# Superimposed processes

## Abstract

Suppose that l different kinds of arrivals are identified. Let *N*_{ i }(*t*) be the number of arrivals of type *i* that occur up to time *t*. Then\(N\left( t \right) = {N_1}\left( t \right) + \cdots + {N_l}\left( t \right) \), the summary process, is the total number of arrivals of all types that occur up to time *t*. Arrival types may, for instance, be failures of *l* components of a series system. *N*(*t*) would then count the system failures. Or in an application to reactor safety, *N*_{ i }(*t*) might be the number of accidents due to the ith cause and *N*(*t*) would be the total number of accidents due to all causes. Writing*M*(*t*)=*EN*(*t*) and *M*_{ i }(*t*)=*EN*_{ i }(*t*), then of course*M*(*t*)=*M*_{ 1 }(*t*)+⋯+*M*_{ l }(*t*) and \(\mu \left( t \right) = \mu \left( t \right) + \cdots + {\mu _l}\left( t \right) \).

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