Extended Phd Thesis Abstract Regenerative Techniques for Estimating Performance Measures of Highly Dependable Systems with Repairs

Consider a system with N components which are subject to failures and repairs. Suppose that each component has a single operating state denoted by 1 and a single failed state denoted by 0. Let X i (t) be the state of component i at time t and let X(t) = (X 1 (t); : : : ; X N (t)) be the state of the system. deened by (x) = (1 if the system operates in state x 0 if the system is failed in state x (For a review of deenitions from reliability, see Barlow and Proschan 2]). Let 1 be the vector with all components equal to one and assume X(0) = 1. Now deene the set F = fx 2 : (x) = 0g of failure states and the following performance measures U = lim t!1 P X(t) 2 F ]; T = E inf(t : t > 0; X(t) 2 F)] : U is the long-run system unavailability whereas T is the expected time to system failure. The exact evaluation of these measures and the computation of tight bounds are diicult problems even for systems of moderate scale (see Ball, Colbourn, and Provan 1]). Consequently, computer simulation frequently becomes the most suitable method for their estimation. The demands put on modern communications and computer systems require them to be highly reliable. In other words, the entrance of X(t) to the set F is a rare event. This property causes crude Monte Carlo simulation to be ineecient since it requires prohibitively long runs to produce precise estimates. We focus on systems with the following properties: (a) They are Markovian, i.e., the time-to-failure of a component and its repair time are exponential random variables; (b) Dependent component failures are allowed but failure propagation is disallowed; (c) Repair policies are allowed to utilize any xed number of repairmen; and (d) The systems can become highly reliable either by making the failure rates much smaller than the repair rates or by introducing component redundancies. For example, in the system depicted in Figure 1 each link l between two nodes represents a set of trunks with cardinality u l. The times to failure for trunks are considered to be independent, exponential random variables with rates l while the respective repair times are exponential with rates l. The Markovian assumption makes the state process of the system fX(t); t 0g a Markov …