PS-G-GERT Effectiveness Evaluation Model of Link-Based GEO Satellite Communication Constellation under Poor Information Background

Ruirui Shao ,Zhigeng Fang ,Su Gao ,Sifeng Liu ,Weiqing You

1 College of Economics and Management,Nanjing University of Aeronautics&Astronautics,Nanjing 211106,China

2 Institute of Telecommunication Satellite China Academy of Space Technology(CAST),Beijing 100094,China

3 National Cryptography Administration of Jiangsu Province,Nanjing 210013,China

Abstract: The effectiveness evaluation of GEO satellite communication constellation not only characterizes the communication capability of the constellation but also provides a basis for optimizing the constellation structure.Whether due to information poverty or the complexity of the system,the impact of uncertain information on the effectiveness evaluation needs to be considered to ensure the accuracy of the evaluation results.To address this issue,this paper develops a model for evaluating the GEO satellite communication constellation’s effectiveness in the context of poor information.Firstly,it analyses the GEO satellite communication constellation plus system based on communication links with an in-depth analysis of the constellation structure.Secondly,an equivalent transfer function algorithm based on the characteristic function and transfer probability is proposed with the help of graphical evaluation and review technique.Then,by analyzing the communication link importance connotation,the algorithm formula of communication link effectiveness and its importance is derived,and the constellation effectiveness and variance are found.Finally,the model constructed in this paper is used to evaluate the effectiveness of a GEO satellite communication constellation,further verifying the accuracy and validity of the model.Through comparative analysis,it is shown that the model can not only solving the effectiveness of the constellation but also analyzing the variation of its effectiveness.It lays a theoretical foundation for the analysis and optimization of the GEO satellite communication constellation effectiveness.

Keywords: poor information;graphical evaluation and review technique (GERT);plus system (PS);effectiveness evaluation

I.INTRODUCTION

Satellite communication refers to the communication between radio communication stations on the earth(including the ground,water surface,and lower atmosphere) using artificial satellites as relay stations.Geostationary earth orbit (GEO) satellite communication has become the first choice of many countries and operators due to its advantages of large coverage area,relative ground static,mature technology,little investment,simple maintenance and management[1],etc.An accurate evaluation of the GEO satellite communications constellation’s effectiveness is important for constellation configuration design,operation and maintenance.It is also the basis for the optimization of the GEO constellation structure.

As early as the 1960s,western countries have begun to study the evaluation of system effectiveness,and have already obtained relatively rich research results.As one of the core problems of effectiveness evaluation,the evaluation method is the basis of effective implementation of effectiveness evaluation.The commonly used effectiveness evaluation methods can be divided into four categories[2].The first is the expert evaluation method.Liu et al.[3] based on the extensive collection of expert opinions,constructed a communication satellite system effectiveness evaluation index system,and then evaluated its effectiveness.This method can evaluate and predict the index values that are difficult to obtain quantitatively such as lack of data,but it will result in a certain subjectivity and tendency inevitably to the evaluation and prediction results.The second is the analytical method.Pirzadeh et al.[4]studied the effectiveness of hybrid ADC architecture in overcoming channel estimation errors caused by rough quantization.Otay et al.[5]proposed a multi-expert fuzzy evaluation method that integrates intuitionistic fuzzy data envelopment analysis(DEA)and intuitionistic fuzzy analytic hierarchy process(IF-AHP)to evaluate the effectiveness of medical institutions.This method is to establish a mathematical model to calculate the evaluation value of the efficiency index.The calculation is simple and easy to understand and apply.The disadvantage is that there are limiting factors to consider in the model construction process.The third is the simulation method.Jia et al.[6] put forward a method for the effectiveness evaluation of the UAV cluster combat system based on the system dynamics model.Gao et al.[7]proposed a new method for weapon equipment effectiveness evaluation based on artificial neural networks.However,the simulation method is constructed based on quantitative information,which is difficult to model for some qualitative information,such as expert opinions and fuzzy data [8].The fourth is complex network analysis,which makes up the deficiency of agent-based modeling and other methods in system information interaction modeling,and is considered to be the theory and method most likely to make a breakthrough in the study of system architecture modeling[9].Abstract various entities in complex systems as nodes and various connection relationships as edges,which has become an important method to study system characteristics in recent years [10,11].The above analysis shows that to measure the system’s effectiveness,it is necessary to characterize the system as a network,based on which the effectiveness of the network can be studied.

