Real Paralyzable Filtering Rejects More Events Than Non-Paralyzable
Two real lmsim runs share the identical true event-arrival stream — same seed 42, true rate 50/s, duration 10 s, tau 0.01 s — and differ only in dead-time model. Chapter 13's real non-paralyzable run kept 349 of the 516 real true events. This lesson's real paralyzable run, on the exact same true stream, keeps only 314. The comparison is real-vs-real rather than real-vs-closed-form: the paralyzable model's asymptotic observed rate is true-rate times e to the minus true-rate-times-tau, an irrational value that cannot be pinned as an exact fraction the way chapter 13's non-paralyzable asymptote could. The real 35-event gap is the paralyzable rule's direct cost: a rejected arrival still restarts the dead-time clock, so clusters of close arrivals lose more real events than a non-paralyzable filter ever would.
Two real lmsim runs over the identical true event-arrival stream (same seed, rate, duration, and tau), compared real-vs-real rather than against a closed form, since the paralyzable asymptotic rate is irrational and cannot be pinned as an exact Fraction. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The identical true stream splits into two different real observed counts
The same real 516 true Poisson events, at the same real rate over the same real 10 seconds with the same real tau=0.0100 seconds, are filtered by two real lmsim runs that differ only in dead-time model. The real non-paralyzable run keeps 349 of them; the real paralyzable run, on the identical true stream, keeps only 314.
516→349(non-paralyzable)
Every arrival extending the window -- not just accepted ones -- costs real events
The real paralyzable count is genuinely lower: 314 observed events against the real non-paralyzable 349, a real difference of 35 events on the identical true stream. That real gap is exactly the paralyzable rule's signature: a rejected arrival still restarts the dead-time clock, so clusters of close arrivals cost more real observed events than a non-paralyzable filter would ever lose.