Mega-Constellations Based TT&C Resource Sharing: Keep Reliable Aeronautical Communication in an Emergency

Haoran Xie ,Yafeng Zhan ,Jianhua Lu

1 Department of Electronic Engineering,Tsinghua University,Beijing 100084,China

2 Beijing National Research Center for Information Science and Technology,Tsinghua University,Beijing 100084,China

Abstract: With the development of the transportation industry,the effective guidance of aircraft in an emergency to prevent catastrophic accidents remains one of the top safety concerns.Undoubtedly,operational status data of the aircraft play an important role in the judgment and command of the Operational Control Center(OCC).However,how to transmit various operational status data from abnormal aircraft back to the OCC in an emergency is still an open problem.In this paper,we propose a novel Telemetry,Tracking,and Command (TT&C) architecture named Collaborative TT&C(CoTT&C)based on mega-constellation to solve such a problem.CoTT&C allows each satellite to help the abnormal aircraft by sharing TT&C resources when needed,realizing real-time and reliable aeronautical communication in an emergency.Specifically,we design a dynamic resource sharing mechanism for CoTT&C and model the mechanism as a single-leader-multi-follower Stackelberg game.Further,we give an unique Nash Equilibrium(NE)of the game as a closed form.Simulation results demonstrate that the proposed resource sharing mechanism is effective,incentive compatible,fair,and reciprocal.We hope that our findings can shed some light for future research on aeronautical communications in an emergency.

Keywords: aeronautical emergency communication;mega-constellation;networked TT&C;resource allocation;stackelberg game

I.INTRODUCTION

Over the past few years,despite the significant advances in commercial jet aircraft,there are still many unexpected situations,such as Lion Air Flight JT610 crash[1]and Malaysia Plane MH370 missing[2],resulting in aircraft accidents remains one of the largest contributors to fatal accidents[3].A safety investigation report is required by the International Civil Aviation Organization (ICAO) after every accident.According to the accident investigation report on the Lion Air Boeing 737 MAX flight JT610 crash[4],the accident was brought out by a combination of an incorrect angle of attack (AOA) sensor input,lack of pilot reporting and training as well as design flaws within the aircraft’s maneuvering characteristics augmentation system (MCAS).In this case,early intervention and effective guidance from OCC can help the pilot remove the fault timely and avoid catastrophic accidents.The MH370 safety investigation report [5] said the lack of the flight recorder(i.e.,“black box”)from the flight made it impossible to determine the real cause of disappearance.For this reason,ICAO called for live streaming of data to the ground in the wake of the accident.It is beyond doubt that various flight data and the cockpit voice(i.e.,TT&C data)play an essential role in the investigation of aviation accidents and incidents.In fact,ICAO has been devoted to promoting airborne radio telephone communication via satellite between commercial aircraft and the OCC since 2003[6],ensuring essential and reliable data link coverage of oceanic and remote continental environments.At present,International Maritime Satellite(Inmarsat)is capable of providing analog voice,fax,and data services for aircraft,keeping cockpit and cabin connected at 35,000 feet [7].In addition,L-band Digital Aeronautical Communications System(LDACS),a key technology of the new air traffic services and operational concepts of the air traffic management(ATM),is undergoing flight trial demonstration [8] and performance analysis [9].Still,given that LDACS is intended to support ATM in the continental airspace,and Inmarsat focuses on the inflight Wi-Fi that enhances the overall passenger experience,current frameworks struggle to ensure globally reliable data link coverage during emergencies.Consequently,how to implement seamless,consistent,reliable,high-speed aeronautical communication in an emergency is our concern.

Mega-constellation,as a topic of significant interest in both academia and industry,is an essential component in sixth-generation (6G) wireless networks [10-12].In order to satisfy the TT&C needs of the megaconstellations,Zhan et al.[13] proposed the concept of the networked TT&C system,as depicted in Figure 1,where the TT&C data of mega-constellations are transmitted from the LEO satellites to the ground station via the Satellite-to-Ground Link (SGL) and the Inter Satellite Links (ISLs) among LEO,MEO,and GEO satellites,realizing the seamless and reliable TT&C[14].What’s more,the low delay of ISLs and the TT&C framework also ensure the real-time performance of information,which is especially important in an emergency [15].Based on the architecture of the networked TT&C system,realizing aeronautical communication in an emergency through the TT&C links of mega-constellations could be envisioned as a promising solution.

