“Please hold the line, your call is important to us.” It’s a sentence we’re all frustratingly familiar with. Just as familiar as we are with standing in line at the supermarket or post office, with the other queues seeming to move much faster than the one we happen to be in. Thankfully, mathematics can help. Queueing theory studies such situations mathematically, and tries to find solutions that minimise the average customer waiting time while also limiting the average time a queue server remains idle. This double constraint makes the problem a difficult one. An additional source of difficulty is the randomness involved. Customers usually do not arrive at regular intervals, but their arrival times are what is called a *stochastic process*. Coming up with a general formula that provides a solution for such stochastic problems is generally difficult, and sometimes even impossible. Recently my uncle Arie Hordijk and I studied such a queueing problem, and came up with a solution based on the movements of a ball on a billiard table…

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