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Vande Vate (1989) described the polytope whose extreme points are the stable (core) matchings in the Marriage Problem. Rothblum (1989) simplified and extended this result. This paper explores a corresponding linear program, its dual and consequences of the fact that the dual solutions have an unusually direct relation to the primal solutions. This close relationship allows us to provide simple proofs both of Vande Vate and Rothblum's results and of other important results about the core of marriage markets. These proofs help explain the structure shared by the marriage problem (without sidepayments) and the assignment game (with sidepayments). The paper further explores “fractional” matchings, which may be interpreted as lotteries over possible matches or as time-sharing arrangements. We show that those fractional matchings in the Stable Marriage Polytope form a lattice with respect to a partial ordering that involves stochastic dominance. Thus, all expected utility functions corresponding to the same ordinal preferences will agree on the relevant comparisons. Finally, we provide linear programming proofs of slightly stronger versions of known incentive compatibility results.

This paper studies the uid models of two-station multiclass queueing networks with deterministic routing. A uid model is globally stable if the uid network eventually empties under each nonidling dispatch policy. We explicitly characterize the global stability region in terms of the arrival and service rates. We show that the global stability region is deÿned by the nominal workload conditions and the "virtual workload conditions," and we introduce two intuitively appealing phenomena-virtual stations and push starts-that explain the virtual workload conditions. When any of the workload conditions is violated, we construct a uid solution that cycles to inÿnity, showing that the uid network is unstable. When all the workload conditions are satisÿed, we solve a network ow problem to ÿnd the coe cients of a piecewise linear Lyapunov function. The Lyapunov function decreases to zero, proving that the uid level eventually reaches zero under any nonidling dispatch policy. Under certain assumptions on the interarrival and service time distributions, a queueing network is stable or positive Harris recurrent if the corresponding uid network is stable. Thus, the workload conditions are su cient to ensure the global stability of two-station multiclass queueing networks with deterministic routing.

We develop a heuristic for a problem motivated by the loading of aircraft or trucks: pack blocks into a bin so that their center-of-gravity is as close as possible to a target point. Our heuristic either produces good solutions or else signals that none is possible. It also works when loading non-homogeneous blocks into a bin of non-zero and possibly nonhomogeneous mass.

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