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To meet the needs of the joint operation under the uncertain information conditions, the description and effectiveness of weapon system-of-systems combat networks has been widely researched. The Timed influence nets(TIN) is an useful tool to describe the uncertain causal relations in the weapon system-of-systems combat networks and has been used for the effectiveness evaluation in alternative military actions. However the uncertain time delay effect is not well considered and computed in the TIN. In this paper, the Random Timed Influence Nets(RTIN) is proposed to introduce the random time delay in the TIN. First, it is introduced that the basic concepts and algorithms used in the RTIN model including the time delay parameter, the belief sequence and the belief-spread algorithm. And then an instance is given in this paper to demonstrate the RTIN modeling process and the result of effectiveness evaluation using the belief spread algorithm. Introduction In the joint operations, it is really important to reach the desired effect of the combat target selection to help the commanders to make decisions. A good way is to model and analyze the uncertain causal relationship between operations and effects. And Bayesian Networks (BN) is frequently used in that issue. However the conditional probability table in the BN is always hard to get. Rosen J A[1] introduced the idea of CAST (causal strength), and came up with the influence nets (IN) modeling method. As the nodes increase, the parameters we need in IN are not exponentially increased. Both BN and IN describe static relationships between parameters, some researchers tried to introduce the time factor and uncertain factor into IN. [2] Wagenhals and his team [3] introduced the time structure in IN modeling, which is called timed influence nets.[4] TIN has been widely used for Effects-based Operations that mainly for the evaluation of operations. There are also disadvantages in the TIN modeling method. And the Dynamic influence nets modeling and some other methods are introduced to make some improvement for influence nets. [5]However, in the military combating process, there are random time delays between the operations and effects, which is unable to describe in TIN. Thus in this paper, we introduced the random time delay and random belief sequence parameters into TIN, which is called random timed influence nets (RTIN). The mathematic description for RTIN and the belief spreading algorithm is elaborated. And an example is given to demonstrate the modeling process and the result of RTIN. Relative Study Bayesian Network is a probability networks model came up by R.Howard and J. Matheson. In Bayesian Networks, the uncertain factors are described as the probabilities which can be deduced and calculated. While modeling for realistic situations, there are also some disadvantages in Bayesian Networks, such as the limitation for calculation time and the amount of given information. To improve the usability of Bayesian networks, Chang introduced CAST parameters and came up Influence Nets. In Influence Nets the causal relationship between nodes. While calculating, the CAST parameters can be converted to the Conditional Probability Table, which can be used in Bayesian Networks probabilities deduce. International Conference on Automation, Mechanical Control and Computational Engineering (AMCCE 2015) © 2015. The authors Published by Atlantis Press 566 Influence Nets is an improved static influence diagram, which cannot reflect the real dynamic relationship between nodes. Thus, Wagenhals introduced the concept of time parameter and came up the modeling method of Timed Influence Nets through setting time parameters in the static influence nets. A typical Timed Influence Nets is shown in Fig 1: Fig 1 Timed Influence Nets Timed Influence Nets is also described as a Directed Acyclic Graph(DAG) for topological structure, in which the nodes reflect the random variable and the directed edges between the nodes reflect the causal relationship. A Timed Influence Networks can be described by the following six elements , , , , , V: The set of the nodes in TIN. E: The set of the edges in TIN. C: The set of CAST parameters between nodes. For ∀e ∈ E, ∃c ∈ C, satisfies : ↦ , in which,C h, g | 1 h, g 1 , (h, g) is the expression of CAST. B: The set of baseline probabilities or prior probabilities. D: The set of time delay on child nodes produced by parent nodes in TIN. A: The input of the TIN, which means the set for the belief sequence of the root nodes. There are some disadvantages of Timed Influence Nets. In real situations, the time delays are always not certain values but random values which also lead to the uncertainty of belief values. In this paper, based on the random factors of time delay and the uncertainty of belief value, an improved Timed Influence Nets is introduced to take the random factors of time into consideration. Description for RTIN The disadvantages of Timed Influence Nets can be seen from the preceding part. Two new concepts are introduced in this paper, Random Time Delay and Random Belief Sequence. An example of the Random Timed Influence Nets is shown in Figure 2. Figure 2 Random Timed Influence Nets Definition 1. Random Time Delay. In Random Timed Influence Nets, the time delay corresponded to any edges is a random variable. Definition 2. Random Belief Sequence. In Random Timed Influence Nets, the taking value process of any node is a random sequence. Here the random belief sequence is the random sequence be t ~ defined in time interval [0, Mt]. In that, be t means the random variable corresponded to the node’s belief at the time of t. Finite set is the range of be t . Any element in means the value that the belief of the nodes can be. is a discrete probability distribution defined on set . During the time [0, A B       0.1@ 0 0.7@3 1.0@5 P B

Advanced Timed Influence Nets with Random Time Delay for Weapon System-of-systems Combat Networks