Fairness for All; Rate Allocation for Mobile Wireless Networks

Fairness for All; Rate Allocation for Mobile Wireless Networks Andreas Loukas∗ , Matthias Woehrle† , Marco Zuniga∗ , Koen Langendoen∗ Delft University of Technology {a.loukas, m.a.zunigazamalloa, k.g.langendoen}@tudelft.nl∗ , [email protected]†

I. I NTRODUCTION The problem of fair rate allocation is one of the fundamental issues in wireless networking. This problem can be succinctly described as follows: each node has to optimize its own channel access rate with the dual objective of maximizing the network throughput and sharing the channel fairly with adjacent nodes. Within the context of wireless networks, the fair rate allocation problem has received significant attention and several notable studies have proposed optimal distributed algorithms for various networking scenarios [1–5]. These solutions, however, are designed for static networks and rely on a key assumption: each node computes its fair allocation on the basis of complete knowledge of its 1hop neighborhood. In this study, we investigate the fair rate allocation problem from a different perspective. We consider mobile and large-scale multi-hop wireless networks. Adding mobility dramatically changes the constraints posed to fair rate allocations algorithms. Since the network topology changes constantly and rapidly, the stateof-the-art algorithms become suboptimal and/or unstable because they are unable to gather complete, up-to-date neighborhood information in real-time. To illustrate the importance of the problem tackled in this paper, let us use the results shown in Fig. 1. We implemented the decentralized algorithm with the

1

node persistence

Abstract—Fair rate allocation deals with the fundamental problem of sharing the channel efficiently and fairly. In wireless networks, several notable works have proposed optimal solutions to this problem. These approaches work well for static networks, but rely on an assumption that renders them sub-optimal when nodes are mobile: at each computation step, nodes must collect the state of all their neighbors (1-hop knowledge assumption). In large-scale mobile networks, nodes need to continuously adapt to changing network conditions. Under these circumstances, it is hard to gather complete 1-hop information accurately and promptly. The key to any efficient solution in mobile networks is fast convergence with limited information. In this paper, we propose a simple decentralized algorithm for fair rate allocation that works well even with partial 1-hop information. The algorithm converges linearly and can be tuned to approach a wide range of trade-offs (from proportional to harmonic fairness). Our evaluation, using real-world mobility traces from 400 taxi cabs, shows that even in the challenging case of highly dynamic and dense networks, the algorithm assigns rate efficiently (mean error of 2.5% from the optimum), while using on average 37% of the 1-hop information.

100%

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rounds Fig. 1. Optimal algorithms for static networks are very sensitive to partial knowledge [3]. Areas annotated in gray show the allocation of a node when operating on the basis of 98% and 95% of the 1-hop information. Even a decrease of 2% of the available information has a devastating effect.

fastest convergence in the literature [3]. This algorithm is shown to be very robust to message loss and to reduce the control signaling with respect to other NUM approaches [6] by a factor of ten. The figure depicts what happens when information is incomplete. The results are shown for a representative node in a 22-node network, but similar behavior was observed in all nodes. When the information is complete (100%) the algorithm converges quite fast to the optimal persistence (≈ 0.55). If some neighborhood information is lost (2% and 5%), the allocation becomes unstable and varies erratically. To work in mobile environments, fair rate allocation algorithms need to be designed considering a different set of premises. Additionally to the requirements for static networks (i.e., decentralized, fast convergence, stability and optimal or near-optimal rate allocation), an algorithm needs to cope with limited knowledge of its surroundings. To handle the high dynamics of the network, the algorithm needs to rely only on partial 1-hop information, while being aware that this partial information can rapidly become stale. Our main contributions: The key distinction of our study with respect to the vast and prolific work on fair rate allocation, is that we propose an algorithm that is designed from inception to be robust against mobility and message loss. These characteristics are central when coping with networks with high dynamics. Considering the new problem presented in this paper, our three main contributions are: Contribution 1: Theoretical foundations (Sec. IV). First, we prove that using aggregate information is

sufficient to have a decentralized algorithm that (i) converges linearly and (ii) successfully trades off fairness for throughput. The ideas and proofs presented in this section are the cornerstone of our work. By using the concept of aggregate information, our algorithm becomes agnostic to the information of specific nodes. The information may or may not be complete. As long as we obtain aggregated information from a small subset of neighbors (any subset), we can provide an efficient allocation. Contribution 2: A practical algorithm (Sec. V). Based on our theoretical results, we implement a scalable and robust algorithm called AutoTune. By employing estimation methods, AutoTune relies only on partial local information. It also provides a single parameter that fine-tunes the trade-off between throughput and fairness according to different optimization goals. Due to its minimalistic communication demands, AutoTune is very suitable for highly mobile and dense networks. Contribution 3: Evaluation based on real-world dynamic traces (Sec. VI). To validate the performance of AutoTune, we use real traces from 400 taxis moving in the San Francisco Bay Area. Our results show that AutoTune is capable of providing a rate allocation that is between 90% and 98% from the optimal solution while using on average only 37% of the neighborhood information.

