Exact analysis of production lines with Coxian-2-distributed processing times and parallel machines

This study considers production lines where each station has parallel machines with 2-phase Coxian processing times. The setting in this paper is designed specifically for scrutiny of the replenishment of raw materials and finished goods inventories with intermediate buffers in-between stations. Each buffer has a limit of capacity. Raw material supply and demand for finished goods are generated according to independent stationary Poisson processes. Coxian-2 processing times can be utilized to model failure-prone machines with exponential service times, times to failure, and repair times. The second phase of Coxian-2 can also be considered as a rework operation visited with a predefined probability. We model the line as a continuous-time Markov chain and propose recursive algorithms to generate the transition rate matrix. Although the general recursive form is specific to 3-station 4-buffer lines, routines for calculating the number of states and generating the states work for any M-station (M + 1)-buffer systems. The developed model allows obtaining steady-state distribution and performance metrics such as throughput, the average number of items in the buffers, and average system cost consisting of production, holding, and shortage costs. Furthermore, we enrich our study with numerical experiments and analyze the impacts of buffer capacities, processing rates of the machines and the number of parallel machines on the system performance. Moreover, the exact analysis provided in this paper can also be used as the decomposition block for the performance analysis of longer lines.