Steady and Fair Rate Allocation for Rechargeable Sensorsin Perpetual Sensor Networks

Kai-Wei Fan Zizhan Zheng Prasun SinhaDepartment of Computer Science and Engineering

The Ohio State University{fank, zhengz, prasun}@cse.ohio-state.edu

ABSTRACTRenewable energy enables sensor networks with the capabil-ity to recharge and provide perpetual data services. Due tolow recharging rates and the dynamics of renewable energysuch as solar and wind power, providing services withoutinterruptions caused by battery runouts is non-trivial. Mostenvironment monitoring applications require data collectionfrom all nodes at a steady rate. The objective of this pa-per is to design a solution for fair and high throughput dataextraction from all nodes in presence of renewable energysources. Specifically, we seek to compute the lexicographi-cally maximum data collection rate for each node, such thatno node will ever run out of energy. We propose a central-ized algorithm and an asynchronous distributed algorithmthat can compute the optimal lexicographic rate assignmentfor all nodes. The centralized algorithm jointly computesthe optimal data collection rate for all nodes along withthe flows on each link, while the distributed algorithm com-putes the optimal rate when the routes are pre-determined.We prove the optimality for both the centralized and thedistributed algorithms, and use a testbed with 155 sensornodes to validate the distributed algorithm.

Categories and Subject DescriptorsC.2.2 [Computer-Communication Networks]: NetworkProtocols; C.2.4 [Computer-Communication Networks]:Distributed Systems

General TermsAlgorithms,Design

KeywordsRechargeable Sensor Networks, Rate Assignment

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.SenSys’08, November 5–7, 2008, Raleigh, North Carolina, USA.Copyright 2008 ACM 978-1-59593-990-6/08/11 ...$5.00.

1. INTRODUCTIONRecent years have witnessed a growing number of proto-

type sensor network deployments for environment monitor-ing such as earthquake monitoring [1], structural monitor-ing [2, 3], soil monitoring [4], and glacial movement moni-toring [5]. Unattended operability for long periods is highlydesirable due to the harshness of the environment, whichis typical in such applications. However, the limitations offixed battery sources often require operation at low datarates, which may even be unacceptable to some applications.

Energy harvesting from natural sources such as solar [6,7], wind [8], thermal [9], and vibration [10, 11], have beenshown to be effective in addressing the above problem. Theamount and source of energy that can be leveraged dependson the environment and the application. For example, in adeployment for monitoring vibration of industrial machines,vibration energy could be readily available. Similarly, for adeployment for monitoring the air-quality in the AC-ducts,thermal energy can be harvested when hot air is blowing.For outdoor deployments, solar energy is readily availablefor a wide range of application scenarios.

Environment monitoring applications often require peri-odic collection of data from all nodes. The objective of thispaper is to design a solution for fair and high throughputdata extraction from all the nodes in the network in pres-ence of renewable energy sources. Specifically, we seek tocompute the lexicographically maximum data collection ratefor each node in the network, such that no node will everrun out of energy. A rate assignment is lexicographicallymaximum if it is impossible to increase the rate of a nodewithout decreasing the rate of another one with a lower rate.

The time varying profile of recharging rates poses chal-lenges in the computation of the optimum data rates. If therate of data collection is high, a node can end up deplet-ing its battery, which can hurt coverage as well as networkconnectivity. If the network gets partitioned due to a noderunning out of energy, the amount of data collected duringthat period also gets impacted. Similarly, if the data col-lection rate is low, the battery may reach the highest levelwhich results in missed recharging opportunities. The de-termination of optimum data rates must take into accountthe profile of recharging, together with the energy cost ofsensing, transmission, and packet reception.

The contributions of the paper are as follows:

• We propose a centralized algorithm to compute theoptimum data collection rate for each node, along withthe amount of flow on each link in the network.

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• We propose an optimum distributed and asynchronousalgorithm assuming that the routing tree is pre-determined.

• We prove the optimality of the centralized and the dis-tributed algorithms.

• Using experimentation with solar panels and a sensornetwork testbed, we evaluate the performance of theproposed approaches under various realistic scenarios.

The organization of the paper is as follows. Section 2presents the centralized algorithm to compute the optimallexicographic rate assignment and corresponding routing paths.Section 3 presents the distributed algorithm. The evaluationof the protocols using experiments with solar energy harvest-ing and experiments on a testbed are presented in Section 4.We discuss some issues not addressed in this paper and ourfuture works in Section 5. Section 6 presents backgroundand related work. Finally, Section 7 concludes the paper.

2. OPTIMAL LEXICOGRAPHIC RATE AS-SIGNMENT

The objective of this paper is to find a fair rate assignmentfor sensors that can best utilize renewable energy sourceswhile providing continuous monitoring services. Consider anetwork with four nodes as shown in Figure 1. Each node isequipped with a solar cell to collect solar energy and storeit in a rechargeable battery. The amount of energy collectedby each node may differ due to their exposure to sunlight.In addition, the size of the solar panels may be different fordifferent nodes. Each node is collecting data and sendingpackets to the sink at a certain rate. If nodes work at lowrates such that their battery will never be depleted, they canprovide continuous service. For example, assuming the fourcurves in Figure 2 are the recharging rates for these four solarcells, Figure 3(a) shows the battery levels of these four nodeswhen they collect readings and send five packets per second.However, the collected energy is not well utilized since wecan increase the rate of each node without losing continuousservice. But if the rate is too high, some nodes may run outof energy before they can be recharged. For example, Figure3(b) shows the battery level of these four nodes when theirworking rates are 10 packets per second. The batteries ofnodes A and C are depleted for 3 to 4 hours. The objectiveis to find a rate assignment for these four nodes that is fair,can best utilize the collected energy, and is able to provideuninterrupted service, as shown in Figure 3(c).

In this section we propose a centralized algorithm to solvethe optimal joint lexicographic rate and flow assignmentproblem given the recharging profiles for all nodes and lim-ited battery capacity. We assume that the link capacity isnot a constraint. For a sensor network to support a contin-uous monitoring service, sensors should not consume moreenergy then they can collect. Therefore, the data rate isconstrained by the amount of energy they can collect. Usu-ally energy collected from a renewable source is too smallto support high data rate applications. Take solar energyas an example. Figure 4 is the measured current collectedfrom a 37mm× 33mm solar cell over a 48 hours period thatincludes a sunny day and a partly cloudy day during theSpring. The total energy collected is 655.15mWh for thesunny day and is 313.70mWh for the partly cloudy day.For a wireless module, such as TelosB from Crossbow [12],

A

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D

SINK

Figure 1: A network with four nodes, each equippedwith a solar cell and a rechargeable battery.

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Figure 2: Recharging profile for nodes in Figure 1

the current drawn in receiving mode is 23mA, in transmit-ting mode is 21mA at 0 dBm, and is 1.8mA for the mi-cro control unit (MCU) in active mode, which can be con-verted to 69mW , 63mW , and 5.4mW , respectively. There-fore, a node can only spend 9.2 hours forwarding packets aday (69mW×t/2 + 63mW×t/2 + 5.4mW×t = 655.15mWh,t = 9.2h), which is about 38% of its time. The energy drawnby the attached sensors can further reduce it. Therefore,sensor nodes can only support low data rates and link ca-pacity is typically not a constraint.

