A class of open queueing networks with Poisson external arrivals, exponential service times at nodes, and fixed probabilistic routing between nodes such that, in steady state, the joint distribution of queue lengths factorizes into the product of individual node distributions (product‑form). Node arrival rates satisfy a system of linear traffic equations.

Principle

Under the stated assumptions the network decomposes: each node behaves like an independent M/M/c or M/M/1 queue fed by an effective Poisson arrival rate obtained from solving traffic equations, and the network steady‑state is the product of node steady‑states (product‑form).

Demonstration

Illustrative scenario → A manufacturing line with multiple machines and probabilistic routing of parts between stations, plus Poisson external arrivals. Recognition → Verify Poisson arrivals, exponential service and fixed routing probabilities. Action → Solve linear traffic equations for node arrival rates, analyze each node's queueing performance independently, and combine results multiplicatively for network metrics. Consequence → Throughput and per‑station performance are computed from node analyses without enumerating joint state space.

Misapplication

Applying Jackson product‑form when routing is state‑dependent, arrivals non‑Poisson, service-time distributions non‑exponential, or when service disciplines violate assumptions. The error assumes node independence and product‑form without verifying conditions, producing incorrect performance estimates.

Consequence

When assumptions hold, Jackson networks provide tractable, exact node‑level performance measures and scalable analysis; if violated, relying on product‑form underestimates correlations and can mislead capacity, delay and throughput planning.

Reversal

If service time distributions are general (non‑exponential), arrivals or routing are state‑dependent, or discipline restrictions exist, the product‑form result may fail; other models (e.g., BCMP networks under specific conditions or approximate/simulation methods) are required.

Boundary

Clearly within: open networks with Poisson external arrivals, exponential service at nodes, fixed routing probabilities and service disciplines compatible with Jackson assumptions, analyzed in steady state. Boundary case: small deviations from Poisson/exponential that can be approximated but reduce exactness. Clearly outside: networks with state‑dependent routing, non‑Markovian service, or closed networks without the open Jackson assumptions.

Semantic Tension

Analytic tractability versus modeling fidelity: Jackson networks' product‑form offers exact, scalable solutions under Markovian assumptions but those assumptions restrict applicability; relaxing them improves realism at the cost of losing closed‑form decomposition.

Synthesis

Jackson networks show that under Markovian and fixed‑routing assumptions network performance reduces to node‑level problems linked by linear traffic equations: the key modeling decision is whether those assumptions are acceptable for the system of interest, because they determine whether decomposition yields valid, actionable metrics.