Question

In: Statistics and Probability

The new Fore and Aft Marina is to be located on the Ohio River near Madison,...

The new Fore and Aft Marina is to be located on the Ohio River near Madison, Indiana. Assume that Fore and 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 4 boats per hour, and that service times follow an exponential probability distribution, with a service rate of 8 boats per hour. The manager of the Fore and Aft Marina wants to investigate the possibility of enlarging the docking facility so that two boats can stop for gas and servicing simultaneously.

Note: Use P0 values from Table 11.4 to answer the questions below.

  1. What is the probability that the boat dock will be idle? Round your answer to four decimal places.

    P0 =  
  2. What is the average number of boats that will be waiting for service? Round your answer to four decimal places.

    Lq =  
  3. What is the average time a boat will spend waiting for service? Round your answer to four decimal places.

    Wq =  hours
  4. What is the average time a boat will spend at the dock? Round your answer to four decimal places.

    W =  hours
  5. If you were the manager of Fore and Aft Marina, would you be satisfied with the service level your system will be providing?

    Yes  

    Why or why not? Round your answers to whole numbers.

    Because the average wait time is  seconds. Each channel is idle  % of the time.

Solutions

Expert Solution

Arrival rate ( λ ) = 4 boats per hour.

Service rate ( μ ) = 8 boats per hour.

a. The boat dock will be idle if there is no boat at the dock.

P( 0 boats at the dock) =

P (0) = 1 - ρ ( where ρ is utilization rate)

= 1 - ( λ / μ )

= 1 - ( 4/8)

= 0.5

Therefore P0 = 0.5

b. Average number of boats waiting for service =

Lq =

=

= 0.5

Therefore the average number of boats that will be waiting for service ( Lq ) = 0.5

c. The average waiting time

Wq =  

=

= 0.125 hours

Therefore the average time a boat will spend waiting for service ( Wq ) = 0.125

d. The average waiting time for a boat at the dock

Ws =  

=

= 0.25 hours

Therefore the average waiting time for a boat at the dock ( Ws ) = 0.25 hours

e. Yes, I would be satisfied with the service.

Because average waiting time is 0.25 hours that is 900 seconds.

Each channel is idle 50% of the time.

orchestra answered 1 year ago
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