2.1 Problem FormulationWe seek to compute the optimal lexicographic rate assign-

ment which is defined as follows:

Definition 1. Let L1 and L2 be two rate assignments.A rate vector RL is a sorted rate vector of a rate assignmentL if RL is the result of sorting L in non-decreasing order,and RL

i is the ith rate in RL. We say L1 = L2 if RL1 =RL2 , L1 > L2 if there exists an i such that RL1

i > RL2

i

and RL1

j = RL2

j , ∀j < i, and L1 < L2 otherwise. L∗ isan optimal lexicographic rate assignment if there is no otherrate assignment L′ > L∗.

The goal of our paper is to find L∗ such that given thebattery constraints and energy recharging profiles for eachnode, no node ever runs out of energy. First, we define

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(a) ~r = (5, 5, 5, 5) pkt/s (b) ~r = (10, 10, 10, 10) pkt/s (c) ~r = (8.6, 8.6, 6.4, 6.4) pkt/s

Figure 3: Battery levels of four nodes in Figure 1 working at different data collection rates.

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Figure 4: Current measured from a 37mm×33mm so-lar cell over a 48 hours period starting from 6:15pm2/16/2008. The first day is a sunny day and thesecond day is a partly cloudy day.

constant parameters that will be used in our formulation inTable 1, and variables for flows and energy constraints inTable 2.

With the given parameters and variables, we can formu-late the lexicographic rate assignment problem as follows:

Problem: LP-LexObjective: Lexicographically Maximize L = {r1, r2, ..., rn}

subject to

V the set of all nodes in the network.n number of nodes in the network.πi,t amount of energy collected by node i in time slot t.Πi maximum battery capacity for node i.Wi initial battery level for node i.λtx energy cost for per packet transmission.λrx energy cost for per packet reception.λsn energy cost for sensing.Ni node i’s neighbors.

Table 1: Constant parameters used in the formula-tion.

ri data collection rate for node i.fij,t flow from node i to node j in time slot t.wi,t amount of energy available for node i at the begin-

ning of time slot t.xi,t total energy consumption for node i in time slot t.li,t a surplus variable that represents the amount of

energy not being collected when the battery is full.

Table 2: Variables used in formulation.

ri +X

j∈Ni

fji,t =X

j∈Ni

fij,t (1)

xi,t = λsnri + λtx

X

j∈Ni

fij,t + λrx

X

j∈Ni

fji,t (2)

wi,t+1 = wi,t + πi,t − xi,t − li,t (3)

TX

k=1

xi,k ≤T

X

k=1

πi,k (4)

0 ≤ wi,t ≤ Πi (5)

wi,1 = Wi (6)

li,1 ≥ 0 (7)

for all i ∈ V and 1 ≤ t ≤ T where T is the number ofslots after which the recharging pattern is expected to repeatitself.

Constraint 1 ensures that the inflow equals the outflow.Constraint 2 specifies total energy consumption in time slott, which includes the energy consumption in sensing, packettransmissions, and packet receptions. Constraint 3 statesthat the available energy in the next time slot is equal tothe available energy in the current slot plus the energy col-lected minus the energy consumed in the current time slot.Constraint 4 ensures that nodes do not consume more en-ergy than they collect. Constraint 5 states that the availableenergy does not go below zero and does not go above the bat-tery capacity. Constraint 6 states that the available energyin the first time slot is the initial battery level Wi. Con-straint 7 states that the surplus variable should be greaterthan or equal to 0.

2.2 Optimal Lexicographic Rate AssignmentThe optimal lexicographic rate assignment problem can

be solved using a similar approach as proposed in [13]. Theapproach in [13] first finds a maximum common rate r∗ that

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is feasible for all the nodes with given constraints. It thencomputes the maximum feasible rate ri for each node i as-suming that the rates of other nodes are fixed at r∗. Ifri = r∗, the rate of node i cannot be increased any furthereven if all other nodes are assigned with the rate r∗; there-fore, it will be assigned with the rate r∗. We call these nodesthe constrained nodes. The algorithm finds all constrainednodes with ri = r∗, fixes their rates to r∗, and repeats theprocess for the rest of the nodes. Due to the uniquenessof the optimal solution, in each iteration at least one con-strained node exists. Therefore, iteratively the rate of eachnode will be determined and the algorithm converges to theoptimal lexicographic rate.

In this paper we show that in our setting the optimallexicographic rate assignment is also unique, and thereforewe can use a similar approach to solve the problem. Thedifferences between our approach and [13] are as follows.First, in [13], the resources are static and do not change whilein our case the recharging rate changes over time. Second,we use a more general proof to show the uniqueness of theoptimal lexicographic rate assignment that is suitable forboth static as well as dynamic resources, as in our scenario.

To solve this problem iteratively, first we need to find themaximum common rate for the nodes. This can be achievedby modifying the objective and constraints of the formula-tion in problem LP -Lex. Instead of using different ri vari-ables for each node, we use the same rate r for all the nodesand try to maximize r. Therefore, the problem becomes:

Problem: LP-MaxminRateObjective: Maximize r

subject to

r +X

j∈Ni

fji,t =X

j∈Ni

fij,t (8)

xi,t = λsnr + λtx

X

j∈Ni

fij,t + λrx

X

j∈Ni

fji,t (9)

< Constraints 3 to 7 >

Using an LP solver we can compute a maxmin rate r∗

for all nodes. After the maxmin rate r∗ is computed, wecan compute Maximum Single Rate (MSR) for each nodeto determine whose rate should be fixed at r∗ in the opti-mal lexicographic rate assignment. The MSR problem fora node u can be modeled as a linear programming problemand be solved by an LP solver [13]. The formulation is simi-lar to LP -MaxminRate, but for nodes other than u, we usethe following constraints to replace constraints 8 and 9:

r∗ +X

j∈Ni

fji,t =X

j∈Ni

fij,t (10)

xi,t = λsnr∗ + λtx

X

j∈Ni

fij,t + λrx

X

j∈Ni

fji,t (11)

Node u still uses constraints 8 and 9. By solving this LPformulation we can get the MSR rate of node u. We useLP -MSR(V, r∗, u) to represent the MSR rate for node u as-suming all other nodes in V are assigned with the rate r∗. Bysolving the LP -MaxminRate and LP -MSR problems iter-atively, the optimal lexicographic rate can be determined.The iterative algorithm is shown in Algorithm 1.

Algorithm 1 Optimal Lexicographic Rate Assignment

procedure LexRateAssignment()

1: S ← V2: while S 6= {} do3: r∗ ← Solve LP -MaxminRate(S)4: for each i in S do5: ri ← Solve LP -MSR(S, r∗, i)6: if ri = r∗ then7: D ← D ∪ {i}8: end if9: end for

10: S ← S −D11: end while12: return < r1, r2, ..., rn >

We will show that the solution to Algorithm 1 is unique.Due to the uniqueness, in each iteration of the while loopthe set D with node i such that ri = r∗ will be non-empty.Therefore, the while loop (Lines 2-11) will have at most n it-erations. In each loop, we solve one LP -MaxminRate prob-lem and at most n LP -MSR problems. Therefore, the com-plexity of LexRateAssignment is O(|N |2CLP (|N |, |E|, T ))where CLP (n, |E|, T ) is the complexity of solving an LPproblem with O(nT ) constraints and O(|E|T ) variables whereE is the set of edges of the network.