type exhibits better characteristics in terms of convergence and stability, but does so by knowing all possible transmission conflicts, commonly referred to as the conflict-graph. This type is clearly not suitable for networks with high dynamics. Non-cooperative algorithms, e.g., [5, 7], sacrifice convergence speed and stability by being agnostic to local information. Even though individual nodes act myopically, they converge to the optimum. The main problem of most non-cooperative algorithms is that they require information implicitly: algorithmic constants should be assigned according to specific network dynamics and density constraints. To deal with this problem, Su et al. [8] employ an outer control loop that slowly pushes adaptation constants towards the right value. The drawback of this approach is the message overhead of adding the feedback signal of the outer loop. Significant work has also been done on attaining max-min fair rates [9, 10] and lexicographic max-min fairness [11]. On the contrary, we are interested in attaining a range of fairness trade-offs. An algorithm that operates optimally without message-passing has also been proposed by Mohsenian-Rad et al. [12]. The applicability of the solution is however limited to complete networks, whereas we require a solution for arbitrary topologies. III. P RELIMINARIES

II. R ELATED W ORK The prevalent approach to the rate allocation problem, namely Network Utility Maximization (NUM), defines a global optimization problem and decomposes the global optimization problem to localized strategies that achieve the same global result [3, 4, 6]. The popularity of the NUM approach is mainly due to allocation optimality and convergence guarantees. However, most NUM algorithms exhibit one or more of the following pitfalls: (i) They require extensive message-passing for each computation step. This stems from their dependence on complete local information of at least one hop. (ii) They have to limit their adaptation speed as to guarantee convergence. This slower convergence hinders the ability of these algorithms to operate in mobile networks. Recent efforts by Mohsenian-Rad et al. [3] have dealt to some extent with the above issues (i.e., lower message overhead and faster convergence), but their algorithm still requires complete 1-hop information. Collecting 1-hop information follows the coupon collector paradigm and exhibits a time complexity of Θ(ni log(ni )) for i ∈ N , a significant overhead for dense mobile networks. In such networks optimality guarantees matter less than message complexity and latency. As such, nodes should be able to adapt with only partial information. A second approach entails the use of game theoretic methods to design local strategies that guarantee convergence to fair equilibria [5, 7, 8]. Two types of algorithms are derived: cooperative and non-cooperative. The first

Before detailing our theoretical framework, let us present the outline of the main steps of our approach. Step 1. In this section, we present the mathematical background required to understand our main contributions. We describe the network model, the channel access model, and the rate allocation problem. Step 2. In Sec. IV, we use a simple abstraction to represent topological changes of a mobile network: we represent mobility as a sequence of static topologies. We introduce the concept of aggregated information and use “static snapshots” of the network as the basis to prove that our method convergences fast to a set of allocations ranging from proportional to harmonic fairness. Step 3. Sec. V relaxes several assumptions of the analytical model to implement a practical algorithm (AutoTune) that copes with partial information. Network model: We model a wireless network as a graph G = (N , L), where N is the set of nodes of the network and two nodes are adjacent, (i, j) ∈ L, if they reside within each other’s interference zone. We denote sets with upper-case (e.g., N ) and set cardinalities with lower-case letters (e.g., n = |N |). For each node i, Ni− = {j ∈ N | (i, j) ∈ L, j 6= i} is the set of i’s neighbors and Ni , referred to as i’s cell, is the set that includes node i, as well as its neighbors. Channel access model: To model the channel access, we use a widely accepted model, which is valid for

wireless random access protocols such as CSMA and Aloha [13]. The model defines the following quantities: persistence probability cell (offered) load node rate cell rate

pi P λi = j∈Ni pj ri P ρi = j∈Ni rj ,

∀i ∈ N

The persistence probability pi represents the frequency that a node i tries to transmit. For example, to satisfy pi = 1, i would need to have the channel allocated only to itself. In p-persistent CSMA, nodes only make an attempt if the channel is idle; while in p-persistent Aloha, nodes transmit information without considering the status of the channel. As it is customary, we assume that two concurrent transmissions within a common interference zone lead to collisions (packet losses), while individual transmissions within the reception zone lead to successful transmissions. The total load observed by a cell centered at node i is the cell load λi . Given that the total load can exceed the channel capacity, the expected number of successful transmissions for each node i is given by the node rate ri . The total rate of the cell is given by the cell rate ρi . It is important to recall that the cell rate is a concave function of the cell load, i.e., the rate increases only up to a point (channel saturation), beyond this point any extra load decreases the rate. We denote the tuple (λ∗ , ρ∗ ) as the point where the maximum rate is achieved. Both λ∗ and ρ∗ are invariants of the specific random-access protocol and are known prior to deployment [13]. Note that the above model has no notion of end-to-end flows; it focuses on the problem of channel contention. Fair rate allocation problem: The goal is to find the optimal pi ’s that maximize simultaneously the fairness and the throughput. Formally, this is represented by two objectives: P Objective 1 (Rate max/tion): max i∈N ri P 1 2 Objective 2 (Fairness): min (i,j)∈L (ri − rj ) In general, the fair rate allocation problem does not admit a unique solution. The solution set is the socalled Pareto front that consists of Pareto-optimal (nondominated) points ranging between two extremes: maximum rate and maximum fairness. Specific applications may require different types of trade-offs between these two extremes. For example, streaming applications may want to favor fairness over rate-maximization, while a routing protocol that establishes paths to avoid congested areas may want to favor rate maximization. In our study, we are interested in values of the Pareto front that strike a balance between rate maximization and fairness. To capture in clear terms the specific range we are interested in, let us use the model proposed by Mo and Walrand [14]. This model provides a wide range of optimal trade-offs, attained by solving the global optimization problem with utility u(r) = log(r) if β = 1 1 This

definition is suitable, but by no means the the only one.

