Price-Based Information Routing in Complex Satellite

Price-Based Information Routing in Complex SatelliteNetworks for Space-Based Situational Awareness

Yue Wang∗, Jingkai Su†, Islam I. Hussein∗, A. M. Wyglinski†

Abstract

Future space-based situational awareness and space surveillance systems are envisioned to include alarge array of satellites that seek to cooperatively achieve full awareness over given space and terrestrialdomains. Given the complexity of the communication networkarchitecture of such a system, in this paperwe propose an efficient, adaptable and scalable price-basedrouting and bandwidth allocation algorithmfor the generation, routing and delivery of information in distributed wireless satellite networks. Due tothe potentially large deployments of these satellites, theaccess points employed in a centralized networkcontrol scheme would easily be overwhelmed due to lack of spectral bandwidth, synchronization issues, andmultiple access coordination. Alternatively, decentralized schemes could facilitate the flow and transferenceof information between data gatherers and data collectors via mechanisms such as (multi-hop) routing,allocation of spectral bandwidths per relaying node, and coordination between adjacent nodes. Althoughthere are numerous techniques and concepts focusing on the network operations, control, and managementof sensor networks, existing solution approaches require the use of information for routing, allocation,and decision-making that may not be readily available to thenetwork nodes in a timely fashion. This isespecially true in the literature on price-based routing, where the approach is almost always game theoreticor relies on optimization techniques. Instead of seeking such techniques, in this paper we present algorithmsthat will (1) be energy-aware, (2) be highly adaptable and responsive to demands and seek delivery ofinformation to desired nodes despite the fact that the source and destination are not globally known, (3) besecure, (4) be efficient in allocating bandwidth, and (5) be decentralized and allow for node autonomy.

1 Introduction

With the increased dependence on the use of satellites for domestic, commercial and national security objec-tives, the United States has the most vested interest in protecting its space assets [1]. Thousands of unidenti-fied objects in space that range from small nuts, bolts, and scraps of paint, and larger spent booster rockets toadversarial satellites may harm such valuable assets [2]. Operational earth-based surveillance systems haveobvious performance limitations that include low resolution capabilities, distortions introduced by the atmo-sphere, and incomplete coverage of space orbits. Space-based space surveillance (SBSS) technologies seekto eliminate these shortcomings by placing observation platforms in orbit. Given the very large number ofobjects in space and the potential limitations on the numberand sensory capabilities of the SBSS satellites,control and decision-making algorithms that seek to managethe available SBSS system resources need to bedeveloped.

In large-scale sensor network applications, individual decision-making actors often rely on data that is notimmediately available to them through onboard sensors. Other sensing nodes with access to the desired datacollect and communicate the data to the target node, which, in turn, makes decisions based on the receivedas well as other data collected using onboard sensors. Centralized network control schemes can easily beoverwhelmed if used to manage such a network due to lack of spectral bandwidth, the large size of theamount of routing information to be processed, synchronization issues, and multiple access coordination.

∗Mechanical Engineering, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609. E-mail:{yuewang,ihussein}@wpi.edu

†Electrical and Computer Engineering, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609. E-mail:{jksu,alexw}@wpi.edu

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This, then, motivates the current interest in decentralized data management schemes, which can facilitate theflow and transference of information between data gatherersand data collectors.

Existing centralized and decentralized techniques on sensor networks operation, control, and management,require the use of information for routing, allocation, anddecision-making that may not be readily availableto the nodes in a timely fashion. There are a few challenges with these techniques. Firstly, optimality,especially when seeking an overall socially optimal solution, requires global information not available to thenode at the decision time. In particular, a relay node may notknow either who the source and the destinationare, nor the absolute significance of the transmitted data tothe operation of the entire network. Secondly,in most optimization problems one ends up with necessary butnot sufficient optimality conditions. Solvingthe necessary optimality conditions does not necessitate solving for the optimal solution. Thirdly, even ifone obtains sufficient optimality conditions, solving these conditions in real-time is often time consumingand computationally burdensome (given the limited onboardcomputing powers of the node), which mayend in creating delayed and suboptimal resource allocationdecisions.Given these realities, this paper seeksto develop secure, adaptable, highly responsive, autonomous space sensor network mechanisms, where theindividual satellites seek, using computationally simple, scalable and local-information based algorithms, tocoordinate sensing and decision-making network capabilities even in a non-cooperative network environmentover a given range of orbits.

