On The Negative Eigenvalues Of The Operator Which Corresponds To The M^[x]/M/1 Retrial Queueing Model With Constant Rate Of Repeated Attempts And Absorbent States Of The Service Failure States

This thesis consists of introduction,chapter 1 and conclusion.In Introduction,we firstly introduce the research works of the M[X]/M/1 queueing model,then we state the problem that we will study in this thesis.Chapter 1 is split into two sections.In Section 1,firstly we introduce the M[X]/M/1 retrial queueing model with constant rate of repeated attempts and absorbent states of the service failure states,then we convert the model into an abstract Cauchy problem in a Banach space by introducing a state space,operators and their domains,lastly we introduce the main results of dynamic analysis for the model obtained by other researchers.In Section 2,when the probability ck that at every arrival epoch a batch of k external customers arrives satisfies Ck=1/2k(k?1)we study eigenvalues of the operator,which corresponds to the model,on the left real axis,and prove that if the arrival rate of customers ?,the service rate of the server ?,the completion rate of the server b and the repeated rate of customers a satisfy one of the following conditions:(?)0<?b-2?(2?+ ?+b-?)?2?2 and ?+b>?(?)2?2+2?? + 2?b+ 2?b?2?2 + 2??+ 2?b<?b+?b + 2?? and ?+b?(?)4??b(2? + b)=[2?2 + 2?? + ?b-?(2? + b)]2,2?2 + 2?? + ?b<?(3b + 2?)and?+b>?then-? is an eigenvalue of the operator with geometric multiplicity one.In Conclusion,we state our main result in this thesis.