Figure 1.The space-ground integrated networked TT&C architecture.

To be able to collaborate with the abnormal aircraft,the resource allocation scenario and problem for the mega-constellations need further modeling and analysis.In particular,there are some challenges to be addressed as follows.

•How to design an agile,resilient network architecture,realizing seamless,consistent,reliable,highspeed aeronautical communication in an emergency?

•How to devise an effective mechanism that enables the satellite to be willing to help the aircraft and satisfy its needs?

•How to design an efficient and stable resource allocation algorithm to meet the requirements of aircraft in emergency scenarios?

Existing studies have been extensive research on resource allocation in satellite networks.Given the exceptional capabilities of Reinforcement Learning(RL) [16] in dealing with complex problems,some studies have adopted RL to tackle this problem [17-19].In [17],He et al.studied a multi-objective deep reinforcement learning-based time-frequency two-dimensional resource allocation (MODRL-TF)algorithm to maximize the number of users and system throughput.In[18],Hu et al.used the model-free multi-objective deep reinforcement learning approach to investigate the optimal policy for beam hopping in the DVB-S2X satellite.In[19],Guo et al.proposed a multi-agent deep deterministic policy gradient(MADDPG) based algorithm to minimize transmission latency in a cognitive satellite-aerial network.However,most RL-based methods have limitations in balancing“exploration”and“exploitation,”leading to challenges in ensuring algorithmic stability.This limitation is particularly unsuitable for emergency scenarios where stable and reliable algorithms are crucial.

To solve the above problems,some researchers have studied efficient resource allocation schemes through rigorous optimization theory[20-22].In[20],Zhang et al.developed a three-stage iterative resource allocation algorithm to solve a highly non-convex resource allocation problem in next-generation multiple-access terrestrial-satellite networks.In[21],Abdu et al.proposed a novel carrier and power allocation method for flexible broadband GEO SatCom systems in order to provide the best possible traffic matching while utilizing the least amount of resources.In [22],Liu et al.studied multi-domain resource allocation for cognitive satellite-UAV network (CSUN) and designed a process-oriented optimization framework that considers the whole flight process of UAVs.These huge efforts have led to the rapid development of resource allocation.However,the current literature is challenging to apply in heterogeneous networks,while mega-constellations and abnormal aircraft are generally managed by different operators,which is a regret.

Game theory is the study of strategic interactions among rational agents,and it has recently attracted considerable attention from researchers [23-26].In[23],Wang et al.employed a Stackelberg Mean Field Game(SMFG)to address the dynamic data offloading problem with large-scale ground users in ultra-dense LEO satellite networks.In [24],Li et al.proposed the two double auction mechanisms named TMF and EMF to allocate resources between IoT devices in satellite Mobile Edge Computing (MEC).In [25],Deng et al.designed a pricing mechanism based on the Stackelberg game to motivate both operators in an integrated ultra-dense LEO-based satellite-terrestrial network for data offloading.In[26],Li et al.studied a generic satellite spectrum pricing scheme,for the dynamic allocation of satellite spectrum.However,none of these works focuses on space-air network communications,especially aeronautical communications in an emergency,where the system architecture and operating mechanism are completely different.Naturally,previous studies have not considered the characteristics of flight data,either.

Motivated by the above issues,we propose CoTT&C,a novel collaborative TT&C communication architecture composed of mega-constellations and abnormal aircraft.In this architecture,each satellite can contribute its TT&C resources to help the abnormal aircraft when needed.For the purpose of studying the resource allocation problem when the abnormal aircraft ask for help,we design a fair and incentive-compatible resource sharing method based on the Stackelberg game.Through rigorous gametheoretic analysis,we give the optimal strategies of all players and derive a unique NE for the Stackelberg game as a closed form.Our contributions are summarized as follows:

•We present a novel network design for collaborative communication in emergency scenarios.This resilient network design could realize seamless,consistent,reliable,high-speed aeronautical communication in an emergency.

•We design a fair,efficient,and incentivecompatible resource sharing mechanism such that both the abnormal aircraft and satellites choose their best strategy and reach their maximum utility.

•We establish a single-leader-multi-follower Stackelberg game model to depict the hierarchical decision-making structure.Further,we prove the NE exists and is unique in the proposed game,and derive a unique Stackelberg equilibrium(SE)as a closed form.

