Iv. Optimizing Availability Provisioning In Nfv . Even

IV. OPTIMIZING AVAILABILITY PROVISIONING IN NFV Even though the efficiency is greatly improved, our framework is still impractical in the sense that the process can be further prolonged with 1) a even higher availability requirement, which is normal in a carrier-class network [18] (usually requires an availability of 99.999% or 99.9999% (5’9s or 6’9s)); 2) more requests; and 3) longer contract period duration. The root cause the process takes such a long time is it depends on the strategy we use to provision backups and how many backups are needed. In other words, if we can accurately estimate the least possible number of backups needed without running the trial-and-error process, a great amount of running time can be saved. The process†¦show more content†¦Let _rk = P[Nr i = kr], kr = 0; 1; : : : ; nri + kr i be the probability mass function (pmf) for the Poisson binomial random variable Nr i , then the cumulative distribution function (cdf) of Nr i , denoted by FNr i (kr) = P[Nr i _ kr], kr = 0; 1; : : : ; nri +kr i , gives the probability of having at most kr successes out of a total of nri + kr i , and can be written as [26]: where Rm is the set of all subsets of m integers that can be selected from f1; 2; : : : ; nri + kr i g, and Ac is the complement of set A. Therefore, we have Thus, the probability that all the VNFs are available can be expressed as However, computing such a function is not easy, which requires one to enumerate all elements in Rm, which requires to consider an exponential number of scenarios. It is huge even when nri +kr i is small. In general, we can have the following theorem: Theorem 1. Verifying if the availability of a given deployed SFC request with backups is above a given threshold is PPcomplete. Proof. The detailed proof is ignored here due to the limited space. In general, we can rewrite Eq. (6) as a boolean formular, where xj and _ xj represent if the j-th VM is available or not, while Q and P are replaced by ^ and _, respectively. Then the proof follows as in [13]. To efficiently compute the availability of an SFC if