Question

Customers arrive to a shoe repair shop according to a Poisson process with a rate of six per hour. John is the only employee that does the repairs, and he completes each repair in an exponentially distributed length of time, with rate of eight per hour. We assume that each customer only has one repair job to be fulfilled, and that John services jobs one at a time on a first-come first-serve basis. In order to keep customer retention high, John gives each customer a discount of $10 per hour that they wait at the shop (either waiting in the queue or waiting to have their shoes repaired.) What is the average cost per hour that John incurs by giving this discount?

a) Determine the answer using a Little’s law quantity

b) Determine the answer by using the steady state distribution in an expected value calculation. Hint: use a power series

Answer 1