a Jackson network
/ JAK-suhn /
A Jackson network is a network of interconnected queues — service stations that customers move between — modeled as a single large continuous-time Markov chain. It is the foundational result of queueing-network theory: despite the stations being coupled (output of one feeds the input of others), the equilibrium distribution factorizes as if the queues were independent. This 'product form' is what makes large networks tractable and is the prototype for a whole family of product-form results.
Consider J single-server stations (an open network). Customers arrive from outside to station i as a Poisson stream of rate r_i, station i serves at rate mu_i, and a customer finishing at i routes to station j with probability P_ij or leaves the network with probability 1 - sum_j P_ij. The state is the vector n = (n_1, ..., n_J) of queue lengths, and the dynamics are a CTMC. First solve the traffic equations for the total arrival rate at each station: a_i = r_i + sum_j a_j P_ji, which simply say total flow in equals flow generated plus flow routed in. Set rho_i = a_i / mu_i. Jackson's theorem states that if every rho_i < 1, the network is positive recurrent and its stationary distribution is the PRODUCT pi(n) = prod_i (1 - rho_i) rho_i^{n_i} — each station behaves in equilibrium exactly like an independent M/M/1 queue with load rho_i. Remarkably this holds even though the internal flows are NOT Poisson and the queue lengths are NOT actually independent at a fixed time in the dynamics; the product form is a statement about the stationary law only.
Jackson networks underpin the analysis of computer systems, communication networks, manufacturing lines and any system of routed flows; the product form lets you size each station almost independently. The honest cautions are sharp. The result needs the rho_i < 1 stability condition at every station (one bottleneck destabilizes the whole network). The product form does NOT mean the stations are independent processes — it is only the equilibrium marginal that factorizes; the internal arrival streams are generally not Poisson (Burke's theorem rescues Poissonity for the departure stream of a single M/M/1, and feed-forward networks, but feedback breaks it). Product form is a delicate consequence of the exponential service and Markov routing assumptions, explained at a deeper level by quasi-reversibility; change the service distribution or add state-dependent routing and it can fail.
Two stations in tandem: external Poisson(r) into station 1, all departures route to station 2, then leave. Traffic equations give a_1 = r, a_2 = r; with rho_i = r/mu_i < 1, the stationary law is pi(n_1, n_2) = (1-rho_1) rho_1^{n_1} * (1-rho_2) rho_2^{n_2} — a product of two independent geometrics.
Jackson's product form: in equilibrium a network of coupled queues factorizes into independent M/M/1 marginals.
Product form is about the stationary distribution only — the stations are NOT independent processes and internal flows are generally not Poisson. The result needs rho_i < 1 at every station and the exponential-service / Markov-routing assumptions; it is explained, and bounded, by quasi-reversibility.