Q: What happens when traffic intensity ρ approaches 1?

As ρ → 1, the average queue length Lq = ρ²/(1-ρ) diverges to infinity. In practice, systems are designed for ρ ≤ 0.8 (80% utilization) to keep wait times manageable. When ρ ≥ 1, the queue is unstable and grows without bound.

Q: How do you find the optimal number of servers?

Minimize total cost TC(c) = c × C_server + Lq(c) × C_waiting, where C_server is the hourly cost per server and C_waiting is the cost per waiting customer per hour. The minimum of this convex function gives the optimal c balancing server cost against customer waiting cost.

Q: What are real-world applications of queuing theory?

Queuing theory applies to: bank teller and hospital reception staffing, call center agent sizing, network router and cloud server capacity planning, manufacturing line bottleneck analysis, logistics warehouse throughput, and traffic intersection design. It provides closed-form answers before committing to full simulation.

Question 1

What is the difference between M/M/1 and M/M/c queue models?

Accepted Answer

M/M/1 has a single server with Poisson arrivals and exponential service. Queue length Lq = ρ²/(1-ρ) where ρ = λ/μ < 1 for stability. M/M/c has c parallel servers; the waiting probability uses the Erlang-C formula C(c,ρ₀) where ρ₀ = λ/(cμ) < 1. Adding servers dramatically reduces wait times when ρ is high.

Question 2

What happens when traffic intensity ρ approaches 1?

Accepted Answer

As ρ → 1, the average queue length Lq = ρ²/(1-ρ) diverges to infinity. In practice, systems are designed for ρ ≤ 0.8 (80% utilization) to keep wait times manageable. When ρ ≥ 1, the queue is unstable and grows without bound.

Question 3

How do you find the optimal number of servers?

Accepted Answer

Minimize total cost TC(c) = c × C_server + Lq(c) × C_waiting, where C_server is the hourly cost per server and C_waiting is the cost per waiting customer per hour. The minimum of this convex function gives the optimal c balancing server cost against customer waiting cost.

Question 4

What are real-world applications of queuing theory?

Accepted Answer

Queuing theory applies to: bank teller and hospital reception staffing, call center agent sizing, network router and cloud server capacity planning, manufacturing line bottleneck analysis, logistics warehouse throughput, and traffic intersection design. It provides closed-form answers before committing to full simulation.

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