Analysis of a manufacturing queueing system using bi-level randomized policy

We propose a GI/M/1 queueing model with $$(\xi _1,N_1,N_2)$$ ( ξ 1 , N 1 , N 2 ) -policy. This means that the server shuts down immediately after serving all customers. The server is activated with probability $$\xi _1$$ ξ 1 as soon as queue size reaches to $$N_1(\ge 1)$$ N 1 ( ≥ 1 ) , or remain turned off with probability $$\xi _2~(=1-\xi _1)$$ ξ 2 ( = 1 - ξ 1 ) . When there are $$N_2(>N_1)$$ N 2 ( > N 1 ) customers in the system, the server begins to provide the service to those customers who are waiting for the service and continues to do so until there are no more customers left in the system. To determine the probability distribution of the system content at various epochs, the steady-state analysis of the model is performed by utilizing two well-known methods, namely, the supplementary variable and the difference equation technique. Various performance measures are also provided along with the cost analysis of the model. In the end, we validate our analytical results by means of a few numerical instances, and we also carry out a few numerical experiments to investigate the influence that various factors have on the performance characteristics of the model. We show that for given waiting time, the presented model provides a less expected cost as compared to the traditional N-policy; further, a small increment in expected cost gives a far lower value of waiting time in the queue which is beneficial from customers’ point of view. With regards to the renewal arrival process in $$(\xi _1,N_1,N_2)$$ ( ξ 1 , N 1 , N 2 ) -policy queueing models, we have provided a significant methodological contribution.