Question

The new Fore& Aft Marina is to be located on the Ohio river near Madison, Indiana. Assume that Fore & Aft decides to build a docking facility where one boat at a time can stop for gas and servicing. Assume that arrivals follow a Poisson probability distribution, with an arrival rate of 8 boats per hour, and that service times follow an exponential probability distribution, with a service rate of 10 boats per hour. Answer the following questions using the computer output.

What is the probability that no boats are in the system?
What is the average number of boats that will be waiting in the line for service?
What is the average time in minutes that the boat will be waiting in line for service?
What is the average time in minutes a boat will spend in the system?
If you were the manager of Fore & Aft Marina, would you be satisfied with the service level your system is providing? Why or why not?Time units applied in the use of computer software is HOURS

Answer 1

The probability of no boats in the system is approximately 0.000335. The average number of boats waiting in the line for service is 4. The average time a boat will spend waiting in line for service is 30 minutes.

Step-by-step explanation:

To find the probability that no boats are in the system, we can use the formula for the probability of zero arrivals per unit time in a Poisson distribution. This probability can be calculated using the formula:

P(0) = e^(-λ)

Where λ is the arrival rate. In this case, λ = 8 boats per hour. So, the probability that no boats are in the system is:

P(0) = e^(-8) ≈ 0.000335

To find the average number of boats waiting in the line for service, we can use the formula for the average number of customers in a queuing system:

Lq = λ / (μ - λ)

Where λ is the arrival rate and μ is the service rate. In this case, λ = 8 boats per hour and μ = 10 boats per hour. So, the average number of boats waiting in line for service is:

Lq = 8 / (10 - 8) = 4

To find the average time in minutes that a boat will be waiting in line for service, we can use the formula for the average waiting time in a queuing system:

Wq = Lq / λ

Where Lq is the average number of customers waiting in line and λ is the arrival rate. In this case, Lq = 4 and λ = 8 boats per hour. So, the average time in minutes that a boat will be waiting in line for service is:

Wq = 4 / 8 = 0.5 hours * 60 minutes per hour = 30 minutes

To find the average time in minutes that a boat will spend in the system, we can use the formula for the average time spent in a queuing system:

W = Wq + 1 / μ

Where Wq is the average waiting time and μ is the service rate. In this case, Wq = 30 minutes and μ = 10 boats per hour. So, the average time in minutes that a boat will spend in the system is:

W = 30 + 1 / 10 = 30.1 minutes

Whether the manager would be satisfied with the service level depends on their expectations and the needs of the customers. However, a waiting time of 30 minutes for each boat and an average number of 4 boats waiting in line indicates that the system may not be very efficient. The manager may want to consider ways to improve the service level, such as increasing the service rate or implementing a system to prioritize certain types of boats.