服务失效状态为吸收状态及重试率为常数的M^([X])/M/1排队模型的主算子的谱分析

Spectral Analysis of the Underlying Operator of the M^([X])/M/1 Queueing Model with Service Failure State as Absorbent State and Constant Rate of Repeated Attempts

研究服务失效状态为吸收状态及重试率为常数的M^([X])/M/1排队模型的主算子在左半复平面中的谱.当顾客的到达率λ,服务员的服务率ν,顾客的重试率α和服务员的服务完成率b满足一定的条件时,证明了实部为-(λ+ν+b)的所有复数都不是该模型主算子的特征值;当λ,ν, α, b满足一定的条件时,证明了区间(-(λ+ν+b),0)中无穷多个点是该主算子的几何重数为1的特征值.

We study spectra of the underlying operator of the M^([X])/M/1 queueing model with service failure state as absorbent state and constant rate of repeated attempts on the left half complex plane.When the arrival rate of customersλ,the service rate of the serverν,the repeated rate of customersαand the service completion rate of the server b satisfy a certain condition,we prove that all complex numbers whose real part-(λ+ν+b)are not eigenvalue of the underlying operator.Whenλ,ν,α,b satisfy some conditions,we prove that infinitely many numbers in the interval(-(λ+ν+b),0)are eigenvalues of the underlying operator with geometric multiplicity one.