To help (or not) to clarify discussions we're having about how to best interpret timing curves, for instance in EPD, I made a very trivial toy model in Mathematica.

• The detector pulse is an infinitely sharp step up (to amplitude 1) at the hit time, followed by exponential decay to baseline 0 with a time constant tau
• The hit time is a uniform random variable in the range 0 to 1
• The gated integrator has a perfect rectangular response, integrating its input from time T to T+2, i.e. the width is 2

Note that on purpose I picked the ratio of gate to hit time distribution to be 2, since that is roughly what we have with the QT's today.
Akio was suggesting the exponential pulse with extreme steep rising edge as a worst-case sort of pulse. It's also of course the simplest pulse to write a decent formula for.
This toy model could be modified for other pulse shapes though.

Please see the attached pdf. Even if you're not a Mathematica-an I think the formulas and plots will be clear enough. (I am certainly neither patient bir careful enough to do this calculation by hand anymore. Maybe when I was young!)

There are three plots (on each the x-axis is the "phase" parameter corresponding to GATE START for QT, and the curves are the mean value of the integral, i.e. the effective gain of this system, and the rms of the integral). First case is for tau=1, then tau=0.3, then tau=0.05.

The plots seem to me to confirm that we should set the timing to be at the sweet spot of the timing curve, for smallest stochasticity in the detector gain. (Both in absolute terms and relative to the gain.)