2 Background

Routing mechanisms in ad hoc networks can generally be divided into six main categories: proactive routing[3], reactive routing [4], hierarchical routing [5], Quality-of-Service (QoS) routing [6], position-based routing[7], and price-based routing [8]. Routing protocols in sensor networks can generally be divided into fourcategories: flat routing protocols [9], hierarchical routing protocols [10], position based routing protocols[11] and QoS routing protocols [12].

The literature on space-based situational awareness and dynamic sensor coverage control is quite large,but is almost entirely focused on terrestrial and aerial applications. For ground-based surveillance of space,in [13] the authors perform a cost/benefit analysis of the Earth-based Space Surveillance Network (SSN) andpropose changes to the SSN’s observation approach to improve the performance of the network in estimatingthe set of orbital elements of about 6,000 (by 1992 counts) man-made objects in orbit. Task allocation forthe SSN was studied in [14] using the method of simulated annealing to optimize the weighting on allocationpriorities in a greedy algorithm.

Currently, the SSN also includes the Space Based Visible (SBV) sensor on the Midcourse Space Experi-ment (MSX) satellite [15]. A future constellation of satellites will replace MSX/SBV under the Air Force’snew SBSS acquisition program. Due to this increase in dynamic resources and the ever increasing number ofman-made objects in space, in [16] the author develops a resource allocation algorithm for the tracking andcataloguing of critical objects within the field of view of anSSN surveillance satellite. The algorithm is basedon a marginal analysis [17] but extended to the case where multiple satellites are available to the system.

3 Price-Based Information Management Mechanism

The basic premise of this research, as in an ideal free-market economy, is that prices can be used as a signalto relay demand for specific data [18, 19, 20]. In economics, the demand for, say, a pencil is communicatedvia a price signal (that is discovered by “entrepreneurs” [20]) that consumers are willing to pay. This givesrise to pencil manufacturers who emerge to cater to that demand. They, in turn, create a price signal toother manufacturers capable of making the components that go into making pencils (e.g., wood, rubber, lead,plastic, etc). And the process repeats until it reaches the basic producers of the raw materials that find theirway into a final pencil product. Those producers of the raw materials possibly have no idea for what theiroutput is being used. All they know is that there is a demand and that that demand is detectable via theprice signal. In economics, this emergence of an economic structure for pencils came about based on pricing

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information and “entrepreneurial” activities [19]. This market process [20] is also known as theeconomiccoordination problem∗. In theory, such economic structures emerge under no central planning authorities.

This paper aims at discovering similar mechanisms that result in emergent information flow structures insensor networks for space situational awareness problem, where the satellites requesting certain awarenessdata are viewed as consumers, the data as the products, the data gathering satellites as the producers ofraw material, and intermediate relay satellites as “entrepreneurs” who receive “higher order goods” (i.e.,unprocessed data) as inputs and produce “lower order goods”(e.g., processed data after data fusion) asoutputs. All satellites are engaged in an “entrepreneurial” activity, where each satellite seeks to discoverdemand for data and, based on the “profit” to be generated if data are relayed or produced, also seeks todeliver that information.

The challenges described in Section 1 for existing methods will be overcome if one is able to adapt anetwork operation model based on the emergent economic structures viewpoint described in the openingparagraph of this section. It is postulated that such a viewpoint, if adopted for the problem of data gatheringand flow control in sensor networks, should be able to overcome most or all of these challenges.

The main considerations in the development of such data management mechanisms include (1) greencommunications, (2) high adaptability and responsivenessto demands, (3) network security, (4) bandwidthallocation, and (5) decentralized decision-making.