Although many achievements based on system effectiveness have been accomplished,little research has been devoted to the effectiveness of satellite communication constellation.Yang et al.[12] discussed the visibility model of the satellite constellation communication network with satellites as nodes and realized the transmission capacity evaluation of the communication network.Taking two satellites as nodes,Yan et al.[13] modeled the storage and transmitter of a satellite as the buffer and server of the queue and proposed a time-sensitive M/G/1 model for intermittent transmission between satellites.Then,through transient analysis,the steady-state expression of the system and the lower bound of performance parameters were obtained.Lin et al.[14] equated the satellite network with the topological structure composed of various nodes and used the corresponding topological characteristic values (such as degree distribution,clustering coefficient,betweenness,etc.) to evaluate the effectiveness of the satellite network.It should be noted that the above effectiveness evaluation of the communication satellite constellation is based on the premise that the information is known.In fact,due to the complexity of the technology and the limitations of people’s cognition,the evaluation of the communication satellite constellation’s effectiveness is often done in the context of poor information.Shao et al.[15]divided the states of the communication constellation,equating the states of the constellation with nodes,and studied the effectiveness evaluation of the low-orbit constellation under the background of poor information through the transfer process between states.The above studies show that to measure the effectiveness of the communication satellite constellation,it also needs to be represented as a network.

Among the existing effectiveness evaluation methods,the expert evaluation method is undoubtedly affected by the subjective cognition of experts.When the system is complex,human cognitive ability is limited,and it is difficult to make accurate judgments.When evaluating effectiveness by analytical method,take the ADC solution method as an example.Due to the actual needs of different systems,different effectiveness indicators will be selected to solve the availability,dependability,and capability.However,the selection of capability indicators,the accuracy of capability score,and the weight coefficient always depend on the experience of experts,so the evaluation accuracy is also limited.The simulation method is a kind of effectiveness evaluation method with high precision,which can analyze various influence factors of effectiveness.However,it is difficult to establish an accurate mathematical model for complex systems,and the accuracy has an important impact on the simulation credibility [16].Complex network analysis methods,such as neural networks [17],are all based on data.However,it is a non-trivial challenge to collect data for the construction of effectiveness evaluation models in complex systems,and the impact of uncertain information needs to be considered.Fuzzy sets,interval numbers,rough sets,and grey sets are usually used to describe uncertain information systems.A fuzzy set is used to describe fuzzy information,the rough set is used to describe inconsistent and incomplete information,interval number is used to define boundary information,and grey degree is used to describe part of known information,which means the certain number could not be known while its potential value set can be defined[18].The Grey system has become an effective system modeling method under the background of known partial information.Unlike interval values and fuzzy sets,the grey set can use the discrete grey number as a characteristic function value [19].In the field of communication engineering,grey number have been widely used [15],[20],[21].In [15],the generalized interval grey number is used to characterize the capability value of the LEO satellite communication constellation,and the effectiveness evaluation model based on poor information background is constructed.In [21],the channel allocation strategy of the satellite communication system is studied,and a dynamic channel reservation strategy based on grey prediction is proposed to obtain better access and handover performance.

Furthermore,the graphical evaluation and review technique (GERT) is highly adaptable and descriptive in characterizing complex logical relationships between activities and various uncertainties in the activity process.Therefore,it is widely used in reliability engineering,uncertainty analysis,complex equipment and other fields[22—25].Manju et al.[22]compared the reliability of the m-continuous-k-out-of-n system and m-continuous-at least-k-out-of-n system by GERT technology.Wang et al[23]proposed a new technique integrating GERT and the Bayesian method,and applied it to sustainable quality management.However,due to the lack and incompleteness of information in the actual operation of the system,the parameters of the GERT network are uncertain.Yong et al.[24]used the multiple transfer parameters graphical evaluation and review technique(MTP-GERT)to model the logistics support process.A global sensitivity analysis method based on covariance was designed to study the uncertainty of the logistics support process.Yang et al.[25] introduced interval grey numbers to describe the uncertainty of GERT statistical data and empirical information in the reliability experiment.The above studies show that the GERT network can reveal the intrinsic characteristics of complex networks,with the help of the transfer relationships of relevant parameters in the network.A theoretical foundation is laid for the networked description of GEO satellite communication constellations in the context of poor information.

Considering the complexity of satellite communication constellations,inspired by existing research,we naturally want to know how to accurately solve the effectiveness of the constellation network.Besides,what is the change range of constellation network effectiveness? For this reason,this paper constructs a PS-G-GERT effectiveness evaluation model with characteristic function as the transfer parameter and defines the variance of the network effectiveness.The main contributions of this article are as follows.

1)Network description of communication constellation: Considering the interaction relationship between communication entities,the communication satellite constellation is networked and described by using GERT theory with the characteristic function as the transfer parameter.

2)Modeling and solution techniques: The problem is modeled as solving the equivalent transfer function of the communication network.The PS-G-GERT effectiveness evaluation model is constructed through the solution of communication link effectiveness,importance and other parameters.

3) Simulation and results: The effectiveness of a GEO satellite communication constellation is evaluated,while the accuracy and validity of the model are verified by comparison.The sensitivity factors and changing trends affecting the constellation effectiveness and variance are analyzed in the simulation.The impact of three types of communication link effectiveness changes on the constellation effectiveness is also analyzed.

To realize the concepts mentioned above,this article is organized as follows.The construction of the PS-G-GERT model of the GEO communication constellation based on effectiveness transfer is given in Section II.Through the parameter solving algorithm,Section III mainly solved the parameters such as link effectiveness,importance,constellation effectiveness,and communication activity variance.Afterward,a case analysis is proposed in Section IV,which further verifies the accuracy and validity of the constructed model.Finally,the conclusion is stated in Section V.

