Queueing Model And Optimization Analysis For Vacation Or Failure Under Bernoulli Mechanism

Queueing theory is widely used in life.With the development of call centers,communication systems,computer networks,neural networks,and manufacturing industries,the research on queueing theory has deepened.Therefore,the classical queueing theory has been extended and expanded,and the research on more extensive and complex retrial queueing systems and repairable queueing systems has received widespread attention.According to the queueing behavior in these fields,this paper considers vacation or failure queueing systems with Bernoulli mechanism,and studies optimization problems from the perspective of customers and managers.First,the retrial queue with two-stage service and Bernoulli vacation is studied.Upon finishing the first-stage service the customers may choose an optional service.After each service completion,the server takes a vacation with probability a and waits for serving the next customer with probability 1-a if there are some customers in the orbit.If the orbit is empty after a service completion,the server takes a vacation.At the end of a vacation,the server waits for serving the customer in the orbit or new arriving customers.Based on a linear reward-cost function,arriving customers decide either to enter the orbit or to balk if the server is busy or on vacation.The customers’expected waiting time and system length are deduced by the method of probability generating function,and the equilibrium strategy of customers and the socially optimal strategy are also discussed.In order to eliminate the difference between individual optimal strategy and social optimal strategy,the pricing strategy is studied.Finally,some numerical examples are used to show the influence of some system parameters on the optimal joining probabilities and the optimal social welfare.Secondly,an M/M/1 queueing model with negative customers and unreliable repair is studied,where the arrival of negative customers can lead to server failures.Once the server experiences a failure,it will be repaired immediately,and the repairer may successfully repair it with the probability p.Otherwise,once the repair fails,all customers in the system will be forced to leave.Using the method of probability generating function,the steady-state probability distribution and some performance measures of the system are obtained.Based on the reward-cost structure,the equilibrium strategic behavior of customers is discussed.The impact of system parameters on performance measures,individual benefit and social benefit is analyzed by numerical examples.Finally,based on the previous model,the M/G/1 queueing system with different types of breakdowns and unreliable repair is considered.Three kinds of failure cases are considered:ordinary failure,disaster failure,and failure caused by the arrival of negative customers.After experiencing an ordinary failure,the server is immediately sent for repair,and continues to provide service to customers after the repair is successful.In case of disaster failure,all customers in the system are forced to leave.The arrival of a negative customer takes away the customer currently receiving service and leads to a server failure.Using the method of supplementary variable and the probability generating function,we obtain the performance measures such as the system’s average queue length and the customer’s sojourn time.In special cases,the average sojourn time of customers in the queueing system is compared.Finally,the influence of system parameters on system length and system reliability measures under three kinds of failures is demonstrated by numerical experiments.