Queueing Theory Calculator
Analyze waiting lines using M/M/1 and M/M/s queueing models with Little's Law and performance metrics
Queue Analysis
M = Markovian (exponential) arrival/service times, 1 or s = number of servers
Number of customers arriving per unit time
Number of customers served per unit time per server
Queue Performance Metrics
System Metrics
Key Formulas Used
Probability Calculator
Little's Law
L = λ × W - The average number of customers in the system equals the arrival rate times the average time in the system.
This fundamental relationship applies to any stable queueing system and connects waiting times with queue lengths.
Example: Bank Teller Analysis
Scenario
Setting: Bank with single teller during lunch hour
Arrival rate (λ): 15 customers per hour
Service rate (μ): 20 customers per hour
Queue type: M/M/1 (exponential arrivals and service times)
Results
• Traffic intensity (ρ): 15/20 = 0.75 (75% utilization)
• Average customers in system: 0.75/(1-0.75) = 3 customers
• Average time in system: 1/(20-15) = 0.2 hours (12 minutes)
• Average time waiting: 0.75 × 0.2 = 0.15 hours (9 minutes)
Kendall Notation Guide
A/B/C Format
Common Distributions
Queue Optimization Tips
Keep utilization below 80%
Higher utilization leads to exponentially longer waits
Add servers strategically
M/M/s queues benefit significantly from additional servers
Monitor arrival patterns
Adjust capacity during peak demand periods
Improve service efficiency
Small increases in service rate have large impacts
Understanding Queueing Theory
What is Queueing Theory?
Queueing theory is a mathematical framework for analyzing waiting lines or queues. It was developed by Agner Krarup Erlang in the early 20th century to analyze telephone networks and has since found applications across many fields.
Key Components
- •Arrival Process: How customers enter the system
- •Service Process: How customers are served
- •Queue Discipline: Order of service (FIFO, LIFO, Priority)
- •System Capacity: Maximum customers allowed
Mathematical Foundations
M/M/1 Queue Formulas
Stability Condition
For a stable queue: λ < μ (M/M/1) or λ < s×μ (M/M/s)
If arrival rate equals or exceeds service capacity, the queue grows infinitely.
Real-World Applications
Service Industries
Banks, restaurants, call centers, healthcare facilities
Computer Systems
CPU scheduling, network traffic, database queries
Transportation
Airport operations, traffic signals, logistics