II.CONSTRUCTION OF PS-G-GERT MODEL OF GEO COMMUNICATION CONSTELLATION BASED ON EFFECTIVENESS TRANSFER

Definition 1.Based on the in-depth analysis of the GEO satellite communication constellation structure relationship,each communication link in the constellation network is the plus subsystem,and all communication links form the plus system.Based on the theory of GERT,the grey number is used to judge and estimate the activity parameters in the PS-G-GERT constellation network,and a link based PS-G-GERT constellation structure diagram is constructed from the perspective of effectiveness.At the same time,combined with the importance of each communication link,the effectiveness evaluation value of GEO satellite communication constellation is obtained by the sum of the link effectiveness and its importance,which is called the PS-G-GERT effectiveness evaluation model.

The basic feature of the GEO satellite mobile communication constellation is the use of synchronous geostationary orbit satellites,which are 35,800km above the ground and relatively stationary.The communication system follows the OSI seven-layer model.According to the different business types,this paper divides the GEO satellite into the following processing modules.Reliable transmission module 4: GEO Transport,which guarantees the completely reliable transmission of standard protocol processes.Routing exchange module 3: GEO_Route,guarantees the route.Resource allocation module 2:GEO_Switch,which ensures the basic access success rate.Baseband processing module 1: GEO_PForward,which guarantees the basic bit error rate.RF forwarding module 0:GEO_TForward,to guarantee that radio waves complete the RF transmission.Commonly used terminology symbols are shown in Table 1.Wireless communication signals (mobile phone signals,satellite signals) are transmitted in the form of electro-magnetic waves.In the design of satellite networks,to save satellite resources,for services with low realtime requirements (such as WeChat,E-mail,etc.),the satellite only adopts the form of transparent forwarding,and all the upper services such as baseband resolution are carried out at the ground station.And for the business needs of emergency communication(such as earthquake and other natural disasters,social emergency events),it is necessary to ensure the complete and reliable transmission of signals.The relationship among emergency,real-time and non-realtime services can be seen in Figure 1.

Figure 1.Topological structure of a instantaneous GEO satellite communication constellation.

Table 1.List of abbreviations.

Table 2.Service processing level of a geo satellite communication constellation.

The essential meaning of constellation effectiveness is the effective degree of the constellation to complete the communication task.The evaluation of constellation effectiveness is based on the effectiveness evaluation of each communication entity in the constellation,or directly based on the performance of each communication entity in the constellation,and by means of modeling and simulation,the constellation effectiveness is comprehensively evaluated.In this paper,GERT technology is used as the evaluation method,entity effectiveness in the GEO constellation is used as the transfer parameter,and activity parameters in the GERT network are characterized by grey numbers.Based on the link effectiveness evaluation,a PS-GGERT effectiveness evaluation model is constructed.

2.1 Construction of Plus Relationship

Definition 2.The plus system is widely used in the fields of communication,control,etc.It means that the specified tasks of the system are composed of N branch tasks which can not be replaced by each other,and the i-th(i=1,2,···,n)branch task is completed by corresponding branch unit Bi and central unit C(collectively referred to as the i-th branch).Its structure is shown in Figure 2.

Figure 2.Schematic diagram of plus system.

Based on Figure 1,the communication network composed ofkGEO satellites is denoted as{k(i),N(v,e)},wherekirepresents each communication satellite,andN(v,e)represents the GEO satellite communication network with UE,AS/RS/GMCC as nodeVand radio signal transmission as sidee.The GEO satellite communication network based on the inter-satellite link is shown in Figure 3.

In Figure 3,the wireless signal via GEO1 formsn1communication linkslinki{i=1,2,···,n1},each communication link is formed by connecting communication devices in series,andn1communication links are connected in parallel according to their importance to form a plus subsystem based on GEO1.Similarly,the plus subsystem based on GEO2,···,GEOkcan be formed.The GEO satellite communications constellation is thus a multi-layered plus connected communications network consisting of various GEO satellites.

2.2 Construction of PS-G-GERT Model of GEO Satellite Communication Constellation

The structural description of the constellation is mainly to model the elements of the constellation(such as communication satellites)and the transfer relationship between the elements.To solve the effectiveness of the constellation,it is necessary to analyze the effectiveness transfer relationship in the constellation based on the accurate description of the constellation structure.Thus,the PS-G-GERT model is composed of three elements: node,arrow and flow.And the probabilities of each communication activity and the effectiveness flow are represented by interval grey numbers.

Definition 3.In the PS-G-GERT constellation,the communication entities represented by satellites constitute the set of network nodes.The influence relationship between the entities constitutes the arrow ofthe network.The flow of effectiveness among the communication entities constitutes the flow of the network.Its basic composition unit is shown in Figure 4.

Figure 3.GEO satellite communication constellation plus diagram based on interstellar link.

Figure 4.Schematic diagram of the basic components of the PS-G-GERT constellation network.

