A fast food chain wants to improve their service. Currently, there is only 1 cashier in line. The consultant noticed that on average, 45 customers come per hour. The mean service rate is 60 customers/hour. The number of customers in line follows a Poisson distribution.
After analyzing the current situation, the consultant proposes to hire a second cashier at a rate of 9$ an hour to serve the same line that the first cashier service. The second cashier is assume to work with the same efficiency as the first cashier and the service time follows the negative exponential distribution. Answer the following questions:
Just to clarify, does the 60/hour and service time of 45/hour stay the same once you add the extra server?
A) What is the probability of finding both cashiers busy?
Would that be the probability of 2 people in the queue? (that would be 12.6%) - I think that this is too simple to be the answer
B) The optimal performance of the system in terms of reduced total waiting time in the queue is important to the owner. If the owner assigns a $0.25 value for each minute reduced in the queue, is the hiring of the second cashier cost justified?
I know here that the amount of time waiting with 1 server = 3 minutes (0.05 hours) and the time waiting if there are 2 servers is 1.16 minutes (0.0193 hours), I just don't know how to connect them to the $0.25