Finite Population And Finite Capacity Single Server Compulsory Working Vacation Queues With Single and Batch Service

A Markovian single server queue with source size finite has been considered. 
The services are given in two models single and batch mode. At each service 
complection point, the server takes vacation for a random period of time. During 
vacation, if customers arrives, the server starts service in a slower rate. As a 
modification, the source size isincreased to infinite and the waiting line size is 
controlled by a finite number. Both the models are analyzed in time dependent 
domain. A cost and profit structure are also defined. Some numerical illustrations 
are presented.