The rest of this paper is organized as follows: Section II describes the architecture,mechanism,and model of the aircraft emergency TT&C and formulates the utility functions for resource requesters and helpers.Section III presents our resource sharing method and provides an analysis of the proposed Stackelberg game,followed by the evaluation results given in Section IV.Section V summarizes the paper and gives the conclusion.

II.SYSTEM MODEL AND PROBLEM FORMULATION

In this section,we detail the network design of CoTT&C and the concrete implementation approach of collaborative communication.Further,we model the utility functions of the requester-side and helperside of resource sharing.To enable both parties to choose their best strategy and reach their maximum utility,the following design goals and problem formulation are provided.

2.1 Scenario and System Architecture

In this study,we consider the future space-air-ground integrated scenario.As shown in Figure 2,the space network is a multi-layered satellite network constructed by integrating several satellite networks,connected by inter-satellite and inter-layer links.These satellites and constellations are in different orbits(e.g.,GEO,MEO,and LEO) and with different characteristics,such as Communication,Navigation,Remote Sensing,etc.Here,we assume that satellites are reconfigurable,enabling flexible upgrades of frequency bands and air interface protocols.In this way,satellites can receive various types of data without requiring diverse payloads and antennas.Meanwhile,we consider the programmability of the satellites,which can be injected into the appropriate algorithm from the ground to serve a wide range of applications.

Figure 2.The proposed CoTT&C network architecture.

To ensure the normal operation of the satellite in orbit,comprehensively monitoring the operating status of the satellite is pivotal.Thanks to the inherent advantages in terms of global coverage,low delay,and high reliability,the networked TT&C system is promising for the upcoming era of space-airground integration.In the networked TT&C system,the network is always connected and that there exists End-to-End (E2E) paths in the networks all the time,ensuring reliable message delivery.It adopts a one-station multi-satellites mode,where a ground station controls the entire satellite network through ISLs and SGL.In this mode,the mission center allocates a certain amount of TT&C resources for each satellite in advance,and informs each satellite via the satellite telecommand system.With the time synchronized in all satellites,the allocation schemes can be coordinately implemented,so as to control the transmission of TT&C data.Subsequently,the operational status data of all satellites are sent back to the ground station in real-time through the SGL and ISLs among LEO,MEO,and GEO satellites,and further processed by the mission center.

Based on the networked TT&C system,we present the network design of CoTT&C.The purpose of CoTT&C is to provide support for stable communications between aircraft and the ground during an emergency.In the CoTT&C network,satellites are equipped with directional antennas,where the steerable directional beams are well-designed to avoid signal interference and directed by GNSS to track the aircraft.As for aircraft,omni-directional antennas and phased array antennas are equipped,where the omnidirectional antenna is to ensure communication continuity under any circumstances.They are connected by a microwave link.Besides,each satellite contains an agent and a certain amount of TT&C resources.The purpose of the agent is to determine the optimal strategy in the interactive game between satellite and aircraft.Due to the flexibility of networked TT&C system,the TT&C resources can be shared,and its capacity can be configured dynamically based on aircraft needs.

Figure 2 illustrates a case study of CoTT&C.Here,we consider utilizing mega-constellations to share their TT&C resources.Mega-constellations can provide global coverage on Earth,and the distance between satellites and aircraft is shorter,resulting in small free space path loss.Since the public opinion that successfully resolves an air crash will bring significant economic benefits,commercial mega-constellations also have the incentive to share their own TT&C resources to rescue the abnormal aircraft.Specifically,during an aircraft emergency,the abnormal aircraft first determines its resource request by referring to its emergency and satellite configuration information.Considering that the satellite orbits are fixed,the abnormal aircraft can deduce the satellites visible in the line of sight (LOS) and inform the overhead MEO/GEO satellites of its selected satellites and corresponding resource request through the broadcast channel.Next,the MEO/GEO satellite notifies the satellites within its coverage and obtains the amount of resources each satellite is willing to share with the aircraft.After that,the satellites direct the steerable directional beams toward the abnormal aircraft and tell it of the resource allocation scheme.Finally,data is transmitted by the networked TT&C system to the ground according to the pre-loaded scheme,as depicted in Figure 3.There are two situations in which a handover occurs.One is handover in satellite networks due to the rapid movement of satellites around the Earth.The other is handover between satellite gateways in case satellite link quality worsens.Since they have been extensively studied in the mobility management of mega-constellations and handover management of networked TT&C,this content is above the scope of this paper.In this study,we focus on how much resource the satellites should offer to the abnormal aircraft and what level of demand the abnormal aircraft should send to the satellites in order to reach sufficient support.To model this,we defineJas the set of satellites,r=（R1,R2,...,R|J|）as the resource request vector of the abnormal aircraft to eachas the resource sharing vector of the satellites.Where |J|=J,andRiandSidenote the TT&C resource amount (packet/s)that the abnormal aircraft request to thei-th satellite and thei-th satellite shares with the abnormal aircraft,respectively.The resource amount is constrained to be nonnegative.

