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 ?,the service rate of the server v,the repeated rate of customers ?,the completion rate of the server b1 and the repair rate of b2 satisfy(?)3(?+b2-?)(3?b1+2?b1+4?2)+?(3?b1+?2)+3?2(?+4v+b1-?)>9??(v+?+b2-?)+?2b1(?)?(v+b2)+b1b2>?(?+b1),v+b1-?>0,?+b2-?>0(?)15?b1(?+b2-?)+3??(b2+?+v+b1-?)+2??b1>4?b1b2+8?2(2?+v+b1+b2-2?)+3?2?(?)2?b1(?+b2-?)[(?+?)b1(?+b2-?)+?vb2]+2?b12(?+b2-?)2(5?+?)+2?o?b1(?+b2-?)(5vb2+3?2)+?3?b1(3?+2v+2b1-2?)+3?2(?+b2-?)2(?+v+b1-?)2+?4?b1+2?3(?++b1-?(?+b2-?)(2?+b2+v+b1-2?)>2?2(?+v+b1-?)[(?+?)b1(?+b2-?)+?vb2+?3]+?2(?+b2-?)2[2b1(2?+?)+?2]+2?4(?+b2-?)(?+b1)+2?(?+b2-?)[2?b12(?-?)+3?vb2(?+v+b1-?)]+2?b1(?+v+b1-?)(?+b2-?)2(6?+?)+?4(?2+2vb2)+?2[?2(?+v+b1-?)2+2v?b1b2l then-? is an eigenvalue of the underlying operator of the model with geometric multiplicity one.(2)When ?>0,?>0,b1>0,b2>0,v satisfy a certain condition,-(?+?)and-(?+b2)are not eigenvalue of the underlying operator of the model.