Cost Parameters-Based Comprehensive Analysis of a New Cost Function Construction for Coxian-k Queueing System Characterized by Customer Service Speed Variability

We investigate cost optimization in an M/Coxk/1 queueing system with phase-dependent service speeds. A unified parametric framework is introduced to model both homogeneous and heterogeneous service regimes, and closed-form expressions for steady-state performance measures are derived. These results are used to construct an expected total cost function explicitly parameterized by the traffic intensity. We prove that the cost function is strictly convex on the stability region, ensuring the existence and uniqueness of the optimal traffic intensity. For the Coxian-2 case, analytical and numerical sweep analyses are conducted with respect to waiting and service-capacity cost parameters. Polynomial response surfaces and nonparametric statistical tests are employed to validate the robustness of the results. The analysis shows that balanced service speeds across phases consistently yield lower optimal traffic intensity levels and reduced expected total costs, whereas heterogeneous service speeds increase congestion and cost sweep. These findings provide practical guidance for the economic design and control of multi-phase service systems.