In Figure 4,i,jrepresent the entities that constitute the communication activity,the arrow(i,j)represents the communication activity.Eij(?) is the communication effectiveness of the activity(i,j),which is measured by availabilityA,credibilityD,and capabilityC,the communication activity effectivenessEij(?)=ADC.pij(?) is the probability of successful execution of communication activity (i,j).EFij(?) denotes the effectiveness flow of activity (i,j),and it is determined by the probabilitypij(?) of successful execution of activity (i,j) and its effectiveness flowEij(?).

In the communication process,users divide the satellite communication network processing level into emergency,real-time communication,and non-realtime communication according to their different business forms.Different business forms have different requirements for data transmission.Based on Figure 1,the process of link performing communication tasks is understood as the process in which the user sends signals through the terminal and transmits them to the access satellite through the uplink.The access satellite analyzes the user’s needs,selects inter-satellite link transmission or GMCC for processing,and reaches the target user through the downlink.Therefore,the constellation link is composed of three types of entities:UE,satellite,and GMCC.According to definition 3,the PS-G-GERT constellation structure diagram can be constructed,as shown in Figure 5.

Figure 5.Link-based GEO satellite communication PS-G-GERT constellation structure diagram.

The information processing level of a satellite in the satellite communication network should follow the communication protocol.Based on this,the PS-GGERT constellation network model is described as follows:

1)Any of the following NRT Communicates represents the UE→GEO1/AS→GEO2/RS→GMCC→GEO2/RS→GEO1/AS→UE process;

2)Any of the following RT1,RT2 or EMG communications represents the process of UE→GEO1/AS→GEO2/RS→GEO1/AS→UE;

3) Communication activity TForward represents a TForward-only process;

4)Communication activity PForward represents the process of TForward→PForward→TForward;

5) Communication activity switch represents the process of TForward→PForward→Switch→PForward→TForward;

6) Communication activity route stands for TForward→PForward→Switch→Route→Switch→PForward→TForward;

7) Communication activity transport represents the process of TForward→PForward→Switch→Route→Transport→Route→Switch→PForward→TForward;

8)In the PS-G-GERT model,nodes with different logical characteristics can be included,and multiple probability branches are allowed at the output of the node;

9) The parameters in the satellite communication constellation (probability and communication activity effectiveness) are incomplete information and are given in the form of interval grey numberpij(?)∈

III.ANALYSIS OF PS-G-GERT NETWORK MODEL

3.1 Characteristic Function Design and Equivalent Transfer Function Solution of PS-G-GERT Model

To accurately measure the effectiveness of constellation,it is necessary to establish the equivalent transfer function of constellation network.The effectivenessEof each communication satellite is a random variable that obeys a certain probability distribution,so the equivalent transfer function of the constellation network is the probability distribution of multiple random variable functions,which is difficult to solve.Considering that some random distributions may not have moment mother functions,while the characteristic function is the Fourier transform of the probability density function,and for any distribution there is a characteristic function.Therefore,the characteristic function is introduced into the process of solving the equivalent transfer function of a constellation network.

Let the random variableEij(?)represent the communication effectiveness of the communication activity(i,j),andpij(?)represent the probability of successful execution of the communication activity(i,j).Thus,the definition of the characteristic function and transfer function of the random variable is given as follows:

Definition 4.Assuming that the effectiveness of each communication activity Eij(?)is a continuous ran-dom variable,and its probability density function is f(x),then its characteristic function φEij(?)(s)is[26]:

Where s ∈R.

Definition 5.Define Wij(?,s)as the transfer function of communication activity(i,j),then

Figure 3 and Figure 5 show that there is a mapping between the effectiveness of the GEO satellite communication constellation and the effectiveness of each communication link.To solve the effectiveness of each communication link,it is necessary to analyze the property of its characteristic function.According to GERT theory,combined with the characteristics of poor information in this paper,the characteristic functioncan be deduced from definition 4 to have the following properties[23].

Proposition 1.If E1,E2,···,En are n independent random variables,and the corresponding characteristic functions all exist,denoted asThe characteristic function of the sum of n independent random variables Y is the product of the characteristic functions of each random variable.Then there is

Proposition 2.The value of each derivative of the characteristic functionwith respect to s at s=0and the origin moment μk of random variable Eij(?)satisfy the relationship:

Proof.

Definition 6.To complete a communication task,the user equipment sends a communication request firstly,and the access satellite analyses the user requirements.After processing by the satellite or GMCC,the wireless communication link is finally formed by the interaction of the constellation network,denoted as l(link).Several communication links form a complex communication network.As shown in Figure 5,UE →AS →RS →AS →UE is a communication link.

To solve the average effectiveness of each communication link in the PS-G-GERT constellation network,it is necessary to first solve its equivalent transfer function.Based on reference [27],combined with the poor information feature of PS-G-GERT constellation structure,the equivalent transfer functionW(?,s)of PS-G-GERT constellation is obtained.

