Analysis of Minimal Path Routing Schemes in the Presence of Faults

Gordon, J.M., Analysis of minimal path routing schemes in the presence of faults, Discrete Applied Mathematics 37138 (1992) 245-263. The design and analysis of fault tolerant message routing schemes for large paraIIe1 systems has been the focus of much recent research. In this paper, we present a framework for the analysis of routing schemes in distributed memory multiprocessor systems containing faulty or unusable components. We introduce techniques for the derivation of the probabilities of successfully routing a single inessage using minimal path routing schemes. Using this framework, we derive closed form solutions for a wide range of routing schemes on the hypercube and on the twodimensional mesh. The results obtained show the surprising resilience of the hypercube to a potentially large number of faults while demonstrating the inability of the mesh to tolerate a comparatively smaller number of faults. component failure e Since the corresponding mean time to failure is likely to be short, off-line fault diagnosis and the subsequent replacement of failed units is not an attractive alternative for dealing with the problem. Rather, we would like to be able to continue system operation in the presence of such failures. A fundamental component of the operation of distributed memory multiprocessor systems is the routing of messages. The large size of these parallel systems mitigates against the selection G; routing algorithms which require the propagation of global fault information to individual processors. Rather, we would prefer to use approaches which require Dnly local fault infxmation, such as the fault status of immediate neighbors. For similar reasons, we would prefer to use distributed (as opposed to centralized) routing schemes, i.e., ones in which routing decisions are made at the individual processors. * Supported in part by NS. r crant No. DCR-8507785 1 and a fellowship from the Unisys Corporation. Current address: IBM Corporation, 11400 Burnet Road, Zip 9641, Austin, TX 78758, USA. 0166-218X/92/$05.00