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A batch arrival retrial queueing model with k optional stages of service, extended Bernoulli vacation and stand-by is studied. After completion of the ith stage of service, the customer may have the option to choose (i + 1)th stage of service with probability θi or may leave the system with probability qi = 1 − θi, I = 1, 2, …, k − 1 and qk = 1. After service completion of each customer, the server may take a vacation with probability v1 and extend a vacation with probability v2 or rejoin the system after the first vacation with probability 1 − v2. Busy server may get to breakdown and the stand-by server provides service only during the repair times. At the completion of service, vacation or repair, the server searches for the customer in the orbit (if any) with probability α or remains idle with probability 1 − α. By using the supplementary variable method, steady state probability generating function for system size, some system performance measures like Lq, Ls, Wq and Ws are discussed. Simulation results are given using MATLAB.

In this queueing model service process is split into two phases called first essential service and second essential service. Here the occurrence of breakdown during first essential service and second essential service are different. When the server got failure during first essential service, service process will be interrupted and sent to repair station immediately. On contrary during second essential service when the server got failure the service will not be interrupted, it performs continuously for current batch by doing some technical precaution arrangements. Server will be repaired after the service completion during renewal period. On service completion, if the queue length is less than 'a' then the server leaves for vacation. During vacation if the queue length reaches the value 'a' then the server breaks the vacation and performs preparatory work to start first essential service. Various performance measures and cost effective model with appropriate numerical solution of the model are presented.

In this paper, server provides service in two service modes, as single or in bulk according to the queue length. The server provides single service, if the queue length reaches value 'a' and batch service, if the queue length reaches the value 'c' (c > a). After completing the service if the queue length is less than 'a' then the server leaves for vacation. During secondary job (vacation) if the queue length attains the value 'a' then the server breaks the vacation and provides primary service. In the service completion epoch if the server is not reliable then the server has undergone renovation of service station. For the proposed model probability generating function of queue size at an arbitrary time epoch is obtained. Various performance measures are derived with suitable numerical illustration. A cost model is also presented for the proposed model.

In this paper, we have investigated bulk retrial queueing system with Bernoulli vacation schedule and variable server capacity. According to the variable server capacity bulk service rule, the server can serve a minimum of one customer and a maximum of B customers in a batch. After completion of a batch of service, if the orbit is empty, then the server will go for a vacation. After completion of each batch of service, if the orbit is non-empty, then the server either go for a vacation with probability p or resumes its service with probability p. If arriving customers finds the server is busy with service or on vacation, they are grouped and it termed as unsatisfied customers or orbit. Unsatisfied customers will retry its appeal for service after some time. We have presented steady state system equations for the proposed model. Also, we have obtained some important performance measures for this system. Further, we have given numerical justification for this queueing system.

This paper deals with the steady state analysis of unreliable batch arrival feedback retrial queue with k stages of service and Bernoulli vacation policy, where the busy server is subjected to breakdown due to the arrival of negative customers. After completion of the ith, (i = 1, 2, ..., k) stage of service, the customer may have the option to choose (i + 1)th stage with probability θi, with probability pi may join into orbit as feedback customer or may leave the system with probability 1 – θi – pi = qi, if the service is successful. The server takes Bernoulli vacation after the service completion. Arriving positive customers may balk (or renege) the system at particular times. After completion of the service, vacation or repair, the server searches for the customer in the orbit or remains idle. The system is solved by using the supplementary variable method and numerical illustrations are discussed.