Theorem 1. LexRateAssignment computes the optimumlexicographic rate assignment.

Proof. See Appendix 9.1.

In the proof we first show that the optimal lexicographicrate assignment is unique. Due to the uniqueness of the solu-tion, we can always find some nodes with MSR rate equal tothe maxmin rate r∗. As we cannot increase the rates of thesenodes even if we assign r∗ to all other nodes, the only wayto increase the rate of these constrained nodes is to decreasethe rate of some nodes to be lower than r∗. Therefore, inthe optimal lexicographic rate assignment, these constrainednodes will be assigned with a rate of r∗.

3. DISTRIBUTED LEXICOGRAPHIC RATEASSIGNMENT

In Section 2 we use a centralized algorithm to solve the lex-icographic rate assignment problem using the global knowl-edge of the network. The algorithm involves solving thisproblem using multiple executions of an LP solver, whichis computation intensive and is not suitable for sensor net-works. In this section, we present a distributed algorithmthat does not require an LP solver to solve this problem forthe case when the routes are known. We leave the moregeneral problem of distributed joint routing and rate com-putation in presence of recharging for future work.

There are many studies on finding the maxmin rate in theliterature. Most of them employ the feedback-based flowcontrol mechanism. In particular, Charny et al. [14] usea technique called Consistent Marking to achieve maxminrate assignment using a distributed algorithm. We use asimilar technique to determine the rates of nodes when theirroutes pass through a node, but the way we compute therate is different. We make an assumption as in [14] thateach flow uses a fixed route to forward packets. This makes

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the problem more tractable for distributed computation. Inthis work, each node has exactly one flow connected to thesink, and we use flow i to represent the flow originating fromnode i.

Each node j maintains two variables, rmaxj (i), and rj(i),

for each flow i passing through it, where rmaxj (i) is the cur-

rent maximum achievable data rate and rj(i) is the com-puted data rate for flow i. For each flow i at any node, itmust satisfy rj(i) ≤ rmax

j (i), ∀j. Each node i starts withcomputing its maximum achievable rate rmax

i (i) accordingto its recharging process assuming it only generates and for-wards its own packets. For a non-steady energy resourcesuch as solar energy, the maximum achievable rate dependson the available energy in the battery and the variations inthe recharging rate. In order to support a perpetual sensornetwork, nodes should not consume more energy than theycan collect. Furthermore, to provide continuous monitoringservices, a node should not deplete its battery before it canbe recharged, i.e., the total energy consumption should notbe greater than the summation of the battery level and thetotal energy it collected up to a given time. Algorithm 2computes the maximum rate at which a node can collectreadings and forward them without running out of energyin any time slot.

Line 1 computes the rate in each time slot by computingthe average energy collected per time slot divided by theenergy consumption for collecting and forwarding a reading(es). Lines 3 to 14 consider cases when battery is depletedor full, given the data collection rate, energy recharging rate,and battery capacity. Variable wt is the amount of energyin the battery in the beginning of time slot t, s is the lasttime slot when the battery is full, and E is the summationof the amount of energy collected from time slot s + 1 to tand the available energy in the battery at the end of timeslot s.

Algorithm 2 Maximum Rate

procedure MaximumRate()

1: r←PT

i←1πu,i

T× 1

es

2: E ←Wu, w1 ←Wu, s← 03: for t← 1 to T do4: wt+1 ← wt + πu,t − res

5: E ← E + πu,t

6: if wt+1 > Πu then7: E ← Πu

8: wt+1 ← Πu

9: s← t10: else if wt+1 < 0 then11: r← E

t−s× 1

es

12: wt+1 ← 013: end if14: end for15: return r

First, we consider the case when the battery is full. Ifat time slot t the battery is full, the extra energy collectedcannot be put into the battery. Therefore, E has to beadjusted and wt+1 is set to the maximum battery capacityΠu (Lines 7 and 8). In addition, we know that if the batteryis full at time slot t when working at rate r, the battery willstill be full at time slot t even if the node works at a ratelower than r, and any rate that is smaller than r will still

be feasible from time slot 1 to t. Observing that r will onlybecome smaller each time it is updated in Line 11 (we willshow it later), we can check if the newly computed rate isfeasible by considering only the time slots after t. Therefore,we set s to t.

If at some time slot t the available energy wt+1 becomesnegative, it means that node u cannot support the rate r. Inthis case, we should evenly distribute the energy collectedfrom time slot s+1, plus the energy originally in the batteryat the end of time slot s, to all slots from s + 1 to t. It isclear that the newly computed rate r will be smaller thanthe previous one because Line 4 can be expressed as

wt+1 = ws+1 +t

X

i=s+1

πu,i − res(t− s) = E − res(t− s)

Because wt+1 < 0, E − res(t − s) < 0, E(t−s)es

< r. We

know that as node u can support the original rate r fromtime slot 1 to t− 1, it can also support the newly computedrate, which is lower than r, from time slot 1 to t − 1, andalso in time slot t.

After rmaxi (i) is computed, node i sends a control packet

containing the flow id i and rate ri = rmaxi (i) to its next

hop. Node j receiving the control packet from node i firstassigns the ri from the control packet to rmax

j (i) and thencomputes a new rate rj(i) based on flows passing throughit. Node j then sends a control packet containing the flowid i and the newly computed rate rj(i) to its next-hop node.The process is repeated at each node from leaves to the sink.Once the control packet reaches the sink, the control packetwill contain the maximum rate achievable for node i in theoptimal lexicographic rate assignment, and the sink can senda feedback packet notifying node i of its assigned rate.

To compute the rate for each flow passing through node j,we define two types of flows for node j: restricted flows (Rj)and unrestricted flows (Uj). A flow f is in Rj if rmax

j (f)is smaller than rj(j), i.e., its computed rate is restrictedby some node before it reaches node j; otherwise f is in Uj .Note that for f ∈ Uj , rj(f) = rj(j) in the optimum solution.Given the sets Rj and Uj , node j can compute the assignedrate rj(j) by evenly allocating the remaining rate not usedby flows in Rj to all flows in Uj and node j itself as:

rj(j) =Cj − ef

P

i∈Rjrmax

j (i)

ef (nj − |Rj |) + es

(12)

where Cj = rmaxj (j)es, ef is the cost of forwarding a packet

for other nodes, which includes the cost of receiving andtransmitting a packet, and nj = |Rj | + |Uj | is the totalnumber of flows, excluding flow j, passing through node j.If rmax

j (i) of any flow i in Rj becomes higher than the newrj(j), or rmax

j (i) of any flow i in Uj becomes smaller than thenew rj(j), Rj and Uj are updated accordingly and Equation12 is repeated until Rj and Uj do not change. Algorithm 3shows the pseudocode for the distributed lexicographic rateassignment algorithm for node j.

