Queueing Theory Calculator

Analyze M/M/1, M/M/c, M/G/1, and M/D/1 queueing systems

Queueing System Analysis

Queueing theory is the mathematical study of waiting lines or queues. This calculator helps analyze different queueing models to understand system performance and optimize service operations.

Key Metrics: L = λW (Little's Law), ρ = λ/μ (Utilization), P₀ = 1 - ρ (Empty system probability)

Where:

  • L = Average number of customers in the system
  • λ = Arrival rate (customers per time unit)
  • W = Average time a customer spends in the system
  • μ = Service rate (customers per time unit)
  • ρ = Server utilization factor

Queueing Model

Select the queueing model you want to analyze:

M/M/1
M/M/c
M/G/1
M/D/1

M/M/1 Queue

Single server queue with Poisson arrivals and exponential service times.

M/M/c Queue

Multiple server queue with Poisson arrivals and exponential service times.

M/G/1 Queue

Single server queue with Poisson arrivals and general service time distribution.

M/D/1 Queue

Single server queue with Poisson arrivals and deterministic (constant) service times.

Results

Utilization (ρ)

Server busy probability

Avg in Queue (Lₙ)

Customers waiting

Avg in System (L)

Total customers

Queue Time (Wₙ)

Time spent waiting

System Time (W)

Total time in system

P(0)

System empty probability

Probability Distribution

n P(n) in System P(≥n) in System

Interpretation Guide

Utilization (ρ) System State Recommendation
0% - 60% Underutilized Consider reducing capacity
60% - 80% Well-utilized Good balance
80% - 100% Overutilized Risk of long queues - add capacity
≥100% Unstable Queue will grow indefinitely

Practical Examples

Example 1: M/M/1 - Small Store Checkout

Arrival Rate: 4 customers/hour | Service Rate: 5 customers/hour

Utilization: 80% | Avg in System: 4 customers | Avg Wait: 48 minutes

Example 2: M/M/3 - Call Center

Arrival Rate: 15 calls/hour | Service Rate: 6 calls/hour | Servers: 3

Utilization: 83% | Avg in System: 5.2 calls | Avg Wait: 6.5 minutes

Example 3: M/G/1 - Repair Shop

Arrival Rate: 2 jobs/hour | Service Rate: 3 jobs/hour | Variance: 0.2

Utilization: 67% | Avg in System: 1.27 jobs | Avg Wait: 0.47 hours

Example 4: M/D/1 - Automated Process

Arrival Rate: 9 units/hour | Service Rate: 10 units/hour

Utilization: 90% | Avg in System: 4.95 units | Avg Wait: 0.45 hours

Queueing Model Comparison

Model Arrivals Service Servers Typical Use Cases
M/M/1 Poisson Exponential 1 Single checkout, simple systems
M/M/c Poisson Exponential Multiple Call centers, multi-server systems
M/G/1 Poisson General 1 Repair shops, variable service times
M/D/1 Poisson Deterministic 1 Automated processes, constant service