In this paper, a Markov chain origin-destination matrix estimation problem is investigated in which the average time between two incoming streams to or outgoing streams from nodes in consecutive time periods is considered as a Markov chain. Along with, a normal distribution with pre-determined parameters in each period is considered for traffic counts on links. A bi-level programming problem is introduced where in its upper level the network flow pattern in the n th period is estimated so that the probability of the estimated traffic counts is maximized, while in the lower level a traffic assignment problem with the equilibrium conditions is solved. We reduce the proposed nonlinear bi-level model to a new one level linear programming problem, where by using a trust-region method the local optimal solutions are obtained. Some numerical examples are provided to illustrate the efficiency of the proposed method.

中文翻译：

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马尔可夫链方法中的起点-终点矩阵估计问题

在本文中，研究了一个马尔可夫链起点-目的地矩阵估计问题，其中将连续时间段内两个流入或流出节点的流之间的平均时间视为一个马尔可夫链。同时，在每个周期中使用具有预定参数的正态分布来考虑链路上的流量计数。双级规划问题，其中在其上的水平引入在所述网络流图案Ñ估计第一个时间段，以使估计的流量计数的概率最大化，而在较低级别，则解决了具有平衡条件的流量分配问题。我们将提出的非线性双层模型简化为一个新的一级线性规划问题，其中通过使用信任域方法获得局部最优解。提供了一些数值示例来说明所提出方法的效率。