and u(r) = (1 − β)−1 r1−β otherwise. In this model, the two optimal extremes are captured by β = 0 (maximum aggregate rate) and β → ∞ (max-min fairness). Our work focuses on the range between β = 1 (proportional fairness) and β = 2 (harmonic mean fairness). AutoTune provides a single parameter to tune the network between these last two points (depending on the needs of the application). Figure 2, which is explained in more detail below, depicts the four specific β points. The challenge posed by dynamic irregular networks. In specific network topologies, such as complete and regular graphs2 , the optimization problem can be trivially solved and there is a single solution that maximizes both rate and fairness (i.e., the Pareto front consists of this single point). Irregular networks pose different challenges because rate maximization and fairness become conflicting objectives. To depict the impact of irregular graphs on the Pareto front, let us use the Mo and Walrand model in a ball-and-chain network. In a balland-chain network, illustrated in the subfigure within Fig. 2, there is a single node adjacent to a complete network of degree k. The δn in the right y-axis represents the degree difference between the single node and the complete graph. Fairness is quantified by Jain’s Fairness index [15] that varies from 1/n (gross unfairness) to 1 (equal rates). Since we are comparing networks of different sizes, we quantify rate maximization using the mean and not the aggregate rate. We can observe that when the network is regular (δn = 0), there is a single optimal solution. However, as the degree difference increases, i.e., a bigger ball is connected to the single node, a Pareto front of several Pareto-optimal values between the max-rate scheme (β = 0) and the max-min fairness (β → ∞) appears. The key challenge of our work is that in mobile networks the local topology changes rapidly (network irregularity), and it can have any layout – not only the ball-and-chain graph depicted in the figure. Hence, we need an algorithm that dynamically adapts to these changes as fast and as close to the Pareto front (of the current topology) as possible. As mentioned before, we focus on the range between β = 1 (red triangles) and β = 2 (black circles). IV. A NALYSIS OF THE A LLOCATION S TRATEGY This section, which presents our mathematical analysis, is composed of three parts. First, we describe our novel allocation strategy, which is based on aggregate information. Second, we prove that this allocation strategy converges linearly. Last, we provide bounds on the range of achieved allocations. A. Allocation strategy Current distributed algorithms can set optimal rates by collecting the state of all neighbors, i.e., by gathering the amount of information that each neighbor wants 2A

network is regular if its degree is constant, ∀ i, j ∈ N , ni = nj .

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mean rate

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greedy, β = 0 proportional, β = 1 harmonic, β = 2 max-min, β → ∞

δn= 0 δn= 1

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δn=k−1

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Algorithm 1 Allocation strategy running on node i In round t + 1, i updates pi according to  ∗  λ − λi (t) pi (t + 1) = pavg (t) + α , (2) i ni where 0 < pi < 1, X X pj (t). p (t)/n , and λ (t) = pavg (t) = j i i i j∈Ni

j∈Ni

δn= 15

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Jain’s fairness index Fig. 2. Pareto-optimal allocations for varying δn . The sub-figure demonstrates a ball-and-chain network with ball of size k = δn + 1.

to transmit. In dense mobile networks, collecting this information promptly and accurately follows the coupon collector problem and is Θ(ni log(ni )) hard. We propose a novel allocation scheme that is based solely on three aggregated metrics: the cell load λi , the number of neighbors ni , and the average amount of information that neighbors want to transmit pavg i . This section assumes that the estimates for these parameters are available and additionally that the execution is synchronous. The following section explains how these estimates can be obtained in constant time. Our evaluation shows that synchrony is not necessary. Each node i has to optimize locally two global objectives: rate and fairness. To maximize the aggregated rate, i should set its own rate to a value such that the total cell load λi is equal to the rate-maximizing load λ∗ . The difference between λi and λ∗ should therefore be minimized. Since the cell load λi accounts for all nodes within a cell, pi should be adapted in proportion to the cell size ni . The minimization of (λ∗ − λi )/ni expresses the first node objective. To maximize fairness, i should simultaneously minimize the difference between its own P pi and those of their neighbors. The minimization of | j∈Ni (pj −pi )| ≡ |pavg i −pi | expresses the second node objective. We derive the allocation strategy by expressing the adaptation rate at round t + 1, ∆pi (t + 1) = pi (t + 1) − pi (t), as a combination of the above node objectives.  ∗  λ − λi (t) avg ∆pi (t + 1) = pi (t) − pi (t) + α (1) ni The trade-off parameter α lies in (0, 1]. Large values of α sacrifice fairness for rate maximization and viceversa. The fact that the α parameter operates in the interval (0, 1] and that it affects only the rate allocation metric is used to ensure convergence (cf. Theorem 1). Rearranging the terms of Eq. (1), we obtain the allocation strategy summarized in Algorithm 1. Since our allocation strategy does not rely on individual neighbor probabilities, our solution is very robust to message loss. It is important to remark that, even though the model presented above is synchronous, synchrony is not an assumption of our work. It is used in this section as it

