The Fork-Join program consisting of K parallel tasks is a useful model for a large number of computing applications. When the parallel processor has multi-channels, later tasks may finish execution earlier than their earlier tasks and may join with tasks from other programs. This phenomenon is called exchangeable join (EJ), which introduces correlation to the task’s service time. In this work, we investigate the response time of multiprocessor systems with EJ with a new approach. We analyze two aspects of this kind of systems: exchangeable join (EJ) and the capacity constraint (CC). We prove that the system response time can be effectively reduced by EJ, while the reduced amount is constrained by the capacity of the multiprocessor. An upper bound model is constructed based on this analysis and a quick estimation algorithm is proposed. The approximation formula is verified by extensive simulation results, which show that the relative error of approximation is less than 5%.
Due to various advantages in storage and implementation, simple strategies are usually preferred than complex strategies when the performances are close. Strategy optimization for controlled Markov process with descriptive complexity constraint provides a general framework for many such problems. In this paper, we first show by examples that the descriptive complexity and the performance of a strategy could be independent, and use the F-matrix in the No-Free-Lunch Theorem to show the risk that approximating complex strategies may lead to simple strategies that are unboundedly worse in cardinal performance than the original complex strategies. We then develop a method that handles the descriptive complexity constraint directly, which describes simple strategies exactly and only approximates complex strategies during the optimization. The ordinal performance difference between the resulting strategies of this selective approximation method and the global optimum is quantified. Numerical examples on an engine maintenance problem show how this method improves the solution quality. We hope this work sheds some insights to solving general strategy optimization for controlled Markov process with descriptive complexity constraint.