4 General Problem Setup

Refer to Figure 1. Consider a set of statesxj , j = 1, . . . , n, that are distributed over a given domain (repre-sented by the blue dots in Figure 1). The statesxj could be discrete or continuous random variables and canbe estimated, depending on the problem, through an application of Bayes’ rule or a filter such as the Kalmanfilter. A state may represent the traffic flow rate on streetj in a major city in urban traffic management prob-lems, the weather conditions at a waypoint in a next generation decentralized air traffic management system,or the discrete state that a victim is present or not at a location in search and rescue missions. Sensors in thenetwork have access to the states with varying degrees of measurement qualities or none at all. The qualityof the state estimate depends on the relative location of sensor i to the individual states and the onboardsensor characteristics. For the sake of the preliminary simulations presented in the Section 6, the statesxj

are assumed to be independent. In this paper this assumptionwill be relaxed, implying that the states areinterdependent through some dynamical process or a Bayesian network [25, 26].

We consider a network ofN sensor nodes, where the ground-based sensors are stationary with respect toan earth fixed frame. While the satellites follow Keplerian motion with the dynamics in polar form given by

ri =

√

µ

ai(1 − ei)eisin(θi − ωi), (1)

θi =

√

µ

ai(1 − ei)

1 + eicos(θi − ωi)

ri

.

whereri andθi are the polar coordinates centered at the earth for satellite i, µ is the gravitational parameterand equals to398600km3/s2, ai is the semi-major axis,ei is the eccentricity, andθi − ωi gives the trueanomaly. Two nodes are connected if they are within communication range of one another. The wirelesscommunication signal is assumed to attenuate inversely proportionally to the distance between the nodes andis also assumed to be bidirectional (that is, the graph formed by the nodes from this communication structureis undirected). The neighbors of a nodei are all nodes within the communication range ofi. The set ofneighbors of nodei will be denoted byN i. At time t, the communication graph is denoted byG(t) andis defined by the node setV = {1, . . . , N} and the edge setE =

{

(i, j) : i ∈ V , j ∈ N i}

. The graph isdynamic since the edge set dynamically varies as sensors move. Bidirectionality implies that the adjacencymatrixAnet ∈ R

N×N is such thatanetij = anet

ji 6= 0 if (i, j) ∈ E and zero otherwise. Since the price-based

∗The rich history of spontaneous economic order is reviewed in [21]. More technical details can be found in [18, 22, 19, 23,20, 24]on the economic coordination problem.

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Magenta Dots: Sensor Nodes

Gray Area: Node

Transmission Range

Blue Dots:

State x locationj

Data

Requester

Requested State

Location

Node with

Access to

Requested State

Red Paths: Incomplete Paths

Yellow Paths: Candidate PathsGreen Path: Efficient Path

Through which Data Mostly Flows

Green Arrows:

Flow Direction of

Pricing Information

Blue Arrow:

Flow Direction of

Data from Data-Gatherer

to Requester

Figure 1: Schematic of network and basic operation of price-based mechanism. Green arrows indicate theflow of pricing information that trigger bandwidth allocations for the desired data and the blue line indicatesthe flow of data from the source to destination.

approach in this paper seeks resource allocation over heterogeneous nodes, the bidirectionality assumptionwill be relaxed.

For each nodei, there are two parameters that can be adjusted in this work: the pricepij offered by node

i to gain access to the estimate of statej, and the control vectorui. The price control vectorpi is usedto control data flow through the node, to discover the route from source to destination and vice versa, andallocate bandwidth resources. The motion control variableui will be a feedback function of the sensor’sutility U i (see below), and is used to move the sensor in order to improvea node’s measurement and estimateof a demanded state variable, maintain connectivity, and/or facilitate efficient data relay across the network.

