Question

Suppose 50 people come to a fast food take-out

restaurant each hour. There are four employees, one at

the register, two making the food, and one packaging the

food. Suppose on average, each order takes 30 seconds

at the register, and it takes one employee an average of

1.5 minutes to make the food, and it takes 1 minute on

average to package the food. Assume exponential arrivals

and service times.

1) How many people typically are waiting in line to place an order?

2) How many people are waiting for their food to be made?

3) How many people are waiting for their food to be packed?

4) Which is the bottleneck of the process? Taking orders, making food, or

packaging food?

5) How many customers per hour can the restaurant serve?

6) If the restaurant hires one more employee, which process should the new

employee be placed?

7) How many customers per hour can the restaurant serve if that's the case?

Assuming the new employee can achieve the same efficiency as existing

employees.

Answer 1

Arrival rate, = 50 per hour

Service rate at the register, 1 = 3600 seconds per hour / 30 seconds = 120 per hour

Service rate at making the food, 2 = (60/1.5)*2 = 80 per hour

Service rate at packaging the food, 3 = (60/1) = 60 per hour

1) Number of people waiting in line to place an order, Lq₁ = ²/(₁*(₁-)) = 50²/(120*(120-50)) = 0.30 (rounded off)

2) Number of people waiting in line for their food to be made, Lq₂ = ²/(₂*(₂-)) = 50²/(80*(80-50)) = 1.04 (rounded off)

3) Number of people waiting in line for their food to be packed, Lq₃ = ²/(₃*(₃-)) = 50²/(60*(60-50)) = 4.17 (rounded off)

4) Bottleneck is Packing food, because its service rate is the lowest.

5) Capacity is decided by the bottleneck process. Packaging is the bottleneck. Therefore, restaurant can serve 60 customers per hour

6) The new employees should be placed in the bottleneck process, i.e. packaging food.

7) After addition of an employee in packaging food, it service rate (capacity) = 2*60 = 120 per hour.

Now making food is the bottleneck with the lowest service rate of 80. Therefore, now the restaurant can serve 80 customers per hour.