Heavier Load and Honesty - Queueing I: Birth-Death Chains

A second example shows how the same formulas respond to heavier traffic. The values are exact for the model, but the model is an idealization with memoryless arrivals, memoryless service, one server, and an infinite buffer. Real systems need measured arrival and service data before these formulas become evidence.

highlighted = computed this step

A busier queue

The second example has arrival rate 2 and service rate 3. Why: the arrival rate is closer to service capacity. This makes the example a clean comparison: the same formulas apply, but the traffic ratio is higher.

\lambda=2\quad \mu=3

Higher utilization

The utilization is 2/3 and the mean number in system is 2. Why: moving closer to capacity increases congestion in the steady-state distribution. More probability mass sits in non-empty states, so the expected count rises.

\rho=2/3\quad L=2

Waiting metrics

The mean time in system is 1 and the mean queue wait is 2/3. Why: exact queueing formulas separate service time from waiting time. That separation keeps the mechanics honest: the model says what each component means before anyone compares it to data.

W=1\quad W_q=2/3

Diagram note

These are exact steady-state values for an idealized memoryless single-server queue with infinite buffer and stable rates; real arrival and service rates must be measured. The lesson compares model mechanics, not transient behavior and not a fitted real queue. Pixel positions are rounded for layout; every number shown is exact.

\text{exact model first, empirical fit later}