Queing Analysis problem?

The First American Bank of Rapid City has one outside drive-up teller. It takes the teller an average of 4 minutes to serve a bank customer. Customers arrive at the drive-up window at a rate of 12 per hour. The bank operations officer is currently analyzing the possibility of adding a second drive-up window, at an annual cost of $20,000. It is assumed that arriving cars would be equally divided between both windows. The operations officer estimates that each minute%26#039;s reduction in customer waiting time would increase the bank%26#039;s revenue by $2,000 annually. Should the second drive-up window be installed?

I know that the arrival rate is 12 and the service rate is 15, but I don%26#039;t know what else to put. Please help.

Queing Analysis problem?credit repair

For single server system:

Let 位 = arrival rate = 12

Let 渭 = service rate =15

Let L = average customers in the queueing system = 位/(渭-位) = 4

Let W = average time in system/customer = 1/(渭-位) = 1/3 hour

For a two-server system:

c = number of servers = 2 (wherever you see a 2 below is the number of servers)

Probability of no customers in system:

Po = 1/[(位/渭) + (1/2)(位/渭)2(2渭/(2渭-位)]

Average number of customers in the system:

L = (位渭(位/渭)2)/((2渭-位)2)Po+(位/渭)

Average time a customer spends in the system waiting AND being served W = L/位

You do the rest of the math.

Queing Analysis problem?

loan

Here%26#039;s a hint, not a complete answer.

First, you didn%26#039;t say anything about the distribution of the service times and arrival rates, so I would guess you are probably supposed to assume that they are exponentially distributed. In the first case you then have an M/M/1 queue with arrival rate 12 per hour and service rate 15 per hour, so using a standard formula, you can compute the average time in the system (including wait and service time) for each customer.

In the second case you have two independent M/M/1 queues, each with arrival rate 6 per hour and service rate 15 per hour, so you can again compute the average time in system for each customer. Comparing the average time in system between the two cases should allow you to compute the savings (if any). Don%26#039;t forget to deduct the cost of the second window.