Honest Network Scope - Queueing II: Networks

The exact tandem calculation is a model result, not a measurement claim. The final lesson marks which assumptions carry the exact values and draws the boundary between a theorem for an idealized open Jackson tandem and evidence about a real operating network.

highlighted = computed this step

Pinned rates

This model uses arrival rate 1, station A service rate 3, and station B service rate 2. Why: the exact conclusions depend on these pinned inputs; changing any rate changes the utilizations, product-form factors, and network metrics. The lesson is not estimating a real system from observations, it is checking what this idealized model implies.

\lambda=1\quad \mu_A=3\quad \mu_B=2

Stability check

The two utilizations are 1/3 and 1/2. Why: product-form steady state is only claimed for stable stations. A utilization below the stability threshold means the node has spare service capacity in the long run; without that, the steady-state formulas would not be honest.

\rho_A=1/3\quad \rho_B=1/2

Exact claim

The exact mean network count is 3/2. Why: this value is recomputed from the exact tandem model, not fitted from samples. The interpretation is a model consequence: if the assumptions hold, this is the steady-state mean number in the network; if the assumptions do not hold, the arithmetic is not evidence by itself.

L=3/2

Model boundary

These are exact steady-state values for an idealized open Jackson tandem with memoryless service, infinite buffers, stable rates, and deterministic routing; real systems need measured arrivals, service times, and routing evidence. Product form is a theorem for this open Jackson tandem class, not a universal claim about every network. Real networks need measured external arrivals, measured service behavior, routing evidence, and a separate empirical fit before these exact formulas should be used as descriptions. Pixel positions are rounded for layout; every number shown is exact.

\text{exact model first, empirical fit later}