The restaurant at the Art Gallery of Ontario serves a hot buffet cafeteria-style daily to visitors and employees. The buffet is self-service. From 7:00 A.M.. to 9:00 A.M., customers arrive at the buffet at a rate of eight per minute; from 9:00 A.M. to noon, at four per minute; from noon to 2:00 P.M., at 14 per minute; and from 2:00 P.M. to closing at 5:00 P.M., at eight per minute (Poisson distributed). All the customers take about the same amount of time to serve themselves and proceed to the buffet. Once a customer goes through the buffet, it takes an average of 0.4 minute (exponentially distributed) to pay the cashier. The gallery does not want a customer to have to wait longer than four minutes to pay. How many cashiers should be working at each of the four times during the day?
To determine the number of cashiers needed at each time during the day, we need to calculate the average number of customers in the system and compare it with the desired wait time. Let's break down the problem step by step.
Step 1: Calculate the arrival rates for each time period:
- From 7:00 AM to 9:00 AM: 8 customers/minute
- From 9:00 AM to 12:00 PM: 4 customers/minute
- From 12:00 PM to 2:00 PM: 14 customers/minute
- From 2:00 PM to 5:00 PM: 8 customers/minute
Step 2: Determine the service time for each customer at the buffet:
- The customers take about the same amount of time to serve themselves.
- Given that the average serving time is 0.4 minutes (exponentially distributed).
Step 3: Calculate the average arrival rate and service rate for each time period:
- The average arrival rate is equal to the Poisson arrival rate multiplied by the average service time:
- From 7:00 AM to 9:00 AM: λ1 = 8 customers/minute * 0.4 minute/customer = 3.2 customers/minute
- From 9:00 AM to 12:00 PM: λ2 = 4 customers/minute * 0.4 minute/customer = 1.6 customers/minute
- From 12:00 PM to 2:00 PM: λ3 = 14 customers/minute * 0.4 minute/customer = 5.6 customers/minute
- From 2:00 PM to 5:00 PM: λ4 = 8 customers/minute * 0.4 minute/customer = 3.2 customers/minute
Step 4: Calculate the average number of customers in the system (L) for each time period using the M/M/1 queuing model:
- The formula for the average number of customers (L) is L = λ / (μ - λ), where λ is the arrival rate and μ is the service rate.
- The service rate μ is the inverse of the average service time: μ = 1 / 0.4 minute = 2.5 customers/minute.
- From 7:00 AM to 9:00 AM: L1 = 3.2 / (2.5 - 3.2) = 3.2 / (-0.7) = -4.57 (Ignore negative result)
- From 9:00 AM to 12:00 PM: L2 = 1.6 / (2.5 - 1.6) = 1.6 / (0.9) = 1.78
- From 12:00 PM to 2:00 PM: L3 = 5.6 / (2.5 - 5.6) = 5.6 / (-3.1) = -1.81 (Ignore negative result)
- From 2:00 PM to 5:00 PM: L4 = 3.2 / (2.5 - 3.2) = 3.2 / (-0.7) = -4.57 (Ignore negative result)
Step 5: Determine the number of cashiers needed at each time period:
- To calculate the number of cashiers needed, we round up the average number of customers in the system (L) to the nearest integer and subtract it from the desired wait time of 4 minutes:
- From 7:00 AM to 9:00 AM: Cashiers needed = ceil(4 - L1) = ceil(4 - (-4.57)) = ceil(8.57) = 9 cashiers
- From 9:00 AM to 12:00 PM: Cashiers needed = ceil(4 - L2) = ceil(4 - 1.78) = ceil(2.22) = 3 cashiers
- From 12:00 PM to 2:00 PM: Cashiers needed = ceil(4 - L3) = ceil(4 - (-1.81)) = ceil(5.81) = 6 cashiers
- From 2:00 PM to 5:00 PM: Cashiers needed = ceil(4 - L4) = ceil(4 - (-4.57)) = ceil(8.57) = 9 cashiers
Therefore, the number of cashiers needed at each time period should be:
- From 7:00 AM to 9:00 AM: 9 cashiers
- From 9:00 AM to 12:00 PM: 3 cashiers
- From 12:00 PM to 2:00 PM: 6 cashiers
- From 2:00 PM to 5:00 PM: 9 cashiers