“Waiting time” results are often paradoxical and counter intuitive. To illustrate using a simple example, here is the advertised schedule of train departures from of a Mumbai Local from Churchgate to Borivalli over a a 24 hour timeframe.

The average inter-departure time between the trains appears to be ~ 6min from the graph on the right. However since it is a discrete distribution, one can easily compute the exact value by taking a weighted average. In this case, it is 5.6 min.

If you land up randomly at Churchgate station at any time during the day, what would be your average waiting time?

Since the mean inter-departure time is around 5.6 minutes, one might intuitively argue that the waiting time is about 2.8 minutes, i.e half of the average . However if the inter-departure times of the trains have a deviation(), then the waiting times actually depend on the square of deviation(). In this case, based on the advertised schedule, the deviation is about 3 minutes. So, the expected long run waiting time is given by

Note that this becomes half of average inter-departure time only when the deviation goes to 0. Hence , in this case, the average waiting time for a person landing up at a random time during the day is about __3.7 minutes__ and not 2.8 minutes. __This is a theoretical result__. There is an __assumption made on the service quality__, i.e., the coefficient of variation of the inter-departure times used is, as per the advertised schedules of the train. Any random event that increases the deviation is going to have a squared effect on the waiting times. To get an idea about the effect on deviations , here is a plot of waiting times for increasing .

So, even if the train schedule is quite frequent over a day, the aspect that matters more for the waiting times is the coefficient of variation of the inter-departure times, i.e. the service quality.

So, what’s the paradox here ? : __Mean waiting time for the next train can be LARGER than mean inter-departure time.! __Not paradoxical , once one understands that one is more likely to hit a longer inter-departure time than a short one, when arriving at the station at random!