Multi-Server Queuing Systems With Negative Customers And Servers' Breakdowns

In recent years, the queuing system with negative customers has been paid more and more attention among scholars and it has gradually developed into a new research direction. At present, the studies about it mostly focuse on single-desk situation, but the multi-server model is more common in our actual applications. At the same time, the phenomena of breakdowns of the servers also has a significant impact. Therefore, to study the multi-server queuing system with above integrated mechanism has important theoretical significance and practical application value.In this paper, we consider a multi-server queueing system with negative customers and breakdowns. We put forward three models and analysis the steady-state probability of each model respectively.Firstly, we investigate a finite waiting room M/M/c/N repairable queuing system with RCH strategy of negative customers. Through the state transition diagram, we can see that the syetem is a level dependent quasi-birth-and-death process (LD-QBD). By Markov and block matrix iterative method, we obtain the steady-state probability distribution. When c =2, we give the performance measures of this limited system. In addition, we give some numerical calculation and model analysis.Secondly, we deal with an infinite waiting room M/M/c/∞repairable queuing system with RCE strategy of negative customers. By using quasi- birth-and-death process (QBD) method, we get the existing condition of steady-state equilibrium. Then, through matrix-geometric mathod, we obtain the rate matrix and steady-state probability vectors. Besides, we deduce some performance measures and reliability indices of the system. Furthermore, the numerical analysis of the performance indicators is also given.Finally, we study an infinite waiting room M/M/c/∞repairable queuing system with negative customer offsetting all the customers being in service. First of all, considering c =2, using the continuous parameter Markov process and matrix-geometric solution, we obtain the system steady-state distribution and queuing and reliability indicators. Then, we extend the case to c >2. Through analyzing the transition process and rate matrix, we get the equilibrium condition and the equations of steady-state probabilities of the system.