Thread: Queueing Problem Help plz

We have a supermarket with customers coming with a mean rate of 60 per hour.The boss pays its cashiers $6.00 per hour .The supermarket estimates the goodwill cost of a customer waiting to be served at $5.00 per hour and the goodwill of a customer being served at only $1.50. Customer service times are exponentially distributed with a mean of 3 minutes.
i) Determine the number of cashier person that should be employed to minimize the hourly cost.

ii) If the clients arrival rate increases by 50% and the goodwill cost of a customer waiting to be served increases by $3.00 per hour, determine the number of clerks the boss should employ.

I would really appreciate any help with that problem.
Thanksx

Originally Posted by Amazed

We have a supermarket with customers coming with a mean rate of 60 per hour.The boss pays its cashiers $6.00 per hour .The supermarket estimates the goodwill cost of a customer waiting to be served at $5.00 per hour and the goodwill of a customer being served at only $1.50. Customer service times are exponentially distributed with a mean of 3 minutes.
i) Determine the number of cashier person that should be employed to minimize the hourly cost.

ii) If the clients arrival rate increases by 50% and the goodwill cost of a customer waiting to be served increases by $3.00 per hour, determine the number of clerks the boss should employ.

I would really appreciate any help with that problem.
Thanksx

Problem (i) is an M/M/m queueing model with lambda = 60, mu = 20 (inverse of 3 minutes) and the number of cashiers m to be determined. It must be that m > lambda/mu = 3 if all customers are to be served. Let W(m) be the mean waiting time (queueing delay) when there are m cashiers. W(m)*lambda is the total average waiting time for the customers arriving in an hour. The hourly cost is

C(m) = $5.00*W(m)*lambda + $6.00*m.

Choose integer m to minimize C(m) given m > 3. I find the minimizing m = 4 with cost $31.64.

The formula for W(m) is too much to write out here without LaTeX. It is shown here with rho = lambda/m*mu.

For (ii) I find the minimizing m = 7 with cost $45.13.