要的系统的可靠性是系统工程中的首要问题，因为在大多数实际情况下，必须计算非串并联网络系统的可靠性。一些计算可靠性的方法使用包含排除的概率原理。当处理复杂的网络时，这会导致很长的数学表达式，这些表达式通常在计算上非常昂贵。在本文中，我们提供了一个新的表达式，以简化针对在非串并联网络系统上计算可靠性时出现的并集交叉点的包含排除公式的概率原理。这个新表达式利用了许多重复事件的存在，并减少了许多项，从而大大降低了计算成本。我们还表明，包含排除规则的概率原理的一般形式具有两倍的指数复杂度，而简化形式仅具有线性指数的指数复杂度。最后，我们将其计算效率与不相乘积方法KDH88的总和进行比较，以一个简单的人工示例和一个门管理系统为例，该系统是飞机工程中的安全关键系统。 （C）2018 Elsevier Ltd.保留所有权利。

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