Optimization Analysis of a Multi-Server Queueing System with Two-Phase Heterogeneous Services and Synchronous Working Vacation Interruptions

Despite extensive research on queueing models in flexible manufacturing systems (FMS), few studies have simultaneously considered the heterogeneity of servers and the impact of synchronous working vacations on system performance. To fill this research gap, this study proposes a novel multi-server queueing model that uniquely integrates two-phase heterogeneous services with synchronous working vacation interruptions. The innovation lies in capturing the complex task processing mechanism where the two-phase service is provided by different servers, and they conduct working vacations synchronously when the system is empty and terminate vacations once the system population reaches a specified threshold. Based on the matrix-analytic approach, this research investigates the stability condition and the stationary distribution of the system. A key numerical finding is the diminishing marginal returns of server numbers, where exceeding an optimal count increases congestion, thereby degrading overall performance. The vacation interruption threshold l is also shown to significantly influence server state allocation. A comparative analysis of three different system configurations, demonstrates that dynamic server number or a vacation interruption threshold adjustment effectively mitigates congestion; and in order to enhance the system clearing capability, when vacations are longer, opting for multiple servers is preferable, whereas a single server is more suitable during shorter vacation periods. For cost optimization, three algorithms (CPSO, JAYA, MLS-JAYA) consistently converge to the same robust optimal solution for server count and vacation rate. Furthermore, to simultaneously minimize waiting time, we apply the MOEA/D algorithm to study the bi-objective optimization problem.