Figure 3.Sequence diagram for the proposed resource sharing mechanism.

Figure 4.Utility of the satellite with different values of ζi.

Figure 5.Utility of the abnormal aircraft with different values of ζi.

Figure 6.Utility of the satellite at different urgency.

Figure 7.Utility of the abnormal aircraft at different urgency.

2.2 Utility Function of Mega-Constellation

TT&C is the lifeline of satellites.Once LEO satellites share their TT&C resources with the abnormal aircraft,their risk of an accident increases due to the lack of enough support from ground TT&C stations for TT&C data analysis.The risk of an accident on satellites can be characterized by well-defined reliability functions as they conflict with each other.

Definition 1.The satellite reliability with missing TT&C data amount n follows the assumption that the reliability is a decreasing function and has an increasing marginal utility[27]. Thus,we define the reliability function Qi(n)in exponential form:

where κi(＞0)and ζi(＞0)are used to set the decreasing ratio with the amount of shared resources. The range of resource sharing amount n is(0,ln((κi+1)/κi)/ζi],whereln((κi+1)/κi)/ζi is the upper bound determined by the reliability of each satellite.

Let random variablesZdenote the failure effects of abnormal aircraft,whereZ～U[0,1].The parameter Θ=s(Z) indexes the urgency of abnormal aircraft,which is a strictly increasing and convex function ofZ.Considers(Z)=-λlnk(1-Z),wheres′,s′′＞0;s(0)=0;s′(0)＜ε(a positive number that is arbitrarily close to zero);s(∞)=s′(∞)=∞.Fori ∈{1,2,...,J},the utility function of thei-th satellite is defined as

where Θ(＞0) andηi(＞0) are the weighting variables used to the performance of payoff and risk,respectively.

In the objective function,the first item ΘSi/Rirepresents the satisfaction that thei-th satellite can obtain from sharing TT&C resources with the abnormal aircraft.The second termηi(1+κi(1 -eζiSi)) is the remaining reliability of thei-th satellite after sharing TT&C resources with the abnormal aircraft.The utility function designed in this way aims for a balance between the satisfaction gained from the abnormal aircraft and the risk of the satellite in sharing finite TT&C resources with the abnormal aircraft.

2.3 Utility Function of Abnormal Aircraft

Our design of the utility function for the abnormal aircraft is to maximize the amount of data it transmits1.According to [14],the received power of the relay satelliteprand the signal-to-interference-plusnoise ratioSINRrican be defined as:

wherepi,porepresent the transmission power of thei-th LEO satellite and another satellite,respectively.hi,r,ho,rdenote the channel power gain andσ2is the noise power.According to Shannon bound,the spectrum efficiencyei,ris given by

In the CoTT&C network,the signal propagates in a single path of the vacuum environment along the ISLs,which can be modeled as the Gaussian channel.Different beams of the same satellite are spatially isolated,and different services within each beam adopt different polarization modes and frequency bands.In this case,interference could be almost neglected.Based on the concept of the data transmission rate,we consider the following utility function:

wherecis the light speed,fis the carrier frequency,N0is the one-sided thermal noise power spectral density,ϱis the resource block weighting factor(map resource requests to power dimensions with a link margin of 3dB),Birepresents the TT&C bandwidth of thei-th satellite,andGtandGrdenote the antenna transmission gain of transmitter and the antenna transmission gain of the receiver,respectively.For simplicity,we denote each term of the utility functionU0(r,s)asGi(Ri,Si).