simplifies our analysis, but is dropped in the following section. The allocation strategy is not optimal: First, it is based on aggregate estimates, which are only partially known. Our evaluation shows that estimation errors do not have a significant effect. Second, it models the cell-rate function as symmetric. In many random-access protocols, optimal solutions can be biased towards large loads as cell-rate is a skewed function of load. The error introduced depends on the severity of the conflict between the fairness and rate maximization objectives, as well as on the specifics of the random-access protocol. Our evaluation shows that it is small. B. Convergence analysis Due to the high dynamics induced by mobility, fast convergence is an essential property of any algorithm that solves the fair rate allocation problem in mobile networks. The continuously changing topology of mobile networks can be seen as a sequence of “static snapshots”. We consider these static snapshots as the initial state to study the convergence properties of the allocation strategy. Notation: Let A be the adjacency matrix of the network at any “snapshot”, where aij = 1 if (i, j) ∈ L or i = j, and aij = 0 otherwise. The adjacency matrix A models channel access contention and thus has ones in its diagonal. Let D be the diagonal degree matrix with Dii = ni and D−1 its inverse, which is itself −1 a diagonal matrix with Dii = 1/ni . The persistence state of each node at round t is given by the vector p(t) = [ p1 (t) . . . pn (t)]> , and the load by vector λ(t) = Ap(t). Closed-form: We will now derive a closed-form expression for the allocation p(t). At round t + 1, the state is p(t + 1) = pavg (t) + α D−1 (λ∗ 1 − λ(t)) = (1 − α) P p(t) + α λ∗ D−1 1,

(3)

where pavg (t) = P p(t), 1 is a vector with all entries equal to one, and P = D−1 A is the random walk probability matrix. Solving iteratively we obtain a closedform expression of the state at round t as a function of the initial state p(0). Note that m is the index in vm but

an exponent everywhere else. p(t) = (1 − α)t P t p(0) + α

t−1 X

λ∗ vm , where

(4)

m=0

vm = (1 − α)m P m D−1 1

(5)

Convergence: We are now ready to focus on the convergence of the allocation strategy. Theorem 1: For α ∈ (0, 1], the allocation strategy converges linearly with convergence rate 1 − α. Proof: Given that 0 < α ≤ 1 and because P is row-stochastic, P m 1 = 1 for m ∈ Z+ , the geometric sequence with common ratio (1 − α) P and scale factor 1 converges to zero, limt→∞ (1−α)t P t = 0. This means that the network forgets its initial state linearly with a rate of 1 − α. It is easy to see that the vm terms also converge toPzero with rate 1 − α and consequently that t−1 the series m=0 vm converges linearly with the same rate. To illustrate the convergence speed, we simulated the allocation strategy in a complete network of 20 nodes. In our simulation, nodes converged close to the optimum allocation within at most 50 rounds for α = 0.1, 20 rounds for α = 0.3, and instantaneously for α = 1. Theorem 1 provides an additional insight: achieving fairness is more challenging than achieving maximum rate. When optimizing for fairness (small α) the allocation strategy requires more rounds to converge since each node needs information over multiple hops. Maximizing rate, on the other hand, is a “greedier” objective that requires less information The geometriP∞ exchange. m cally decaying series (1 − α) determines the m=0 maximum number of hops considered, as well as the importance of each term to the overall allocation. Since the series converges to 1/α, the multiplication with α in Eq. (4) is a normalization. C. Steady state analysis We proceed to examine p = limt→∞ p(t), which is the steady state allocation. In the following, all round indexes t are dropped. First, we show why p lies between the proportional (α = 1) and harmonic mean fairness (α → 0). We then prove bounds that capture the influence of the topology and α to the achieved allocation. We will exploit the following decomposition to explain the effect of α (the derivation is found in the Appendix). Assuming that the network’s links are symmetric, p = U M U −1 D−1 1, where U and U −1 are the right and left matrices of eigenvectors u and u−1 of P , M is a diagonal matrix with Mii = −1 λ∗ α (1 − (1 − α) µi (P )) and µi (P ) is the i’th eigenvalue of P . We explain p by examining the above decomposition. We distinguish three cases: (i) α
0+ : p approaches the harmonic mean allocation. Only the first element of M remains non-zero

and p depends on the principal eigenvectors 1 and π, p(t) = 1π T λ∗ D−1 1. If one ignores λ∗ D−1 1 the strategy behaves like a random walk. By multiplying with λ∗ D−1P 1 however, all nodes are assigned the same value λ∗ n/ i∈N ni , which is the harmonic mean of the inverse network degrees normalized to λ∗ . The harmonic mean is a natural choice for the assignment of fair rates because of its tendency to favor more constrained nodes. (ii) α = 1: p achieves the proportional allocation. All diagonal elements of M are equal to λ∗ and p(t) = λ∗ D−1 1 = λ∗ [1/n1 1/n2 . . . 1/nn ]> . This is an optimal allocation, known as the proportionally fair (w.r.t. to degree) allocation. (iii) α ∈ (0, 1): The closed-form solution is not as easy to interpret. To understand the behavior of the strategy, we focus on bounding the fairness and load objectives in the following. Fairness and load control: Our analysis ties the achieved allocation to the spectral properties of the underlying graph. The results demonstrate the effect of the parameter α on the steady state. Since our analysis is agnostic to the specific random-access protocol (i.e., cell-rate function), we analyze the optimization objectives in terms of the local node objectives. Theorem 2 (p-fairness): The aggregate difference of the persistence probabilities of the allocation strategy (2) among neighbors is bounded by X

