Correlated multivariate shock models associated with a renewal sequence and its application to analysis of browsing behavior of internet users

A correlated multivariate shock model is considered where a system is subject to a sequence of J different shocks triggered by a common renewal process. Let (Y (k))∞k=1 be a sequence of independently and identically distributed (i.i.d.) nonnegative random variables associated with the renewal process. For the magnitudes of the k-th shock denoted by a random vector X(k), it is assumed that [X(k), Y (k)] (k = 1, 2,…) constitute a sequence of i.i.d. random vectors with respect to k while X(k) and Y (k) may be correlated. The system fails as soon as the historical maximum of the magnitudes of any component of the random vector exceeds a prespecified level of that component. The Laplace transform of the probability density function of the system lifetime is derived, and its mean and variance are obtained explicitly. Furthermore, the probability of system failure due to the i-th component is obtained explicitly for all i ∈ J = {1,…, J}. The model is applied for analyzing the browsing behavior of internet users.