Ternary State Circular Sequential $k$-out-of-$n$ Congestion System

A ternary state circular sequential k-out-of-n congestion (TSCSknC) system is presented. The system is an extension of the circular sequential k-out-of-n congestion (CSknC) system which consists of two connection states: a) congestion (server busy), and b) successful. In contrast, a TSCSknC system considers three connection states: i) congestion, ii) break down, and iii) successful. It finds applications in some reliable systems to prevent single-point failures, such as the ones used in (k,n) secret key sharing systems. The system further assumes that each of the n servers has known connection probabilities in congestion, break-down, and successful states. These n servers are arranged in a circle, and are made with connection attempts sequentially round after round. If a server is not congested, the connection can be either successful, or a failure. Previously connected servers are blocked from reconnecting if they were in either states ii), or iii). Congested servers are attempted repeatedly until k servers are connected successfully, or (n-k+1) servers have a break-down status. In other words, the system works when k servers are successfully connected, but fails when (n-k+1) servers are in the break-down state. In this paper, we present the recursive, and marginal formulas for the system successful probability, the system failure probability, as well as the average stop length, i.e. number of connections needed to terminate the system to a successful or failure state, and its computational complexity.