具有可选服务的M/M/1排队模型的进一步研究

Further Research on the M/M/1 Queueing Model with Second Optional Service

研究具有可选服务的M/M/1排队模型的主算子在左半实轴上的点谱.当顾客的到达率λ,必选服务的服务率μ_1与可选服务的服务率μ_2满足λ/μ_1+λ/μ_2<1时,证明区间(η,-λ)中的所有点都是该主算子的几何重数为1的特征值,其中η=max{-μ_1,-μ_2,-4/3λ,-2λμ_2/(μ_1+μ_2)-λ,-μ_1μ_2(μ_1μ_2-λμ_1-λμ_2)+λ~3μ_1(1-λ)/[μ_1~2(μ_2-λ)+μ_1μ_2(μ_1-λ)](1-γ)+λ~2μ_1-λ}r表示顾客选择可选服务的概率.

We study the point spectrum of the operator, which corresponds to the M/M/1 queueing model with second optional service, on the left real line and prove that if the arrival rate of customers λ, the service rate of the first essential service μ1 and the service rate of the second optional service μ2 satsify λ/μ1＋λ/μ2〈1, then all points in the interval （η,-λ） are eigenvalues of the operator with geometric multiplicity 1, here η =max{-μ1,-μ2,-4/3λ,-2λμ2/μ1＋μ2-λ,-μ1μ2（μ1μ2-λμ1-λμ2）＋λ3μ1（1-r）/[μ12（μ2-λ）＋μ1μ2（μ1-λ）]（1-r）＋λ2μ1-λ},r represents the probability that customers opt the second optional service.