(pi − pj )2 ≤ µmax (L) tr(M 2 )

n X

n−2 i ,

i=1

(i,j)∈L

where µmax (L) is the largest eigenvalue of the graph Laplacian L = D − A, and kpk2 = Pn 2 −1 2 M 1, u−1 ii hD i i . i=1 Proof: It is well known that for any symmetric > matrix B, µmin (B)kxk2 ≤ xP Bx ≤ µmax (B)kxk2 rep3 resent tight bounds . Due to (i,j)∈L (pi −pj )2 = p> Lp and L being a symmetric matrix, µmin (L)p> p ≤ p> Lp ≤ µmax (L)p> p. From Parseval’s identity, p> p = kpk2 =

n n X X 2 |h Mjj hD−1 1, u−1 j iuj , ui i| i=1

=

n X

j=1

2 |Mii hD−1 1, u−1 i i|

i=1

≤

n X i=1

2 Mii2 kD−1 1k2 ku−1 i k

=

n X i=1

n−2 i

n X

Mii2 ,

i=1

where in the third step we exploited that eigenvectors u form a bi-orthonormal basis of Rn and in the last step we used the Cauchy-Schwarz inequality. 3 Lecture

Notes for EE263, Stephen Boyd, Stanford 2007.

Theorem 3 (λ-gap): The aggregate difference between the rate-maximizing load and the achieved load Ap is bounded by kλ? 1 − Apk2 ≤ n(λ? )2 − µ2min (A)kpk2 ,

(6)

where µmin (A) is the smallest eigenvalue of matrix A. Proof: Because kApk2 = p> (A> A)p is a quadratic form and A> A is symmetric, we can reuse the inequality shown in the beginning of Theorem 2. µmin (A> A)kpk2 ≤ kApk2 kλ? 1k2 − kApk2 ≤ kλ? 1k2 − µmin (A> A)kpk2 = n(λ? )2 − µ2min (A)kpk2

(7)

Inequality (6) follows then from the fact that kλ? 1 − Apk2 ≤ kλ? 1k2 − kApk2 . The expansion of kpk2 can be found in the proof of Theorem 2. Let us examine the above bounds on p-fairness (Theorem 2) and load (Theorem 3). First, α has a direct effect on the optimization trade-off. kpk2 , which is a quadratic function of α, increases the upper bound of p-fairness, while it decreases the upper bound on load. Small α therefore favor the fairness objective over load. Second, there is a strong influence of the graph spectrum on the optimization trade-off. The graph topology influences the bounds on fairness and load by means of L and A eigenvalues, respectively. Unfortunately, we can not compare the two bounds as no closed-form relation exists between the eigenvalues of L and A. Still, we can interpret the role of the random-walk transition matrix. Our analysis indicates that well connected graphs favor fairness over load. The influence is expressed by Mii , which is a function of µi (P ) and is larger for sparse graphs. V. AUTOT UNE A LGORITHM This section presents AutoTune, an approximation algorithm for fair rate allocation that is suitable for highly dense and mobile networks as it operates using only partial one-hop information. A. Estimating aggregated values ˆ i ): In state-of the-art algorithms, Load estimation (λ calculating load requires a complete knowledge of the individual pj , j ∈ Ni . If some pj are not taken into account, node i will underestimate the channel utilization which in turn will hamper the operation of the allocation algorithm. We use indirect methods to calculate the load. AutoTune borrows ideas from NetDetect [16], a node degree estimation algorithm that, as an intermediate step (before degree estimation), performs load estimation. NetDetect works for various traffic patterns and randomaccess protocols, such as CSMA, Aloha, and simple duty-cycling MACs. Clearly, NetDetect is not the only alternative, but it is one that suits our purposes. Below we describe succinctly the way in which NetDetect

