Calculating exit times for series Jackson networks
Calculating exit times for series Jackson networks
复制标题
计算系列 Jackson 网络的退出时间
DOI：
10.2307/3214073
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发表时间：
1985
影响因子：
1
通讯作者：
W. A. Massey
中科院分区：
文献类型：
--
作者：
W. A. Massey
我们定义了一个新的家庭的特殊功能，我们称之为格贝塞尔函数。它们由N维整数格索引，使得它们在N = 1时简化为修改的贝塞尔函数，并且在N = 0时简化为指数函数。M/M/1排队在变为空闲（在0退出）之前从一个状态到另一个状态的转移概率可以用修正的贝塞尔函数来求解。本文利用格Bessel函数解决了N维串联杰克逊网络从格的正正交点内部退出时间的类似问题。这些特殊的功能使我们能够获得禁忌转移概率的渐近展开，以及退出时间分布的尾部。
We define a new family of special functions that we call lattice Bessel functions. They are indexed by the N-dimensional integer lattice such that they reduce to modified Bessel functions when N = 1, and the exponential function when N = 0. The transition probabilities for an M/M/1 queue going from one state to another before becoming idle (exiting at 0) can be solved in terms of modified Bessel functions. In this paper, we use lattice Bessel functions to solve the analogous problem involving the exit time from the interior of the positive orthant of the N-dimensional lattice for a series Jackson network with N nodes. These special functions allow us to derive asymptotic expansions for the taboo transition probabilities, as well as for the tail of the exit-time distribution.