Asymptotic distributions for downtimes of monotone systems

Consider a monotone system with independent alternating renewal processes a s component processes, and assume the component uptimes are exponentially d istributed. In this paper we study the asymptotic properties of the distrib ution of the rth downtime of the system, as the failure rates of the compon ents converge to zero. We show that this distribution converges, and the li miting function has a simple form. Thus we have established an easy computa ble approximation formula for the downtime distribution of the system for h ighly available systems. We also show that the steady state downtime distri bution, i.e. the downtime distribution of a system failure occurring after an infinite run-in period, converges to the same Limiting function as the f ailure rates converge to zero.