  Ambient energy harvesting is touted as a low cost solution to prolong the
life of low-powered devices, reduce the carbon footprint, and make the system
self-sustainable. Most research to date have focused either on the physical
aspects of energy conversion process or on optimal consumption policy of the
harvested energy at the system level. However, although intuitively understood,
to the best of our knowledge, the idea of self-sustainability is yet to be made
precise and studied as a performance metric. In this paper, we provide a
mathematical definition of the concept of self-sustainability of an energy
harvesting system, based on the complementary idea of eventual outage. In
particular, we analyze the harvest-store-consume system with infinite battery
capacity, stochastic energy arrivals, and fixed energy consumption rate. Using
the random walk theory, we identify the necessary condition for the system to
be self-sustainable. General formulas are given for the self-sustainability
probability in the form of integral equations. Since these integral equations
are difficult to solve analytically, an exponential upper bound for eventual
outage probability is given using martingales. This bound guarantees that the
eventual outage probability can be made arbitrarily small simply by increasing
the initial battery energy. We also give an asymptotic formula for eventual
outage. For the special case when the energy arrival follows a Poisson process,
we are able to find the exact formulas for the eventual outage probability. We
also show that the harvest-store-consume system is mathematically equivalent to
a $GI/G/1$ queueing system, which allows us to easily find the outage
probability, in case the necessary condition for self-sustainability is
violated. Monte-Carlo simulations verify our analysis.
