Queueing Theory (M/M/1) Calculator

Calculate M/M/1 queue performance metrics: utilization, queue length, wait time, and system time with step-by-step solutions.

Enter Queue Parameters

Average arrivals per time unit
Average services completed per time unit
Calculate P(n) = probability of exactly n customers in system

Results

Server Utilization (ρ)
--
--
Avg Customers in System (L) --
Avg Customers in Queue (Lq) --
Avg Time in System (W) --
Avg Wait Time in Queue (Wq) --
P(idle) = P(0) --
P(n=3) --

Step-by-Step Solution

What Is M/M/1 Queueing Theory?

The M/M/1 queue is the simplest and most fundamental model in queueing theory. It describes a single-server queue where arrivals follow a Poisson process (rate λ) and service times are exponentially distributed (rate μ). The system is stable when the utilization ρ = λ/μ is less than 1.

M/M/1 Queue Formulas

Utilization

Fraction of time the server is busy.

ρ = λ / μ

Avg Customers in System

Expected number of customers including the one being served.

L = λ / (μ - λ)

Avg Queue Length

Expected number waiting in queue (not being served).

Lq = λ² / [μ(μ - λ)]

Avg System Time

Expected total time a customer spends in the system.

W = 1 / (μ - λ)

Avg Wait Time

Expected time waiting in queue before being served.

Wq = λ / [μ(μ - λ)]

Probability of n Customers

Probability of exactly n customers in the system.

P(n) = (1 - ρ) × ρ^n

Understanding the M/M/1 Notation

In Kendall notation, M/M/1 means:

  • First M: Markovian (Poisson) arrival process
  • Second M: Markovian (exponential) service time distribution
  • 1: Single server

Stability Condition

The M/M/1 queue is stable (reaches steady state) only when ρ = λ/μ < 1, meaning the service rate must exceed the arrival rate. If ρ >= 1, the queue grows without bound.

Real-World Applications

  • Call centers: Predicting hold times and staffing needs.
  • Network traffic: Modeling packet queues in routers and switches.
  • Retail: Optimizing checkout lane staffing.
  • Healthcare: Emergency room wait time estimation.
  • Manufacturing: Production line bottleneck analysis.

Little's Law

Little's Law (L = λW) connects the average number of customers in the system to the arrival rate and average system time. It holds for any stable queueing system and is used to verify the M/M/1 results: L = λ / (μ - λ) and W = 1 / (μ - λ), confirming L = λW.