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The next generation of capability-class, massively parallel processing (MPP) systems is expected to have hundreds of thousands of processors. For application-driven, periodic checkpoint operations, the state-of-the-art does not provide a solution that scales to next-generation systems. We demonstrate this by using mathematical modeling to compute a lower bound of the impact of these approaches on the performance of applications executed on three massive-scale, in-production, DOE systems and a theoretical petaflop system. We also adapt the model to investigate a proposed optimization that makes use of "lightweight" storage architectures and overlay networks to overcome the storage system bottleneck. Our results indicate that (1) as we approach the scale of next-generation systems, traditional checkpoint/restart approaches will increasingly impact application performance, accounting for over 50% of total application execution time; (2) although our alternative approach improves performance, it has limitations of its own; and (3) there is a critical need for new approaches to fault tolerance that allow continuous computing with minimal impact on application scalability.