5 General Solution Approach

5.1 Subjective Utility and Pricing.

At any given point in timet, a nodei assigns a subjective valuesij(t) ≥ 0 to statej. A strictly positive value

of sij is a realization of nodei’s request for an estimate of statexj . It reflects the value earned by nodei if it

gains access to an estimate of statexj . The revenuerij is defined as the total income received by nodei if it

would relay an estimate for statej to its neighborsN idj who demand statej:

rij =

∑

k∈N idj

pkj . (2)

Let cij be the total cost incurred on nodei if it were to relayxj . The costci

j has four components: (1)communication costci

cj , (2) processing costcipj , (3) measurement costci

mj , and (4) running costcir. Hence,

we havecij = ci

cj + cipj + ci

mj + cir. Depending on whether the node takes measurements, relays,or processes

data or not, a node may not necessarily be incurred all four types of costs. The running costcir is an energy

cost and depends on whether the node is in active or sleep mode. The decisions to measure, relay, process

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data and where to move the sensor depend on the net subjectiveutility given by:

U ij = si

j + rij − pi

j − cij . (3)

5.2 A Secure Network Architecture and Information Flow Scheme.

When a new demand for an estimate of a statexj with a desired signal quality measureσij arises at node

i, the node changes the subjective valuesij corresponding to this state fromsi

j = 0 to somesij > 0. The

exact value ofsij will depend on several factors, chief of which are the desired quality of the requested state

estimate, its bit rate and bandwidth. It may be assumed that the node will seek to decrease the price offered.However, this action is not in the interest of the node since alower price implies that it is less probable thatthe requested data be delivered since intermediary relay nodes may prefer to assign more bandwidth to otherrequests. A pricing mechanism will adjust the price to higher values if no delivery is made and lowers it iftoo many relay/source nodes are offering the requested information.

A nodei assigns a pricepij to stimulate neighboring nodes with access to the desired information to relay

their information. This choice ofpij will be dynamically adjusted over time depending on the actual quality

and bit rate of the delivered data. As a simple example, it could be made to adjust according to the first orderdynamics:

pij(t) = α1

(

bij(t) − bi

dj(t))

+ α2

(

σidj(t) − σi

j(t))

, (4)

where0 ≤ bij(t) ≤ 1 is the normalized bandwidth allocated by nodei for data flow for statej, bi

dj isthe actual bandwidth of the data delivered by nodei, andσi

dj is the actual quality of the delivered stateestimate measured by entropy. The entropy function of each state for every node is derived from the posteriorprobability of object presence through Bayes rule. The parametersα1 andα2 are constant gain parametersset by the designer. As can be seen, whenever the difference between the actual quality and/or data rate arehigher than the desired values, the price decreases. Otherwise, the price increases to encourage relay nodesto allocate more bandwidth and sensing nodes to increase thequality of their measurements.

The above outlines the general pricing scheme. The scheme can be split into two stages: (1) price propa-gation (green arrows in Figure 1) and (2) information propagation and delivery (blue arrows in Figure 1). Inthe first stage, prices are propagated across the network from the source to the destination node that has accessto the requested information. Once the destination node receives the price signal that the data it has access tois in demand, and if the offered price warrants an exchange (of, depending on the implementation, virtual orreal money for data†), the data is transmitted to its neighboring node that made the request. This triggers thesecond stage. The data is then sequentially relayed until the data is delivered to the original requester. Oncethe link is established between source and destination, prices are continuously being adjusted until the data isdelivered to the requester with the desired quality.

Note that a node only releases its prices to indicate interest or lack thereof in the various states that couldbe relayed. Hence, the above algorithm is secure since no node releases any information identifying it or itsposition.

5.3 Bandwidth Allocation.

Bandwidth relates to the transmitted data rate. The higher the transmission bandwidth, the higher the com-munication quality is in terms of the overall throughput (which potentially could be very high for applicationswith digital video signals). Let nodek transmit signalxj at anactual bandwidth ofbk

aj . However, the nodewill allocate anominal bandwidthbk

j based on the utilityUkj . The actual bandwidth it operates at can then

be chosen to bebkaj = min

{

bkj , bk

dj

}

, wherebkdj is the bandwidth of the delivered signal (assuming a single

version of the same signal is delivered). In other words, it can either down-sample the incoming signal (in

†In a cooperative network with no cheating nodes, virtual money is used as a tool to convey price signals and no actual financialtransactions are necessarily carried out.