Remark 1.Regarding the effectiveness of the proposed utility function,as the Logarithmic functionshave diminishing marginal effects

when xi ≫1,the abnormal aircraft prefers to make fewer TT&C resource requests to multiple satellites than to make more TT&C resource requests to a few satellites for more data transmission. This is a realistic expectation since as more requests are distributed to multiple satellites,the lower impact on the reliability of each satellite.

Remark 2.The data transmission rate in the channel is determined by the bandwidth of the channel and the signal-to-noise ratio of the channel. In practical applications,satellite systems would adopt an appropriate resource allocation scheme for different services based on the rarity of on-board resources. Actually,we have also considered the situation that the satellite shared resource is bandwidth. It can be demonstrated that the following utility function:

also has diminishing marginal effects,suggesting that the statement in Remark 1 holds as well. Meanwhile,the above utility function will have no impact on the meaningful conclusions drawn in the subsequent analysis,except for slight differences in the expression of the optimal solution.

2.4 Problem Formulation

Our goal is to design a collaborative mechanism that is efficient,fair,and incentive-compatible,as well as enable all participants to achieve their respective maximum utility.Consider the participants are individualrational and risk-neutral.All of the above is common knowledge.

Definition 2.Bi(a-i)is the set of player i’s best reply given a-i,i.e.,

where Ai be the strategy set of the player i and a-i be the strategy of the other players.

LetAairdenote the strategy set of the abnormal aircraft anddenote the strategy set of satellitei,With a slight abuse of notation,we usea1,a2to denote the strategy of the abnormal aircraft and satellites,respectively.The problem of interest can be formulated as finding a pairsuch thatwhereg*is a function of?,andwithandg*∈AAair,satisfying that:

•for eacha1∈Aair,g*(a1)∈B(a1)and

•for eacha1∈ Aair,≥U(a1,g*(a1)).

III.RESOURCE SHARING MECHANISM

We formulate the TT&C resource allocation problem among mega-constellation and abnormal aircraft as a Stackelberg game.Stackelberg game is a special type of full-information extensive game where players are classified as leaders and followers [28].In our proposed mechanism,we consider the abnormal aircraft as the leader who acts first and the mega-constellation as the followers.On the one hand,after the megaconstellations determine their actions,the abnormal aircraft will choose its own optimal response strategy for the actions.On the other hand,the abnormal aircraft also know that the mega-constellations will choose the best reaction according to their strategy.We use a Backward Induction to solve the game.First,calculate the optimal resource sharing amount for a given request on the satellite side.Subsequently,taking the optimal response function of the satellite side into the utility function Eq.(2),the abnormal aircraft finds the optimal request amount that maximizes the utility function Eq.(2).When the game converges,both abnormal aircraft and mega-constellations will get their optimal strategy.Also,we prove that the NE of the game exists and the closed-form optimal solution we give is the unique SE of the game.

3.1 Analysis of the Helper-Side Game

Each satellite individually sets its optimal resource sharing amount to maximize its utility function by considering the satisfaction of abnormal aircraft and its own reliability.The optimization problem for each satelliteiis given by

where,the constraint(3b)indicates that the resources provided by satellites should be no more than the abnormal aircraft actual need.

Subsequently,we have the following theorem to determine the optimal solution in Eq.(3)using the optimal solution.

Theorem 1.The optimal solutions*=for Eq.(3)is determined as follows:

Proof.The Lagrange function of the optimization problem determined by Eq.(3)is

In Eq.(5),µi,υi ∈R+is Lagrange multiplier.Note that the optimization problem is convex and satisfies the Slater condition.Thus,there exists an optimal solutionthat satisfies the following Karush-Kuhn-Tucker(KKT)conditions:

Following the preceding discussion,the optimal solution for each satelliteiis shown in Eq.(4).

3.2 Analysis of the Requester-Side Game

The preceding discussion gives the solution on how much TT&C resource a satellite shall share with the abnormal aircraft to maximize its utility function.Now,we provide a resource request policy for abnormal aircrafts to set the TT&C data request vector over mega-constellation.The optimization problem for abnormal aircraft to find the optimal TT&C data request vector is given by

where the constraint(10b)indicates that the abnormal aircraft can not request more resources than the boundaries determined by the reliability of each satellite.

It is obvious that the optimization problem determined by Eq.(10) is also convex.Subsequently,we have the following theorem to determine the optimal solution in Eq.(10) using the optimal solution describe in Eq.(4).

