If events of type
occur at an average rate
with each type of event independent of any other and each can be modelled by a Poisson process with parameter
respectively, then the occurrence of any type of event will occur at a rate
and it seems plausible to model the occurrences of these events by a Poisson process with parameter
More concisely let
be the waiting time for the occurrence of any event. If
is the time for the event I to occur then
so
Since each process is Poisson,
hence
Then