M/M/1Repairable Queueing System With Variable Srevice Rate Caused By Negative Customers

Recently, the queueing system with negative customers has been paid more and moreattention among scholars and it has gradually developed into a new research direction. Atpresent, the studies about it mostly focus on the offset strategies, but that negativecustomers' arrival influence to the service rate has not been researched. it is a new topic.The service rate becoming more and more slower by some factors is more common in ouractual applications. Therefore, to study the model that variable service rates caused bynegative customers has important theoretical significance and practical application value.In this paper, we consider M/M/1repairable queueing system with variable servicerates caused by negative customers. We put forward four models and respectively analysisthe steady-state probability of each model.Firstly, we deal with an infinite waiting room M/M/1repairable queueing systemwith variable service rates caused by negative customers. By using the quasi-birth-and-death process method we get the existing condition of steady-state equilibrium. Thus usingthe matrix-geometric method we get the steady-state probability vectors. Besides, wededuce some queueing and reliability indices of the system. Finally, using Matlab, we givesome numerical calculation and model analysis.Secondly, we investigate a finite waiting room M/M/1repairable queueing systemwith variable service rates caused by negative customers. By using the matrix-geometricmethod, we obtain the steady-state probability distribution and some queueing andreliability indices respectively. Furthermore, the numerical analysis of the performanceindicators is also given.Thirdly, we deal with an finite waiting room M/M/1repairable queueing system withimpatient and variable service rates caused by negative customers. Through the statetransition diagram, we can see that the system is a level dependent quasi-birth-and-deathprocess (LD-QBD). By using the Markov process and the block matrix iterative methodwe obtain steady-state probability vectors. Besides, we deduce some queueing andreliability indices of the system. Finally, When N=4, using Matlab, the numerical analysisof the performance indicators is given.Finally, we study an infinite waiting room M/M/1repairable queueing system withvariety variable service rates caused by negative customers. By using the quasi-birth-and- death process method and the matrix-geometric method we get the existing condition ofsteady-state equilibrium and the steady-state probability vectors. Besides, we deduce somequeueing and reliability indices of the system. Furthermore, the numerical analysis of theperformance indicators is also given.