Question

A worker at the unloading dock of a warehouse takes, on average, 15 minutes to unload a truck. On average, three trucks arrive every hour at the unloading dock. Assume there is enough space to accommodate any number of waiting trucks at the warehouse and the average inter-arrival time and the unloading time are both exponentially distributed.

a. What is the average number of trucks in the waiting line before being unloaded?

b. What is the total time (waiting and unloading) that a truck spends in the system?

Management is considering adding an additional worker as there is sufficient space on the loading dock to allow two trucks to unload at the same time. Each unloading dock worker is paid $200 per hour while the hourly waiting cost (incurred while waiting in line and being unloaded) of a truck is $500.

c. With a crew of two workers what is the average number of trucks in the system?

d. With the crew of two workers, what is the total average cost per hour (including the waiting cost and the wage of the two workers)?

e. Assume that having two workers at the unloading dock costs $720 per hour including the cost of waiting and the wages paid to the workers. (We are making this assumption so that you do not need to know the answer to the previous question to answer this one.) Does adding another worker to have in total of 3 unloading dock workers reduce the total average cost per hour compared to the case with 2 workers?

Answer 1

The average number of trucks in the waiting line before being unloaded is 45. The total time that a truck spends in the system is 30 minutes.

Step-by-step explanation:

To determine the average number of trucks in the waiting line before being unloaded, we can use the formula for the average number of customers in a waiting line system, which is given by the formula = / , where is the average arrival rate and is the average service rate.

In this case, the average arrival rate is 3 trucks per hour, so = 3.

The average service rate is given as 15 minutes per truck, so = 1/15 trucks per minute.

Plugging these values into the formula, we get = 3 / (1/15) = 45 trucks.

Therefore, the average number of trucks in the waiting line before being unloaded is 45.

To calculate the total time that a truck spends in the system, we need to consider both the waiting time and the unloading time. Since the average service rate is 1/15 trucks per minute, the average service time is 15 minutes per truck. Therefore, the total time a truck spends in the system is 15 minutes for unloading and the average waiting time, which can be calculated using = , where is the average number of trucks in the waiting line and is the average arrival rate. Plugging in the values = 45 and = 3, we get = 45 / 3 = 15 minutes.

Therefore, the total time that a truck spends in the system is 15 minutes for unloading and 15 minutes for waiting, giving a total of 30 minutes.