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Joint Approximation of Information and Distributed Link-Scheduling Decisions in Wireless Networks
arXiv:1201.2575v1 [cs.LG] 12 Jan 2012
Sung-eok Jeon, and Chuanyi Ji
[email protected],
[email protected]
Abstract—For a large multi-hop wireless network, nodes are preferable to make distributed and localized link-scheduling decisions with only interactions among a small number of neighbors. However, for a slowly decaying channel and densely populated interferers, a small size neighborhood often results in nontrivial link outages and is thus insufficient for making optimal scheduling decisions. A question arises how to deal with the information outside a neighborhood in distributed link-scheduling. In this work, we develop joint approximation of information and distributed link scheduling. We first apply machine learning approaches to model distributed link-scheduling with complete information. We then characterize the information outside a neighborhood in form of residual interference as a random loss variable. The loss variable is further characterized by either a Mean Field approximation or a normal distribution based on the Lyapunov central limit theorem. The approximated information outside a neighborhood is incorporated in a factor graph. This results in joint approximation and distributed link-scheduling in an iterative fashion. Link-scheduling decisions are first made at each individual node based on the approximated loss variables. Loss variables are then updated and used for next link-scheduling decisions. The algorithm repeats between these two phases until convergence. Interactive iterations among these variables are implemented with a message-passing algorithm over a factor graph. Simulation results show that using learned information outside a neighborhood jointly with distributed link-scheduling reduces the outage probability close to zero even for a small neighborhood. Index Terms—Distributed Link-Scheduling, Wireless Networks, Mean-field approximation, Lyapunov central limit theorem, Message Passing, Factor Graph.
I. I NTRODUCTION Efficient scheduling and channel assignment are fundamental for optimal resource utilization in wireless networks [3][12][17]. Due to infeasible centralized control, distributed link-scheduling is inevitable where individual nodes decide when and which channel to access. Most of the existing distributed algorithms make scheduling decisions at individual nodes assuming all management information in the network is available [5][7][22]. In large wireless networks, this assumption does not hold as it is non-scalable for nodes to possess and exchange complete information across the network. Distributed scheduling with limited information is thus desirable. There exist lots of prior works on distributed optimal control using only local interactions among neighbors (see [16][28] and references therein). However, an objective function used is often local by nature; and thus distributed control using local information is naturally optimal. In this work, we focus on situations where an objective function is non-local by nature. Thus, distributed management
with limited information is not guaranteed to be optimal[8]. Scheduling in multi-hop wireless networks is such a scenario, where interference can cause long-range spatial dependence [13][23]. Consider residual interference which is the total interference from interferers outside a certain distance (i.e., a neighborhood) from a node. It has been shown that the residual interference cannot be always ignored depending on channel conditions, density of interferers, and transmission power [2][5][13]. This implies that a node needs to possess information outside its neighborhood range, e.g., on residual interference, when making cooperative scheduling-decisions. Otherwise, decisions on channel-access made by a node may cause significant interference even at faraway nodes in a network. In our prior work, we have developed a distributed scheduling framework with information exchange only among neighbors. Information, i.e., on residual interference, is entirely ignored outside a neighborhood range. We have derived sufficient conditions that such distributed algorithm is nearoptimal when the aggregated spatial dependence outside a neighborhood is sufficiently weak. However, the scenarios are not studied that the algorithm can be non-optimal when the spatial dependence is of long-range and strong. In this work, the residual interference is not ignored but approximated by each node outside its neighborhood. The approximated residual interference is then used jointly with information exchange from neighbors in making distributed scheduling decisions. Our approximation is derived from an exact model of distributed scheduling decisions. As the first step, we provide a network model for link-scheduling. The model is based on a Boltzmann distribution and probabilistic graphical models in our prior work [13]. The model is exact under given assumptions, and thus includes long-range spatial dependence among all nodes in a network. We then consider two approximations to the long-range dependence, i.e., residual interference, outside a neighborhood range. The first approximation views the aggregated residual interference as a deterministic mean-field parameter, and obtains the parameter value through solving coupled mean-field equations. The second approximation treats the aggregated residual interference as a random variable, referred to as loss variable. We show that the loss variable can be approximated by a normal distribution through the Lyapunov central limit theorem (CLT) [27]. The mean and the variance of the normal distribution are then learned using information within a neighborhood range.
