In this paper, we discuss about the steady state behaviour of M/G/1 retrial queueing system with two phases of services and immediate feedbacks under working vacation policy where the regular busy server is affected due to the arrival of negative customers. Upon arrival if the customer finds the server busy, breakdown or on working vacation it enters an orbit; otherwise the customer enters into the service area immediately. After service completion, the customer is allowed to make finite number of immediate feedback. The feedback service also consists of two phases. At the service completion epoch of a positive customer, if the orbit is empty the server goes for a working vacation. The server works at a lower service rate during working vacation (WV) period. Using the supplementary variable technique, we found out the steady state probability generating function for the system and in orbit. System performance measures and reliability measures are discussed. Finally, some numerical examples are presented to validate the analyticalresults.

In this review article, we have discussed about energy of a graph and energy of a fuzzy graph. Further we presented some standard results on these graphs. Since most real world networks such as communication networks, data organization, computational devices, the flow of computation, web graphs etc are directed graphs, we have analyzed energy of an intuitionistic fuzzy directed graph through real time example.

In this paper, we have considered an M^X / (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.

The objective of this paper is to analyse state dependent arrival in bulk retrial queueing system with immediate Bernoulli feedback, multiple vacations, threshold and constant retrial policy. Primary customers are arriving into the system in bulk with different arrival rates λ _a and λ _b . If arriving customers find the server is busy then the entire batch will join to orbit. Customer from orbit request service one by one with constant retrial rate γ. On the other hand if an arrival of customers finds the server is idle then customers will be served in batches according to general bulk service rule. After service completion, customers may request service again with probability δ as feedback or leave from the system with probability 1 − δ. In the service completion epoch, if the orbit size is zero then the server leaves for multiple vacations. The server continues the vacation until the orbit size reaches the value 'N' (N > b). At the vacation completion, if the orbit size is 'N' then the server becomes ready to provide service for customers from the main pool or from the orbit. For the designed queueing model, probability generating function of the queue size at an arbitrary time will be obtained by using supplementary variable technique. Various performance measures will be derived with suitable numerical illustrations.