  All kind of networks, e.g., Internet of Things, social networks, wireless
sensor networks, transportation networks, 4g/5G, etc., are around us to benefit
and help our daily life. The multistate flow network (MFN) is always used to
model network structures and applications. The level d reliability, Rd, of the
MFN is the success probability of sending at least d units of integer flow from
the source node to the sink node. The reliability Rd is a popular index for
designing, managing, controlling, and evaluating MFNs. The traditional indirect
algorithms must have all d-MPs (special connected vectors) or d-MCs (special
disconnected vectors) first, then use Inclusion-Exclusion Technique (IET) or
Sum-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 Rd
for all d at the same time A novel algorithm based on the binary-addition-tree
algorithm (BAT) is proposed to calculate the Rd directly for all d at the same
time without using any of the above four procedures. The time complexity and
demonstration of the proposed algorithm are analyzed, and examples are
provided. An experiment is also conducted to compare the proposed algorithm and
existing algorithms based on d-MPs, d-MCs, IET, and/or SDP to validate the
proposed algorithm.
