Answer First
The M/M/s queue models a system with Poisson arrivals, exponential service times, and s identical servers. The Erlang C formula gives the probability that an arriving customer must wait. Once we know this probability, we can compute expected waiting time, queue length, and system performance.
Model Setup
- Arrivals: Poisson with rate λ
- Service times: Exponential with rate μ per server
- Servers: s identical servers
- Queue discipline: FIFO
- Stability condition: λ < sμ
Traffic Intensity
\[ \rho = \frac{\lambda}{s\mu} \]
ρ measures how busy the system is relative to total capacity.
Erlang C Formula (Probability of Waiting)
\[ P(W>0) = \frac{\frac{(s\rho)^s}{s!(1-\rho)}}{\sum_{k=0}^{s-1} \frac{(s\rho)^k}{k!} + \frac{(s\rho)^s}{s!(1-\rho)}} \]
This gives the probability that all servers are busy and an arriving customer must wait.
Performance Measures
Expected Waiting Time in Queue
\[ W_q = \frac{P(W>0)}{s\mu – \lambda} \]
Expected Time in System
\[ W = W_q + \frac{1}{\mu} \]
Expected Number in Queue
\[ L_q = \lambda W_q \]
Expected Number in System
\[ L = \lambda W \]
Step-by-Step Solution
1. Compute traffic intensity
Check stability: λ < sμ.
2. Compute the Erlang C numerator
\[ \frac{(s\rho)^s}{s!(1-\rho)} \]
3. Compute the Erlang C denominator
\[ \sum_{k=0}^{s-1} \frac{(s\rho)^k}{k!} + \frac{(s\rho)^s}{s!(1-\rho)} \]
4. Compute the waiting probability
\[ P(W>0) \]
5. Compute Wq, W, Lq, and L
Use the formulas above.
Intuition
With multiple servers, customers only wait when all servers are busy. Erlang C captures this probability exactly. As traffic intensity approaches 1, waiting times explode—even with many servers.
Common Exam Mistakes
- Using ρ = λ/μ instead of λ/(sμ).
- Forgetting the stability condition λ < sμ.
- Mixing up W and Wq.
- Ignoring the Erlang C denominator sum.
- Assuming more servers always eliminate waiting (they reduce it but do not remove it).
Why This Matters
The M/M/s queue is the backbone of call center staffing, hospital scheduling, IT support, and service operations. Erlang C is used in workforce management software worldwide to determine staffing levels and service quality.
Final Summary
The M/M/s queue models multi‑server systems with random arrivals and service times. The Erlang C formula gives the probability of waiting, which leads to closed‑form expressions for waiting time and queue length. This model is essential for staffing, capacity planning, and service optimization.
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