Theorem 1.In the constellation communication network,if there are n series activities between the terminal sending node and the terminal receiving node,the transfer function is Wi(?,s)(i=1,2,···,n);if there are m parallel activities and the transfer function is Wj(?,s)(j=1,2,···,m),then the equivalent transfer function from the terminal sending node to the terminal receiving node is

Proof.1) When the constellation communication structure isnactivities in series,the probability of the occurrence of constellation communication activity isAccording to proposition 1,the effectiveness of the constellation network is the superposition of the effectiveness of each communication satellite,the equivalent characteristic function ofnseries activities isBy definition 5,the equivalent transfer function for a network withnseries activities

2)When the constellation communication structure ismactivities in parallel,according to the sum formula for probability,the probability of the constellation’s communication activity being successfully executed is

Due to the XOR node characteristics of the GERT constellation network,the communication activity 1,2...,mhas one and only one occurrence,then all the items in (5)are zero except for the first term,so

In addition,the probability of occurrence of communication activity 1,2,···,misp1(?),p2(?),···,pm(?) respectively.When communication activity 1 occurs,its characteristic function isSimilarly,when communication activitymoccurs,its characteristic function isMoreover,since the GERT of the XOR node is only considered here,only one communication activity occurs when the link runs once,and the communication activity between node 1 and nodemmust be implemented,so:

Thus,combined with definition 5,the equivalent transfer function of a network consisting ofmparallel activities is

3)Analyzing Figure 5,it can be found that the linkbased GEO satellite communication constellation is a multi-layer plus system.Therefore,the equivalent transfer function of the PS-G-GERT network is the function of the equivalent transfer function of the series structure and the parallel structure.

3.2 Solution of Equivalent Probability and Equivalent Characteristic Function of PSG-GERT Model

The equivalent transfer function is obtained by theorem 1.To obtain the average effectiveness of each communication link in the PS-G-GERT constellation network,it is necessary to express the equivalent transfer probability of communication entitiesitojin the PS-G-GERT constellation network.Taking the literature[23,28]as the reference,and combining the poor information characteristics of the PS-G-GERT constellation,the equivalent transfer probabilitypij(?)of the communication entitiesitojin the PS-G-GERT constellation is obtained.

Theorem 2.Assumes that the equivalent transfer function of communication entities i to j in the PSG-GERT constellation network is Wij(?,s),then the equivalent transfer probability from i to j is pij(?)=

Proof.It is proved that if the equivalent transfer function of communication entitiesitojin PS-G-GERT constellation network isWij(?,s),then according to definition 5,there is

Combined with definition 4,we know that

3.3 The Solution of PS-G-GERT Constellation Network Parameters

3.3.1 Solution of Communication Link Effectiveness in PS-G-GERT Constellation Network

The equivalent transfer function and equivalent transfer probability of the PS-G-GERT constellation network can be obtained through theorem 1 and theorem 2.According to the above analysis,it can be known that there is a mapping relationship between the effectiveness of the GEO constellation and the effectiveness of each link.Based on the classical GERT theory[24,26,23],combined with the solution of the average device lifetime in the grey gert network model[29],the average effectiveness of each communication link in the PS-G-GERT constellation can be represented.

Theorem 3.Under the description of PS-G-GERT constellation network,the average effectiveness of each communication link is

Proof.If the equivalent transfer function of the communication link in the PS-G-GERT constellation network isWl(?,s),as known from Theorem 2 and Definition 5,the equivalent characteristic functionis

According to the properties of the characteristic function,the value of each order derivative of the characteristic function atS=0 has a functional relationship with the origin moments of random variables.The average effectiveness of each communication link in the PS-G-GERT constellation network is calculated as follows:

Corollary 1.If the equivalent transfer function of communication entities i to j in PS-G-GERT constellation network is Wij(?,s).The k-th moment E[Ek(?)]of grey effectiveness parameter E(?)that from communication entities i to j is equal to Wij(?,s),the value of the k-order partial derivative of s in effectiveness parameter E(?)is 0,as shown in equation(13).

3.3.2 PS-G-GERT Constellation Network Communication Link Importance Analysis

From Figure 3 and Figure 5,it can be seen that the PSG-GERT communication constellation is a multi-layer plus communication network based on the communication link.The average effectiveness of each communication link is known from Theorem 3.To solve the effectiveness of the PS-G-GERT communication constellation,the importance of each communication link must first be solved[30].Through the investigation of the existing literature,researchers quantify the importance of the research object as the change rate of the system related parameters from the presence to the absence[10,31].From the understanding of link importance of satellite communication constellation,that is,the influence degree of link effectiveness variation on communication constellation effectiveness.From the differential point of view,the effectiveness equivalent transfer function of the PS-G-GERT constellation network can be derived for a certain communication link effectiveness parameter.And combining the properties of the characteristic function,the communication link importance assessment value is obtained[27].

