  We consider a network consisting of $n$ components (links or nodes) and
assume that the network has two states, up and down. We further suppose that
the network is subject to shocks that appear according to a counting process
and that each shock may lead to the component failures. Under some assumptions
on the shock occurrences, we present a new variant of the notion of signature
which we call it t-signature. Then t-signature based mixture representations
for the reliability function of the network are obtained. Several stochastic
properties of the network lifetime are investigated. In particular, under the
assumption that the number of failures at each shock follows a binomial
distribution and the process of shocks is non-homogeneous Poisson process,
explicit form of the network reliability is derived and its aging properties
are explored. Several examples are also provided
