Monte Carlo simulation offers a valuable tool for capturing the complex stochastic behavior of distributed, interconnected systems. To reduce the associated computational burden, it is possible to resort to biasing techniques which improve the efficiency of the simulation. In this paper, a biasing method is proposed for improving the efficiency of the unreliability estimate of complex multi-state network systems, in which the arcs and the nodes can stay in various states of different performance. The biasing is founded on a sample strategy tailored to encourage the multi-state system to enter failed configurations with respect to the required demand at the network target node. This is achieved by forcing the arcs to visit their lower performance states. The performance of the method is tested on a literature case study and a sensitivity analysis is carried out with respect to the parameter controlling the intensity of the bias.