Proceedings of the 2010 Winter Simulation Conference B. Johansson, S. Jain, J. Montoya-Torres, J. Hugan, and E. Y¨ ucesan, eds. COMBINATION OF CONDITIONAL MONTE CARLO AND APPROXIMATE ZERO-VARIANCE IMPORTANCE SAMPLING FOR NETWORK RELIABILITY ESTIMATION Hector Cancela Universidad de la Rep´ ublica Montevideo, URUGUAY Pierre L’Ecuyer DIRO, Universit´ e de Montreal C.P. 6128, Succ. Centre-Ville Montr´ eal (Qu´ ebec), H3C 3J7, CANADA Gerardo Rubino INRIA Rennes Bretagne Atlantique Campus Universitaire de Beaulieu 35042 Rennes Cedex, FRANCE Bruno Tuffin INRIA Rennes Bretagne Atlantique Campus Universitaire de Beaulieu 35042 Rennes Cedex, FRANCE ABSTRACT We study the combination of two efficient rare event Monte Carlo simulation techniques for the estimation of the connectivity probability of a given set of nodes in a graph when links can fail: approximate zero-variance importance sampling and a conditional Monte Carlo method which conditions on the event that a prespecified set of disjoint minpaths linking the set of nodes fails. Those two methods have been applied separately. Here we show how their combination can be defined and implemented, we derive asymptotic robustness properties of the resulting estimator when reliabilities of individual links go arbitrarily close to one, and we illustrate numerically the efficiency gain that can be obtained. 1 INTRODUCTION In the design of telecommunication networks, an historically important topic from the reliability point of view has been the study of connectivity properties of the network topology. This study can be accomplished by building a probabilistic model of the network focusing on the possible failure of its components, from which we can compute the probability of still supporting the specified communications when taking these failures into account. Today, these problems are of high importance with the technological arrival of different types of wireless architectures. In these contexts, network components (and particularly, links) are more prone to fail because of changes in the environment, changes which are difficult to control. Network reliability is the branch of Operations Research where these problems are studied. In this area, exact computations are usually hard (i.e., highly computationally expensive), and Monte Carlo techniques provide the most powerful tools for quantitative evaluations of the systems under consideration. In this paper, we consider the most representative model in network reliability. One can think of the system as a communication network, but there are several other applications of that same model. The model is represented by a non-oriented graph G =(N , L ) where N is the set of m nodes, and L = {1,...,ℓ} is the set of links connecting nodes. Nodes are assumed perfect, in the sense that they never fail. On the other hand, links may fail, with probability q i for link i (1 ≤ i ≤ ℓ), and failures are assumed to occur independently across links. Our goal is to compute the probability q(G ) that a given subset K of the set of nodes is not connected, that is, that these nodes are not in the same connected component of G after removing the failed links. In this context, the graph together with the probabilistic model is often called a network, and q(G ) is the network unreliability. The most frequent case is when estimating the two-terminal or source-to-terminal unreliability, where K is comprised of only two nodes. Formally, define X i = 1 if link i works, and X i = 0 otherwise. The random configuration of the graph is the vector X =(X 1 ,..., X ℓ ). Let us denote by φ (·) the following structure function of the network: φ (·) is defined on the set of all configurations {0, 1} ℓ and takes values in {0, 1}; for each configuration x =(x 1 ,..., x ℓ ), we have φ (x)= 1 if the set K is not included in the same connected component in the graph that contains only the links i