London, Jan 24 : How many times has it happened that you've waited for a bus for a long time and chosen to walk to the next stop instead, only to see the bus running past you?
The best strategy to avoid the same frustrating situation is to just wait at the first stop for the bus as it is the quickest way to reach your destination, says American mathematicians, who have developed a new formula to prove this fact.
Justin Chen, from the California Institute of Technology, and Scott Kominers and Robert Wyatt Sinnott, both from Harvard, reported this formula in their paper, Walk Versus Wait: The Lazy Mathematician Wins.
"Many mathematicians probably ponder this on their way to work, but never get round to working it out. It certainly has changed the way I travel," New Scientist magazine quoted Kominers, as saying.
The researchers wrote in their paper: "Recently, the first author [Justin] had to walk to the second author's [Scott's] house to work on a problemset. There is a bus route which covers the distance directly, but the bus arrives sporadically."
They added: So, he faced a conundrum: walk the distance or wait for the bus? Being lazy, he would always rather ride the bus, if possible. Being punctual, however, he will always choose the option which gets him to his destination as quickly as possible."
After working out the variables for the journey, the researches turned them into a formula.
The formula takes into account d, the distance in miles Dr Chen has to travel; n, the number of bus stops; vw, Dr Chen's walking speed; and vb, the bus's speed. The time when the journey starts - t - equals 0, and time spent waiting is tw.
It was shown that it is never worth him starting to walk and then boarding the bus at the next stop if it happens to be passing: all he does is waste energy or see a bus speeding past him between stops.
It was seen that the time spent waiting added to the time the bus would take is almost always lower than the time it would take him to walk.
"'Eureka!', Justin shouted upon realising that the laziest possible waiting strategy would prevail. He promptly sat down to wait for the bus," concluded the paper.
However, the researchers said that only when no bus is due for an hour or more and your destination is less than a kilometre away, is it worth walking. They suggested that starting your journey on foot after waiting for a long time will only make you late.