You and two other customers arrive simultaneously at a bank

You and two other customers arrive simultaneously at a bank. At the moment of your arrival no other customers were being served and only two tellers were available. In a display of civility, you kindly let the other two customers get ahead of you while you wait in line for your turn. Suppose the service time for each of the two customers is exponentially distributed with mean A (in units of time). In addition, assume that the service times for the two customers are independent. Determine the probability density function (PDF) of the time you wait in line before a teller is available to serve you.

Solution

Time to serve one customer = lemda

If both customers go to the same clerk, then time will be 2 lemda other wise it would be lemda

Hence your waiting time varies from lemda to 2 lemda

Minimum waiting time lemda when each customer goes to each clerk with prob = 0.5

But if they go to the same clerk with p =0.25+0.25 = 0.5

Expected waiting time = 0.5lemda +0.5 lemda =lemda

Hence your waiting time is exp with mean = lemda