estimates load, the interested reader is referred to the original work for more details [16]. In NetDetect, nodes periodically track the activity of the wireless channel events, i.e., the length of the idle and busy periods. The probability distribution of event inter-arrivals is used to identify the load λi that best approximates the observed distribution. The load is estimated using maximum-likelihood-estimation (MLE). Average persistence estimation (ˆ piavg ): During each round, AutoTune collects as many pj ’s as possible and uses them to estimate pavg i . Depending on the round length %, as well on the neighborhood degree and channel saturation, the amount of information received can be small compared to the complete 1-hop knowledge. In the evaluation section, we show that Autotune performs very well with as little as 12% of the neighboring information. A drawback of this estimation method, is that it results in small inefficiencies in neighborhoods with large degree-irregularity δn . Since the sampling is not uniform, pavg is slightly over-estimated. As illustrated i in our evaluation, this phenomenon causes errors of less that 3% for δn = 13. Node degree estimation (ˆ ni ): The node degree estimaˆ i /ˆ tion is obtained by ni ≈ λ piavg . We conclude this section with two comments on the above estimation methods. A criticism of NetDetect is that its accuracy decreases when the node degree is very small. However, so does the need for estimation. In very sparse networks there is little or no congestion and nodes can count their neighbors (and compute λ) sufficiently fast by gathering beacons. One of the benefits of NetDetect is that it computes load by taking into account all nodes in both the reception and interference zone of the radio transceiver. State-of-the-art algorithms usually do not consider the difference between transmission and interference zones, which is problematic as interference zones can be more than twice as large as reception zones. Remember that, if a node i resides within node j’s interference but not within its reception zone, i can not receive j’s messages, even though they are necessary for the computation of load. Any direct computation of load therefore fails to consider all pj and results in a suboptimal allocation. On the other hand, AutoTune uses simple CCA mechanisms to track channel events. We thus hypothesize that it is better equipped to consider transmitting and interfering nodes in its computation. Validating this hypothesis would require a thorough evaluation on real testbeds. We leave this evaluation as part of our future work. B. AutoTune algorithm AutoTune is shown in Algorithm 2. During initialization each node chooses pi ∈ (0, 1) randomly and instantiates a data structure Si− that stores the information sent by neighboring nodes (line 1). In contrast to our analysis, AutoTune does not rely on any form of synchronization.

A

Algorithm 2 AutoTune running on node i Require: trade-off parameter 0 < α ≤ 1, round length % > 0, optimal load λ∗ > 0, locally-unique id. 1: Randomly choose 0 < pi ≤ 1, and create data structure Si− indexed by node id. 2: loop 3: Process any received msg: Si− = Si− ∪ msg j . 4: do EVERY % SLOTS ˆ i and 5: Use NetDetect to estimate λ 6: adjust it to the 1-hop load, X ˆi = λ ˆ i /2 + ˆ j /2s− . λ λ i

20

9: 10: 11: 12: 13:

Add msgi to the P rx-set Si = ∪ msg i and compute pˆiavg = j∈Si pj /si .  ˆ i )/λ ˆi . Update pi = pˆ avg 1 + α (λ∗ − λ i

Si−

Purge received messages, = ∅. end do ˆ i } with persistence pi . Transmit msgi = {pi , λ end loop

rate allocation error (%)

7: 8:

VI. E VALUATION This section, complements our analytical results with simulations. First, we evaluate AutoTune’s performance using real world mobility traces. To gain further under-

α = 0.1

10 0 20

α = 0.3

10 0 20

α=1

10 0

The round duration is % ts , where ts is the length of the time slot (usually equal to the transmission delay of a packet) and the round length % is a positive real number. The exact effect of % to the trade-off between stability and responsiveness is studied in the evaluation. At the end of each round t, nodes independently and asynchronously adapt their probability allocations ˆ i (t) is estimated (line 6) as (line 4). First the local load λ outlined above. Instead of considering the estimated load directly, AutoTune uses the average over the neighbors’ and the local load. Averaging is a modification of the allocation strategy, which decreases the error of the load estimation. The next step is the the estimation of pˆ avg (line 8). For the duration of round t, each node i transmits a message containing a tuple of its transmission probability and estimated load msg i (t) = { pi (t), λˆi (t)}. Si− (t) contains a single tuple from each node from which node i received a message in round t, i.e., we assume that nodes have unique identifiers and duplicate messages are dropped. Since i ∈ Ni , we are interested in the superset Si (t) = Si− (t) ∪ msg i (t) that also contains pi . Finally, nodes update their probability (line 9) and purge all messages from the current round (line 10). Even though nodes try to acquire as much information about their neighbors as possible, the algorithm is designed to operate with information about only a subset of the neighborhood, si (t) ≤ ni . Our simulations confirm that partial one-hop knowledge is sufficient for convergence.

C

Fig. 3. Summary of San Fransisco mobility trace. The lines plot the trajectory of each taxi. White and black circle markers indicate the initial and final taxi positions, respectively.

j∈Si−

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time (sec) Fig. 4. Allocation error δ over time for 400 taxis in San Fransisco. From top to bottom, the (min, mean, max) errors over time expressed in percentages were: for α = 0.1 (0.77, 2.98, 11.78), for α = 0.3 (0.86, 2.24, 9.99), and for α = 1 (1.30, 2.28, 9.87).

