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All kind of networks, e.g., Internet of Things, social networks, wirelesssensor networks, transportation networks, 4g/5G, etc., are around us to benefitand help our daily life. The multistate flow network (MFN) is always used tomodel network structures and applications. The level d reliability, Rd, of theMFN is the success probability of sending at least d units of integer flow fromthe source node to the sink node. The reliability Rd is a popular index fordesigning, managing, controlling, and evaluating MFNs. The traditional indirectalgorithms must have all d-MPs (special connected vectors) or d-MCs (specialdisconnected vectors) first, then use Inclusion-Exclusion Technique (IET) orSum-of-disjoint Product (SDP) in terms of found d-MPs or d-MCs to calculate Rd.The above four procedures are all NP-Hard and #P-Hard and cannot calculate Rdfor all d at the same time A novel algorithm based on the binary-addition-treealgorithm (BAT) is proposed to calculate the Rd directly for all d at the sametime without using any of the above four procedures. The time complexity anddemonstration of the proposed algorithm are analyzed, and examples areprovided. An experiment is also conducted to compare the proposed algorithm andexisting algorithms based on d-MPs, d-MCs, IET, and/or SDP to validate theproposed algorithm.

Novel Direct Algorithm for Computing Simultaneous All-Levels Reliability of Multi-state Flow Networks