Definition 7.The derivative function of the effectiveness equivalent transfer function of the PS-G-GERT network is derived from the equivalent transfer function Wl(?,s)of link l,and the derivative function of the effectiveness parameters of link l is obtained.The first order origin moment is the influence degree of link l on the PS-G-GERT network effectiveness.Let the proportion of link l’s influence on constellation effectiveness over the influence of all links on constellation effectiveness be the link l importance,denoted asimp(l).That is

Where,l is the communication link(l=1,2,···,n);?is the set of all communication links in GEO satellite communication constellation.

Theorem 4.The importance of communication link imp(l)has the following properties:

(1)0

3.3.3 Effectiveness Solution of PS-G-GERT Communication Constellation

Definition 8.The effectiveness of GEO satellite communication constellation can be obtained by combining the effectiveness El(?)of each communication link with its imp(l).

3.3.4 Effectiveness Variance of PS-G-GERT Communication Network

To further analyze the sensitivity and change trend of each communication activity and communication link,it is also necessary to measure the effectiveness variance of the constellation.According to the literature[32,33],combined with grey GERT theory [28,29],the variance of the communication activity effectiveness distribution function is defined as the measure of the variance of the communication activity effectiveness.

Theorem 5.In the PS-G-GERT constellation network,the variance measure of communication activity effectiveness is

Proof.From the variance calculation formula combined with corollary 1,we can get

3.3.5 Steps of PS-G-GERT Constellation Network Effectiveness Evaluation

The basic idea of GEO satellite communication constellation effectiveness evaluation is shown in Figure 6,which is a process of transformation,collect,solve,aggregate,and explain.

Figure 6.The basic idea of effectiveness evaluation of GEO satellite communication constellation.

IV.CASE ANALYSIS

The link establishment relationship and working sequence of the inter-satellite link of the satellite communication constellation are determined by the link planning table,which is generated according to the visible relationship between satellites and service requirements.It is known that one GEO satellite(height about 35786km)can theoretically cover 42.4%of the earth’s surface area,and three GEO satellites can cover the whole world [34].Suppose a GEO satellite is located at 80?E,200?E,and 320?E,marked as GEO1,GEO2,GEO3.In the actual communication process,because the GEO constellation is suitable for long-distance communication,it is generally considered that the two user terminals are at most within the coverage area of two GEO satellites.In the communication process,the user divides the satellite communication network processing levels into emergency,realtime communication,and non-real-time communication according to their different business forms.Different business requirements have different requirements for data transmission.The processing levels of different services in the constellation transmission process are shown in Table 2.

Table 3.A GEO satellite communication link and its link effectiveness.

4.1 Construction of PS-G-GERT Framework for a GEO Satellite Communication Constellation

The communication logic between the various entities of this GEO constellation is described in terms of a PSG-GERT network by analyzing the background of the case and determining the delivery hierarchy of the two user devices according to their different operational requirements.The network is mainly composed of satellites,GMCC and user equipment,and the communication entities form the vertices of the PS-G-GERT network.Among them,it can be seen from Table 2 that satellites have different processing layers for different services during the transmission mission.The four layers processing modules of each GEO satellite are represented by GERT nodes according to the signal transmission process in the communication layer.The grey information transfer relationship between each communication entity constitutes the edge of the network,and the uncertain probability flow between each communication service in the three entities constitutes the flow in the network.From the former analysis,it is clear that in the actual communication process two user terminals are in the coverage area of at most two GEO satellites.When the two user equipment is in thesame satellite coverage area,regardless of under which GEO satellite coverage,there are three cases of emergency communication,real-time communication and non-real-time communication.The constellation has a total of three GEO satellites,so it can be known that when the two user equipment are in the same satellite coverage area there are a total of nine communication links.When the two user equipment is in different satellite coverage areas,such as GEO1 and GEO2 as access and relay satellites respectively,considering three different communication scenarios,a total of six communication links exist.Similarly,the GEO1 and GEO3,GEO2 and GEO3,as access and relay satellites respectively,six communication links can be established.In summary,there are at most 27 links in this PS-G-GERT network at a given instant,and the link establishment situation is shown in Figure 7.From Figure 1 it can be seen that the link establishment process of the GEO satellite communication constellation is a series connection of each communication device.From Figure 3 it is known that the GEO satellite communication constellation is a multi-layer plus relationship PS-G-GERT network.This GEO satellite communications constellation is used as an example to analyze its effectiveness and variance.

Table 4.Importance of a GEO satellite communication link.

4.2 Effectiveness Evaluation of a GEO Satellite Communication Constellation

4.2.1 Effectiveness of Each Link

The probability and effectiveness of communication activities are affected by factors such as weather and the invulnerability of communication equipment,and there is a certain degree of uncertainty.Among them,the probability of each communication activity is obtained according to the statistical law,and the effectiveness distribution of each communication activity is evaluated by using the ADC model according to its capability performance and performance data in each link [35].According to theorem 1,the equivalent transfer function of each communication link in the GEO satellite communication constellation can be obtained,and the average effectiveness of each communication link can be obtained from theorem 3,as shown in Table 3.

4.2.2 Solving the Importance of Each Link

According to the equivalent transfer function of a GEO constellation,PS-G-GERT network,combined with Definition 7,the importance of each communication link can be obtained as shown in Table 4.

