Asymptotic exponentiality of the tail of the waiting-time distribution in a Ph/Ph/C queue
Asymptotic exponentiality of the tail of the waiting-time distribution in a Ph/Ph/C queue
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Ph/Ph/C 队列中等待时间分布尾部的渐近指数
DOI：
10.2307/1426788
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发表时间：
1981
影响因子：
1.2
通讯作者：
Yukio Takahashi
中科院分区：
文献类型：
--
作者：
Yukio Takahashi
结果表明，在到达间隔和服务时间分布为相位类型（PH/PH/c）的多服务器队列中，等待时间分布W(x)具有一个渐近指数型尾部，即1 - W(x)∽Ke-c kx。参数k是满足T*(ck) S*(-k) = 1的唯一正数，其中T*(S)和S*(S)是到达间隔和服务时间分布的Laplace-Stieltjes变换。还证明了队列长度分布具有衰减速率η = T*(ck)的渐近几何尾。这些结果的证明是基于系统稳态状态概率的矩阵几何形式。k的等式显示了单服务器队列和多服务器队列在等待时间尾部衰减率和队列长度分布之间的有趣关系。通过求解代数方程可以很容易地计算出参数k和η。乘法常数K不容易计算。为了得到它的数值，我们必须解平衡方程或通过模拟来估计它。
It is shown that, in a multiserver queue with interarrival and service-time distributions of phase type (PH/PH/c), the waiting-time distribution W(x) has an asymptotically exponential tail, i.e., 1 – W(x) ∽ Ke–c kx . The parameter k is the unique positive number satisfying T*(ck) S*(–k) = 1, where T*(s) and S*(s) are the Laplace–Stieltjes transforms of the interarrival and the service-time distributions. It is also shown that the queue-length distribution has an asymptotically geometric tail with the rate of decay η = T*(ck). The proofs of these results are based on the matrix-geometric form of the state probabilities of the system in the steady state. The equation for k shows interesting relations between single- and multiserver queues in the rates of decay of the tails of the waiting-time and the queue-length distributions. The parameters k and η can be easily computed by solving an algebraic equation. The multiplicative constant K is not so easy to compute. In order to obtain its numerical value we have to solve the balance equations or estimate it from simulation.