The matrix-form solution for Geo X/G/1/N working vacation queue and its application to state-dependent cost control

This paper studies a finite buffer size GeoˆX/G/1/N queue with single working vacation and different input rates. By combining two classic methods of supplementary variable and embedded Markov chain, the matrix-form solutions of queue-length distributions at departure, arbitrary, pre-arrival and outside observer's observation epochs are obtained. Based on these queue-length distributions, some performance measures are also derived and some numerical experiments are carried out. Then, an optimal control on state-dependent operating cost of a production to order is presented.