Abstract
Widening the exit speeds evacuation only when there are many people; with few, the walking distance sets the time (Paper 895). Which exit people head for, the choice of exit, was not included there. With the walking speed of 1.3 m/s and exit flow of 1.3 persons/(m s) of Paper 895, this paper places exits at both ends of a 100 m corridor and compares the time for the last person to leave under a rule of heading for the nearer exit and under the best choice of boundary. No new theorem or law is claimed.
First, with few people the two rules take almost the same time: with 60 people, averaged over 20 arrangements, the best time was $\mathbf{0.994959}$ (widths 1 to 3) to $0.995980$ (1 to 2) of the time heading for the nearer exit, staying in this range for all three width ratios (Section 3).
Second, the more people, the more time heading for the nearer exit loses: for widths 1 to 3 the mean ratio was $0.994959$, $0.674894$ and $\mathbf{0.497270}$ for 60, 200 and 600 people (Section 3).
Third, with many people the ratio approaches a value set by the widths alone: if only queueing sets the time, the ratio is twice the narrow width over the sum of widths, $0.666667$ for 1 to 2 and $0.500000$ for 1 to 3. The means for 600 people, $0.662493$ and $0.497270$, differed from these by $-0.004173$ and $-0.002730$ (Section 4).
Fourth, with many people the best boundary moves to the point dividing by width: for 600 people the mean best boundary was $\mathbf{33.398}$ m for 1 to 2 (width point $33.333$ m) and $\mathbf{25.274}$ m for 1 to 3 ($25.000$ m). For 200 people at 1 to 3 it was $32.862$ m, between the width point and the middle (Section 4).
Fifth, the separator is whether people are divided by distance or by exit width: with 60 people, the middle boundary dividing by walking distance took almost the same time as the best. With 600 people, a boundary within 1 m of the point dividing by exit width was best (Section 5).
Keywords: evacuation, exit choice, queueing, exit width, evacuation time, corridor.
Scope (what is not claimed)
No new theorem or law is claimed: that evacuation time splits into walking time and time through exits, that people crowding to the nearest exit lengthen its queue and slow the whole, and that dividing people according to what each exit can pass speeds evacuation, are all known in evacuation calculations. These are set values: the walking speed of 1.3 m/s and exit flow of 1.3 persons/(m s) are the model values of Paper 895; the 100 m corridor, exits at both ends, uniformly placed people, widths of 1 m and 1, 2, 3 m, 60, 200 and 600 people and 20 arrangements are set here. They correspond to no building or code. Human movement is not included: pushing, poor visibility from smoke, time before starting to walk, people choosing exits for themselves, and switching exits on the way are not included. The boundary is set from outside. No signage is recommended, which rule should guide people is not treated. Only the times of the two rules are compared. Relation to earlier papers: Paper 895, in its scope note, did not include smoke or human movement, leaving out poor visibility from smoke, pushing, the choice of exit and time before starting to walk; this paper sits in the seat of the choice of exit (no earlier occupant). Paper 895 showed that widening the exit speeds evacuation only when there are many people, and that with few the walking distance sets the time. Here the change in time from how people are divided between two exits of different widths is counted. What is added: comparing the two rules over 20 arrangements for three numbers of people and three width ratios, deriving the ratio when only queueing matters from the widths and comparing it with 600 people, measuring that the best boundary moves to the width point, checking the queue formula against person-by-person queueing, and placing the separator on “divided by distance, or by width”.
1. Notation and the Question
Setting: exits A and B are at 0 m and 100 m of a 100 m corridor, with widths $w_A=1$ m and $w_B=1,2,3$ m. People are placed uniformly at random along the corridor. The walking speed is $v=1.3$ m/s and the exit flow is $q=1.3$ persons/s per metre of width (Paper 895). If the $n$ people heading for one exit arrive at times $a_1\le\cdots\le a_n$, the last one leaves at
$$T=\max_{k}\Bigl(a_k+\frac{n-k+1}{q\,w}\Bigr)$$which agreed within $10^{-9}$ with person-by-person queueing for four random sets. People left of a boundary $x$ go to A and those right of it to B, and the total time is the larger $T$ of the two exits. Heading for the nearer exit is $x=50$ m; the best rule tries every midpoint between neighbouring people and both ends and takes the $x$ with the shortest time. For 60, 200 and 600 people, 20 arrangements were made (random seeds 97900 to 97919).
