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This article aims to present the steady-state analysis in finite-buffer M/G/1 queues with starting failures under a randomized control of N-policy. When the system empties, the server is turned off. If the system size reaches the threshold N, the server is turned on with probability p or turned off with probability 1 − p. The server needs a startup time before providing service. If the server is started successfully (with probability θ), customers are served immediately. When the server is started unsuccessfully, he is started again. It is assumed that the restart is always successful. Using the supplementary variable method, we obtain the stationary distribution of the number of customers in the system. We develop a number of performance measures. A cost model is constructed, and we then determine numerical values of the optimal service rate to minimize the average cost per unit time. Finally, numerical examples are provided to show the effects of system parameters on performance measures and the optimal service rate. [ABSTRACT FROM AUTHOR]

We consider the distributions of operating characteristics of an M[x]/G/1 queue under vacation policies, where the first customer of each busy period receives an exceptional service. When all the customers are served in the system exhaustively, the server deactivates and operates one of two vacation policies: (1) multiple vacation policy and (2) single vacation policy. We develop the performance measures for both systems. Finally, some numerical illustrations are also given. These two vacation models have potential applications in day-to-day life, such as post offices, banks, hospitals, etc. [ABSTRACT FROM AUTHOR]