On The Point Spectrum Of The Underlying Operator Of An M^[x]/M/1 Queueing Model With Server Breakdowns And Constant Rate Of Repeated Attempts

Retrial queuing system is widely used in many fields such as communication,computer network,communication network,etc.So,study of structures of the time-dependent solutions of retrial queueing models is important in view of theory and practice.And,the structures of the time-dependent solutions of the models are decided by the spectral distributions of the corresponding operators.We study point spectrum of the operator corresponding to an M[X]/M/1 retrial queueing model with sever breakdowns and constant rate of repeated attempts and prove that:(1)if the arrival rate of customers λ,the service rate of the server v,the repeated rate of customers α,the completion rate of the server b1 and the repair rate of b2 satisfy(ⅰ)3(λ+b2-α)(3αb1+2λb1+4λ2)+λ(3αb1+λ2)+3λ2(λ+4v+b1-α)>9λα(v+λ+b2-α)+λ2b1(ⅱ)λ(v+b2)+b1b2>α(λ+b1),v+b1-α>0,λ+b2-α>0(ⅲ)15αb1(λ+b2-α)+3λα(b2+λ+v+b1-α)+2λαb1>4λb1b2+8λ2(2λ+v+b1+b2-2α)+3λ2α(ⅳ)2αb1(λ+b2-α)[(λ+α)b1(λ+b2-α)+λvb2]+2αb12(λ+b2-α)2(5α+λ)+2λoαb1(λ+b2-α)(5vb2+3λ2)+λ3αb1(3λ+2v+2b1-2α)+3λ2(λ+b2-α)2(λ+v+b1-α)2+λ4αb1+2λ3(λ++b1-α(λ+b2-α)(2λ+b2+v+b1-2α)>2λ2(λ+v+b1-α)[(λ+α)b1(λ+b2-α)+λvb2+λ3]+λ2(λ+b2-α)2[2b1(2α+λ)+λ2]+2λ4(λ+b2-α)(λ+b1)+2λ(λ+b2-α)[2αb12(α-λ)+3λvb2(λ+v+b1-α)]+2λb1(λ+v+b1-α)(λ+b2-α)2(6α+λ)+λ4(λ2+2vb2)+λ2[λ2(λ+v+b1-α)2+2vαb1b2l then-α is an eigenvalue of the underlying operator of the model with geometric multiplicity one.(2)When λ>0,α>0,b1>0,b2>0,v satisfy a certain condition,-(λ+α)and-(λ+b2)are not eigenvalue of the underlying operator of the model.