Counting statistics distorted by two dead times in series which end with an extended type dead time

The distorted counting statistics resulting from two dead times occurring in series are discussed. The cases studied are those of series combinations of non-extended–extended (NE–E) dead times and of extended–extended (E–E) dead times under a Poisson input distribution. Three choices of time origin of the counting processes are considered, leading to the distinct statistics of three distinct processes—ordinary, equilibrium, and shifted processes. A set of formulae is presented for the event interval densities, corresponding Laplace transformations, the expected number and variance of counts in a given duration and the associated asymptotic expressions. Results are validated by comparison with previously published Monte Carlo simulations and checking the mathematical expressions in certain reduction limits. A possible application of the derived formulae is discussed.