5

the casebkdj > bk

j ) or keep it as it is (in the casebkdj ≤ bk

j ). It then transmits the data to a demanding neigh-boring nodei at a bandwidthbk

aj . Nodei will perform a similar computation until the data-requesting node isreached. During this process, all relay, data gathering andrequesting nodes keep adjusting their prices untilthe allocated and actual bandwidths match for every node across the network. This defines an equilibriumstate. Note that this equilibrium is unknown beforehand since solving for it requires information from othernodes involved in the complex transactions, which no node has access to since interactions are only local.Hence, any mechanism that assumes equilibrium or near-equilibrium conditions is to some extent futile. Thefutility of such analyses is increased in light of the fact that demand for data in real-world situations is usuallyhighly dynamic and every new request that is made or canceled, a new (and unknown) equilibrium point iscreated.

The algorithms sought in this paper will not require any pre-knowledge of equilibria, and will be shownto tend to converge to some equilibrium state when all data demands are held fixed. Here we give a possiblesuch mechanism that is inspired by a model from evolutionarybiology and population dynamics. Assume thatthe maximum transmission bandwidth across the network is limited to a normalizedωmax = 1. Hence, anyallocation algorithm must guarantee the constraint

∑n

j=1bij(t) = 1 for all i and for allt. The assignment of

bandwidth for each state transmission should also depend onthe relative utility gained from the transmissionof each state. One particular algorithm that can meet these two desired properties is one where the bandwidthallocation is made to satisfy a replicator-mutator dynamicmodel (see for example [27, 28]):

bij(t) =

n∑

k=1

bik(t)f i

k(t)qikj(t) − φi(t)bi

j(t), i = 1, . . . , N, j = 1, . . . , n, (5)

wheref ik is the “fitness” (as evaluated by nodei) of allocating bandwidth to the transmission of statexk,

[

qikj

]

is, for a fixed nodei, a row stochastic matrix that represents the rate at which nodei is willing to assign

more or less bandwidth allocated to statexk for an allocation for statexj , andφi =∑n

j=1f i

jbij is the average

fitness as assessed by nodei. The above dynamic model will guarantee that∑n

j=1bij(t) = 1 for all i andt.

The “fitness” variablesf ij and the mutation matricesQi =

[

qikj

]

can assume different forms. For the sake

of the preliminary results we present below, we will assume that the fitness variables depend linearly on thecurrent allocations [29, 28]:

f ik = ai

kjbij , (6)

where the matrixAi =[

aikj

]

is the subjective reward matrix associated with nodei. The elementaikj

represents the reward earned in switching a bandwidth allocation from the transmission of statexj to statexk.The quantityai

kj can be regarded as the gain in utility from transmission of statexj to statexk. Therefore,if there is some reward gained by transmitting from statexj to xk, we assignai

kj as some positive valueamax > 1. If there is no reward after the transmission, we assignai

kj to 1. However, if there is loss bytransmitting from statexj to xk, ai

kj is set to0. That is, we have

aikj =

amax > 1 U ik > U i

j

1 U ik = U i

j

0 U ik < U i

j

.

The mutation matrix could be any one from a wide range of possibilities, though here we will use the samemodel as that used in [29, 28]. We letQi = I− µi(I−Wi), whereµi is themutation parameter associated

with node i andWi =(

Di)−1

Ai, with Di = diag(di1, . . . , d

in) and wheredi

k =∑n

j=1ai

kj . The mutationparameterµi is a parameter that describes how easily the node in questiontends to change its allocationbetween the various state transmissions. Higher values ofµi result in fast responses and high adaptiveness ofthe nodei to changes in the utilities, but that comes at the cost of highfluctuations in the allocations. Lowervalues ofµi result in a more sluggish response to changes in utilities [29, 28].