Question: how much time does heading for the nearer exit lose against the best boundary, and how does that change with the number of people and the width ratio?
2. One Arrangement
Computation 2.1: for the first single arrangement made, the time heading for the nearer exit and the best time.
| people | widths | nearer (s) | best (s) | best boundary (m) | ratio |
|---|---|---|---|---|---|
| $60$ | 1 to 3 | $38.085$ | $38.085$ | $50.585$ | $1.000000$ |
| $200$ | 1 to 1 | $80.235$ | $77.158$ | $47.792$ | $0.961651$ |
| $200$ | 1 to 2 | $80.235$ | $54.081$ | $33.009$ | $0.674035$ |
| $200$ | 1 to 3 | $80.235$ | $52.543$ | $31.841$ | $0.654861$ |
| $600$ | 1 to 1 | $235.588$ | $231.026$ | $51.034$ | $0.980637$ |
| $600$ | 1 to 2 | $226.411$ | $154.103$ | $35.490$ | $0.680635$ |
| $600$ | 1 to 3 | $226.411$ | $115.642$ | $26.793$ | $0.510760$ |
With 60 people all three width ratios gave $38.085$ s, the two rules being the same. This is shorter than the $38.462$ s to walk from the middle, 50 m, to an exit.
3. Means over 20 Arrangements
Computation 3.1: ratio of times (best over nearer) and the best boundary.
| people | widths | mean ratio | min and max ratio | mean boundary (m) | min and max boundary (m) |
|---|---|---|---|---|---|
| $60$ | 1 to 1 | $\mathbf{0.995728}$ | $0.963705$ and $1.000000$ | $49.994$ | $48.471$ and $51.169$ |
| $60$ | 1 to 2 | $0.995980$ | $0.962986$ and $1.000000$ | $49.380$ | $46.876$ and $50.974$ |
| $60$ | 1 to 3 | $0.994959$ | $0.962986$ and $1.000000$ | $49.380$ | $46.876$ and $50.974$ |
| $200$ | 1 to 1 | $0.960589$ | $0.870442$ and $1.000000$ | $50.085$ | $44.342$ and $56.267$ |
| $200$ | 1 to 2 | $0.694901$ | $0.638604$ and $0.789772$ | $33.849$ | $30.194$ and $40.109$ |
| $200$ | 1 to 3 | $0.674894$ | $0.625249$ and $0.740598$ | $32.862$ | $29.825$ and $37.235$ |
| $600$ | 1 to 1 | $0.979916$ | $0.943416$ and $1.000000$ | $49.706$ | $47.867$ and $52.444$ |
| $600$ | 1 to 2 | $0.662493$ | $0.629064$ and $0.694669$ | $\mathbf{33.398}$ | $30.826$ and $35.961$ |
| $600$ | 1 to 3 | $\mathbf{0.497270}$ | $0.472005$ and $0.521186$ | $\mathbf{25.274}$ | $22.465$ and $28.177$ |
Observation 3.2 the more people, the more time heading for the nearer exit loses. For widths 1 to 3 the mean ratio fell as $0.994959$, $0.674894$ and $0.497270$ for 60, 200 and 600 people: with 60 people the mean ratio stayed between $0.994959$ and $0.995980$ whatever the width ratio. The 60-person mean boundary was the same $49.380$ m for 1 to 2 and 1 to 3 (when several boundaries gave the shortest time, the smallest was taken).
