An improved approximation ratio for the minimum latency problem
Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms
Minimizing service and operation costs of periodic scheduling
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
The data broadcast problem with non-uniform transmission times
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
The Complexity of Optimal Queuing Network Control
Mathematics of Operations Research
A new greedy approach for facility location problems
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Dynamic Programming and Optimal Control
Dynamic Programming and Optimal Control
Approximating the Stochastic Knapsack Problem: The Benefit of Adaptivity
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Adaptivity and approximation for stochastic packing problems
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
AdWords and Generalized On-line Matching
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
The pipelined set cover problem
ICDT'05 Proceedings of the 10th international conference on Database Theory
Approximation algorithms for restless bandit problems
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Approximation algorithms for restless bandit problems
Journal of the ACM (JACM)
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We study the stochastic machine replenishment problem, which is a canonical special case of closed multiclass queuing systems in Markov decision theory. The problem models the scheduling of processor repairs in a multiprocessor system in which one repair can be made at a time and the goal is to maximize system utilization. We analyze the performance of a natural greedy index policy for this problem. We first show that this policy is a 2 approximation by exploring linear queuing structure in the index policy. We then try to exploit more complex queuing structures, but this necessitates solving an infinite-size, non-linear, non-convex, and non-separable function-maximization program. We develop a general technique to solve such programs to arbitrary degree of accuracy, which involves solving a discretized program on the computer and rigorously bounding the error. This proves that the index policy is in fact a 1.51 approximation. The main non-trivial ingredients of the proof are two folds: finding a way to analyze the complex queuing structure of the index policy, and bounding the error in discretization when numerically solving the non-linear function-maximization. We believe that this framework is general enough to be useful in the analysis of index policies in related problems. As far as we are aware, this is one of the first non-trivial approximation analysis of an index policy a multi-class queuing problem, as well as one of the first non-trivial example of an approximation ratio that is rigorously proven by numerical optimization via a computer.