The above model will conserve the property that∑n

j=1bij(t) = 1. Based on the relative utilities, it is

also very appropriate for deciding on how to allocate bandwidth in a very responsive and adaptable fash-

6

ion. Note that each node implements the above decision-making evolutionary model in an autonomous anddecentralized behavior. To compute the allocations, a nodeonly needs pricing information from its neighbors.

5.4 Bayes Update and Entropy

In this work, we use a sensor model with Bernoulli distribution, which gives binary outputs: object “present”or “absent” for each state (cell). The probability mass function f of a Bernoulli distribution is given by

fY (y; β) =

β if y = 1,1 − β if y = 0,0 otherwise.

(7)

whereY is a binary random variable of the possible observation results, with 1 indicating a positive obser-vation, and0 a negative one. The quantityy is the actual realization of the random variableY andβ is thedetection probability of the sensor.

Let X be a binary random variable for the real state in each cell, whereX = 1 corresponds to objectpresent, andX = 0 corresponds to no object.

The actual observationY is taken according to the probability parameterβ of the Bernoulli distribution.Since there are two states in all, two Bernoulli distributions are used and the conditional probability matrix isgiven by

B =

[

Prob[Y = 0|X = 0] Prob[Y = 0|X = 1]Prob[Y = 1|X = 0] Prob[Y = 1|X = 1]

]

(8)

where Prob[Y = i|X = j], i, j = 0, 1 describes the probability of measuringY = i given stateX = j. Forthe sake of simplicity, we can assume that the sensor probabilities of making a correct measurement are thesame. That is, we have Prob[Y = 0|X = 0] = Prob[Y = 1|X = 1] = β.

We employ Bayes’ theorem to update the probability of objectpresence at each cell. Given an observa-tion Yt = 1 at time stept, Bayes’ theorem gives, for each cell, the posterior probability of object presentP (Xt+1 = 1|Yt = 1):

P (Xt+1 = 1|Yt = 1) = α1P (Yt = 1|Xt = 1)P (Xt = 1) (9)

whereP (Yt = 1|Xt = 1) is the probability of the particular observationYt = 1 being taken given that theobject state type isXt = 1, which is given by the conditional probability matrix (8) and β, P (Xt = 1) is theprior probability ofXt = 1 being correct att, andα1 serves as a normalizing function that ensures that theposterior probabilitiesP (Xt+1 = j|Y = 1), j = 0, 1 sum to one over the state typesXj , j = 0, 1. Note thatthe probability of object absence at a cell is given by

P (Xt = 0|Yt = 1) = 1 − P (Xt = 1|Yt = 1). (10)

For brevity, we letP (t + 1) denoteP (Xt+1 = 1|Yt = 1) andP (t) denoteP (Xt = 1). By the law oftotal probability, we obtain the expression forα1 and the posterior probabilityP (t + 1) for object presencegiven the observation that there is an object present at the cell:

α1 =1

2βP (t) − β − P (t) + 1(11)

and

P (t + 1) =βP (t)

2βP (t) − β − P (t) + 1. (12)

Similarly, one can derive an expression for the normalizingfactor α2 which ensures that the posteriorprobabilitiesP (Xt+1 = j|Y = 0), j = 0, 1 sum to one over the state typesXj , j = 0, 1 and obtain theupdate equation for the probability of object present atq given that the observation isY = 0:

α2 =1

−2βP (t) + β + P (t)(13)

7

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

P (t)

Hs

Figure 2: Information entropy functionH .

P (t + 1) =(1 − β)P (t)

−2βP (t) + β + P (t). (14)

Thus, the object presence probability update equation at a cell regardless of observation is given by

P (t + 1) = yβP (t)

2βP (t) − β − P (t) + 1+ (1 − y)

(1 − β)P (t)

−2βP (t) + β + P (t), (15)

and the probability of object absent is1 − P (t + 1).

