关于排队过程GI/E_k/l的若干結果

<正> §1.引言 采用Kendall的記号,所謂GI/E_k/1是指由下述条件規定的一个排队过程: (i)若用t_n表第n个顾客来到服务系统的时刻,而用ui=ti-t_(i-1)山表示相紕两顾客到达时刻間的間隔(簡称到达間隔),則这些u互相独立,并且服从同一分布

SOME RESULTS ABOUT THE QUEUEING SYSTEM GI/E_k/1

Let GI/E_k/1 be a queueing process as defined in Kendall.Applying the method introduced by Conolly, we obtained the distribution of the queue length at any finite time. The main result cap be described as follows:Theorem. Let p_n(t) be the probability that the system is in state n at dine t, and let be its Laplace transform, then where λ_1,…, λ_h are the distinct roots of the equation in the unit circle |λ|< 1, and l_1,…, l_h are their multiplicities respectively; the coefficients α_(ii) are determined by the matrix equation and A_(ki) is the cofactor of a_(ki) in A; and fo(λ) =1/1-λ,f_r(λ)=λf′_(r-1)(λ) (r≥1), F_o(λ,m)=b(1-λ~k)λ~m/bz+k(1-λ)’By calculation,we have For A_(ki), four particular cases have been considered here, namely (Ⅰ) h = k, l_1 =… =l_k= 1; (Ⅱ) h = 1, l = k; (Ⅲ) h=2,l_1=k-1,l_2=1; (Ⅳ) h=k- 1, l_1= 2,l_2=…= l_(k-1)= 1.We concluded this paper by considering the Busy Period in relation to GI/E_k/1 as Conolly in.