Analytical Study of Priority-Feedback Queue Network Model Comprising of Three Parallel Servers within Stochastic Conditions

This study presents a comprehensive analytical investigation of a four-server queueing system, modeled as a priority feedback queue network operating under stochastic conditions. The system architecture comprises a primary server that incorporates a priority mechanism and is connected to three parallel servers. Customers, divided into low- and high-priority groups, arrive at the primary server according to a Poisson process. Service times at all servers are assumed to follow exponential distributions. After completion of service at any server, a customer will either exit the system or re-enter for further service with a different feedback probability, which reflects a realistic service satisfaction mechanism. This analysis evaluates explicit expressions for time-independent state probabilities and derives steady-state performance metrics of the network, including server utilization, average queue lengths, waiting times, and throughput by application of generating function techniques and the calculus-based resolution of the associated differential-difference equations. Extensive numerical simulations are carried out to validate the theoretical results and to delve into the impact of different parameters on various performance indicators, which gives information to system administration in order to reduce congestion. This study yields actionable insights into the optimal allocation of resources, aiming to minimize delays and maximize throughput under varying stochastic conditions. Furthermore, the behavioral analysis of the model underscores the notable effects of priority and feedback mechanisms on the system achievement measures and provides a foundation for optimization strategies in multi-class service environments. The findings have significant implications for the design and optimization of complex queueing networks in telecommunications, manufacturing systems, and other service-oriented industries where feedback and priority-based processing are prevalent.