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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.

The objective of the paper is to analyse two-level control policy of an MX∕G(a, b)∕1 queueing system with fast and slow vacation rates and vacation disruption. In the service completion epoch, if the queue length is less than ‘a’, then the server leaves for a vacation. In this model depending upon the queue length, the server is allowed to take two types of vacation called fast vacation and slow vacation. Addressing this in the service completion epoch, if the queue length ψ(say) is less than β where β ζ, where a − 1 ≥ ζ > β during service completion, then the server leaves for fast vacation. During slow vacation if the queue length reaches the value ζ, then the server breaks the slow vacation and switches over to fast vacation. Also if the queue length attains the threshold value ‘a’ during fast vacation, then the server breaks the fast vacation too and moves to tune-up process to start the service. After tune-up process service will be initiated only if ψ ≥ N(N > b). For the designed queueing system probability, generating function of the queue size at an arbitrary time epoch is obtained by using supplementary variable technique. Various performance characteristics will also be derived with suitable numerical illustrations. Cost-effective analysis is also carried out in the paper.

This paper concentrates on single server bulk arrival queue system with batch size dependent service and working vacation. The server provides service in two service modes depending upon the queue length. The server provides single service if the queue length is at least ‘a’. On the other hand the server provides fixed batch service if the queue length is at least ‘k’ (k > a). Batch service is provided with some fixed batch size ‘k’. After completion of service if the queue length is less than ‘a’ then the server leaves for working vacation. During working vacation customers are served with lower service rate than the regular service rate. Service during working vacation also contains two service modes. For the proposed model probability generating function of the queue length at an arbitrary time will be obtained by using supplementary variable technique. Some performance measures will also be presented with suitable numerical illustrations.This paper concentrates on single server bulk arrival queue system with batch size dependent service and working vacation. The server provides service in two service modes depending upon the queue length. The server provides single service if the queue length is at least ‘a’. On the other hand the server provides fixed batch service if the queue length is at least ‘k’ (k > a). Batch service is provided with some fixed batch size ‘k’. After completion of service if the queue length is less than ‘a’ then the server leaves for working vacation. During working vacation customers are served with lower service rate than the regular service rate. Service during working vacation also contains two service modes. For the proposed model probability generating function of the queue length at an arbitrary time will be obtained by using supplementary variable technique. Some performance measures will also be presented with suitable numerical illustrations.

In this paper, we have considered an MX / (a,b) / 1 queueing system with server breakdown without interruption, multiple vacations, setup times and N-policy. After a batch of service, if the size of the queue is ξ (< a), then the server immediately takes a vacation. Upon returns from a vacation, if the queue is less than N, then the server takes another vacation. This process continues until the server finds atleast N customers in the queue. After a vacation, if the server finds at least N customers waiting for service, then the server needs a setup time to start the service. After a batch of service, if the amount of waiting customers in the queue is ξ (≥ a) then the server serves a batch of min(ξ,b) customers, where b ≥ a. We derived the probability generating function of queue length at arbitrary time epoch. Further, we obtained some important performance measures.

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.