quasi-reversibility
Quasi-reversibility is the structural property of a queueing station that explains WHY product-form networks (like Jackson networks) have their beautiful factorized stationary distributions. Developed by Frank Kelly, it is a weakening of full reversibility tailored to networks with several customer classes and routing: a station need not look the same backwards in time, but its arrival and departure streams must have a particular Markov structure. It is the unifying principle behind a wide class of product-form results.
A queue, viewed as a Markov process with marked arrival and departure events of various classes, is quasi-reversible if, at any fixed time, the state is independent of the future arrival times of each class AND independent of the past departure times of each class — equivalently, both the arrival processes and the (time-reversed) departure processes are Poisson with rates not depending on the state. The key payoff: for a quasi-reversible station fed Poisson input, the DEPARTURE process of each class is also Poisson (this is the abstract form of Burke's theorem), so quasi-reversible stations can be hooked together and the Poisson-in / Poisson-out property propagates. When you assemble a network of quasi-reversible stations, the global stationary distribution factorizes into a product over stations, each station carrying the stationary law it would have in isolation. Quasi-reversibility is implied by, but strictly weaker than, full reversibility (detailed balance); it relies on a 'partial balance' condition where flows balance class-by-class rather than edge-by-edge.
Quasi-reversibility matters because it greatly enlarges the world of solvable networks: multiclass queues, certain processor-sharing and symmetric service disciplines, and BCMP networks (Baskett-Chandy-Muntz-Palacios) all fit, well beyond the single-class M/M Jackson setting. The honest caveat: quasi-reversibility is a real hypothesis, not a free lunch. It pins down the allowed service disciplines and class structures; many natural queueing features — first-come-first-served with non-exponential service, state-dependent routing that creates correlations, blocking and finite buffers with overflow — break quasi-reversibility, and then product form fails and the network must be solved by other (usually numerical) means.
An M/M/1 queue is quasi-reversible: by Burke's theorem its departure process is Poisson(lambda), the same rate as its Poisson arrivals, and at any time the queue length is independent of future arrivals and past departures. That is exactly the property that lets you chain M/M/1 stations into a Jackson network with product-form equilibrium.
Quasi-reversibility = Poisson-in / Poisson-out plus the right independence; chaining such stations gives product form.
Quasi-reversibility is strictly weaker than reversibility (detailed balance) but still a genuine hypothesis: it constrains the service discipline and class structure. FCFS with non-exponential service, correlation-inducing state-dependent routing, and finite-buffer blocking typically break it, and then product form does not hold.