1. What is Queuing Theory?

Queuing theory is the mathematical study of waiting lines or queues. This calculator focuses on the M/M/1 queue model which assumes Poisson arrivals, exponential service times, and a single server.

2. How Does the Calculator Work?

The calculator uses the queuing theory formula:

Where:

• \( L \) — Average number of customers in the system (unitless)
• \( \lambda \) — Arrival rate (customers per unit time)
• \( \mu \) — Service rate (customers per unit time)

Explanation: The formula calculates the average number of customers in the system (both waiting and being served) based on arrival and service rates.

3. Importance of Queuing Theory

Details: Queuing theory helps optimize service systems in telecommunications, traffic engineering, computing, and customer service by predicting queue lengths and waiting times.

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 are typical units for λ and μ?
A: Common units include customers per hour, requests per second, or calls per minute - as long as both use the same unit.

Q2: What if μ ≤ λ?
A: The queue becomes unstable and grows infinitely long. The system needs higher service capacity.

Q3: What other metrics can be calculated?
A: From L, you can derive average wait time (W = L/λ), queue length (Lq), and waiting time (Wq).

Q4: What are limitations of M/M/1 model?
A: It assumes Poisson arrivals, exponential service times, single server, infinite queue capacity, and FCFS discipline.

Q5: How can I extend this to multiple servers?
A: The M/M/c model handles multiple servers with more complex formulas.