The restricted h-connectivity of the data center network DCell

Xi Wang, Jianxi Fan, Jingya Zhou, Cheng-Kuan Lin

Abstract: Traditional data center networks (DCNs) are faced with many challenges with the development of cloud computing. This fact makes design of new DCNs represented by DCell networks become a hot research topic. For any integers k≥0<math><mi is="true">k</mi><mo is="true">≥</mo><mn is="true">0</mn></math> and n≥2<math><mi is="true">n</mi><mo is="true">≥</mo><mn is="true">2</mn></math>, the k<math><mi is="true">k</mi></math>-dimensional DCell with n<math><mi is="true">n</mi></math>-port switches and tk,n<math><msub is="true"><mrow is="true"><mi is="true">t</mi></mrow><mrow is="true"><mi is="true">k</mi><mo is="true">,</mo><mi is="true">n</mi></mrow></msub></math> nodes, Dk,n<math><msub is="true"><mrow is="true"><mi is="true">D</mi></mrow><mrow is="true"><mi is="true">k</mi><mo is="true">,</mo><mi is="true">n</mi></mrow></msub></math>, has been proposed for an important DCNs as a server-centric DCN structure. In this paper, we prove that under the condition that each fault-free node of the Dk,n<math><msub is="true"><mrow is="true"><mi is="true">D</mi></mrow><mrow is="true"><mi is="true">k</mi><mo is="true">,</mo><mi is="true">n</mi></mrow></msub></math> has at least h<math><mi is="true">h</mi></math> fault-free neighbor(s) its restricted h<math><mi is="true">h</mi></math>-connectivity is (h+1)(k−1)+n<math><mrow is="true"><mo is="true">(</mo><mi is="true">h</mi><mo is="true">+</mo><mn is="true">1</mn><mo is="true">)</mo></mrow><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">−</mo><mn is="true">1</mn><mo is="true">)</mo></mrow><mo is="true">+</mo><mi is="true">n</mi></math> (resp. (n+k−h−1)th−n+1,n<math><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">+</mo><mi is="true">k</mi><mo is="true">−</mo><mi is="true">h</mi><mo is="true">−</mo><mn is="true">1</mn><mo is="true">)</mo></mrow><msub is="true"><mrow is="true"><mi is="true">t</mi></mrow><mrow is="true"><mi is="true">h</mi><mo is="true">−</mo><mi is="true">n</mi><mo is="true">+</mo><mn is="true">1</mn><mo is="true">,</mo><mi is="true">n</mi></mrow></msub></math>) with 0≤h≤n−1<math><mn is="true">0</mn><mo is="true">≤</mo><mi is="true">h</mi><mo is="true">≤</mo><mi is="true">n</mi><mo is="true">−</mo><mn is="true">1</mn></math> (resp. n≤h≤n+k−2<math><mi is="true">n</mi><mo is="true">≤</mo><mi is="true">h</mi><mo is="true">≤</mo><mi is="true">n</mi><mo is="true">+</mo><mi is="true">k</mi><mo is="true">−</mo><mn is="true">2</mn></math>), which is almost as (h+1)<math><mrow is="true"><mo is="true">(</mo><mi is="true">h</mi><mo is="true">+</mo><mn is="true">1</mn><mo is="true">)</mo></mrow></math> (resp. th−n+1,n<math><msub is="true"><mrow is="true"><mi is="true">t</mi></mrow><mrow is="true"><mi is="true">h</mi><mo is="true">−</mo><mi is="true">n</mi><mo is="true">+</mo><mn is="true">1</mn><mo is="true">,</mo><mi is="true">n</mi></mrow></msub></math>) times as traditional connectivity of Dk,n<math><msub is="true"><mrow is="true"><mi is="true">D</mi></mrow><mrow is="true"><mi is="true">k</mi><mo is="true">,</mo><mi is="true">n</mi></mrow></msub></math>. When the DCell network is used to model the topological structure of a large-scale DCN, this result can provide a more accurate measure for the fault tolerance of the network.