Circular sequential k-out-of-n congestion system

A circular sequential k-out-of-n congestion (CSknC) model is presented. This model finds use in reliable systems to prevent single-point failure, such as (k,n) secret key sharing systems. The model assumes that each of the n servers has a known congestion probability. These n servers are arranged in a circle, and are connected sequentially. A server is connected successfully if it is not congested. Previously connected servers are blocked from reconnecting. Congested servers have to be reconnected until k servers are connected successfully. We present the recursive, and marginal formulas for the probability of success, as well as the average number of connection attempts needed with k successfully connected servers. The optimal arrangement of servers is specified to minimize the average number of connection attempts. These formulas can be used to rearrange the servers adaptively when the initial congestion probabilities are not known. Compared to the circular CknG systems, the CSknC systems are shown to have a higher success probability, and require less connection attempts with k successfully connected servers.