This research aims to investigate convexity conditions for a single server discrete-time queueing system with setback (disaster). The system operates under a randomized threshold policy \(\left(p,\;N\right)\) , where the server activates with probability \(p\) when queue length reaches threshold \(N\) , or remains idle with probability \(1-p\) , providing flexible control over system congestion and resource utilization. This policy is particularly important in systems prone to sudden disruptions, as it helps optimize service efficiency while managing system recovery after setbacks. First, we perform the convexity analysis analytically for the discrete parameter \(N\) . Then the optimal queue length for the best value of \(N\) is determined. Also, as the optimization problem for finding the optimal value of continuous parameter \(p\) is a linear fractional programming problem thus Charnes and Cooper method is used to get the best value of \(p\) . Furthermore, by constructing a cost function and using the direct search method and firefly algorithm, the minimum cost is estimated. The goal of this work is to show how convexity in a discrete-time work frame can provide fresh perspectives on existing problems and lead to significantly simpler analyses and algorithm modifications. Also we compare analytical results with that of ANFIS (Adaptive Neuro-Fuzzy Inference System) results which validate our findings.
