A waiting-line system that meets the assumptions of M/M/1 has λ = 1,μ = 4.For this system,P_o is ________ and utilization is ________. A)0.75;0.25 B)0.80;.20 C)-3;-4 D)3;4 E)none of the above

Question

Quizplus · Accepted Answer

Using the M/M/1 queuing model, we have: λ = 1 (arrival rate) μ = 4 (service rate) Utilization (ρ) = λ/μ = 1/4 = 0.25 Probability of no customers in the system (P0) = 1 - ρ = 1 - 0.25 = 0.75 Probability of exactly 1 customer in the system (P1) = ρ(1 - ρ) = 0.25(1-0.25) = 0.1875 Probability of at least 1 customer in the system (P1 or more) = 1 - P0 = 0.25 Probability of a customer being served without any delay (Pw) = 1 - ρ = 0.75 Using Little's Law, we can find the average number of customers in the system (L): L = λ * W where W is the average time a customer spends in the system We know that μ > λ, so the system is stable and W can be found using: W = 1/(μ - λ) = 1/(4 - 1) = 1/3 Therefore, L = 1 * (1/3) = 0.333 Putting it all together, we get: P_o = P1 or more * Pw = 0.25 * 0.75 = 0.1875 Utilization = ρ = 0.25 Therefore, the best choice is A.

A Waiting-Line System That Meets the Assumptions of M/M/1 Has