Theorem 2. The UpdateRate rate computation using Equa-tion 12 converges and computes the optimal lexicographicrate assignment.

Proof. We show the convergence by showing that afterthe first round of computation, the rj(j) will only increaseand this moves at least one flow from Uj to Rj . As the

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Algorithm 3 Distributed Lexicographic Rate Assignment

Require: π1..T : recharging rate from time slot 1 to T

procedure InitializeRate()

1: rmaxj (j)←MaximumRate()

2: rj(j)← rmaxj (j)

// i: received flow id// ri: maximum achievable rate of flow iprocedure UpdateRate(i, ri)

1: rmaxj (i)← ri

2: R′j ← Rj \ {i}

3: U ′j ← Uj ∪ {i}

4: repeat5: Rj ← R′

j

6: Uj ← U ′j

7: Compute rj(j) using Equation 128: R′

j ← {i|rmaxj (i) < rj(j)}

9: U ′j ← {i|r

maxj (i) ≥ rj(j)}

10: until Rj = R′j

11: rj(i)← rmaxj (i), ∀i ∈ Rj

12: rj(i)← rj(j), ∀i ∈ Uj

number of flows is finite, UpdateRate will eventually con-verge. For the optimality of the UpdateRate algorithm, wefirst assume that there is an optimal solution that is betterthan the result computed by the UpdateRate algorithm, andthen prove that this can not happen. See Appendix 9.2 fordetails.

200r

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Figure 5: Distributed lexicographic rate computa-tion for four nodes. The numbers on links representthe order of packet transmissions in this example.They also represent the rows that are transmittedand the instances that trigger the recomputation ofrates of corresponding nodes. The R field is T ifthe corresponding flow is a restricted flow, other-wise it is an unrestricted flow. The items that aremarked grey are data actually transmitted in thepacket. Other fields are for internal reference only.The final solution is (100, 80, 60, 60).

Figure 5 shows an example of the distributed rate com-putation for four nodes in steps. For ease of understandingwe assume es = ef . Nodes first compute their maximumrate rmax using the InitializeRate procedure in Algorithm 3(The first table for each node). The maximum rates for nodeA through D are 300, 80, 120, and 200, respectively. After

the nodes compute rmax, they send a control packet contain-ing the flow id and the maximum achievable rate r = rmax

to their next-hop nodes. Note that the control packets donot need to be transmitted in a synchronous fashion fromleaves to the root. However, transmitting control packetsin this order can reduce the number of control packets, aswhen the rate of a flow is updated, the next-hop node hasto update and compute a new rate accordingly.

When a node receives the control packet from its children,it uses UpdateRate to compute the rates. For example, whennode C receives the rate 200 from node D in the first step,node C sets rmax

C (D) = 200, RD = {} and UD = {D}.Therefore, rC(C) = 120es/(ef + es) = 60. When node Areceives the rate 60 from node C for flow D at step 3, node Asets rmax

A (D) = 60, RA = {B}, and UA = {D}. Therefore,rA(A) = (300es − 80ef )/(ef + es) = 110. Since 110 >rmax

A (D), D will be put into RA, and in the next roundrA(A) = (300es − (80 + 60)ef )/(es) = 160, and RA(D) =Rmax

A (D) = 60.

4. EVALUATIONWe evaluate our distributed algorithm, called DLEX, on

MoteLab [15], a testbed with more than 150 TmoteSky sen-sor motes deployed over a 3-floor building. TmoteSky con-sists of TI MSP430 processor running at 8MHz, has 10KBRAM, and uses CC2420 radio operating at 2.4GHz. Sincethe TmoteSky nodes in the testbed are not equipped withsolar cells, we use the recharging model collected on a sunnyday, as shown in Figure 4, as a baseline, and generate arecharging model in which the whole recharging profile isvaried by a random amount that is −10% to 10% of thebaseline for each sensor. Each sensor node stores the ran-domly generated charging rate for 24 hours, and uses onehour as the length of a slot to compute the rate.

4.1 OptimalityFigure 6 shows the rate assignments computed by an LP

solver and our distributed algorithm for a shortest path rout-ing tree. The X-axis is node id sorted in non-decreasing or-der according to their assigned rates. The two curves overlapand therefore we only see one curve in Figure 6. Figure 7shows the difference between these two rate assignments.The difference between these two rate assignments is lessthan 0.03%. The difference comes from the difference inprecision in arithmetic operation between sensor nodes andthe LP solver running on a PC. The CPU of the sensor nodeshas limited computation power and floating point operationsare extremely slow on them. Therefore, we use integer op-erations to replace the floating point operations, and thisleads to the loss of precision resulting in the difference.

4.2 Recharging ProfileWe evaluate the effect of different recharging profiles on

the rate assignment. Based on the data shown in Figure 4,we extract two different recharging profiles, one for a sunnyday and one for a partly cloudy day.

Figure 8 shows the rate assignments obtained from ourdistributed algorithm using the two different recharging pro-files as shown in Figure 4. We start the rate assignmentprotocol at 12:00am in the midnight and use 24 hours as thelength of the recharging profile. The total energy collectedon the sunny day is 655.15mWh, which is about twice of313.7mWh, the energy collected on the partly cloudy day.

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Figure 6: Rate assignments computed by an LPsolver and our distributed protocol on a short-est path tree (These two curves overlap with eachother).

0.1%

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-0.1% 0 20 40 60 80 100 120 140

(DLE

X-L

P)/L

P

Figure 7: The difference between rate assignmentscomputed by a centralized LP solver and our dis-tributed protocol.

Since in our protocol nodes do not consume more energythan they can collect, the rates assigned to nodes are di-rectly related to the amount of energy that they collect.

Next, we assign nodes with solar profiles obtained from[16] that are collected from six locations in California. Thisexperiment represents the scenario where different sensorsmay have different recharging patterns. The solar radiationis sampled every two seconds. Every five minutes these two-second samples are averaged to obtain five-minute values.Each hour the average, maximum, and minimum values arerecorded. To be consistent, we only use one day, or 24 hours,of data, from these six locations because the weather condi-tion might be different on different days.

For each sensor node, we randomly select one solar profile,and randomly choose a number between the maximum andminimum energy collected from the solar cell as the recharg-ing rate for that sensor. The result is shown in Figure 9.

4.3 Protocol OverheadFigure 10 shows the number of control packets of the dis-

tributed protocol and also the size of subtrees rooted at the

0

2

4

6

8

10

12

14

16

0 20 40 60 80 100 120 140

Rat

e(pk

t/s)

SunnyCloudy

Figure 8: Rate assignments for energy collected ona sunny day and partly cloudy day shown in Figure4.

0

5

10

15

20

25

30

35

40

45

0 20 40 60 80 100 120

Rat

e (p

kt/s

)

Node

DLEX

Figure 9: The rate assignments for six solar profilesobtained from [16].

corresponding nodes. The X-axis is node id sorted in non-decreasing order according to their assigned rates. The num-ber of control packets for the entire network is 1285, includ-ing the retransmitted packets due to packet losses. AssumeTi is the subtree rooted at node i. In our distributed algo-rithm, node i has to forward control packets from all nodesin Ti to the sink, and the responses from the sink to all nodesin Ti −{i}. For a node that is the root of a large subtree, ithas to forward more control packets than other nodes, andthis can be seen clearly in Figure 10 (nodes 27, 59, 62, 178,160, 124). Ideally, the number of control packets sent by anode i is 2|Ti| − 1. However, due to asynchronous operationand packet losses, the number of control packets is higher.

