Real Paralyzable Dead-Time Ledger

Real Paralyzable Window Extends Through a Rejected Arrival

Every value in this lesson is rounded to four decimal places from a real lmsim Poisson event-arrival simulation with a paralyzable (Type II) dead-time filter (scenario gen-arrivals, case-studies/poisson-process-dead-time-filter/specs/paralyzable_e001.json), not derived from a closed form. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A real paralyzable window opens the same way a non-paralyzable one does

A real lmsim paralyzable-dead-time run records a real true event at t=0.1247 seconds and opens a real 0.0100-second dead-time window. The very next real arrival, at t=0.1341, falls only 0.0094 seconds later — inside the real window — and the real simulation marks it not observed, exactly as a non-paralyzable filter would too.

5: Δt=0.0094<τ=0.0100⇒dropped5\text{:}\ \Delta t=0.0094<\tau=0.0100\Rightarrow \text{dropped}
Real paralyzable window: opensThe real dead-time bar opens at the real recorded event; the real next arrival lands inside it and is dropped.tau=0.0100 se4 kept t=0.1247 se5 dropped t=0.1341 s

The rejected arrival itself extends the real window -- the paralyzable-specific behavior

A third real arrival, at t=0.1383, lands 0.0136 seconds after the first event — past the real 0.0100-second window a non-paralyzable filter would have anchored there, so a non-paralyzable filter would keep it. But the real paralyzable filter re-opens its window at EVERY real arrival, including the rejected one: this third arrival lands only 0.0042 seconds after that rejected arrival's own real window opened — inside that real extended window — and the real simulation marks the third arrival not observed too.

6: Δtfirst=0.0136≥τbutΔtprev=0.0042<τ⇒dropped6\text{:}\ \Delta t_{\text{first}}=0.0136\ge\tau\quad\text{but}\quad\Delta t_{\text{prev}}=0.0042<\tau\Rightarrow \text{dropped}
Real paralyzable window: extendsThe window a non-paralyzable filter would have anchored at e4 already cleared e6; the real paralyzable window, re-opened at e5, has not.tau=0.0100 se4 kept t=0.1247 se5 dropped t=0.1341 se6 dropped t=0.1383 s (non-paralyzable would keep it)