Question

A department has 10 machines that each run for an average of 8 hours (exponential) before service is required. Service time average is 2 hours (exponential). N = T = U = X = T / ( T + U) = Using the information from above go to the Finite Queue Table and fill in the following: If M = 5 D = F = If M = 4 D = F = If M = 3 D = F =

Answer 1

To find the values for M, D, and F for different values of M, we use the formulas D = M - X and F = X / (T + U), where X is the average number of machines available at a given time. For M = 5, D = 4.5 machines and F = 0.05 machines. For M = 4, D = 3.6 machines and F = 0.04 machines. For M = 3, D = 2.7 machines and F = 0.03 machines.

Step-by-step explanation:

The given problem deals with a department that has 10 machines. Each machine runs for an average of 8 hours before service is required. The service time average is 2 hours. We need to find the values for M, D, and F for different values of M.

For M = 5:

D: The average number of machines requiring service at a given time is calculated using the formula D = M - X, where X is the average number of machines available at a given time. Here, X = M / (T + U) = 5 / (8 + 2) = 0.5 machines. So, D = 5 - 0.5 = 4.5 machines.

F: The average number of machines waiting for service in the queue can be calculated using the formula F = X / (T + U) = 0.5 / (8 + 2) = 0.05 machines.

Similarly, we can calculate the values of D and F for M = 4 and M = 3.

For M = 4:

D = 4 - 0.4 = 3.6 machines

F = 0.4 / (8 + 2) = 0.04 machines

For M = 3:

D = 3 - 0.3 = 2.7 machines

F = 0.3 / (8 + 2) = 0.03 machines

Answer 2

For M = 5, D ≈ 0.52 and F ≈ 0.4019.
For M = 4, D ≈ 0.44 and F ≈ 0.4096.
For M = 3, D ≈ 0.375 and F ≈ 0.4219.

To fill in the Finite Queue Table, we need to calculate the values for D and F for different values of M.

D represents the average delay time in the queue, while F represents the probability that a customer has to wait in the queue.

Let's calculate the values for M = 5:

D = (M^2) / (2 * (M - 1) * (T - U))
= (5^2) / (2 * (5 - 1) * (8 - 2))
= 25 / (2 * 4 * 6)
= 25 / 48
≈ 0.52

F = (M / (M + 1))^M
= (5 / (5 + 1))^5
= (5 / 6)^5
≈ 0.4019

Now, let's calculate the values for M = 4:

D = (4^2) / (2 * (4 - 1) * (8 - 2))
= 16 / (2 * 3 * 6)
= 16 / 36
≈ 0.44

F = (4 / (4 + 1))^4
= (4 / 5)^4
≈ 0.4096

Finally, let's calculate the values for M = 3:

D = (3^2) / (2 * (3 - 1) * (8 - 2))
= 9 / (2 * 2 * 6)
= 9 / 24
≈ 0.375

F = (3 / (3 + 1))^3
= (3 / 4)^3
≈ 0.4219