The size of a control packet payload is 9 bytes, whichinclude 4 bytes for rate, 2 bytes for flow id, 2 bytes forforwarder id, and 1 byte for control message. The size ofthe control packet, including 7 bytes of packet header, is 16bytes. For a network with 155 nodes, the maximum over-head for a node is around 2.5KB. Note that the overheadcan be further reduced by combining multiple control pack-ets into one packet to save the overhead of packet header.

From Figure 10 we can also observe a trend that amongthose nodes that have larger subtrees (nodes 27, 59, 62, 178,

245

0 5

10 15 20 25 30

0 20 40 60 80 100 120 140

Size

of S

ubtre

e

Node

Size of Subtree2759

62 178160

124

0 20 40 60 80

100 120 140 160 180

# of

Pac

kets

# of Control Packets

Figure 10: The number of control messages forDLEX and the number of children of correspond-ing nodes.

160, 124), their rates are lower if their subtrees are larger.From Figure 11 we can clearly see the trend. This is becausethe nodes closer to the sink are usually the bottleneck nodessince they have to forward packets for others. From the logsof experiments we do find that the nodes under the subtreerooted at these nodes are assigned with the same rate asthese nodes. If we want to increase the total throughput orimprove the lexicographic rate assignment, it is sufficient toincrease the recharging rate of first hop nodes.

0

10

20

30

40

50

0 20 40 60 80 100 120 140

6

4

2

0

Size

of S

ubtre

e

Rat

e (p

kt/s

)

Node

Size of SubtreeRate

Figure 11: The size of a subtree v.s. the rate.

From our experiments, we observed that the running timeof our distributed protocol varies from 50 seconds to 224seconds. Figure 12 shows the CDF of the number of nodesassigned with the optimal rate versus protocol running timefrom one set of results where the total running time is 78seconds. We observed that most of the time is wasted ontimeouts (0.25 second + 0.1 second random backoff in ourexperiment) before retransmission due to packet losses be-cause of unstable link quality. If the link quality can be con-sidered while creating the routing tree, we should be able toreduce the running time significantly.

4.4 Initial Battery LevelIn this section we study the performance of rate assign-

ment when the battery level is low. When the battery level

0

20

40

60

80

100

0 10 20 30 40 50 60 70

CD

F (%

)

Time (s)

Figure 12: CDF of the number of nodes assignedwith the optimal rate versus protocol running time.

is low, how the maximum achievable rate is determined hasbig impact on the system performance since the battery doesnot have enough energy to serve as buffer when the energycollected is not sufficient.

We evaluate our protocol and compare it with a variationand a naive approach. In the variation, called DLEX-A, in-stead of using Algorithm 2 to compute the maximum achiev-able rate, we simply use the average recharging rate per timeslot to compute the maximum achievable rate, and use thedistributed algorithm to compute the rate assignment. Inthe naive approach, NAVG, nodes use the average rate pertime slot to compute the maximum achievable rate, and usethat rate as their working rate. We set the initial batterylevel to 30mAh for this experiment.

Figure 13 shows the actual rate assigned to each node.The X-axis is the node id, sorted in non-decreasing orderaccording to their rates assigned by DLEX. The rates com-puted by DLEX are slightly smaller than DLEX-A becauseusing Algorithm 2, DLEX has to lower the rate to preventa node from running out of energy when the battery level islow. NAVG has the highest rate since it uses the maximumrate as the working rate. However, the higher working ratedoes not necessarily result in higher system performance.

Figure 14 shows the number of packets received at thesink for each node in the simulation. Because of the policyof the testbed, we are not allowed to run the experimentfor 24 hours, we use a simulator and plug the rates we ob-tained from the testbed into the simulator to simulate packettransmissions and energy consumption in the rechargeablenetwork. From the figure we can see that the number ofpackets received at the sink in NAVG does not correspondto the high working rate in Figure 13. This is because nodesmay run out of energy before their battery is recharged dueto the dynamics of the recharging process. Nodes that runout of energy can not generate packets anymore. In additionthey can not forward packets for others either.

Figure 15 shows the percentage of time each node runsout of energy in 24 hours. We can see that in NAVG, allnodes run out of energy for some duration, and over 30%of nodes run out of energy for over 50% of the time duringthe simulation (13% of nodes run out of energy over 90% ofthe time during the simulation). DLEX-A only has 28 nodesthat have reached 0 available energy for 10% to 15% of the

246

0

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6

8

10

12

14

16

18

0 20 40 60 80 100 120 140

Rat

e (p

kt/s

)

Node

DLEXDLEX-A

NAVG

Figure 13: Rates of nodes in different rate assign-ment approaches.

0

2

4

6

8

10

12

14

0 20 40 60 80 100 120 140

Thro

ugpu

t (M

pkt

s)

Node

DLEXDLEX-A

NAVG

Figure 14: Number of packets received for eachsource.

simulation time. DLEX has 24 nodes that run out of energyfor less than 3% of the simulation time.

Ideally, DLEX should not have any node running out ofenergy. The reason that nodes run out of energy in DLEX isbecause we use one hour as the unit to store the rechargingprofile, and use that to compute the maximum achievablerate. However, the recharging rate may vary within onehour, and the variation results in occasional energy runouts.We have conducted experiments using one minute as theunit to store the recharging profile and no nodes have everrun out of energy. However, storing recharging profile with afiner resolution will consume more memory or storage space,and therefore it is a design trade-off. To prevent nodes fromrunning out of energy using only coarse grained rechargingprofile, nodes may reserve a small amount of energy as thebuffer when the battery level is low. We leave this as futurework and do not investigate it further in this paper.

Figure 16 shows the number of packets received at the sinkand the percentage of nodes that ran out of energy duringthe 24 hours of simulation. NAVG can hardly receive anypackets for about 2.5 hours of the time, from 5:50 to 8:15due to many nodes running out of energy during that period.DLEX-A is better, however, its throughput is also affectedseverely during the same period because the nodes that run

100

50

0 0 20 40 60 80 100 120 140

Node

DLEX

100

50

0Perc

enta

ge (%

)

DLEX-A

100

50

0

NAVG

Figure 15: Percentage of time a node runs out ofenergy.

out of energy are usually close to the sink. When they runout of energy, they cannot forward packets for other nodes,thus resulting in severe throughput degradation. DLEX isaffected only for a small portion of the time because it hasfewest nodes running out of energy for very short duration.Again, if we store finer grained recharging profile on sensors,we can maintain a stable throughput for the entire simula-tion for all nodes. This shows that DLEX performs betterin terms of uniformly collecting data across time.

