Boarding time statistics
In the models the time taken for each boarding event is determined by a Weibull distribution function, driven by a single random variable, which gives a distribution of boarding times likely to be experienced in reality. The time taken to board will depend on the type of passengers as shown in the table
| Passenger type | boarding time, secs |
| Fit, unencumbered | >10 |
| Fit with luggage | 10 - 15 |
| Families | 15 - 20 |
| Elderly | 20 - 40 |
| Disabled | 40 - 60 |
We used four different mixes of passengers as shown in the table:
The graph shows the result of running the boarding time function a number of times large enough to average out the random input. The random input variable has been rounded of to the nearest iteration time interval of 2.5 seconds:
Run the spreadsheet User_dist_W.xls and open the sheet named typical. It should look like this:
The cells highlighted in blue are a typical mix[1] of passengers that might be boarding some form of transport.
Column I is the number of seconds from a start time and column P is the proportion of the passengers that take this length of time to board. In the case of a PAT vehicles it is the proportion of the number of vehicles that take this time to be ready to leave.
Column Q is the cumulative proportion of the passengers that have boarded by this time. The last figure in this column should be a 1, as all the passengers have boarded.
This is the input data. The figures along row 6, highlighted in pink, are the proportion of the total number of passengers in each category divided by the time interval over which they board. These figures in row 6 need to be copied by hand into the appropriate cells below.
Run Tools > Solver and you should see the solver window:
Solver window
Set Target Cell: to the cell highlighted in purple
Select Equal To: Min
Set By Changing Cells: to the cells highlighted in yellow
Click on solve.
The graph of Cumulative distribution function should show a reasonable fit of the the curve to the data and look something like this:
Weibull fit for typical passenger boarding times.
The curve is the cumulative Weibull distribution function[2]:
Where 
lm in the spreadsheet. Units are seconds
which is dimensionless
Set in Subject to Constraints window
which has a value between 0 and 1
Cell G7 is the sum of the squares of the differences between the
points and the function which is minimised by the Excel solver to find
the best fit values of 
and
.
The curve in the probability density function graph is:
The inverse Weibull cumulative distribution function is:
The total time for the boarding is then:
Where 
tmin in the spreadsheet
The mean boarding time is calculated from:
and the Variance from:
[1] This particular mix was provided by Graham Bowles of Calver Marketing Associates Ltd.