Bernoulli's vacation of M^x/G/1 queue with two-tier service based volatile server

Abstract

This article describes a Bernoulli's vacation of M^x/G/1 queue with two-tier service based volatile server. When the server is at running stage, the server may fail/breakdown and the service line will no longer be available due to downtime and latency.  If the client does not come when the server is available, then the server will remain inactive on the system until the queue size increases to the threshold value . If the client does not come when the server is unavailable, the server will remain inactive, but the threshold of  will decrease. When the queue size is greater than the threshold value of , then the server immediately starts to do the service of its pending works of its clients. When the queue size is lesser than the threshold value of , then the server immediately starts to do the re-service of its clients. In general, the distribution of queue sizes by random and departure periods, as well as various performance indicators of the system. After that the server can go on vacation or stay in the system to service the next device if necessary.  Stationary analysis of the system is extended including the existence of stationary regime, queue size distribution of idle period process, embedded Markov chain steady state distribution along with some system characteristics.