4.2.3 Constellation Effectiveness

The effectiveness of the GEO satellite communication constellation is plused according to the importance of each link and the effectiveness of the link,that is

4.2.4 Constellation Effectivenes Variance

Combined with theorem 1,the equivalent transfer functionW(?,s) of the PS-G-GERT network of the GEO satellite communication constellation can be obtained.According to theorem 5,the effectiveness variance of the GEO satellite communication constellation can be obtained by MATLAB,which is

4.3 Discussion on PS-G-GERT Effectiveness Evaluation Results of a GEO Satellite Communication Constellation

4.3.1 Sensitivity Analysis

In Figure 8 and Figure 9,the original variance of satellite communication constellation is taken as the baseline,and the effectiveness flow of each communica-tion activity and the communication link is increased by 0.1 and 0.2 respectively,to see the deviation degree from the original effectiveness value under the same change range.As can be seen from Figure 8,communication activities 1 and 46 have the highest impact on constellation effectiveness,and communication activities 18 and 19,12 and 15,7 and 52,11 and 14 also have a significant impact on constellation effectiveness.This is consistent with the solution of the influence amplitude of each communication link on the constellation effectiveness in Figure 9.From Figure 9,link 24 has the most obvious impact on constellation effectiveness,followed by link 27,6,and 25.The communication activities that have a great impact on constellation effectiveness in Figure 8 constitute the communication links with an obvious impact on constellation effectiveness in Figure 9.Moreover,the sensitivity analysis results are consistent with the solution results of the importance of each link,which further verifies the credibility of the calculation results of the importance.

Figure 7.Construction of PS-G-GERT network model for a GEO satellite communication constellation.

In Figure 10 and Figure 11,the original variance of satellite communication constellation is taken as the baseline,and the variance of each communication activity and the communication link is increased by 0.1 to see the deviation degree from the original variance under the same change range.It can be seen from Figure 10 that the variance of communication activities 8 and 53 have the highest impact on constellation effectiveness variance,followed by communication activities 2 and 47,5 and 50,9 and 54,6 and 51.This is consistent with the calculated result of the influence range of each communication link variance on constellation effectiveness variance in Figure 11.From Figure 11,the variance of link 12 has the most obvious effect on the constellation effectiveness variance,followed by links 10,11,3,and link 2.The communication activities in Figure 10 which have a great influence on the constellation effectiveness variance constitute the communication link with obvious influence on the constellation effectiveness variance in Figure 11.The sensitivity analysis results further verify the correctness and effectiveness of the model.

Figure 8.The relationship between communication activity effectiveness and constellation effectiveness.

4.3.2 Change Trend Analysis

The effectivenessEof the GEO satellite communication constellation is a monotonic decreasing function of the equivalent transfer probability (Figure 12,13),a monotonic increasing function of the link effectiveness (Figure 12,14),and a function of the link variance(Figure 13,14)first increasing and then decreasing.Therefore,when the equivalent transfer probability is lower,the link effectiveness is higher,and the constellation effectivenessEis higher(Figure 12);when the variance of link effectiveness changes,constellation effectivenessEpresents a fluctuating state(Figure 13,14).The effectiveness variance of the GEO satellite communication constellation is a monotonic increasing function of the equivalent transfer probability(Figure 15,16),but the increase is very slow and has nothing to do with the fluctuation of the link effectiveness(Figure 15,17).The effectiveness variance is also a monotone increasing function of the link effectiveness variance (Figure 16,17).Therefore,when the equivalent transfer probability is lower,the link effectiveness variance and the constellation effectiveness variance are smaller(Figure 16).

Analysis of Figure 13 and Figure 14 shows that with the increase of link variance increment,the effectiveness value of constellation appears at an inflection point when the increment of link variance is about 0.24,and the effectiveness value changes from increase to decrease.In Figure 12,constellation effectiveness presents a downward trend.When the equivalent transfer probability increases to about 0.05,the growth of constellation effectiveness slows down.From Figure 12 and Figure 14,it can be seen that when the effectiveness of each link increases (up to 0.1),the constellation effectiveness value is greater than 1.Therefore,reasonably assigning tasks and improving the effectiveness of each link can effectively improve the effectiveness of the constellation.But when the link effectiveness increment reaches 0.1,the enhancement of link effectiveness increment has little effect on the constellation effectiveness.Analysis of Figure 16 and Figure 17 shows that the increase in link variance has an obvious effect on the variance of the constellation.When the link variance increases from 0 to 0.1,the variance of the constellation increases from 0.05 to 0.25.The effect of the increment of equivalent transfer probability on the variance of the constellation is limited,and when it is greater than 0.03,the impact on the variance of the constellation can be negligible(Figure 15,16).Figure 15 and Figure 17 show that the change of link effectiveness has little effect on constellation variance.

Figure 9.The relationship between the effectiveness of each communication link and the effectiveness of constellation.

Figure 10.The relationship between communication activity variance and Constellation variance.

