1. What is the M/M/1 Queue Formula?

The M/M/1 queue formula calculates the average time a customer spends waiting in a queue for a single-server system with Poisson arrivals and exponential service times. It's a fundamental model in queueing theory.

2. How Does the Calculator Work?

The calculator uses the M/M/1 queue formula:

Where:

• \( \lambda \) — Arrival rate (customers per unit time)
• \( \mu \) — Service rate (customers per unit time)

Explanation: The formula assumes a single server, Poisson arrival process, exponential service time distribution, and infinite queue capacity.

3. Importance of Queue Time Calculation

Details: Calculating average queue time helps in system design, resource allocation, and customer service optimization across various industries including telecommunications, healthcare, and retail.

4. Using the Calculator

Tips: Enter arrival rate (λ) and service rate (μ) in customers per unit time. The service rate must be greater than the arrival rate for a stable system.

5. Frequently Asked Questions (FAQ)

Q1: What does M/M/1 represent?
A: The notation represents: M (Markovian/Memoryless arrivals), M (Markovian service times), and 1 (single server).

Q2: What are the assumptions of the M/M/1 model?
A: Poisson arrivals, exponential service times, single server, infinite queue capacity, and first-come-first-served discipline.

Q3: When is this model not appropriate?
A: When arrivals don't follow Poisson process, service times aren't exponential, multiple servers exist, or queue capacity is limited.

Q4: What if μ ≤ λ?
A: The queue becomes unstable and grows infinitely over time. The formula doesn't apply in this case.

Q5: Can this be used for multi-server systems?
A: No, for multi-server systems you would need the M/M/c model with different formulas.