Theorem 2.The optimal solutionr*=which is a Nash equilibriumsolution of the Stackelberg Game for Eq.(2),is determined as follows:

where W(x)2denotes the principle branch of the Lambert function[29].

Proof.WhenWe can find that thei-th term of the utility functionalways holds.At this time

Subsequently,consider the case whereThen

whereN=BiN0(4πfd/c)2/GtGr.It is important to notice that by differentiatingwith respect toRi,we can obtain

This indicates that at the optimal solution,the helping rate of mega-constellations decreases as the abnormal aircraft sets a higher requesting rate.If the abnormal aircraft wishes to obtain the maximized utility,it has no incentive to overstate its request level.Note that the optimal solutionS*iis upper bounded byRi.Thus,the abnormal aircraft shall understate its requestRitill the best response from mega-constellation meets the requested amount,which is equivalent to

Substituting Eq.(12)into Eq.(13),we have

Letz=＞0.The equation can be represented as

whereeis the natural constant andW-1denotes the inverse function ofW.Solve Eq.(15)with respect to.Then,we can deduce

3.3 Mechanism Design with Imperfect Discrimination

In the previous derivation,a strategy for playeriis a functiongi(θi),where for each urgencyθiin Θi,gi(θi)specifies the action from the feasible setAi.However,not all aircraft are capable of performing fault diagnosis and accurately determining their urgency.In this section,we proposed the solution to the resource allocation problem in the case of imperfect discrimination.

Let Θidenote the set of possible states for aircraftkand Θ:=Πi∈KΘi.Consider that aircraft and satellites have common prior knowledge:ρ ∈Δ(Θ×Ω),where Ω is the utility-relevant states of the system,and Θ is the set of all possible states of the aircraft.In an emergency,aircraft and satellites only know the distribution of aircraft’s urgencyθrather than the exact value ofθ,whereDefineas the action profiles andas the utility functions,where,for eachi,ui: Ω×Θ×A →R.

1)Problem Reformulation for Followers:Now,each satellite only has beliefs about the abnormal aircraft’s urgency and should anticipate that the amount of resource requests for the abnormal aircraftr*(θi),depending on the abnormal aircraft’s urgency.Hence,each satelliteichooseto solve

so as to maximize its expected profit.Wherep(θ)represents the satisfaction andH(Si(θ)represents the remaining reliability.

Corollary 1.When the constraint(16b)is inactive,the best response for satellite i is determined as follows:

Proof.Please refer to Appendix A.

2)Problem Reformulation for Leaders: Similarly,given the beliefs about the abnormal aircraft’s urgency and the utility-relevant states of the system,i.e.,ρ ∈Δ(Θ×Ω),the optimization problem for aircraft is reformulated as

Similar to Theorem 2,we have the following corollary to determine the optimal solution in Eq.(18)using the optimal solution

describe in Eq.(17).

Corollary 2.When the constraint(18b)is inactive,a Nash equilibrium solution of the proposed game can be derived as follows,

Proof.Please refer to Appendix B.

3.4 Stackelberg Game Equilibrium Analysis

In this section,we prove that the SE exists and is unique,ensuring that our proposed solution is stable.First,we define NE as follows.

Definition 3.A profileis a pure strategy NE point of a game if and only if no player can improve its utility by deviating unilaterally [30],i.e.,

WhereAibe the strategy set of the playerianda-ibe the strategy of the other players.Subsequently,we have the following theorem.

Theorem 3.The pure strategy NE point(r*,s*(r*))of the proposed game exists and is unique.

Proof.LetAibe the strategy set of the satellitei(helper-side) anda-ibe the given strategy of the abnormal aircraft.Considering that the bounded closed sets and compact sets are equivalent in Rk,the strategy setAiis a nonempty,convex,and compact set of Rk.Moreover,Ui:Ai →R is continuous,andUi(a-i,.)is quasi-concave onAifor eacha-i.Then,for each satellitei ∈J,the best reply is an upper hemicontinuous,nonempty-valued,closed-valued,and convexvalued correspondence.So does the requester-side.According to the Kakutani’s Fixed Point Theorem introduced in[31],the NE of the proposed game exists.Furthermore,since the solution in Eq.(4)uniquely exists for∀i ∈Jwhen given all non-decision variables,the solution of the abnormal aircraft can be uniquely determined according to Eq.(11),either.Therefore,the NE is unique for the proposed game.