We then use an information-based approach to evaluate measurement quality for every cell within thespace. The information entropy function of a probability distribution is used to evaluate the quality. Let theprobability of the occurrence of an event beP , then the information is measured as

I(P ) = logb(1/P ) ≥ 0, (16)

whereb is the base (we useb = e in simulations). If an event is bound to happen, i.e.,P = 0 or P = 1,then no new information can be obtained:I(0) = I(1) = 0. In our case, there are2 distinct state valuesfor the discrete probabilities. Therefore, the probability distribution is given by{P (t), 1 − P (t)}. We definethe weighted average of informationI(P ) as the information entropy distribution for discrete probabilitydistribution at a cell at each time stept:

H(P, t) = −P (t) lnP (t) − (1 − P (t)) ln(1 − P (t)). (17)

WhenP (t) = 1 or 0, there is no uncertainty about object existence or lack thereof and, therefore,H = 0(highest measurement quality). Maximum uncertainty (lowest measurement quality) is whenP (t) = 0.5.Figure (2) shows the information entropy function (17) as a function ofP (t).

6 Application to the Space Situational Awareness Problem

Figure (3) illustrates the schematic of the price-based mechanism as applied to space situational awareness.The figure follows the same color code as Figure 1.

Figure (4) shows the space system architecture used in this simulation. The radii of Earth and the geosta-tionary orbit are indicated by the black circles. There aren = 120 cells (states) in the space between Earthand the GEO orbit shown by the dotted black lines. The4 ground-based sensors are indicated by the yellowdots on Earth. The communication range of each ground-basedsensor is as far as the GEO orbit and hasan angular range of±π/2 radians. Its sensory range is as far as the GEO orbit and has anangular range of±π/6. The trajectory and communication/sensory range of the3 satellites are demonstrated by the magenta,green and cyan ellipses and circles, respectively. We assume that the satellites could always communicatewith the ground-based sensors. We have a total ofN = 7 nodes in the network.

Node1 is the requesting node and transmits price information to its neighboring nodes. Node5 is thesensing satellite that has access to the state requested by node1 and transmits data measurements of the states

8

42164

50000

120

150

180

210

240

270

300

330

0

30

60

90

Figure 3: Schematic of network and basic operation of price-based mechanism with applications on SSAproblems (following the same color code as the general case).

10000

20000

30000

40000

50000

30

210

60

240

90

270

120

300

150

330

180 0

12

3 4

5

6

7

Figure 4: Space system architecture used in the simulation.

9

to its neighboring nodes. The remainder of the nodes are relay nodes. The blue dot on GEO orbit in Figure(4) is cell number90, which is the state that we are interested in. In the simulation, we assume only cell90contains an object.

We set the initial bandwidth allocation to bebkj = 1/n, j = 1, 2, · · · , 120, k = 1, 2, · · · , 7. The

communication costsckcj are proportional to the relative distances between each pair of nodes. All other costs

and all initial values are set to zero. The constants in the bandwidth and price dynamics are:µi = 0 (for alli = 1, . . . , 7), α1 = 1, andα2 = 0.1. Initial subjective values are all zero except fors90

1 = 5000, whichmeans that the requesting node1 is interested in state90 throughout the simulation. The sensor parameterβis set as0.8.

Let Zkj = 1 if an estimate of statej is available to nodek, andZk

j = 0 if not. Hence, each node isequipped with a vectorZk, which is a vector of zeros and ones. At every time step,Zk

j is determined asfollows: If nodek is a neighbor of nodei, and we have bothZk

j = 1 andpij > 0, then we setZi

j = 1 whichmeans nodei also has information about statej through communication with its neighbor nodek.

In parallel with the bandwidth allocation process, each sensor makes measurements about the states withinits sensory range. Bayes rule is used to update the posteriorprobability of object presence in each cell (state).We then use the entropy functionH over the posterior probability as a metric of measurement quality.

Figures (5-8) show the bandwidthbkj , j = 1, · · · , 120, k = 1, · · · , 7, pricepk

j , the vectorZ, and entropyfunctionH for all states of each node (with state90 in red line).

