The measurement of reliability in stochastic transport networks

This paper proposes a method which identifies those links or nodes whose failure would impair network performance most. It is assumed that all links have two costs, a normal cost and a failed cost, both of which may be traffic-dependent. A 2-player, non-cooperative, zero sum game is envisaged between a router, seeking a least cost path, and a network tester, with the power to fail one link. At the mixed strategy Nash equilibrium, link choice probabilities are optimal for the router and link failure probabilities are optimal for the network tester. Finding the equilibrium involves solving a maximin programming problem. When link costs are fixed (not traffic-dependent), the maximin problem may be solved as a linear programming problem. Two forms of the linear programming problem are presented, one requiring path enumeration and the other not. Where link costs are traffic-dependent, for example where queuing is a feature, the mixed strategy Nash equilibrium may be found by the method of successive averages. A numerical example is presented to illustrate the approach on a stochastic network with queuing. While the example relates to single commodity flows, it is noted that the method of successive averages approach may also applied where flows are multi-commodity, for example where there are multiple origins and destinations.