You are planning the new layout for the local branch of the Sixth Ninth Bank. You...
You are planning the new layout for the local branch of the Sixth Ninth Bank. You are considering separate cashier windows for the three different classes of service. Each class of service would be separate with its own cashiers and customers. Oddly enough, each class of service, while different, has exactly the same demand and service times. People for one class of service arrive every 3 minutes and arrival times are exponentially distributed (the standard deviation is equal to the mean). It takes 12 minutes to service each customer, and the standard deviation of the service times is 6 minutes. You assign 6 cashiers to each type of service.
a) On average, how long will each waiting line be at each of the cashier windows?
b) On average how long (in minutes) will a customer spend in the bank? Assume they enter, go directly to one line, and leave as soon as service is complete.
Homework Answers
The number of cashier. m= 6
Activity time, p = 12 minutes
Arrival time, a = 3 minutes
Utilization , u = p / (a*m) = 12 / (3*6) = 0.6667
CVp = Coefficient of variation of service times = Std deviation of service time / Service time = 6/12 = 0.50
Similarly , CVa = 3/3 =1
Avg length of queue = Lq = ((utiltization^(sqrt(2(m+1)) / (1 - utilization)) * ((CVa^2 + CVp^2)/2) = =(0.6667 ^ SQRT(14)) / (1-0.6667)*(1^2 + 0.5^2)/2 = 0.411
Total in system, Ls = Lq + Number of servers * Utilizaton = 0.411 + 6*0.6667 = 4.411
Average time in queue = Total in system (Ls) / (1/a) = 4.411 / (1/3) =13.233 minutes
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