Question

A petroleum company is considering the expansion of its one unloading facility a

refinery. The ships arrive according to a Poisson process to unload crude oil at a Tale

of 5 ships per week. The service is carried out according to an exponential probability

distribution with a rate of 10 ships per

week

The company has under consideration a second unloading berth which can be rented for Rs 5,000 a weak. The service rate for this berth will also be 10 per week For each week a ship remains idle waiting in line or for unloading the company loses Rs. 20,000.

1.What is the average time a ship must wait before beginning to deliver its cargo to the refinary?
2. If a second berth is rented, what will be the average number of ships waiting?
3. What will be the average waiting time of a ship with a second berth?
4. Is the benefit of reduced waiting time worth the rental cost of second berth?

Answer 1

Average Time a Ship Must Wait Before Beginning to Deliver Cargo:

The average waiting time in the queue can be calculated using the formula for the average waiting time in an M/M/1 queue:

Average Waiting Time (W) = (1 / Service Rate) / (1 - (Arrival Rate / Service Rate))

Given the arrival rate (λ) = 5 ships per week and service rate (μ) = 10 ships per week, we can calculate:

W = (1 / 10) / (1 - (5 / 10)) = 0.1 / 0.5 = 0.2 weeks

So, the average time a ship must wait before beginning to deliver its cargo is 0.2 weeks.

Average Number of Ships Waiting with a Second Berth:

The average number of ships waiting in a queue can be calculated using the formula for the average number of customers in an M/M/1 queue:

Average Number of Ships Waiting (Lq) = (Arrival Rate^2) / (Service Rate * (Service Rate - Arrival Rate))

Given the arrival rate (λ) = 5 ships per week and service rate (μ) = 10 ships per week, we can calculate:

Lq = (5^2) / (10 * (10 - 5)) = 25 / 50 = 0.5 ships

So, the average number of ships waiting with a second berth is 0.5 ships.

Average Waiting Time of a Ship with a Second Berth:

The average waiting time of a ship with a second berth can be calculated using Little's Law, which states:

Average Waiting Time (Wq) = Average Number of Ships Waiting (Lq) / Arrival Rate (λ)

Given the average number of ships waiting (Lq) = 0.5 ships and arrival rate (λ) = 5 ships per week, we can calculate:

Wq = 0.5 / 5 = 0.1 weeks

So, the average waiting time of a ship with a second berth is 0.1 weeks.

Benefit of Reduced Waiting Time vs. Rental Cost of Second Berth:

To determine if the benefit of reduced waiting time is worth the rental cost of the second berth, we need to compare the cost savings due to reduced waiting time with the cost of renting the second berth.

Cost Savings = (Average Waiting Time * Cost per Idle Ship) * Arrival Rate

Cost of Renting Second Berth = Rs 5,000

Given the calculated average waiting time (0.2 weeks), cost per idle ship (Rs 20,000), and arrival rate (5 ships per week), we can calculate the cost savings:

Cost Savings = (0.2 * 20,000) * 5 = Rs 20,000

Since the cost savings (Rs 20,000) is greater than the cost of renting the second berth (Rs 5,000), the benefit of reduced waiting time is worth the rental cost of the second berth.

Overall, the analysis suggests that renting the second berth is beneficial in terms of reducing waiting time and cost savings.