Analytical model for computing hypercube availability

An analytical model is presented for computing the availability of an n-dimensional hypercube. The model computes the probability of j connected working nodes in a hypercube by multiplying two probabilistic terms. The firsts term is the probability of x connected nodes (x ≥ j) working out of 2ⁿ fully connected nodes. This is obtained from the numerical solution of the well-known machine repairman model, modified to capture imperfect coverage and imprecise repair. The second term, which is the probability of having j connected nodes in a hypercube, is computed from an approximate model of the hypercube. The approximate model, in turn, is based on a decomposition principle, where an n-cube connectivity is computed from a two-cube base model using a recursive equation. The availability model studied here is known as a task-based availability, where a system remains operational as long as a task can be executed on the system. Analytical results for n-dimensional cubes are given for various task requirements. The model is validated by comparing the analytical results with those from simulation.