Question

A goat herder in a remote undeveloped village rents out her goat to locals who need the goat to graze down their poison ivy and other weeds. The goat is available 50 weeks per year (a herder likes to keep her goat home for the Holidays). Each year there are 40 villagers who randomly call requesting the goat. Villagers have different sizes of yards, so the amount of time they keep the goat is variable with an average of one week and a standard deviation of one week. How long does a villager have to wait for the goat, on average, after making a request?

1 week

2 weeks

4 weeks

There is no wait since the goat is only busy 40 out of the 50 weeks.

Answer 1

Average arrival rate, L = 40 villagers per year = 40/50 villagers per week = 0.8 villagers per week.
Average service rate, M = 1 villager is 1 week = 1 villager per week

Using an M/M/1 queuing formula, Average waiting time = L / (M*(M - L) = 0.8/(1*(1 - 0.8)) = 4 weeks

[Note: for the service time, the mean and stdev were same, so it can be approximated to an exponential distribution. For the arrival rate, assume a Poisson arrival]