Observation 3.3 even with equal widths the best is faster than the nearer rule, depending on the arrangement. For widths 1 to 1 the mean ratio was $0.960589$ for 200 people and $0.979916$ for 600. This comes from moving the boundary away from 50 m, with a minimum of $0.870442$ (200 people).
4. The Value Set by Widths and the Best Boundary
Computation 4.1: if only queueing sets the time, heading for the nearer exit sends half the people to the narrow exit A, taking $(N/2)/(q w_A)$, while the best equalises the two queues at $N/(q(w_A+w_B))$, so the ratio is
$$\frac{2\,w_A}{w_A+w_B}$$which is $1.000000$, $0.666667$ and $0.500000$ for 1 to 1, 1 to 2 and 1 to 3. The means for 600 people differed from these by $-0.020084$, $-0.004173$ and $-0.002730$.
Observation 4.2 with many people the best boundary moves to the point dividing by width. The width point is $100 w_A/(w_A+w_B)$ m, $33.333$ m for 1 to 2 and $25.000$ m for 1 to 3: the mean best boundaries for 600 people were $33.398$ m and $25.274$ m, both within 1 m. For 200 people at 1 to 3, $32.862$ m lay between the width point $25.000$ m and the middle $50$ m.
5. The Separator
Observation 5.1 the separator is whether people are divided by distance or by exit width. With 60 people, the best time was on average $0.994959$ to $0.995980$ of the time with the middle boundary that equalises walking distance: with 600 people a boundary within 1 m of the point dividing people by width became best, and for 200 people at 1 to 3 the boundary fell between the two points. Which of the two terms of $T$, the arrival time $a_k$ or the queue term $(n-k+1)/(q w)$, is larger was not measured term by term here.
Observation 5.2 what is not said. Pushing, smoke, time before starting to walk, people choosing exits for themselves, switching exits on the way, and which rule should guide people are not treated. What is shown is how, in one corridor model, the ratio of times between heading for the nearer exit and the best boundary changes with the number of people and the width ratio.
6. Roll Call and Closing
- the queue formula (Section 1): agrees with person-by-person queueing.
- mean ratio $\mathbf{0.994959}$ to 0.995980 for 60 people (Computation 3.1): the two rules are almost the same.
- 0.994959, 0.674894, $\mathbf{0.497270}$ at 1 to 3 (Computation 3.1): more people, more loss.
- the ratio approaches twice the narrow width over the sum (Computation 4.1): difference $-0.002730$ for 600 people.
- boundaries $\mathbf{33.398}$ m and $\mathbf{25.274}$ m (Observation 4.2): moved to the width point.
- divided by distance, or by width (Observation 5.1): the separator; the scope note of Paper 895.
Closing: With two exits of different widths, heading for the nearer one loses more time the more people there are. With the walking speed and exit flow of Paper 895 and exits at both ends of a 100 m corridor, the best time was on average $0.994959$ to $0.995980$ of the nearer-exit time for 60 people, but for widths 1 to 3 it was $0.674894$ for 200 people and $0.497270$ for 600: for 600 people the ratio was close to the queue-only value of twice the narrow width over the sum of widths, and the mean best boundary was within 1 m of the width point ($33.398$ m and $25.274$ m). The separator is whether people are divided by distance or by exit width. Placed among the earlier papers: to the walking time and queueing that Paper 895 counted for one exit, the choice of exit that Paper 895 did not include was added. To be honest. This is one corridor without pushing or smoke, with the boundary set from outside and only times compared. How much heading for the nearer exit loses with the number of people and the width ratio, and no further.
References
- Yamagishi, Y. (2026). Widening the Exit Speeds Evacuation Only When There Are Many People; With Few, the Walking Distance Sets the Time [Paper 895]. YOI: https://www.yamagishiyuuki.com/yoi/895
Reference
Yamagishi, Y. (2026). With Two Exits of Different Widths, Heading for the Nearer One Loses More Time the More People There Are [Paper 979]. YOI: https://www.yamagishiyuuki.com/yoi/979
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 979. This paper is a preprint and has not been peer reviewed.