Pollaczek-Khinchin (P-K) Formula Visualization

Pollaczek-Khinchin (P-K) Formula Visualization

P-K Formula Parameters

Arrival Rate (λ): 1.0 customers/second

Service Rate (μ): 1.5 customers/second

Service Time Variance (σ²): 0.50 seconds²

Utilization Range (ρ): 0.0 to 0.67

Pollaczek-Khinchin Formulas

Average Queue Length (L)	L = ρ + ρ² + λ²σ² 2(1-ρ)	L = 0.83
Average Waiting Time (W)	W = ρ/μ + λσ² 1-ρ	W = 0.56s

Explanation:

The Pollaczek-Khinchin formulas describe the relationship between queue characteristics in an M/G/1 queueing system:

• ρ (Utilization) = λ/μ (must be < 1 for stability)
• λ: Arrival rate (customers per unit time)
• μ: Service rate (customers per unit time)
• σ²: Variance of service time distribution

The graphs show how queue length and waiting time change as utilization approaches 100%.