M/G/1Repairable Queueing System With Service Performance Influenced By Negative Customers

Recently, the queueing systems with negative customers have been paid more andmore attention among scholars and it has developed into a new research direction，but themost studies on this topic focus on its different offset strategies. The paper mainly studieswhat the negative customers’ arrival may influence the service rate of the system. On theone hand, we consider that the negative customers’ arrival will cause the system serviceabnormal failure. On the other hand, we consider that the negative customers’ arrival willcause the service rate slow down. In real life, the system failure or the slow service ratecaused by some factors is common. Therefore, it has theoretical significance and practicalapplication value for studying this issue.In this paper, we consider M/G/1repairable queueing system with serviceperformance influenced by negative customers. We put forward three models and analyzethe steady-state probability of each model, respectively.Firstly, we deal with a repairable queueing system with abnormal breakdown causedby negative customers and normal breakdown caused by the life time of the server. Weestablished a Markov Process with an absorbent state to obtain the steady-stateequilibrium condition of the system. Then, using the supplementary variable method,L-transform technology and generating function method, we obtain the probability ofnormal work and two kinds of breakdown states as well as the calculation expression ofthe mean length. Finally, we give some numerical analysis for the performance measure ofthe system.Secondly, we investigate retrial queueing systems with non-exhaustive randomvacations and abnormal breakdown caused by negative customers. By constructing anabsorbing state of the Markov chain, the steady-state equilibrium condition of the systemwas obtained. By using the supplementary variable method and generating functionmethod, we obtain the expressions of the probability of each state and the probabilitygenerating function of the queue length under the steady-state condition. Then, wenumerically analyze the probability of each state and the mean queue length caused byarrival of negative customers, vacation rate and customers retry rate.Thirdly, we deal with repairable queueing system with variable service rates causedby negative customers. By using the supplementary variable method and state transferanalysis, we obtain the probability for each state and the specific expression of the probability generating function of the queue length under the steady-state condition. Inaddition, we solve numerically the mean length of the system and investigate the effect ofthe system parameters on the mean length of the system.