MACHINE REPAIR PROBLEM WITH MULTIPLE WORKING VACATIONS AND TWO REMOVABLE SERVERS OPERATING UNDER THE TRIADIC (��,��,��,��) POLICY

Abstract

The main objective of this study was to investigate the 𝑀/𝑀/2 machine repair problem with 𝐿 operating machines and two removable servers that repair the upcoming failed machines under the multiple working vacations and triadic (0,𝑄,𝑁,𝑀) policies. By using the idea of Markov process for the transition of each state, set of difference differential equations of the model have been derived. The existence of steady state of the model was detected. After steady state analysis of the model has been guaranteed, limiting distribution was used to derive the steady state equations from the obtained difference differential equations of the model. These steady state equations of the model have been solved recursively and expressions for the steady state probabilities of the number of failed machines in the system were derived and obtained in a closed-form. By taking the obtained expressions of the probabilities as the building blocks, important performance measures of the system were formulated. The formulated performance measures of the system were used to establish the total expected cost function per machine per unit time and formulate a cost optimization problem. Grid search algorithm was applied to determine the optimal operating (0,𝑄,𝑁,𝑀) policy wherein minimizing the cost function is the criteria for the operating policy’s optimality. The overall behaviour of the system was detected with the influences of model’s parameters and MATLAB was used in this aspect. Recursive method was a powerful and easy method for solving the present model. It was further seen that multiple working vacations has played a significant role in minimizing the waiting time of the failed machines in the system which can minimize the loss of the system associated with the waiting time. But there is still a production or system breakdown and to avoid this loss, standby units can be assigned. Therefore, the present model can be extended with an additional condition called standby units.