We investigate the optimal allocation of effort to a collection of n projects. The projects are ‘restless' in that the state of a project evolves in time, whether or not it is allocated effort. The evolution of the state of each project follows a Markov rule, but transitions and rewards depend on whether or not the project receives effort. The objective is to maximize the expected time-average reward under a constraint that exactly m of the n projects receive effort at any one time. We show that as m and n tend to ∞ with m/n fixed, the per-project reward of the optimal policy is asymptotically the same as that achieved by a policy which operates under the relaxed constraint that an average of m projects be active. The relaxed constraint was considered by Whittle (1988) who described how to use a Lagrangian multiplier approach to assign indices to the projects. He conjectured that the policy of allocating effort to the m projects of greatest index is asymptotically optimal as m and n tend to∞. We show that the conjecture is true if the differential equation describing the fluid approximation to the index policy has a globally stable equilibrium point. This need not be the case, and we present an example for which the index policy is not asymptotically optimal. However, numerical work suggests that such counterexamples are extremely rare and that the size of the suboptimality which one might expect is minuscule.

We consider a memoryless single station service system with servers S = {m 1 , . . . , m K }, and with job types C = {a, b, . . .}. Service is skill-based, so that server m i can serve a subset of job types C(m i ). Waiting jobs are served on a first-come-firstserved basis, while arriving jobs that find several idle servers are assigned to a feasible server randomly. We show that there exist assignment probabilities under which the system has a product-form stationary distribution, and obtain explicit expressions for it. We also derive waiting time distributions in steady state.

We consider an infinite sequence of items of types C = {c1, . . . , cI }, and another infinite sequence of items of types S = {s1, . . . , sJ }, and a bipartite graph G of allowable matches between the types. Matching the two sequences on a first come first served basis defines a unique infinite matching between the sequences. For (ci, sj) ∈ G we define the matching rate rc i ,s j as the long term fraction of (ci, sj) matches in the infinite matching, if it exists. We assume that the types of items in the two sequences are i.i.d. with given probability vectors α, β. We describe this system by a Markov chain, obtain conditions for ergodicity, and derive its stationary distribution which is of product form. We show that if the chain is ergodic, then the matching rates exist almost surely, and give a closed form formula to calculate them.

Reentrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or near-optimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai (1995) which states that a scheduling policy is stable if the corresponding fluid model is stable, we study the stability and instability of fluid models. To do this we utilize piecewise linear Lyapunov functions. We establish stability of First-Buffer-First-Served (FBFS) and Last-Buffer-First-Served (LBFS) disciplines in all reentrant lines, and of all work-conserving disciplines in any three buffer reentrant lines. For the four buffer network of Lu and Kumar we characterize the stability region of the Lu and Kumar policy, and show that it is also the global stability region for this network. We also study stability and instability of Kelly-type networks. In particular, we show that not all work-conserving policies are stable for such networks; however, all work-conserving policies are stable in a ring network.