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The approximations are combined with iterative and distributed link-scheduling decisions. In particular, a messagepassing algorithm on the factor graph [18] is combined with the approximations of the long-range residual interference for the distributed and statistical link-scheduling. The approximated mean-field parameters and the normal residual random variable form particular local functions in the factor graph. The Mean Field approximation or the distribution of the loss variables are learned from the available information within the neighborhood range. Distributed decisions are then implemented through alternating between approximation and message passing over the factor graph. This process repeats until convergence. Simulation results show that distributed link-scheduling with an arbitrarily small neighborhood can satisfy the global SINR requirements while maximizing the channel reuse. The rest of the paper is composed as follows. In Section II, we discuss the prior works. Section III provides motivating examples on distributed scheduling using complete, partial and approximated information. In Section IV, we provide a problem formulation. Section V provides an accurate probabilistic model of distributed link-scheduling decisions at individual nodes. In Section VI, we obtain two approximations on residual interference, and the simplified probabilistic models. In Section VII, we derive distributed message-passing algorithm coupled with the approximation. In Section VIII, we provide simulations to show the performance of the algorithms using two approximations. Finally, Section IX concludes this study and points to open questions. II. P RIOR W ORK The link-scheduling has been studied as integer programming and graph coloring in a centralized fashion [1]. For example, spatial-reuse TDMA (STDMA) link-scheduling is accomplished using centralized optimal linear programming [1]. The complexity of this approach in terms of information from a network is exponential to the number of links. To reduce the complexity, heuristic centralized algorithms (e.g., column generation methods) are proposed. However, most centralized heuristic algorithms still do not scale well. Distributed link-scheduling methods are thus studied as an alternative. Using simple contention methods such as the protocol model in [10], scalable and distributed scheduling algorithms are developed. For example, scalable algorithm are developed based on graph coloring in [7][22]. The protocol model in [10] is shown to be useful for distributed algorithm but insufficient to reflect a realistic communication environment [13]. For example, each node maintains the configuration information from an entire network, which is infeasible for distributed link-scheduling. There are distributed link-scheduling methods [1] that use an approximated model by simply ignoring the residual interference outside a pre-defined distance. Such approximation imposes locality to distributed link scheduling, and thus only requires information exchange with neighbors. However, it is known that the residual interference cannot always be ignored especially for strong interference [2][13][23]. Furthermore,
systematic study is lacking on how large a sufficient neighborhood is for optimal distributed link-scheduling in a wireless network. In our prior work [13], we show that residual interference outside a neighborhood cannot be ignored especially for slowly decaying channels and densely populated nodes. However, how to incorporate the information outside a neighborhood is not studied there. III. D ISTRIBUTED L INK -S CHEDULING VARIABLES
WITH
L OSS
In this section, we provide motivation on the importance of information outside a neighborhood. We first present an example for distributed scheduling with complete information. We then discuss performance degradation for distributed scheduling with partial information. This motivates two approximation schemes that take into consideration neglected information in form of loss variables. An overview is then given on the approximation schemes to motivate the subsequent sections. A. Distributed Decisions with Complete Information First, we consider distributed scheduling in an ideal setting, where complete information is available for individual nodes to make link-scheduling decisions independently [6][25]. The complete information includes positions, transmission statuses (i.e., who are transmitting to whom), transmission power, and scheduling decisions of links for all nodes in a network. A node then determines whether to access a channel at a timeslot. For simplicity, this work does not consider power control and assumes fixed transmission power. Consider an example of a linear network in Figure 1. Assume that the channel-contention constraint requires any active link between two adjacent nodes to be separated by one neighboring link. The interference constraint requires any active link to be separated by at least two links. Assume also that the network achieves the spatial channel-reuse maximization where the total number of concurrent active links is maximized. 1 Fig. 1.
