Distribution and reliability evaluation of max-flow in dynamic multi-state flow networks

In this article, each arc in multi-state flow networks, in addition to multi-valued capacities and associated operation probabilities, is weighted with a transit time. As a result, the value of max-flow from source to sink within specified time horizon is multi-valued. Existing literature evaluates the transit time integrated reliability with a restriction that data is transmitted through k disjoint minimal paths. This article considers the same reliability problem. However, the transmission of data is extended from k disjoint minimal paths to a flow that includes all disjoint and non-disjoint minimal paths simultaneously. This paper presents an algorithm to evaluate the probability distribution of the values of dynamic max-flow. The expectation of dynamic max-flow as a representation of the distribution can be induced directly. The transit time integrated reliability Rd,T is then computed, which is the probability that at least d unit of data can be transmitted from source to sink within time horizon T. This study is the first that discusses transit time integrated reliability in terms of flow. Finally, computational experiments are conducted on a benchmark network to explore the properties of the proposed algorithms.