Convexity analysis and cost optimization of a retrial queue with Bernoulli vacation and delayed phase mending

A recurrent queueing system under multi-optional and state dependent arrivals has been explored wherein the customers/units join in cluster form. The service provided to the units is carried out in duet junctures. The server is unreliable and may collapse. Thereafter, it needs to be mended in d requisite phases. Delay in the mending procedure is also considered. Whenever the system is free of customers, then server either departs for a vacation with a certain probability or pursues to serve the next customer with the complimentary probability. Eminent results are derived by employing Embedded Markov Chain (EMC) approach. Further, convexity of expected system size is also discussed. Also, the results are verified via Adaptive Neuro Fuzzy Interface System (ANFIS). Lastly, cost optimization is carried out and optimal cost is compared via different methods, namely: Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Artificial Bee Colony (ABC) algorithm and Cuckoo Search Algorithm (CSA).