1. What is the M/M/1 Queuing Model?

The M/M/1 queuing model represents a single-server queue where arrivals follow a Poisson process (exponential inter-arrival times) and service times are exponentially distributed. It is one of the most fundamental models in queueing theory.

2. How Does the Calculator Work?

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

Where:

• \( L \) — Average number of customers in the system (queue length)
• \( \lambda \) — Arrival rate (customers per hour)
• \( \mu \) — Service rate (customers per hour)

Explanation: This formula calculates the expected number of customers in the system (both waiting and being served) under steady-state conditions.

3. Importance of Queue Length Calculation

Details: Understanding queue length helps in designing efficient service systems, optimizing resource allocation, and improving customer satisfaction in various service industries.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What does M/M/1 stand for?
A: The first M stands for Markovian (exponential) inter-arrival times, the second M for Markovian service times, and 1 indicates a 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 the M/M/1 model applicable?
A: It's suitable for systems with random arrivals and service times, such as call centers, retail checkout lines, and simple computer systems.

Q4: What is the stability condition for M/M/1?
A: The system is stable only when λ < μ (arrival rate less than service rate).

Q5: Can this model handle multiple servers?
A: No, this is specifically for single-server systems. For multiple servers, you would need M/M/c models.