2
3
4
5
6
7
8
9
10
A Linear Topology with 10 Nodes and 9 Directional Links.
Figure 2 shows all possible configurations of active links. Each row corresponds to a snapshot of active links at a time epoch, where the active links satisfy the aforementioned contention and interference requirements. For example, the first row denotes a configuration where both link (3, 4) and (8, 9) are active. There are 2 configurations with two active links. The remaining 8 configurations have three active links. For distributed decisions where each node has complete information on the activities of the others, only the configurations with three active links are possible. For distributed decisions where each node only knows the activities of its neighbors, the patterns for two active links are also possible. In our prior work [13], we show that an optimal configuration,
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e.g., with three active links, can be obtained through fully distributed node decisions with only local information. For example, if two links (2, 3) and (7, 8) are active at the beginning, a pattern of 3 active links can be obtained if the distributed algorithms are randomized, i.e., nodes make probabilistic decisions for transmission. However, our prior work [13] does not study sufficiently when partial information in distributed scheduling is insufficient for an obtaining optimal configuration.
Under the above assumption on the neighborhood size, it is legitimate for node 3 to transmit to node 4 when links (1, 2), (6, 7) and (9, 10) are active (see the eighth row in Figure 2). Those active links are outside the neighborhood system and thus invisible to node 2. Active link (3, 4) thus generates interference, violating the SINR constraints at the existing active links.
C. Importance of Residual Interference (9 ,10)
Possible configurations. “→”: Active link
B. Distributed Decisions with Partial Information Now consider a realistic scenario where a node only has partial information, i.e., from its neighbors [1][13] but not faraway nodes. Specifically, the partial information available to a node is on positions and channel-access schedule of links (who are transmitting to whom) only within a neighborhood. The actual shape of a neighborhood for each node does not need to be circular nor the same but can be bounded by a circular area with a constant radius. Hence a node determines its channel access schedule, i.e., whether to access a channel at a time slot, using all the information within the neighborhood. The decision made is cooperative and based on two criteria. The first is whether the SINR requirements are satisfied at the neighboring links if the node uses the channel. The second is to maximize the spatial channel-reuse, i.e., the number of concurrently active links in the network. Decisions are made in a distributed fashion. A decision made by a node is sent to its neighbors. The neighbors use the information to make their own link-scheduling decisions accordingly, and then send the decisions to their neighbors. Distributed decisions are made iteratively and collectively by all nodes in a network until convergence is achieved. Refer back to the example in Figure 2. Assume the neighborhood for each node is the adjacent link. This means that the neighborhood is only determined by the contention constraint where a node would not access the channel if its neighbor is transmitting. However, the SINR constraint may be violated. This is because the neighborhood size is smaller than the desired one. When determining the channel access, the node does not know whether SINR is satisfied at its neighbors’ neighbors.
The above example shows that for a finite (and often small) number of neighbors, the SINR requirement is not guaranteed [13][23]. In other words, information within a neighborhood can be insufficient for making correct distributed decisions when the information outside the neighborhood is neglected. To further illustrate the importance of information outside a neighborhood, we consider another example of a multi-hop wireless network. The network has nodes located in a square area of 10 x 10 square meters, L = 200 links, α = 4 as the channel attenuation factor, and SINR threshold SINRth = 10. Consider the information outside of a neighborhood is aggregated as residual interference. Suppose we ignore the residual interference outside a neighborhood range, which is often inevitable in large wireless networks. We then calculate the link outage probability which is the probability that the SINR requirement is violated at a link. The outage probability is shown to be significant and not negligible as in Figure 3.