40 20

0

24:0018:0012:006:000:00

100

0

Time

DLEX 40 20

0

24:0018:0012:006:000:00

100

0

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DLEX

40 20

0 100

0

Thro

ugpu

t (K

pkt

s)

Perc

enta

ge (%

)

DLEX-A 40 20

0 100

0

Thro

ugpu

t (K

pkt

s)

Perc

enta

ge (%

)

DLEX-A

100 50

0 100

0

NAVG 100

50 0 100

0

NAVG

Figure 16: Total number of packets received at thesink (top) and the percentage of nodes running outof energy (bottom) in different time slots.

4.5 TopologyIn this experiment we vary the transmission power to cre-

ate networks with different sizes and densities. With lowertransmission power, nodes have few choices for selecting therouting paths in shortest path tree, and the diameter, i.e.,the maximum hop count of the network, will increase too.Figure 17 shows the results of three different transmissionpowers. The higher number represents higher transmissionpower. We can see that with higher transmission power, wecan get better lexicographic rate assignment. This is due tothe higher density of the network. In a high density net-work, there will be more nodes that are one hop away from

247

the sink. The size of subtrees rooted at these one hop nodeswill be smaller, compared with the subtrees of one hop nodesin a low density network which has fewer number of one hopnodes. Since the first hop nodes are usually the bottlenecknodes, fewer nodes in the subtree implies higher share of theavailable energy, thus resulting in higher achievable rate.

0

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6

8

10

12

0 20 40 60 80 100 120 140

Rat

e(pk

t/s)

-10 dBm-5 dBm0 dBm

Figure 17: Rate assignment in different topology.

5. DISCUSSION

5.1 Impact of Tree ImbalanceOur distributed algorithm finds the optimal lexicographic

rate assignment when each source has a fixed route and therouting path is known. As the optimal rates of nodes dependon which nodes their flows pass through, different routingpaths may result in different optimum rate assignments. Ifsome nodes have much larger subtree than others, nodesunder those subtrees will typically be assigned with smallerrates. As the shortest path tree does not consider the loadon the nodes, some nodes might have many more nodes intheir subtrees than others. Nodes under such subtrees canonly share a small portion of resources, and therefore will beassigned only a small rate. If we can create a more balancedtree such that each subtree has roughly the same number ofnodes, we can improve the lexicographic rate assignment.

As computing a load balanced tree is an NP-hard problem[17], we use an offline heuristic to create a balanced treefrom the connectivity graph we obtained from the testbed.The heuristic starts at any arbitrary spanning tree T of theconnectivity graph G and greedily checks each edge of Gconnecting two different subtrees of T . An edge connectingtwo different subtrees is added to T if, by removing certainedge of T , the resulting set of subtrees has a lower standarddeviation in terms of their sizes. The heuristic is applied tothe subtrees of first hop nodes, and recursively applied tolower levels of subtrees.

We compare the rate assignment on the balanced tree re-sulting from our heuristic with the shortest path tree. Wealso compare it with the optimal solution where routes arenot fixed and flows can be split over multiple routes.

Figure 18 shows the results we obtained from an LP solverfor a dense network and Figure 19 shows the results for asparse network. In both cases, the balanced tree does im-prove the lexicographic rate assignment over the shortestpath tree, though in the sparse network the improvement is

marginal since nodes do not have many choices for selectinga parent. This shows that in a dense network, a tree maynot be the best routing structure, and multipath routingschemes are preferable for balancing the traffic load.

0

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4

6

8

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16

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Rat

e(pk

t/s)

balanced treeoptimal

shortest-path tree

Figure 18: Rate assignment on shortest path tree,balanced tree, and the optimal in a dense network.

0

2

4

6

8

10

12

14

16

0 20 40 60 80 100 120 140

Rat

e(pk

t/s)

balanced treeoptimal

shortest-path tree

Figure 19: Rate assignment on shortest path tree,balanced tree, and the optimal in a sparse network.

We seek to design a better algorithm to create a balancedtree distributedly and also investigate algorithms that canexploit multi-path routing to achieve better lexicographicrate assignment. We leave these as our future work.

5.2 Granularity of the Recharging ProfilesIn this paper we assume that the recharging profile of the

renewable resources are known by each sensor. This can beachieved by keeping track of the recharging rate the sensorsobserved in the previous recharging cycles. To maintain asteady data collection rate, the recharging rate can be com-puted using the average of previous recharging cycles. Aslong as the average recharging rate does not change dramati-cally, sensors do not need to invoke the distributed algorithmto compute new rate assignments.

In experiments we use one day as one recharging cycle tocompute the optimal rate. It is possible to use one month,one quarter, or one year as the recharging cycle to lowerthe dynamics. However, when the length of the recharg-

248

ing cycle increases, the initial battery level and when to in-voke the protocol become more important. For example, ifthe recharging cycle is one year, during the winter season ahigher battery level is desired so the sensors will have largerenergy buffer to boost the data collection rate.

5.3 Link Quality and Battery LeakageIn our algorithm we use the cost of sensing, receiving,

and transmitting data to compute the optimal rate, andassume there is no battery leakage. However, in realitypacket loss due to poor channel conditions, interference, andcollisions are inevitable, and battery leakage is a commonphenomenon. Therefore, packet retransmission and batteryleakage have to be considered.

For the centralized algorithm, we can take the retransmis-sion probability into consideration when deciding the valueof es and ef to reflect the expected cost. For example, if theretransmission probability is pij for link (i, j), the cost offorwarding on link (i, j) can be expressed as erx + etx

pijwhere

erx is the receiving cost and etx is transmitting cost (noteerx + etx = ef ). In the distributed algorithm, the retrans-mission probability of the link can be included in the controlpacket and be forwarded to the sink along with the maxi-mum achievable rate. However, this will slightly increase thecontrol overhead. Similarly, battery leakage can be modeledby an extra variable for the discharging rate. In each slota fixed amount of energy is discharged from the battery.More complex models based on battery level, humidity, andtemperature can also be used.

However, when the change of link quality results in discon-nection of some links, rates have to be reassigned. For thecentralized algorithm, rates have to be recomputed. For thedistributed algorithm, first, a new routing tree is required.However, rates do not need to be recomputed for all nodes.Only nodes that change their parent node need to initiatethe rate recomputation. If a change results in a new rateassignment, only nodes whose rates are different need to benotified with their new rates, though it is possible that achange of link quality may change the rate assignment of allnodes.

6. RELATED WORKThere are many works on developing sensors with ca-

pability of harvesting energy from solar or wind resources[8, 18–20]. There are also many studies on exploiting re-newable energy to increase system performance or networklifetime [21–28]. In [22], the authors consider solar-awarerouting in rechargeable sensor networks. They use a sim-ple heuristic that preferably routes packets through solar-powered nodes, and the extra harvested energy is only ameans to boost the network lifetime. In [24], the authorsmeasure the environmental energy properties and renewableopportunities, and use the information to schedule tasks toincrease the network lifetime. In [25] and [26], the authorsfurther consider maximizing the system performance whilemaintaining Energy-Neutral operation, i.e., the energy usedis always less than the energy harvested so that the systemcan continue to operate perennially. In [27], the authorsstudy how sensor nodes should be activated dynamically soas to maximize a utility function based on the coverage areaof the sensors. In [28], in addition to adjusting the duty cy-cle of the sensors to achieve Energy-Neutral operation, theauthors also consider the variability of environmental energy

resource and attempt to reduce the variation of the duty cy-cle using adaptive control theory. However, these works ei-ther only consider the workload of individual sensors and donot consider the influence on overall network performance bythe individual decisions, or only try to maximize the systemperformance but do not consider the impact on individualsensors.

