Difficult Questions

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Sheridan College *

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Industrial Engineering

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May 18, 2024

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1) In an M/M/2 queuing system, if the arrival rate (λ) is 4 customers per minute and the service rate (μ) is 5 customers per minute, calculate the utilization (ρ) of the system. Then, determine the average number of customers in the system (L). Finally, find the average number of customers waiting in the queue (Lq). What is the value of Lq? A) 0.5 customers B) 0.8 customers C) 0.6 customers D) 1.2 customers Answer: C) 0.6 customers Rationale: Utilization (ρ) = λ / (c * μ) = 4 / (2 * 5) = 4 / 10 = 0.4. Average number of customers in the system (L) = ρ / (1 - ρ) = 0.4 / (1 - 0.4) = 0.4 / 0.6 = 0.666. Average number of customers waiting in the queue (Lq) = λ * W = 4 * (0.666 / 4) = 0.666. 2) In an M/M/3 queuing system, the arrival rate (λ) is 10 customers per hour, and the service rate (μ) is 5 customers per hour. Calculate the utilization (ρ) of the system. Then, determine the average number of customers in the system (L). Finally, find the average number of customers waiting in the queue (Lq). What is the value of Lq? A) 2 customers B) 3 customers C) 4 customers D) 1.5 customers Answer: A) 2 customers Rationale: Utilization (ρ) = λ / (c * μ) = 10 / (3 * 5) = 10 / 15 = 2 / 3. Average number of customers in the system (L) = ρ / (1 - ρ) = (2 / 3) / (1 - 2 / 3) = (2 / 3) / (1 / 3) = 2. Average number of customers waiting in the queue (Lq) = λ * W = 10 * (2 / 10) = 2. 3) In an M/M/4 queuing system with c = 4 servers, if the utilization (ρ) is 0.7 and the average number of customers waiting in the queue (Lq) is 5, calculate the arrival rate (λ). Then, determine the average time a customer spends in the system (W). What is the value of W? A) 0.4 B) 0.5 C) 0.6

D) 0.8 Answer: A) 0.4 Rationale: Arrival rate (λ) = ρ * c * μ = 0.7 * 4 * μ = 2.8 * μ. Average time a customer spends in the system (W) = L / λ = (Lq + ρ) / λ = (5 + 0.7) / 2.8 = 5.7 / 2.8 = 2.035. Approximately, W = 0.4. 4) In an M/M/2 queuing system, the arrival rate (λ) is 6 customers per hour, and the service rate (μ) is 8 customers per hour. Calculate the utilization (ρ) of the system. Then, determine the average number of customers in the system (L). Finally, find the average number of customers waiting in the queue (Lq). What is the value of Lq? A) 1.25 customers B) 2.0 customers C) 0.75 customers D) 1.5 customers Answer: A) 1.25 customers Rationale: Utilization (ρ) = λ / (c * μ) = 6 / (2 * 8) = 6 / 16 = 0.375. Average number of customers in the system (L) = ρ / (1 - ρ) = 0.375 / (1 - 0.375) = 0.375 / 0.625 = 0.6. Average number of customers waiting in the queue (Lq) = λ * W = 6 * (0.6 / 6) = 0.6. 5) In a single-server M/M/1 queuing system, if the arrival rate (λ) is 12 customers per hour and the average time a customer spends in the system (W) is 0.1 hour, calculate the service rate (μ). Then, determine the utilization (ρ) of the system. What is the value of ρ? A) 0.8 B) 0.9 C) 0.6 D) 0.5 Answer: A) 0.8 Rationale: Service rate (μ) = 1 / W = 1 / 0.1 = 10 customers per hour. Utilization (ρ) = λ / μ = 12 / 10 = 1.2 / 1 = 0.8. 6) In an M/M/3 queuing system, if the arrival rate (λ) is 8 customers per minute and the service rate (μ) is 10 customers per minute, calculate the utilization (ρ) of the system. Then, determine the average number of customers in the system (L). Finally, find the average number of customers waiting in the queue (Lq). What is the value of Lq? A) 1.2 customers B) 1.6 customers C) 2.4 customers

D) 2.0 customers Answer: A) 1.2 customers Rationale: Utilization (ρ) = λ / (c * μ) = 8 / (3 * 10) = 8 / 30 = 4 / 15. Average number of customers in the system (L) = ρ / (1 - ρ) = (4 / 15) / (1 - 4 / 15) = (4 / 15) / (11 / 15) = 4 / 11. Average number of customers waiting in the queue (Lq) = λ * W = 8 * (4 / 11 * 8) = 32 / 11. 7) In an M/M/4 queuing system, the arrival rate (λ) is 20 customers per hour, and the service rate (μ) is 8 customers per hour. Calculate the utilization (ρ) of the system. Then, determine the average number of customers in the system (L). Finally, find the average number of customers waiting in the queue (Lq). What is the value of Lq? A) 3 customers B) 4 customers C) 2 customers D) 5 customers Answer: A) 3 customers Rationale: Utilization (ρ) = λ / (c * μ) = 20 / (4 * 8) = 20 / 32 = 5 / 8. Average number of customers in the system (L) = ρ / (1 - ρ) = (5 / 8) / (1 - 5 / 8) = (5 / 8) / (3 / 8) = 5 / 3 = 1.67. Average number of customers waiting in the queue (Lq) = λ * W = 20 * (5 / 3 * 20) = 100 / 3 = 33.33. 8) In an M/M/3 queuing system, if the utilization (ρ) is 0.6 and the average number of customers waiting in the queue (Lq) is 4, calculate the arrival rate (λ). Then, determine the average time a customer spends in the system (W). What is the value of W? A) 0.5 B) 0.6 C) 0.7 D) 0.8 Answer: B) 0.6 Rationale: Arrival rate (λ) = ρ * c * μ = 0.6 * 3 * μ = 1.8 * μ. Average time a customer spends in the system (W) = L / λ = (Lq + ρ) / λ = (4 + 0.6) / 1.8 = 4.6 / 1.8 = 2.56. Approximately, W = 0.6. 9) In an M/M/2 queuing system, the arrival rate (λ) is 5 customers per minute and the service rate (μ) is 6 customers per minute. Calculate the utilization (ρ) of the system. Then, determine the average number of customers in the system (L). Finally, find the average number of customers waiting in the queue (Lq). What is the value of Lq? A) 0.5 customers B) 0.6 customers

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