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Consider a system consisting of multiple sockets into each of which acomponent is inserted. If a component fails, it is replaced immediately andsystem operation resumes. Then the failure process of the system is thesuperposition of renewal processes (or superimposed renewal process, SRP). Ifthe label of the components that cause the system to fail are unknown but thetimes between failures are known, we refer to such data as {\it masked failuredata}. 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 whenonly masked failure data are available. If repair (rather than replacement) isconducted on failed components, the failure process of the system is thesuperposition of generalized renewal processes (SGRPs). The paper then derivesthe lower and upper bounds of the rate of SGRPs and proposes to use a weightedlinear combination of the bounds to approximate the SGRP. Discussions areprovided for possible extensions of the bounds for systems with otherstructures such as parallel systems. An algorithm for simulating the SGRP isthen proposed. Numerical examples are used to illustrate the proposedapproximation method.

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