To maximize the system performance while balancing theworkload, fairness has to be considered. Maxmin fairness,or lexicographic fairness, has been widely used to define thefairness of a system. In [14], a distributed maxmin rate com-putation algorithm for fixed flows routed through capacityconstrained switches in wired networks is proposed. Theproposed algorithm computes a maxmin rate assignment foreach flow and guarantees quick convergence. In [29], the au-thors generalize the problem by adding maximum and min-imum rate requirement for each flow, and propose a central-ized algorithm similar to the one proposed in [30] that iden-tifies bottleneck links first, and assigns rates equally to allflows passing through these bottleneck links. A distributedalgorithm is also proposed that is based on the algorithmproposed in [14]. In [13], a centralized algorithm is pro-posed that iteratively uses linear programming to find lexi-cographic rate assignment for all sensor nodes that period-ically report readings to the sink. The proposed algorithmdoes not require these flows to be forwarded through fixedroutes. In [31] a distributed congestion control scheme isproposed to achieve maxmin rate allocation through over-hearing and propagating congestion announcement, but itrequires sophisticated parameter tuning to achieve stable op-eration and the rates oscillate up and down after convergingeven when the topology does not change. All these workssolve the fairness problem with static constraints, such asbased on switches or battery capacity, and do not considerdynamic resources such as changing harvested energy.

7. CONCLUSIONIn this paper we study the rate assignment problem for

rechargeable sensor networks. We propose a centralizedalgorithm and a distributed algorithm for optimal lexico-graphic rate assignment. The centralized algorithm com-putes the optimal rate assignment along with determiningthe amount of flow on each link. The distributed algorithmcomputes the optimal rate when the routing tree is pre-determined. We prove the optimality of the centralized andthe distributed algorithms. To evaluate the proposed dis-tributed algorithm, we conduct experiments using a testbedwith 155 sensor nodes under various scenarios. The ratescomputed by the distributed algorithm are dependent onthe routing tree. As part of our future work, we plan toexplore distributed solutions that can jointly determine theoptimum rates for all nodes and the flows on each link.

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9. APPENDIX

9.1 Proof of Theorem 1

Theorem 1. LexRateAssignment computes optimal lex-icographic rate assignment.

250

First, we define some terms that will be used in the proof.

Definition 2. Let G(V, E, S) be a network with nodesV = {v1, v2, ..., vn}, wireless links E, and node constraintsS = {s1, s2, ..., sn}. A node constraint si is a 2-tuple of(Πi, Ri) that specifies the maximum battery capacity andenergy recharging rate for vi. We define G′(N ′, E′, S′) =G2(N, E, S) where N ′ = N , E′ = E, S′ = {s′1, s

′2, ..., s

′n},

and s′i = 2 × si = (2Πi, 2Ri), 1 ≤ i ≤ n. In short, G2 isa network with the same topology as G, but the maximumcapacity and energy recharging rate of each node is doubledin G2. We also define G = G2/2.

Definition 3. Let L1 and L2 be any two rate assign-

ments for nodes {v1, v2, ..., vn} where L1 = {r(1)1 , r

(1)2 , ..., r

(1)n }

and L2 = {r(2)1 , r

(2)2 , ..., r

(2)n }. We define L′ = L1 +L2 where

L′ = {r′1, r′2, ..., r

′n} and r′i = r

(1)i + r

(2)i , 1 ≤ i ≤ n. We also

define L′′ = L/2 where L′′ = {r′′1 , r′′2 , ..., r′′n} and r′′i = ri/2,1 ≤ i ≤ n.

Lemma 1. The lexicographic optimal rate assignment isunique.

Proof. We prove the uniqueness by contradiction. Weshow that if there are more than one optimal solutions, sayL1 and L2, there must exist a rate assignment that is betterthan L1 and L2 so they can not be optimal.

Suppose there are two different optimal rate assignments,L1 and L2, for a network G. L1 and L2 are identical assorted vectors but their rate assignments for some nodesdiffer. Let M be the set of nodes whose rates are differentin L1 and L2. Among all the nodes in M , let vk be a node

with the smallest rate in L1. Let S1 = {vi | r(1)i < r

(1)k },

Q1 = {vi | r(1)i = r

(1)k }, and X1 = {vi | r

(1)i > r

(1)k }, i.e.,

S1 contains all nodes whose rates are smaller than vk’s ratein L1, Q1 contains all nodes whose rates are equal to vk’srate in L1, and X1 contains all nodes whose rates are larger

than vk’s rate in L1. Next, we define S2 = {vi | r(2)i < r

(1)k },

i.e. S2 contains all nodes whose rates in L2 are smaller thanvk’s rate in L1. We know that

r(1)i = r

(2)i , ∀vi ∈ S1 (13)

because vk is the node with the smallest rate in L1 thatdiffers in its rate assignment in L2. Therefore,

S1 ⊆ S2 (14)

Furthermore,

|S1| = |S2| (15)

otherwise L1 and L2 will have different number of nodeswhose rates are smaller than r

(1)k , which contradicts that

L1 = L2. By (14) and (15), we have

S1 = S2 (16)

Now we construct a new network G2. Observe that L2

= L1 + L2 is a feasible rate assignment for G2. MoreoverL2/2 is a feasible rate assignment for G2/2. Therefore, L′ =L2/2 = (L1 +L2)/2 is a feasible rate assignment for G sinceG2/2 = G. We show that L′ is lexicographically greaterthan L1, and therefore L1 can not be optimal.

First, consider the rates in L′ for nodes in S1. By Defini-tion 3 and (13) we have

r′i = (r(1)i + r

(2)i )/2 = r

(1)i < r

(1)k , ∀vi ∈ S1 (17)

Considering the rates in L′ for nodes in Q1 and X1. First

r(2)i ≥ r

(1)k , ∀vi ∈ Q1 ∪X1 (18)

otherwise vi would be in S2, which indicates vi ∈ S1 and isa contradiction that vi ∈ Q1 ∪X1. Therefore, by Definition3 and (18) we have

r′i = (r(1)i + r

(2)i )/2 ≥ (r

(1)k + r

(1)k )/2 = r

(1)k , ∀vi ∈ Q1 (19)

r′i = (r(1)i + r

(2)i )/2 > (r

(1)k + r

(1)k )/2 = r

(1)k , ∀vi ∈ X1 (20)

Now we define S′ = {vi | r′i < r

(1)k }, Q′ = {vi | r

′i = r

(1)k },

and X ′ = {vi | r′i > r

(1)k }. By (17), (19), and (20) we know

S1 ⊆ S′ (21)

X1 ⊆ X ′ (22)

Because S1 ∪Q1 ∪X1 = S′ ∪Q′ ∪X ′, by (21) and (22)

Q1 ⊇ Q′ (23)

By (19) and (20)

Q1 ∪X1 ⊆ Q′ ∪X ′ (24)

Because S′ ∩ (Q′∪X ′) = φ and S1∩ (Q1∪X1) = φ, we have

|S′| + |Q′ ∪X ′| = |S1| + |Q1 ∪X1| (25)

From (21) and (24) we know that |S1| ≤ |S′| and |Q1 ∪

X1| ≤ |Q′ ∪ X ′|. By (25), we know |S1| = |S′| and this

together with (21) gives us

S1 = S′ (26)

Furthermore, r(2)k > r

(1)k otherwise vk would be in S2 which

indicates vk ∈ S1, a contradiction of the definition of S1.

