Mx/G/1 retrial vacation queue for multi-optional services, phase repair and reneging

The present paper deals with a batch arrival general retrial queue with multi-optional services, vacation and impatient customers undergoing reneging. The incoming customer (or a batch of customers) on finding the server in busy or in non-working condition forms a virtual pool of customers called orbit otherwise undergoes service if the server is free. The service is provided at two levels; first essential service or second optional service depending upon customer's choice to opt for it or not. Moreover, the server may go for vacation if he finds no customer waiting for service and returns back to the service center again as soon as the customer approaches for the service. The server is unreliable as such subject to breakdown during the service; as soon as the server fails, it is immediately sent for repair so as to restore its functionality as before failure. The repairman assigned for the repair of the server also takes some setup time before commencing the repair process. By using supplementary variables technique and probability generating function method, the explicit expressions for the queue size distribution and other performance measures have been derived. The sensitivity of different parameters on various performance indices has been examined numerically by taking an illustration. Also, the neuro fuzzy approach has been used to approximate the analytical results.