A generalization of little's law to moments of queue lengths and waiting times in closed, product-form queueing networks
A generalization of little's law to moments of queue lengths and waiting times in closed, product-form queueing networks
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DOI：
10.2307/3214322
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发表时间：
1988-05
影响因子：
1
通讯作者：
J. McKenna
中科院分区：
文献类型：
--
作者：
J. McKenna
利特尔定理指出，在非常一般的条件下，L = λW，其中L是系统中的时间平均值，W是系统中的期望逗留时间，λ是系统的平均到达率。对于某些系统，已知形式为E((L) L) = λ lE（(W) L）的关系也是成立的，其中(L) L = L（L - 1）···（L - L + 1）。本文证明了在封闭的乘积型排队网络中存在紧密的类似关系。类似的表达式涉及Nji和Sji，其中Nji是中心i的第j类作业的总数，Sji是中心i的第j类作业的总逗留时间，当中心i是单服务器FCFS中心时。当中心i是c服务器时，FCFS中心、Qji和Wji是这样关联的，其中Qji是中心i排队的第j类作业的数量，但在中心i没有服务，Wji是中心i的第j类作业的队列等待时间。更值得注意的是，将这些结果推广到队列长度和停留时间沿无超限路径的联合力矩是成立的。
Little's theorem states that under very general conditions L = λW, where L is the time average number in the system, W is the expected sojourn time in the system, and λ is the mean arrival rate to the system. For certain systems it is known that relations of the form E((L) l ) = λ lE((W) l ) are also true, where (L) l = L(L – 1)· ·· (L – l + 1). It is shown in this paper that closely analogous relations hold in closed, product-form queueing networks. Similar expressions relate Nji and Sji, where Nji is the total number of class j jobs at center i and Sji is the total sojourn time of a class j job at center i, when center i is a single-server, FCFS center. When center i is a c-server, FCFS center, Qji and Wji are related this way, where Qji is the number of class j jobs queued, but not in service at center i and Wji is the waiting time in queue of a class j job at center i. More remarkably, generalizations of these results to joint moments of queue lengths and sojourn times along overtake-free paths are shown to hold.