0.30
(8 , 9)
0.25
(7 , 8)
0.20
(6 , 7 )
0.15
(5 , 6)
0.10
(4 , 5)
0.05
(3 , 4)
0.00
Fig. 2.
(2 , 3)
Link Outage Probability
(1 , 2)
3
4
5
6
7
8
9
Neighborhood range γf
Fig. 3.
Link Outage probability.
Note that the link outage due to the ignored residual interference is different from the transient outage due to the fading channel. The link outage here is persistent; thus, the network connectivity may not be guaranteed. Some nodes can even be permanently disconnected if adjacent links are persistently outaged. This example clearly shows that the residual interference cannot be arbitrarily ignored in a large wireless network.
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D. Approximation of Residual Interference Now consider that each node accounts for information outside its neighborhood in the form of the aggregated residual interference. The aggregated residual interference is referred to as a loss variable for each node. The loss variable treats the impact from the rest of a network outside of the neighborhood of a node in aggregation. Two approaches are used for the approximation. The first is to characterize the expected value of a loss variable through the deterministic mean-field approximation [19]. That is, the expected value of the loss variable can be obtained iteratively through updating deterministic mean-field equations. The second is to regard the loss variable as random and characterize that by a probability distribution. The probability distribution can be learned iteratively using the samples from nodes and links within the neighborhood. Obtained approximation is then incorporated into distributed scheduling decisions. First, each node determines either the distribution or the expected value of its loss variable. The node then updates its channel-access schedule using both information within its neighborhood and on the loss variable. The algorithm is to be described in Section VII. IV. P ROBLEM F ORMULATION Distributed scheduling with loss variables requires a problem formulation. A. Notations Consider a wireless multi-hop network with n nodes. Xi and Pi denote the position and the transmission power of node i, respectively. Let X = {X1 , · · · , Xn }, where bold characters are used to denote vectors in the rest of the paper. Let L denote a set of communication links that access a common wireless channel. Let σij represent the activity of link (i, j) ∈ L at a time instance: σij =1 if link (i, j) is active, i.e., using the common wireless channel; σij =0, otherwise. Let σ = {σij } where (i, j) ∈ L. We assume that each node maintains complete information within a neighborhood range. The neighborhood range does not need to be the same for all nodes but can be bounded by a radius γf . Hence, without loss of generality, γf is assumed to be the neighborhood range for each node. For link (i, j), the channel gain from node i to j is denoted as G(i, j). For simplicity, G(i, j) = |Xi − Xj |−α , where α is the channel power attenuation factor and 2 ≤ α ≤ 6. G(i, j) is random when positions Xi ’s are assumed random. Rij is the inverse of SINR of an active link (i, j) at a time instance, P mn6=ij Pm G(m, j)σmn + ηj , (1) Rij = Pi G(i, j) where ηj denotes the noise power at the receiver j. For simplicity, all nodes assume to have the same (random) noise power ηj = Nb . To satisfy the SINR constraint, Rij ≤ Rth holds, where Rth = 1/SINRth is the inverse of required SINR threshold SINRth . Let Rlij be the approximation of Rij that
includes complete information within the neighborhood of node i plus the residual interference outside, P |Xm −Xj |≤γf Pm G(m, j)σmn + ηj + Resij l , (2) Rij = Pi G(i, j) where Resij denotes the approximated value of residual interference outside the neighborhood range experienced by the receiver j of active link (i, j). In this work, we model Resij with h∗ij by Mean-Field approximation or with nij by a normal variable approximation. For an active link, the distance between the closest interferer and the receiver of the active link is referred to as the contention range of the active link. We denote the contention range as rc . B. Objectives The objective of link scheduling is to maximize the spatial channel-reuse at a time instance, while satisfying the SINR requirement of active links, i.e., X (R0 − Rij )σij (3) maximize ij
