State‐space partition techniques for multiterminal flows in stochastic networks

Abstract

This article develops state-space partition methods for computing performance measures for stochastic networks with demands between multiple pairs of nodes. The chief concern is the evaluation of the probability that there exist separate, noninteracting flows that satisfy all demands. This relates to the multiterminal maximum flow problem discussed in the classic article of Gomory and Hu. The network arcs are assumed to have independent, discrete random capacities. We refer to the probability that all demands can be satisfied as the network reliability (with the understanding that its definition is application dependent). In addition, we also consider the calculation of secondary measures, such as the probability that a particular subset of demands can be met, and the probability that a particular arc lies on a minimum cut. The evaluation of each of these probabilities is shown to be NP-hard. The proposed methods are based on an iterative partition of the system state space, with each iteration tightening the bounds on the measure of interest. This last property allows the design of increasingly efficient Monte Carlo sampling plans that yield substantially more precise estimators than the standard Monte Carlo method that draws samples from the original capacity distribution.

Decomposition algorithms for system reliability estimation with applications to interdependent lifeline networks

“…The decomposition approach we use is inherently computationally hard, meaning that full decompositions are only attainable for relatively small systems; however, the algorithms we introduce here aim to produce tight bounds within a few iterations of such algorithms. Furthermore, if slow convergence of bounds is unavoidable, we adopt an importance and stratified sampling scheme on the remaining unexplored sets that has proven far more efficient than MCS . Our more direct and decoupled treatment of the system's probabilistic state space with respect to previous studies allows us to recycle all computations throughout the seismic catalog as well.…”

“…We may use the notation S = [ α , β ] to refer to the hyper‐rectangular subset S with lowest and highest capacity levels α and β , respectively. We can compute the probability of v ∈ S as where p i ( j ) are component's discrete state probabilities (pmf) as described previously. Note that for valid pmfs, Equation yields .…”

“…In particular, is not guaranteed to be hyper‐rectangular, and thus, we need to further partition it into disjoint hyper‐rectangular sets. We can partition into disjoint hyper‐rectangular subsets U ( i , j ) as follows: …”

“…If storage in external memory is not possible, one can simply store lower and upper bounds. In particular, one can initialize lower bound P F = 0 and upper bound P I = 1 and replace steps 9 and 14 in Algorithms 1 and 2 with corresponding update rules and . However, when considering many FMPs, discarding subsets in F and I can result into a considerable amount of recomputation.…”

“…It has been pointed out that SSP algorithms work in a branch‐and‐bound fashion. () Therefore, a way to better describe the previous issue is to use a tree‐like representation among unexplored subsets. For example, when removing the first input subset U i − 1 from U and decomposing it (Equation ), we expect to generate at most a new subproblems U ( i , j ) (Equations and ) to be pushed into U .…”