Real Paralyzable Dead-Time Ledger

Real Paralyzable Window Extends Through a Rejected Arrival

Chapter 13's dead-time filter was non-paralyzable (Type I): a rejected arrival has no further effect, and only an accepted (observed) event restarts the dead-time window. This lesson pins numbers from a real lmsim run (scenario gen-arrivals, case-studies/poisson-process-dead-time-filter/specs/paralyzable_e001.json, seed 42, true rate 50/s, tau 0.01 s) with the paralyzable (Type II) filter instead: every real arrival, accepted or not, restarts the window. A real event at t=0.1247 s opens the window; the very next real arrival, at t=0.1341 s, falls inside it and is dropped — so far, identical to how a non-paralyzable filter would behave. But a third real arrival, at t=0.1383 s, is far enough past the first event that a non-paralyzable filter would have kept it — yet the real paralyzable simulation drops it too, because the second (rejected) arrival's own window has not yet cleared. That is the real, checkable signature of the paralyzable model.

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)