Cost optimal analysis for a differentiated vacation machining system with discouragement under the two threshold control policies

Abstract

The present study explores the cost analysis for machine repair problem of queueing system using two-threshold control policies with discouragement and differentiated vacation. The way repairmen offer service has been established by two-threshold control policies based on queue size, with randomly distributed vacations of type $A_1$ (full vacation) and type $A_2$ (working vacation). Prior to the queue length being less than the lower threshold, both repairmen are on $A_1$ type vacations. One repairman remains on $A_1$ type vacation while the second repairman takes $A_2$ type vacation if the queue length is in the intermediate range of the two thresholds. If the queue length is at or exceeds the upper threshold, both repairmen serve the queue to the best of their abilities. Meanwhile, during $A_1$ and $A_2$ types of vacations, failing machines exhibit impatience, which is a common occurrence in getting the service. It is supposed that vacation and repair time are exponentially distributed. The recursive method is utilized to calculate steady-state system probabilities. Furthermore, several metrics have been defined to measure the effectiveness of the system. A cost model is developed using system performance data for cost analysis. The Fibonacci search algorithm (FSA) and an artificial bee colony (ABC) are used to find the optimal value for the choice of variables with the lowest expected cost. A real-world instance demonstrating the implementation of the proposed model in practice will be shown at the end.