standing of the algorithm’s performance, we then focus on convergence properties and error. Experimental setup: Simulations were performed using LapSang4 , a discrete-time simulator for mobile ad-hoc networks. Radio communication used a unit-disc graph model with transmission radius of 250 meters. As a well understood random-access protocol, we chose Aloha (λ∗ = 1). The duration of each time slot (the time necessary for the transmission of a packet) was 10 ms. NetDetect’s estimation history size is set to equal to AutoTune’s round length %. We quantified allocation optimality over time using error metric δ = |J (ropt ) − J (r)| /2 + P the opt i∈N ri − ri /2n. For each “static snapshot” of the mobile topology, δ expresses the relative difference of AutoTune’s network-wide rate allocation r to ropt in terms of mean rate and fairness index J (r). The optimal allocation ropt was the closest solution of the global optimization problem, computed for each snapshot. Even though the error metric becomes zero when the allocation is optimal, a continuous error of zero is unrealistic for any algorithm as it would require an instantaneous optimal allocation adaptation in each time slot, where AutoTune only adapts every % time slots. A. San Francisco cab traces We evaluated our algorithm using real world traces of 400 taxi cabs moving in San Francisco’s Bay area in an experiment that simulated car-to-car communication (http://cabspotting.org). The trajectories of 4 http://code.google.com/p/lapsang

B. Convergence, stability, allocation We now study the trade-off between convergence speed and stability by focusing on “static snapshots”. We give guidelines towards the choice of the tradeoff parameter α and round length %. We conclude by showing that achieved allocations are close to optimal. Stability vs responsiveness: Performance was characterized by two metrics: convergence time tc – shown in Fig. 5(a) – is the mean number of slots that a node requires to converge close to the steady state and normalized standard deviation – shown in Fig. 5(b) – is the standard deviation from the point a node converged to, normalized by the mean value the node converges to. Both metrics were averaged over all nodes and over 60 random networks. The network diameter ranged from 2 to 9 and average degree from 10.9 to 96.5. First, observe that when α is large and % small the algorithm convergences faster. As shown in Corollary 1, the convergence time is exponential to 1 − α, as well as proportional to %. However, small % also result in less stable operation. If % is small, nodes have less

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a 10 minute interval – illustrated in Fig 3 – resulted in networks with average degree of 22.5 and average diameter of 21.0. The traces exhibited not only fast node movement, but also dramatic changes in density. In regions A, B, and C (red circles) in particular, the average node degree was often ten times higher than in other regions. These sudden and severe connectivity variations constituted a very challenging scenario for any rate allocation algorithm. Fig. 4 shows AutoTune’s error over time for three representative values of α. When optimizing for fairness (α = 0.1), both the error mean, max and variance, were larger than when favoring rate-maximization (α = 1). This is due to an interesting phenomenon. Upon their entrance to regions A-to-C, taxis that were venturing in sparse regions suddenly had to contend with dozens of other taxis for the channel. The interval annotated in gray in Fig. 4 depicts such an example. In this example, a taxi entered region C with very high persistence and thus suddenly increased pavg . Favoring fairness, nodes already in region C tried to make their probabilities equal to pavg before realizing that the resulting allocation was not optimal (λ  λ∗ ) and thus decreasing their probabilities again. For large α, errors due to sudden density variations were much smaller. As suggested by our analysis, enforcing fairness takes longer as it requires information over multiple hops (convergence is exponential to 1 − α). In the presence of frequent density variations, optimizing for fairness creates shortterm contention. On the contrary, the minimum error was smaller when favoring fairness. The error was 0.86% for α = 0.1 as compared to 1.3% for α = 1. This is a side-effect of estimation. Even though the allocation strategy was proven to be optimal for α = 1, the estimation techniques used by AutoTune (see lines 6to-8) introduce small inefficiencies.

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ˆi. time to estimate aggregate quantities pˆiavg , n ˆ i , and λ Furthermore, there is a clear dependency between α and stability. When α is small, nodes adapt mostly according to pˆ avg which is not affected by estimation errors. For this reason, when optimizing for fairness AutoTune converges even knowing very little information about its surroundings (% is small). For α = 0.1, less than 12% of neighborhood information was sufficient at each round to guarantee a small standard deviation. For larger α, nodes achieved the same normalized standard deviation by adapting to larger percentages of neighbor probabilities. The results are shown in Table I. Parametrization: AutoTune can be optimized by selecting a suitable round size % dependent on α. We experimentally found that choosing a round length % for which the normalized standard deviation is smaller than 0.1 yields a good balance between stability and convergence speed. The chosen value pairs are shown with white markers in Fig. 5 and are summarized in Table I. These optimized settings were used in all other experiments. α % tc %

0.1 20 480 12

0.3 30 327 19

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0.7 70 429 44

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1 120 569 76

TABLE I. Convergence time tc (in slots) and upper bound on the percentage (%) of neighborhood information used at each round, for selected (α, %) pairs.

