Effective Arrival Rates - Queueing II: Networks

Before computing congestion, each station needs its effective arrival rate. In this deterministic tandem, every external arrival visits both stations, so the node arrival rates stay exact. The lesson separates external arrivals from routed arrivals because that distinction is what makes queueing networks different from isolated queues.

highlighted = computed this step

Arrival to station A

The effective arrival rate at station A is 1. Why: outside jobs enter there first, so station A sees the external stream directly. This is the easy node; the subtle part of a tandem is that the next node's input is created by routing completed work forward.

\lambda_A=1

Route to station B

The routing probability from station A to station B is 1. Why: this scoped network is a deterministic tandem, meaning no completed job branches away before station B. That route probability is what converts station A departures into station B arrivals in the traffic equations.

r_{AB}=1

Arrival to station B

The effective arrival rate at station B is 1. Why: all station A completions become station B arrivals, so the downstream arrival rate is not guessed independently. It is recomputed from the external rate and routing rule, which keeps the network accounting honest.

\lambda_B=\lambda_A r_{AB}=1

Diagram note

The table shows the recomputed node arrival rates and deterministic route probability. Interpreting these entries correctly is the bridge from a path diagram to queueing formulas: each node receives its own effective arrival rate before utilization or waiting time is computed. 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. Pixel positions are rounded for layout; every number shown is exact.

\text{effective arrivals are exact fractions}