Therefore, r′k = (r(1)k +r

(2)k )/2 > (r

(1)k +r

(1)k )/2 = r

(1)k . Since

vk ∈ Q1 and vk ∈ X ′, using (23) we further have

Q1 ⊃ Q′ (27)

X1 ⊂ X ′ (28)

By (26), (27), and (28), L′ > L1, which is a contradictionthat L1 is an optimal lexicographic rate assignment.

9.2 Proof of Theorem 2Theorem 2. The UpdateRate rate computation using Equa-

tion 12 converges and computes optimal lexicographic rateassignment.

Proof. First we show the convergence of the UpdateRate

algorithm. Let rj(j)(x) be the rj(j) at the end of round x.

There are only three possibilities considering Rj and Uj :(1) Rj and Uj do not change. (2) Some flows in Uj becomerestricted and are moved to Rj . (3) Some flows in Rj becomeunrestricted and are moved to Uj . For case 1, the algorithmterminates, so we only consider cases 2 and 3.

• Case 2: If a non-empty subset of flows, say Z, in Uj

is moved to Rj , the rj(j)(2) will be computed in next

round. Since

rj(i)(1) =

Cj − ef

P

i∈Rjrmax

j (i)

ef (nj − |Rj |) + es

rj(i)(2) =

Cj − ef

P

i∈Rj∪Zrmax

j (i)

ef (nj − (|Rj |+ |Z|)) + es

251

Therefore,

rj(j)(2)(ef (nj − (|Rj | + |Z|)) + es)

= Cj − ef

X

i∈Rj∪Z

rmaxj (i)

= Cj − ef

X

i∈Rj

rmaxj (i) − ef

X

i∈Z

rmaxj (i)

= rj(j)(1)(ef (nj − |Rj |) + es)− ef

X

i∈Z

rmaxj (i)

> rj(j)(1)(ef (nj − |Rj |) + es)− rj(j)

(1)ef |Z|

= rj(j)(1)(ef (nj − (|Rj |+ |Z|)) + es)

and rj(j)(2) > rj(j)

(1). The above argument couldbe generalized for round x and x + 1 to show that

rj(j)(x+1) > rj(j)

(x). Since the number of flows in Uj

is finite, the process will eventually converge.

• Case 3: If a non-empty subset of flows, say Z, in Rj

are moved to Uj , the rj(j)(2) will be computed as

rj(j)(2) =

Cj − ef

P

i∈Rj−Z rmaxj (i)

ef (nj − |Rj − Z|) + es

=Cj − ef

P

i∈Rjrmax

j (i) + ef

P

i∈Zrmax

j (i)

ef (nj − |Rj |+ |Z|) + es

>Cj − ef

P

i∈Rjrmax

j (i) + ef

P

i∈Zrj(j)

(1)

ef (nj − |Rj |+ |Z|) + es

= rj(j)(1)

Using the same argument as in case 2, we know rj(j)(x+1) >

rj(j)(x), and the process will converge.

Now we show that the UpdateRate algorithm computesthe optimal lexicographic rate assignment. We show thisby assuming that there is an optimal rate assignment D∗

which is better than the rate assignment D computed byUpdateRate, and show that given D∗, UpdateRate can com-pute a rate assignment better than D∗, and hence a contra-diction.

Let D = (r(1), ..., r(n)) be the rate assignment computedfrom the distributed algorithm and D∗ = (r∗(1), ..., r∗(n))be the optimal lexicographic rate assignment for flows 1 ton, and D∗ > D. Among those flows whose rates are differentin D and D∗, let f be the node that is assigned with thesmallest rate in D; therefore, r(f) < r∗(f).

First, as r(f) < r∗(f) ≤ rmaxf (f), flow f must be re-

stricted at some node j, i.e., r(f) = rj(f) = rj(j) < rmaxf (f)

(it is possible that f = j). We define Rj and Uj as the setof flows passing through node j and whose rates are smallerthan and greater or equal to r(f) in D respectively. Thusf ∈ Uj . We also define R∗

j and U∗j as the set of flows that

pass through node j and whose rates are smaller than andgreater or equal to r∗(f) in D∗ respectively.

We first show the following three properties.

rj(i) = r∗(i), ∀i ∈ Rj (29)

r(f) ≤ r∗(i), ∀i ∈ Uj (30)

r(f) ≤ r∗(j) (31)

Because r(f) < r∗(f), it is clear that

Rj ⊆ R∗j (32)

Furthermore, since flow f is the flow whose rates are differentin D and D∗ and is the smallest one in D, rj(i) = r∗(i), ∀i ∈Rj . This proves Equation 29.

Second, for i ∈ Uj , if r∗(i) < r(f), from (29) and the factthat i ∈ Uj , the number of flows whose rates are smaller thanr(f) in D∗ will be greater than the number of flows whoserates are smaller than r(f) in D, which contradicts that D∗

is the optimal lexicographic rate assignment, and this provesEquation 30. Similarly if r∗(j) < r(f), it contradicts D∗ isthe optimal lexicographic rate assignment and this provesEquation 31.

Note that Cj = ef

P

i∈Rj∪Ujrj(i)+esrj(j). Because r(j)

= r(f), and together with (29), (30), and (31), if f 6= j, wehave

r(f) =Cj − ef

P

i∈Rj∪Uj−{f} rj(i)− esrj(j)

ef

=Cj − ef

P

i∈Rjrj(i)− ef

P

i∈Uj−{f} rj(i)− esrj(j)

ef

=Cj − ef

P

i∈Rjrj(i)− ef

P

i∈Uj−{f} rj(j) − esrj(j)

ef

≥Cj − ef

P

i∈Rjr∗(i)− ef

P

i∈Uj−{f} r∗(i)− esr∗(j)

ef

=Cj − ef

P

i∈Rj∪Uj−{f} r∗(i)− esr∗(j)

ef

= r∗(f)

which contradicts that r(f) < r∗(f).If f = j, we have

r(f) =Cj − ef

P

i∈Rj∪Ujrj(i)

es

=Cj − ef

P

i∈Rjrj(i)− ef

P

i∈Ujrj(i)

es

=Cj − ef

P

i∈Rjrj(i)− ef

P

i∈Ujrj(j)

es

≥Cj − ef

P

i∈Rjr∗(i)− ef

P

i∈Ujr∗(i)

es

=Cj − ef

P

i∈Rj∪Ujr∗(i)

es

= r∗(f)

which also contradicts that r(f) < r∗(f). Therefore, Dmust be the optimal lexicographic rate assignment, and thiscompletes the proof.

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