Allocation error: Finally, we compared the allocations achieved by AutoTune to the Pareto front of optimal allocations. We tested our algorithm with increasing severity of constraints by using ball-and-chain networks with increasing δn . Figure 6 shows the trade-off between rate and fairness. AutoTune achieved allocations that were very close to the optimal even in the difficult case of δn = 13. For the rightmost AutoTune markers (α = 0.1), the achieved allocation approached the harmonic

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mean fair allocation. For the leftmost markers (α = 1), AutoTune approached the proportionally fair allocation. The mean and maximum rate allocation errors after convergence over all networks and α were 0.78% and 2.8%, respectively. VII. C ONCLUSIONS This paper proposed a decentralized algorithm for rate allocation, tailored to mobile and large-scale networks. Unlike solutions aimed towards static networks, AutoTune operates with aggregate quantities – estimated in constant time. Analysis and evaluation showed that AutoTune converges fast and with limited knowledge of its surroundings. To the best of our knowledge, this has not been achieved before. We should point out however that the main trade off of our work is optimality. While stateof-the-art algorithms for static networks make optimality guarantees, AutoTune does not. In mobile networks, allocating rate optimally is a very hard problem as it requires continuous and instantaneous knowledge of the nodes’ neighborhood. Extending algorithms designed for static networks to cope with the challenges of highly dynamic environments is in the best case not straightforward and in the worst case in-feasible. Even though AutoTune was shown to be a good candidate for mobile networks, we cannot claim its superiority until it is compared to related state-of-theart approaches. This is however non-trivial; the overwhelming majority of rate allocation algorithms are designed for static networks and operate under different assumptions. Porting them to mobile scenarios is a major endeavor. Nevertheless, this comparison in an important milestone that should be addressed in future work. Acknowledgements. Andreas Loukas and Matthias Woehrle were supported by the Dutch Technology Foundation STW and the Technology Program of the Ministry of Economic Affairs, Agriculture and Innovation (D2S2 project). R EFERENCES [1] K. Kar, S. Sarkar, and L. Tassiulas. Achieving proportional fairness using local information in Aloha networks. Trans. on Automatic Control, pages 1858–1863, 2004. [2] D. D. Vergados, D. J. Vergados, A. Sgora, D. Vouyioukas, and I. Anagnostopoulos. Enhancing fairness in wireless multi-hop networks. In Proc. of MobiMedia, 2007.

[3] A. H. Mohsenian-Rad, J. Huang, M. Chiang, and V. W. S. Wong. Utility-optimal random access: Reduced complexity, fast convergence, and robust performance. Trans. on Wireless Communications, pages 898–911, 2009. [4] P. Gupta and A. Stolyar. Optimal throughput allocation in general random-access networks. In Proc. of CISS, pages 1254–1259, 2006. [5] Z. Fang and B. Bensaou. Fair bandwidth sharing algorithms based on game theory frameworks for wireless ad-hoc networks. In Proc. of Infocom, pages 1284–1295, 2004. [6] JW Lee, M Chiang, and AR Calderbank. Utility-Optimal Random-Access Control. IEEE Transactions on Wireless Communications, pages 2741–2751, 2007. [7] Z. Fang and B. Bensaou. A novel topology-blind fair medium access control for wireless LAN and ad-hoc networks. In Proc. of ICC, pages 1129–1134, 2003. [8] Y. Su and M. van der Schaar. Towards efficient, stable, and fair random access networks: A conjectural equilibrium approach. In Proc. of GlobeCom, pages 1–6, 2010. [9] L. Tassiulas and S. Saswati. Maxmin fair scheduling in wireless ad-hoc networks. IEEE Selected Areas in Communications, pages 163–173, 2005. [10] X. L. Huang and B. Bensaou. On max-min fairness and scheduling in wireless ad-hoc networks: analytical framework and implementation. In Proc of MobiHoc, pages 221–231, 2001. [11] X. Wang, K. Kar, and J. Pang. Lexicographic maxmin fairness in a wireless ad-hoc network with random access. In Proc. of CDC, 2006. [12] A.H.M. Rad, H. Jianwei, C. Mung, and V.W.S. Wong. Utility-optimal random access without message passing. IEEE Trans. on Wireless Communications, pages 1073– 1079, 2009. [13] L. Kleinrock and F. Tobagi. Packet switching in radio channels: Part I–Carrier sense multiple-access modes and their throughput-delay characteristics. IEEE Trans. on Communications, pages 1400–1416, 1975. [14] J. Mo and J. Walrand. Fair end-to-end window-based congestion control. Trans. on Networking, pages 556– 567, 2000. [15] R. Jain, D. Chiu, and W. Hawe. A quantitative measure of fairness and discrimination for resource allocation in shared computer systems. Technical report, 1998. [16] V.G. Iyer. Adaptability in Dynamic Wireless Networks. PhD thesis, Delft University of Technology, 2012.

A PPENDIX Let Pij = 1/ni a random walk matrix. The corresponding Markov chain, which is time-homogeneous, irreducible, aperiodic, and ergodic, converges to a unique stationary distribution π. According to the PerronFrobenius theorem, P has a unique largest eigenvalue µ1 (P ) = 1 and |µi6=1 (P )| < 1. Since 0 ≤ 1 − α < 1, the largest eigenvalue of (1 − α) P is always smaller P∞ than 1 and (1 − α)t P t = (I − (1 − α)P )−1 . t=0 F = I − (1 − α)P has the same eigenvectors u as P and that its eigenvalues µ(F ) = 1 − (1 − α)µ(P ). From definition, P u = 1 − µ(F )/(1 − α)u = µ(P ) u. Applying eigenvalue decomposition, Eq. (4) becomes p(t) = U M U −1 D−1 1, where U and U −1 are the right and left matrices of eigenvectors u and u−1 , respectively, and M is a diagonal matrix with Mii = λ∗ α/ (1 − (1 − α) µi (P )).