To estimate the stage completion time for a two-stage flexible flow shop with dynamic job arrival

Considering the two-stage flexible flow shop with dynamic job arrival, each stage contains identical parallel machines, and the second stage receives jobs with stochastic arriving time from first stage. To estimate the job completion time at each stage is useful to enhance the customer service or production performances. The completion time of jobs at each stage can be determined with the sum of the stage arriving time, the expected stage waiting time, and the expected stage service time of jobs. The expected stage waiting time of job relates to the utilization rates of machines, which should be corrected due to the idleness of machine. The idle time depends on the number of jobs in the system at specific time. The birth-and-death process can be used to describe probabilistically how the number of jobs in the system changes as time increasing. Note the stage service time of jobs relates to the estimated setup time and the processing time of jobs at each stage. To analyze the accuracy on the estimated stage completion time of jobs, a computational test through simulation is conducted to verify the proposed probabilistic model.