Customers enter a single-server service desk with exponential interarrival and service times. A reproducible event sequence contrasts observed queue length and waiting with stationary M/M/1 predictions.
• A 3D laboratory scene with: Arrival doorway; FIFO waiting positions; Single service desk; Departure doorway; Utilization and waiting indicators. • Controls: Mean arrival rate λ (0.5-6 customers/min); Mean service rate μ (1-8 customers/min); Random sequence seed (1-20). • Live readouts: Customers waiting; Server occupied · 1=yes; Completed customers; Utilization ratio λ/μ; Observed completed waits; Stationary Wq · −1 means unstable. • Guided experiments: Light traffic; Near capacity; Overloaded desk.
• ρ=λ/μ; stable only for ρ<1 • For stable M/M/1: Wq=ρ/(μ−λ), Lq=λWq • Interarrival=−ln(U)/λ; service duration=−ln(U)/μ • One animation second = 0.5 minutes of queue time.
Single FIFO server with unlimited waiting capacity and independent exponential times. The seed makes replay repeatable; a short transient sample need not match stationary means. No balking, abandonment, priorities or walking-time contribution. Overload has no finite stationary waiting-time prediction.
No. Random arrivals and service times create temporary queues.
No. The unbounded M/M/1 queue has no stable stationary mean in overload.
The desk often empties, though random bursts can still cause waits.
Single FIFO server with unlimited waiting capacity and independent exponential times. The seed makes replay repeatable; a short transient sample need not match stationary means. No balking, abandonment, priorities or walking-time contribution. Overload has no finite stationary waiting-time prediction.