0 20 40 60 80 100 120 1400

0.5bandwidth of sensor 1 for all states

0 20 40 60 80 100 120 1400

0.2

0.4price of sensor 1 for all states

0 20 40 60 80 100 120 1400

0.5

1Z vector of sensor 1 for all states

0 20 40 60 80 100 120 1400

0.5

1entropy of sensor 1 for all states

(a)

0 20 40 60 80 100 120 1400


0 20 40 60 80 100 120 140−0.5

0


0 20 40 60 80 100 120 1400

0.5


0 20 40 60 80 100 120 1400

0.5


(b)

Figure 5: Bandwidth, price, the vectorZ, and entropy functionH of (a) node1 and (b) node2 for all states(with state90 in red line).

Figure 7(a) shows the results for node5. The prices of node5 for all states are always zero because node5is the one that owns the required information and does not need to offer a price for it. When the object is withinnode5’s sensory range, the vectorZ for state90 (shown by red line) jumps to1. Since now node5 obtains theawareness information about state90, the entropy functionH of node5 for this state becomes zero. Figure5(a) shows the results for the requesting node1. The price of this node for state90 keeps increasing becausenode1 is interested in this cell and willing to offer more price to get the awareness information. Figures 5(b),6(a), 6(b), 7(b), and 8 show the behavior of the relay nodes. Note that a node will only measure the stateswithin its sensory range or get the awareness information from its neighbors, hence only part of the states willend up with zero entropy (highest measurement quality). Viathe routing and bandwidth allocation algorithm,these relay nodes deliver the desired awareness information to their neighbors. Therefore, we can see fromthe Figure 5(a) that node1 finally gets the information about cell90 with zero measurement uncertainty.

Figure (9) compares the difference in bandwidthbk90, k = 1, · · · , 7 between all nodes. For the requesting

node1, it can be seen that most of the bandwidth has been assigned tostate90 since∑

j b1j = 1, j =

1, · · · , 120. Moreover, the response of node1 is fastest among all the nodes.

10

0 20 40 60 80 100 120 1400

0.2


0 20 40 60 80 100 120 140−0.5

0


0 20 40 60 80 100 120 1400

0.5


0 20 40 60 80 100 120 1400

0.5


(a)

0 20 40 60 80 100 120 1400

0.2


0 20 40 60 80 100 120 140−0.5

0


0 20 40 60 80 100 120 1400

0.5


0 20 40 60 80 100 120 1400

0.5


(b)


0 20 40 60 80 100 120 1400

0.2


0 20 40 60 80 100 120 140−1

0

1price of sensor 5 for all states

0 20 40 60 80 100 120 1400

0.5


0 20 40 60 80 100 120 1400

0.5


(a)

0 20 40 60 80 100 120 1400

0.2


0 20 40 60 80 100 120 1400

0.1


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0.5


0 20 40 60 80 100 120 1400

0.5


(b)


7 Conclusion

In this paper, we developed a decentralized price-based routing and bandwidth allocation algorithm for thegeneration, routing and delivery of space situational awareness information using ground and space sensors.The pricing mechanism is based on both the nominal bandwidthand sensor measurement quality. The band-width allocation algorithm satisfies a replicator-mutatordynamic model from the literature on evolutionarybiology and population dynamics, which does not require anypre-knowledge of equilibria. A Bayesian prob-ability update process is carried out at each cell, and information entropy was used to determine the qualityof the estimates. Simulation results for a space situational network are provided to illustrate the operation ofthe proposed mechanism.

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0 20 40 60 80 100 120 1400

0.2


0 20 40 60 80 100 120 1400

0.2


0 20 40 60 80 100 120 1400

0.5


0 20 40 60 80 100 120 1400

0.5


Figure 8: Bandwidth, price, the vectorZ, and entropy functionH of node7 for all states (with state90 in redline).

0 20 40 60 80 100 120 1400

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