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

Counting Statistics Modified by Two Dead Times in Series

“…The features of the counting statistics were discussed for each case of the E-NE and NE-NE series DTs. Through the present and previous studies [40], all the TID functions and total density functions were known for the series of two DTs. Therefore the advanced task of providing the mean counting rates for the series of three DTs was achieved.…”

“…The TID function and total density for the NE-E DTs were derived in a previous study [40]. The total density, NE-E 2D(t), is given by the following: This is simply the input total density NE 1D(t) translated by the magnitude of τ2-τ1 and modified by a factor of exp(-ρτ2+ρτ1).…”

“…Even if the TID is measured, it normally indicates the magnitude of the dominant DT in the counting arrangement, whereas modeling the counting behavior by either a single E or NE DT is insufficient for a realistic system to explain the counting rate or variance, especially at higher counting rates. DT models based on a series combination of two or more DTs are more suitable because a real counting system is normally composed of a detector and several electronic circuit modules combined in series [1,14,27,33,[36][37][38][39][40][41]. An exact analysis of the DT behavior of the counting arrangement requires a general theoretical treatment of many DTs occurring in series.…”

“…In a previous study, the event TID functions, the exact mean and variance of the counts in a measurement interval, and the associated asymptotic expressions were given in a full set for a series of two DTs ended by an E-type DT, which were denoted as NE-E and E-E cases, respectively [40]. Most of the derivations were based on the existing formalism arising from renewal theory and operational calculus with associated tedious algebra [1,2,[9][10][11][12][13][14][15][16][17].…”

“…Most of the derivations were based on the existing formalism arising from renewal theory and operational calculus with associated tedious algebra [1,2,[9][10][11][12][13][14][15][16][17]. An explicit derivation can be avoided for the first event TID function, g1(t), of the equilibrium process and the probability distribution, Wk(t), of exact k counts in time t, in order to obtain the exact mean and variance of the counts [40]. Instead, these values could be obtained from an integral formula that leads to a closed form or a numerical one.…”