4.3.3 Three Forms of Communication for Effectiveness Sensitivity

The analysis of Figure 18 to Figure 20 reveals that when the effectiveness of the emergency link,the realtime link and the non-real-time link all vary in increments between 0 to 0.05,the trend of constellation effectiveness varies as follows.When the effectiveness of the emergency link varies,the lower limit of constellation effectiveness varies in the range of 0.69-0.72 and the upper limit of constellation effectiveness varies in the range of 0.9-0.96(Figure 18,19).When the effectiveness of the real-time link varies,the lower limit of constellation effectiveness varies in the range of 0.67-0.69 and the upper limit of constellation effectiveness varies in the range of 0.87-0.94(Figure 18,20).When the non-real-time link effectiveness varies,the lower limit of the constellation effectiveness varies between approximately 0.69-0.72 and the upper limit of the constellation effectiveness varies between approximately 0.9-0.97 (Figure 19,20).The above trends show that the effectiveness of the GEO constellation is significantly influenced by the emergency and non-real-time links.The main reason is that emergency communications have the highest reliability requirements and the establishment of non-realtime communication links requires the collaboration of various devices.Both of which have high reliability and resource requirements for the constellation and are prone to weaknesses.The results of this analysis further support the conclusions of the sensitivity analysis and the changing trend of effectiveness,as well as being consistent with the results of solving for the importance of each link.

Figure 11.The relationship between the variance of each communication link and the variance of constellation.

Figure 12.Relationship between equivalent transfer probability increment and link effectiveness increment and constellation effectiveness.

Figure 13.Relationship between the increment of equivalent transmission probability and the variance of link and the effectiveness of constellation.

Figure 14.Relationship between link effectiveness increment and link variance increment and constellation effectiveness.

Figure 15.Relationship between equivalent transfer probability increment and link effectiveness increment and constellation effectiveness variance.

Figure 16.Relationship between increment of equivalent transmission probability and increment of link variance and variance of constellation effectiveness.

Figure 17.Relationship between link effectiveness increment and link variance increment and constellation effectiveness variance.

Figure 18.Relationship between emergency link effectiveness increment and real-time link effectiveness increment and constellation effectiveness.

Figure 19.Relationship between emergency link effectiveness increment and non-real-time link effectiveness increment and constellation effectiveness.

Figure 20.Relationship between real-time link effectiveness increment and non-real-time link effectiveness increment and constellation effectiveness.

4.4 Comparative Analysis of Effectiveness Model

This paper constructs the PS-G-GERT effectiveness evaluation model for the GEO satellite communication constellation.To further verify the effectiveness of the model,the method in this paper is compared with the existing effectiveness evaluation models.In the comparison,based on the effectiveness values and importance obtained in the PS-G-GERT model,thenetwork effectiveness values are calculated by reference[36,37]and the PS-G-GERT model respectively,as shown in Table 5.

Table 5.Comparison of effectiveness results.

It can be seen from Table 5 that,the system effectiveness obtained by the existing effectiveness model is within the effectiveness range obtained by the PSG-GERT effectiveness evaluation model,which further verifies the validity of the PS-G-GERT effectiveness evaluation model.Among them,the effectiveness value obtained by using the fuzzy relation information system in reference [37] is 0.971,but 0.971/∈[0.6946,0.9364].However,when PS-G-GERT adds the change of effectiveness variance,the right endpoint of its effectiveness interval is [0.9737,0.9831].That is,the effectiveness value 0.971 obtained by using fuzzy relation information system in reference[37]is actually within the effectiveness range obtained by the PS-G-GERT effectiveness evaluation model.The PS-G-GERT effectiveness evaluation model can not only solve the system’s effectiveness under certain information conditions but also can be used to measure the system’s effectiveness in the context of poor information.At the same time,the existing effectiveness evaluation methods can not solve the effectiveness variance and the method is affected by subjective factors.The PS-G-GERT model proposed in this paper not only can avoid such problems,but also can further adjust the evaluation parameters by summarizing the historical operation data of the system.Thus,the accuracy of the system effectiveness solution can be further enhanced.

V.CONCLUSION

In this paper,the interaction relationship between communication entities is characterized by GERT,using the characteristic function as the transfer parameter.The effectiveness and variance of the constellation are obtained by pulsing the effectiveness of each communication link and its importance.In particular,to consider the influence of uncertain information on the evaluation results,we constructed the GEO satellite effectiveness evaluation model in the context of poor information for the first time.By solving and analyzing the effectiveness of a GEO constellation and comparing it with existing effectiveness evaluation models,the validity of the model in this paper is verified.In future research,we will consider constructing the effectiveness evaluation model for communication satellite constellation with a large number of satellites and solving the network effectiveness through computer programming.

ACKNOWLEDGEMENT

This work was supported by the National Natural Science Foundation of China under Grant 71671091 and in part by the Major Project Cultivation Fund and Major Science and Technology Achievements Development Fund of Nanjing University of Aeronautics and Astronautics under Grant NP2019104 and NC2019003 respectively,and in part by the National Development and Reform Commission (High-Tech[2017]1069).