Remark 3.We get a unique NE for the proposed game except the point whenNext,we will discuss this point.Supposethenwill certainly hold because of the problem optimality. Apparently,if the i-th satellite has decided not to share its TT&C resources with aircraft,the abnormal aircraft will not make resource requests to the i-th satellite,either.

3.5 Complexity Analysis

It is meaningful to analyze the time complexity of the proposed mechanism for resource sharing since this study is an emergency plan in emergency scenarios.Considering that there areNsatellites in the CoTT&C network that have decided to share their TT&C resources with aircraft,the computational complexity of the proposed architecture relying on the Stackelberg game isO(N),while the computational complexity of centralized optimization iskO(N3),wherekmeans the number of iterations.Based on the above analysis,the proposed mechanism has low computational complexity,which is important for emergency applications.

IV.PERFORMANCE EVALUATION

In this section,we present our detailed numerical results for evaluating the proposed collaborative TT&C network architecture and analyze the effect of different parameters on the utility of the abnormal aircraft and satellites.The parameters are set as follows:|J|=15,κi=0.1,ηi=1 fori=1,2,...,|J|,ζi ∈{0.03,0.05,0.08},Θ∈{0.5,1.0,1.5}.The link propagation loss and the antenna gain are set the same as that in Ref.[14].First,we consider the case of perfect discrimination.

In Figures 4 to 7,we demonstrate the utility of abnormal aircraft and satellites concerning the different request strategies for abnormal aircraft.When the amount of resource requests is low,satellites tend to provide sufficient support,while the abnormal aircraft’s utility is still very low even if it obtains the amount of resources requested.With the increase in the amount of resource requests,the abnormal aircraft’s utility will increase gradually,while the reliability of the satellite will decrease,resulting in a gradual decline in the utility of the satellite.After the request reaches a certain level,there is no way for the satellite to slow down the decline of its utility by sharing more resources.In the sense of the abnormal aircraft’s utility,there is a maximum number of points beyond which the abnormal aircraft cannot increase its own utility by increasing the amount of resource requests.When resource requests are high,the willingness of satellites to share resources approaches 0.This follows that the abnormal aircraft’s utility also drops to 0.Accordingly,we confirm that there is an optimal resource request corresponding to the optimal amount of resource sharing,i.e.,the NE of the proposed game as proved by Theorem 3.

In Figures 4 and 5,we demonstrate the impact of satellite reliability in Definition 1 on the utility of both the abnormal aircraft and satellites.Under smallerζi,the utility of satellites is maximized at a higher amount of resource sharing.That is,if the satellite has more TT&C resources margin,it can maximize its utility by sharing more resources.In Figures 6 and 7,we demonstrate the impact of aircraft urgency on the utility of both the abnormal aircraft and satellites.Specifically,as the urgency increases,the abnormal aircraft has higher utility.The reason is that the satellite could accept a higher amount of resource requests,that is,it has incentive to share more resources under the same resource request.Correspondingly,the linear relationship between the amount of resource sharing and the satisfaction of abnormal aircraft also results in an increase in the utility of satellites.

To evaluate the proposed hierarchical request algorithm(HRQA),it is compared with the random selection algorithm (RSA),in which the abnormal aircraft randomly selects one resource request within the upper bound of the satellite resource sharing amount at each time.In the first experiment,we consider a simple scenario including 5 satellites as an example and give the resource allocation results of two algorithms in different abnormal aircraft urgency and satellite reliability requirements.To ensure a fair comparison among the algorithms,we assume that RSA selects half of the upper bound as the amount of resource requests for each satellite.Additionally,for simplicity without loss of generality,we normalize the satellite reliability requirement.In Figure 8,we demonstrate that our proposed method has the capability to dynamically adapt the amount of resource requests based on the satellite reliability requirements,maximizing the resources available to the abnormal aircraft.Specifically,for Satellites 1 to 3,due to relatively lower reliability requirements,our proposed method selects higher resource requests.In this context,satellites tend to provide sufficient support.Accordingly,although the baseline algorithm also obtains the amount of resources requested,a noticeable performance gap is observed compared to our proposed algorithm.For Satellites 4 and 5,our proposed method chooses relatively lower resource requests due to higher reliability requirements.Actually,in the case of Satellites 4 and 5,the satisfaction obtained from resource sharing is very small when resource requests are high,while the risk it bears is very high.Consequently,compared to the baseline algorithm,the proposed method obtains more resources even if it requests less.Different from the results of a certain experiment given in the previous experiment,in the following experiment,we employ the Monte Carlo method to calculate the expected utility of the requester-side.Figure 9 shows the performance comparison of user’s utility for different solutions in different influencing factors,where Figure 9a sets the same value for the reliability requirements across all satellites but considers the different reliability requirements of the satellites;Figure 9b considers a scenario where the reliability distribution of satellites is non-uniform,that is,each satellite has varying reliability requirements with an average value of 5Figure 9a indicates that the proposed HRQA algorithm yields higher utility of the abnormal aircraft than the RSA algorithm,especially for the relatively higher satellite reliability requirements.Figure 9b indicates that even if the non-uniform reliability brings opportunities to RSA so that the resource request of RSA is more likely to meet the reliability requirements of the satellites,there is still a certain performance gap from HRQA.The reason is that the proposed HRQA algorithm can converge to a desirable solution,whereas the RSA algorithm is an instinctive approach.

