Lower and upper bounds of the superposition of renewal processes and extensions

Consider a system consisting of multiple sockets into each of which a component is inserted. If a component fails, it is replaced immediately and system operation resumes. Then the failure process of the system is the superposition of renewal processes (or superimposed renewal process, SRP). If the label of the components that cause the system to fail are unknown but the times between failures are known, we refer to such data as {\it masked failure data}. To estimate the SRP based on masked failure data is challenging. This paper obtains the lower and upper bounds of the rate of the SRP when only masked failure data are available. If repair (rather than replacement) is conducted on failed components, the failure process of the system is the superposition of generalized renewal processes (SGRPs). The paper then derives the lower and upper bounds of the rate of SGRPs and proposes to use a weighted linear combination of the bounds to approximate the SGRP. Discussions are provided for possible extensions of the bounds for systems with other structures such as parallel systems. An algorithm for simulating the SGRP is then proposed. Numerical examples are used to illustrate the proposed approximation method.