stall-solver

Use when checking whether a venue's restroom count actually survives a halftime rush, planning fixtures for an event space/coffee-shop/school, or auditing fixture counts against plumbing-code minimums — Erlang-C queue math on toilet demand (arrival rates by venue type, burst-break simulation, IPC fixture tables) plus verdicts (OK / TIGHT / FAIL) and queue-clearing times.

Stall Solver — Restroom Queue Math

Restroom planning has a hard split personality: plumbing codes size fixtures for sustained occupancy, but the pain people remember is a burst — halftime, intermission, the session break. A venue can be fully code-compliant and still have a 25-minute women's-room line, because 40% of 400 attendees hit the toilets in the same 15-minute window. This skill runs both checks: queueing math on arrivals vs fixtures, and fixture-count audits against code tables.

When to Use

• Evaluating a venue (theater, bar, conference hall, school) for an event — will the break queue be survivable?
• Planning a cafe/restaurant/office buildout — how many stalls and urinals to install
• Reviewing an architect's fixture schedule against expected use
• Diagnosing an existing restroom-line problem (is it fixture count, or is it dwell time / layout?)
• All-gender restroom conversions (stalls-only) — how many stalls replace a given WC+urinal split

Don't use for: residential bathrooms, portable-toilet logistics for outdoor festivals (different tables entirely), or ADA clearance/ layout questions (those are geometry, not counts).

The Model

Three layers, each answering a different question:

1. Demand rates — visits/person/hour by venue type (concert 0.45, conference 0.35, office 0.25...) and median occupancy per fixture (urinal ~35 s, stall ~95 s, sink ~20 s).
2. Queue math — M/M/c with Erlang-C expected wait. Sustained mode gives steady-state utilization and average wait; event mode takes a burst share (default 40% of attendees) over a window (default 15 min) and reports burst utilization, expected wait, and total drain time.
3. Code minimums — IPC-style assembly fixture ladder by occupancy bucket (water closets per sex, urinals, lavatories), with linear scaling beyond the table.

Key asymmetry the math exposes: urinals are throughput machines (3x the turnover of a stall) but only serve half the population; female/stall demand is where queues actually form. In queue terms, stalls need more fixtures for the same arrival rate because service time is 3x longer.

Workflow

1. Establish the worst case: peak headcount AND the break structure (is there a simultaneous break — halftime, intermission, plenary gap?).
2. Sustained check:

   bash

   python3 scripts/stall_detector.py demand --venue concert \
       --attendees 600 --fixtures 8 --urinals 6
   

3. Burst check (the one that matters at halftime):

   bash

   python3 scripts/stall_detector.py event --venue sports \
       --attendees 400 --fixtures 6 --urinals 4 \
       --burst-share 0.5 --window-min 15
   

   Verdicts per fixture class: OK (ρ<0.8), TIGHT (0.8-1.0), FAIL (ρ≥1 — the line grows faster than it drains).
4. Code audit (buildout/permit stage):

   bash

   python3 scripts/stall_detector.py audit --venue conference \
       --occupancy 350 --f-wc 8 --m-wc 3 --m-ur 2 --f-lav 6 --m-lav 4
   

5. Fix FAILs by, in order of cost-effectiveness: staggering the break (free), converting WCs to urinals in men's rooms, adding stalls (esp. all-gender), adding a fixture bank at a remote location to split arrival streams.

Reading the numbers

• Utilization ≥100% in event mode = the queue never clears during the window; the drain-time figure tells you how long the line runs after the break ends.
• A 56-minute "avg wait" in a 15-minute window means the math says the burst saturates the room — people self-adjust (leave early, skip), so read it as "severe," not literal.
• Code-minimum compliance ≠ no lines. The audit is a floor, not a comfort guarantee; run the demand/event checks separately.

Common Pitfalls

1. Sizing only from code tables. Tables assume steady use; a single 15-minute burst can require 2-3x the sustained fixture load.
2. Ignoring the female/stall asymmetry. Same arrival rate, 3x service time = ~3x the fixtures needed for the same wait. The classic "huge men's room, huge women's line" design error.
3. Treating all-gender conversion as a pure swap. Removing urinals without adding ~1 stall per 2 urinals lost typically FAILs the burst check — urinals were carrying men's throughput.
4. Forgetting sinks. Lavatory demand peaks with the restroom exit wave; hand-wash occupancy (~20 s) usually clears fine but code requires the counts anyway.
5. Miscounting "fixtures." A double-lav counts as 2; a WC in a stall counts as 1 regardless of stall size; urinals with privacy partitions are still urinals for throughput.
6. Blindly trusting the bundled code table. Local amendments (UPC jurisdictions, city addenda) shift minimums; always verify against the adopted edition before permitting.

Verification Checklist

• Peak headcount is the real one (fire-marshal occupancy, not RSVP)
• Break simultaneity assessed (halftime = everyone; staggered sessions = lower burst share)
• Both sustained AND burst checks run
• Fixture counts match actual architectural counts (not plans from an earlier revision)
• Code table cross-checked against locally adopted plumbing code

One-Shot Recipe: is the theater's intermission survivable?

bash

python3 scripts/stall_detector.py event --venue theater \
    --attendees 800 --fixtures 12 --urinals 8 --window-min 20

If stalls read TIGHT/FAIL: recommend adding 2-4 all-gender stalls OR holding the house 5 minutes before intermission ends (staggering the return wave does nothing — the arrival burst is the problem; lengthen the intermission instead).