Figure 8.The comparison of resource allocation results between the proposed scheme and baseline algorithm.

Figure 9.Performance comparison of the utility of requester-side for different solutions.

Next,let us suppose that thei-th satellite has a valuationwhereθiis the actual urgency)for the urgency of the aircraft malfunction -that is,thei-th satellite shares the TT&C resource amountand gets the satisfactionp(θi),then there exists a gap betweenandUi(Ri,Si,θi).We consider using the functionto represent the utility loss,which can adjust utility function values on different scales to a common scale for analysis.Figure 10 shows the utility loss of the satellite and the probability density function ofθunder differentλandk.When the actual urgency is lower than the valuation,the reason satellite utility has declined is that the satellite cannot get the desired benefit from the airlines,which is realistic,but at the same time its risk cost will increase rapidly with the increase in the amount of resources provided,as the satellite itself is at risk of failure at any time.In other words,the satellite will realize in hindsight that there is no need to give so many resources.Correspondingly,when the actual urgency is higher than the valuation,the satellite will regret not giving more resources in hindsight.Overall,resource sharing is always profitable,especially for smallerλandk.Nevertheless,only if the exact fault diagnosis is realized the satellite achieves an efficient resource allocation.Beyond that,all satellites have a performance penalty that increases with belief bias.

Figure 10.Utility loss of the satellite at different urgency.

V.CONCLUSION

In this paper,we propose CoTT&C,a megaconstellations-based collaborative TT&C architecture in an emergency scenario for aircraft,where each satellite can flexibly contribute its TT&C resources to help the abnormal aircraft in the network when needed.This solution can bring the actual improvement in flight safety.Specially,we focus on the resource sharing mechanism that determines how much resource helpers should offer to the requester and what level of demand the requester should send to the helpers in order to reach sufficient support.To model a hierarchical decision-making structure in the mechanism,we build a single-leader-multi-follower Stackelberg game between mega-constellations and abnormal aircraft.The evaluation results demonstrate that CoTT&C can effectively support abnormal aircraft to transmit operational status data back to the OCC in an emergency and the proposed resource sharing mechanism can meet the design goal.In light of the revelation that more information can benefit satellites,fault diagnosis of on-orbit spacecraft will be further studied in the future.

ACKNOWLEDGMENT

This work was supported by the National Natural Science Foundation of China under Grant 62131012/61971261.

NOTES

1In CoTT&C,the data of the abnormal aircraft are first transmitted to the overhead satellite and then relayed from other satellites to the ground station along the ISLs among LEO,MEO,and GEO satellites.What we are interested in is the TT&C resources pre-allocated to overhead satellites.The more TT&C resources the overhead satellites share,the more data the abnormal aircraft can transmit back to the ground.

2W(x)denotes the branch satisfyingW(x)eW(x)=xforW(x)≥-1.

APPENDIX

A Proof for Corollary 1

We first derive the distribution of random variableθ.According to the definition,Θ=-λlnk(1-Z)andZ～U[0,1].We apply the cumulative distribution function principle and obtain

Then,we derive the integral term in Eq.(16).Definewe have

B Proof for Corollary 2

By taking derivative with respect toRionwe can obtainas well,i.e.,the aircraft reaches the maximum utility whenSubstitutinginto Eq.(12),the optimalcan be given by Eq.(14).