Random events are typically encountered at both the start and stop inputs when it is not possible to periodically stimulate the process to be measured. An example is the measurement of the lifetime of a excited state in a nucleus when the excited state is populated as the result of radioactive decay. For example, consider the emission of an alpha particle from a radioactive sample signaling the decay which forms the excited state, followed by the emission of a gamma ray marking the decay of the excited state to the ground state. The alpha particle detector supplies the pulse for the TAC start input, and the gamma ray detector feeds the stop input. Since the detection probability for both types of radiation is modest, there is a moderate probability that 1) a start event will be detected without detecting the correlated stop pulse, 2) a stop pulse will be detected without detecting the correlated start event, and 3) an uncorrelated pair of start and stop events will be recorded. These actions can cause dead time or uncorrelated background in the measured time spectrum.

If it is sufficient to measure the correct shape of the decay curve to extract the lifetime, then equations (4) through (11) of Case 1 provide an adequate description of the measurement. If the absolute probability of detecting a particular start-to-stop time interval is also required, the effect of dead time losses for the start input must be accounted for.

If the start events are randomly and uniformly distributed in time (constant counting rate), the throughput relationship is expressed by^{1, 2}

where N₁ is the number of start events at the detector (before dead time losses) and n₁ is the number of start pulses accepted by the TAC/MCA combination. U(T_d –T_e) is the previously defined step function, and R₁ is the counting rate of start events at the detector, i.e.,

Normally T_e << T_d, and equation (26) simplifies to the form for non-extending dead time.

where

The simplest way to account for the relation in equation (28) is to use a simple livetime clock that turns off for the combined dead time of the TAC and MCA. The relationship between live time, t_L, and real time, t, is given by¹

Consequently, the joint probability of detecting a start pulse and a stop pulse such that the start-to-stop time interval is destined for channel i is

The division by t_L and Δt expresses both the start and stop probabilities on a per-unit-time basis.

If the live time, t_L, required to record n₁ accepted start pulses is measured, the relative standard deviation in t_L is given by¹

Applying a propagation-of-errors calculation leads to the expression for the relative standard deviation in P_i

Because q_i << n₁, the relative standard deviation in equation (33) will be dominated by q_i.

¹Ron Jenkins, R. W. Gould, Dale Gedcke, Quantitative X-Ray Spectrometry, Marcel Dekker, New York, 1981, First Edition, Chapter 4.