Markovian Queueing System with Bulk Arrival and Three Types of Essential and Optional Services

Abstract

This paper analyses the single server bulk arrival Markovian queueing system where customers arrive according to the general bulk rule and follows the Poisson distribution. Customers are served with three types of services, one is essential service and two of them are optional services. Server is giving service to the customers according to Exponential distribution with the bulk service rule. After providing service to the bulk, server goes for a vacation according to the exponentially distribution. Server returns from the vacation when there are sufficient number of customers is available for receiving the service. Again server go back for a vacation if arrived number of customers is not enough to get the essential service. The server returns when arrived bulk is sufficient. After receiving the essential service, customers have the choice that they take first essential service, second essential service or take both. We have also include the impatience behaviour of customers i.e. the bulk of customers also balks from the system if the server is busy to giving service to other customers or server is on the vacation. We have computed the time dependent probability generating functions with the help of corresponding steady state results. Also we determine the number of customers waiting in the queue for this system.