Ivo’s travel time to work

You’re right — your travel time is not exactly normal. But you can’t model it with a Poisson distribution either.

By the way, this is your example:

 

Let’s define a generic function to calculate total delay time due to some reason, with the following parameters:

• n — number of “opportunities” for delay to occur (e.g. number of signals between the first and last stations);
• p — probability of delay given an opportunity;
• mean — average delay time;
• standard deviation (we assume delay time is normally distributed).

delay <- function(n, p, mean, sd) sum(
    rtruncnorm(n, 0, Inf, mean, sd)[
        sample(
            c(FALSE, TRUE),
            size = n,
            replace = TRUE,
            prob = c(1 - p, p)
        )
    ]
)

 

A simple model of your travel time — sum of three components:

• “base” time — normally distributed, with \(mean = 40\) and \(sd = 1\);
• 10 signals with probability of being red of 20%, and delay time with \(mean = 5\) and \(sd = 1\);
• 10 “opportunites” of a big delay (sick people, etc.) with probability of 1%, and delay time with \(mean = 30\) and \(sd = 5\).

n <- 1100
plot_ly(
    data.frame(
        model = c(
            rep("example", 220),
            rep("model", n)
        ),
        t = c(
            rep(45, 90), rep(40, 40), rep(50, 40), rep(55, 30), rep(60, 15), rep(90, 5),
            rnorm(n, 40, 1)
                + replicate(n, delay(10, .2, 5, 1))
                + replicate(n, delay(10, .01, 30, 5))
        )
    ),
    alpha = 0.5,
    colors = "Set1",
    width = 900
) %>% 
add_histogram(
    color = ~model,
    x = ~t,
    xbins = list(
        size = 1,
        start = .5, end = 120
    )
) %>% layout(
    barmode = "overlay",
    xaxis = list(
        dtick = 15,
        rangemode = "tozero",
        title = "minutes"
    ),
    yaxis = list(
        title = "occurences"
    )
)

 

Not bad.

By the way, this appears to be a serious topic of research. For example: Modeling Distribution of Travel Time in Signalized Road Section Using Truncated Distribution.