1. What is Queueing Theory?

Queueing theory is the mathematical study of waiting lines or queues. This calculator computes the average number of customers in the queue (Lq) for a simple M/M/1 queue system.

2. How Does the Calculator Work?

The calculator uses the queueing theory formula:

Where:

• \( L_q \) — Average number of customers in the queue
• \( \lambda \) — Arrival rate (customers per unit time)
• \( \mu \) — Service rate (customers per unit time)

Explanation: The formula assumes Poisson arrivals, exponential service times, and a single server (M/M/1 queue).

3. Importance of Queueing Theory

Details: Queueing theory helps optimize service systems in telecommunications, traffic engineering, healthcare, and customer service by analyzing wait times and system efficiency.

4. Using the Calculator

Tips: Enter arrival rate (λ) and service rate (μ) in the same time units. The service rate must be greater than the arrival rate for stable queues.

5. Frequently Asked Questions (FAQ)

Q1: What does M/M/1 mean?
A: The notation stands for Markovian arrivals/Markovian service/1 server, meaning both arrival and service times follow exponential distributions.

Q2: What are typical units for λ and μ?
A: Common units are customers per hour, per minute, or per second - just ensure both rates use the same units.

Q3: What if μ ≤ λ?
A: The queue would grow infinitely long as the system cannot keep up with arrivals. This is an unstable condition.

Q4: What other queue metrics can be calculated?
A: From Lq you can derive average wait time (Wq = Lq/λ) and other performance measures.

Q5: When is this model not appropriate?
A: When arrivals aren't Poisson, service times aren't exponential, or there are multiple servers or limited queue capacity.