Reliability evaluation of capacitated-flow networks with budget constraints

Reliable and economic resource allocation in an unreliable flow network

“…The path vectors to demand d with the transmission cost limit c may not be (d; c)-MPs because there may be a directed cycle present within each of them. As shown in [13], the x q 's are truly (d ; c)-MPs in an acyclic ow network, but they may not be (d; c)-MPs in a cyclic ow network. Thus, when considering a cyclic network, we need to test every x q .…”

“…The evaluation schemes of network reliability in a ow network with unreliable nodes and arcs were examined in [8,11,12]. Other factors such as budget constraints and ow time were also incorporated into the evaluation of network reliability in a ow network in [13,14].…”

Single minimal path based backup path for multi-state network

“…According to equations (9) and (10), we get the following results in the first main MP failure case Pr SMPÀBP (SjP 1 P 2 , P 4 ) À Pr DMPÀBP (SjP 1 P 2 , P 4 P 5 ) = 0:145227973 Let P 5 be the backup path in the SMP-BP algorithm. In the same way, we have Pr(SjP 1 P 5 ) = 0:8834393906 and Pr(SjP 2 P 5 ) = 0:697868875, so we obtain the following results in the first main MP failure case Pr SMPÀBP (SjP 1 P 2 , P 5 ) = Pr(P 1 )Pr(SjP 2 P 5 ) + Pr(P 2 )Pr(SjP 1 P 5 ) = 0:2255340914 Pr SMPÀBP (SjP 1 P 2 , P 5 ) À Pr DMPÀBP (SjP 1 P 2 , P 4 P 5 ) = 0:1430968414…”

Generate upper boundary vectors meeting the demand and budget for a p-commodity network with unreliable nodes

“…It can be calculated by applying methods such as inclusion-exclusion method [7,[14][15][16][17][18][19][20][21], disjoint subsets [14,21,24] and state-space decomposition [10,12,13]. Note that Pr{Y 6 X} = Pr{y 1 6 x 1 } · Pr{y 2 6 x 2 } · Á Á Á · Pr{y n+q 6 x n+q } by assumption 2 if Y = (y 1 , y 2 , .…”