Variables
Description
What is this formula?
The Single-Server Queue Length formula calculates the average number of customers waiting in line in an M/M/1 queueing system.
It is one of the primary performance indicators used to evaluate congestion and service efficiency.
When to use it
Use this formula when:
- Evaluating queue congestion.
- Designing service facilities.
- Estimating waiting populations.
- Optimizing staffing levels.
- Analyzing bottlenecks.
Example
A service desk receives:
λ = 18 customers/hour
The service rate is:
μ = 24 customers/hour
Formula:
Lq = λ²/[μ(μ−λ)]
Substitution:
Lq = 18²/[24(24−18)]
Lq = 324/144
Lq = 2.25
Result:
On average, 2.25 customers are waiting in line.
Applications
- Queueing theory
- Call centers
- Manufacturing systems
- Retail operations
- Healthcare services
- Transportation systems
Note
This formula is valid for M/M/1 queueing systems with Poisson arrivals, exponential service times, and a single server. As utilization approaches 100%, queue length grows rapidly and may become extremely large.