This paper discusses a system of difference-differential equations (DDE) that is satisfied by the time-dependent state probabilities of open Markov queueing networks with various features. The number of network states in this case and the number of equations in this system is infinite. Flows of customers arriving at the network are a simple and independent, the time of customer services is exponentially distributed. The intensities of transitions between the network states are deterministic functions depending on its states.To solve the system of DDE, we propose a modified method of successive approximations, combined with the method of series. The convergence of successive approximations with time to a stationary probability distribution, the form of which is indicated in the paper has been proved. The sequence of approximations converges to a unique solution of the system of equations. Any successive approximation can be represented as a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for calculations on a computer. Examples of the analysis of Markov G-networks with various features have been presented.

中文翻译：

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寻找具有多类客户和各种特征的开放马尔可夫网络的非平稳状态概率

本文讨论了一个差分微分方程（DDE）系统，该系统由具有各种特征的开放马尔可夫排队网络的时间相关状态概率满足。这种情况下的网络状态数和该系统中的方程数是无限的。到达网络的客户流是简单且独立的，客户服务时间呈指数分布。网络状态之间的转换强度是取决于其状态的确定性函数。为了求解DDE系统，我们提出了一种改进的逐次逼近法，并结合了级数法。逐次逼近随时间收敛到一个平稳的概率分布，其形式在论文中已经得到证明。近似序列收敛到方程组的唯一解。任何逐次逼近都可以表示为具有无限收敛半径的收敛幂级数，其系数满足递推关系，便于计算机计算。已经介绍了具有各种特征的马尔可夫 G 网络的分析示例。