Construction and Optimization of a Cost Function in a Single‐Server Multiphase Queueing System With Hypoexponential‐ k Customer Service Time

In this paper, a new generalized cost function construction for a single‐server multiphase queueing system with a Poisson input stream, hypoexponential‐ k customer service time, and FIFO discipline is presented. A proposed relationship between service parameters in k phases is based on a geometric progression with rate α , 0 < α < 1. In this way, an approach with a constantly decreasing speed of servicing customers through the phases is introduced. Several auxiliary lemmas about the performance measures of the proposed model are proved. Then, an explicit form of the cost function is obtained in such a construction, expressed as a function of the traffic intensity ρ . Also, an analysis of the cost function is made, and a theorem for the existence and uniqueness of the absolute minimum of the cost function is proved. Furthermore, a sensitivity analysis of the optimal solutions with different changes in the cost values, input stream parameter, and progression rate α is additionally made.