Exact reliabilities of most reliable double‐loop networks

A double‐loop network with hop constants h 1 , h 2 , DL ( n , h 1 , h 2 ) may be represented as a directed graph with n nodes 0, 1, …, n − 1 and 2 n links of the form i → i + h 1 mod n and i → i + h 2 mod n (referred to as h 1 ‐links and h 2 ‐links). They have been proposed as architectures for local area networks and for data alignment in SIMD processors, among other applications. Three reliability models of double‐loop networks have been studied in the literature. In the link model , nodes always work and each link fails independently with probability p . Hwang and Li showed that for p small DL ( n , 1, 1 + n /2) is most reliable for n even, and DL ( n , 1, 2) is most reliable for n odd. In the node model , links always work and each node fails independently with probability p . Hu et al. showed that for p small DL ( n , 1, 1 + ⌈ n /2⌉) is the most reliable. However, no nonenumerative algorithms were given to compute the reliabilities of these most reliable networks except DL ( n , 1, 1 + n /2) for even n under the node model. Recently, Hwang and Wright proposed a novel approach to compute the reliabilities of double‐loop networks under the uniform model that each node fails with probability p , each h 1 ‐link with probability p 1 , and each h 2 ‐link with probability p 2 , and the failures are independent. In particular, they obtained the reliabilities for DL ( n , 1, 2). In this paper, we applied their approach to compute the reliabilities of DL ( n , 1, 1 + ⌈ n /2⌉) under the uniform model, except that for n odd we need the assumption that h 1 ‐links always work. Note that even under this additional assumption our reliability model is more general than is the node model, the original model under which DL ( n , 1, 1 + ⌈ n /2⌉) is found to be most reliable for n odd. We also used this approach to obtain the reliabilities of DL ( n , 1, n − 2), known as the daisy chain in the literature. © 1997 John Wiley & Sons, Inc. Networks 30 : 81–90, 1997