subject to
Rth ≥ Rij , for ∀ σij = 1,
where Rij is the inverse of the interference at receiver j from (1). R0 is a large positive constant where R0 > Rij for ∀ (i, j) ∈ L. R0 is used to guarantee the objective function to be maximized whenever a new link becomes active. The first P σ is the total number of active links, and used term ij ij to maximize the channel reuse in the network. The second P term ij Rij σij represents the effect of total interference from active links. The objective function in (3) satisfies the following three properties: (i) constrained to satisfy the SINR requirement of any active links; (ii) monotonically increasing as the number of active links increases; and (iii) monotonically increasing as the inverse SINR of an active link decreases. Therefore, the two terms in the objective function make trade-offs between the channel reuse and the SINR requirements. To measure the optimality of the objective function, we use the link outage probability P (SINRij < SINRth ). As soon to be shown, a limited neighborhood size is a major cause of significant link outage probability considered in this work. C. Probabilistic Decisions The optimization problem is implemented through distributed decisions at individual nodes. That is, nodes determine whether and when σij ’s become active. Decisions made by a node need to account for randomness in loss variables and decisions by other nodes. Furthermore, deterministic distributed and localized decisions can be easily trapped in local minima. Hence, we use probabilistic, i.e., randomized decisions (see [8] [11] [13] and references therein). A node makes linkscheduling decisions iteratively. At each time epoch, nodes make the link-scheduling decisions based on a probabilistic model of the neighboring nodes’ link-scheduling decisions at the previous time epoch and the loss variable. This requires a probabilistic decision model.
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V. P ROBABILISTIC M ODEL In this section, we derive a probabilistic model of distributed link-scheduling decisions and the residual interference in form of Boltzmann distribution and a dependency graph. The model is based on our prior work [13], and included here for completeness.
A. Boltzmann Distribution Our model considers σij ’s as a random field of link activities. Such a random field resembles a particle system with interacting particles (spins) [29]. At a high-level, particles can be viewed as general random variables. A configuration ω of interacting binary-state particles can be described with a system potential energy H(ω). The probability distribution of H(ω) obeys the Boltzmann distribution P (ω) = Z0−1 · P exp (−H(ω)) [8][29], where Z0 = ω exp (−H(ω)) is a normalizing constant. We now regard the objective (3) as the system potential energy and appeal to Boltzmann law in statistical physics [8][13][20][28][29]. In particular, the system potential energy of binary link-scheduling decisions is defined as, X X H(σ)
=
(−R0 + Rij )σij + β
ij
=
σ12
σ23
σ34
σ45
σ56
σ67
σ78
U (Rij − Rth )σij (4)
ij
X ij
B. Dependency Graph The probability model corresponds to a graphical representation [13][14][18]. The graphical representation shows the statistical spatial dependence among links. For simplicity of illustration, consider a directional linear topology of eight nodes and seven directional links, where the directions correspond to a data flow. The corresponding dependency graph of link-scheduling decisions σ is shown in Figure 4. Nodes in the graph represent binary random variable {σij } for σij ∈ {0, 1} and 1 ≤ i ≤ 7 and j = i + 1, and a link between two nodes represents the spatial dependence. The spatial dependence results from interference in the wireless channel and corresponds to coupled terms in the system potential energy H(σ). The dependency graph can be further represented in form of a factor graph shown in Figure 5. A factor graph is composed of variable nodes and functional nodes [18]. In Figure 5, σij denotes a variable node for link (i, j) ∈ L, and a functional node ψij = Rij denotes a potential function of σij .
Nb (−R0 + −α )σij Pi lij
+
−α X X Pm lmj
+
X hij
ij
ij
mn∈Nij
−α Pi lij
−α σij Pi lij
+β
Fig. 4.
σij σmn
X
U (Rij − Rth )σij ,
ij
where β is a large positive constant, βU (Rij − Rth ) is a penalty term for the SINR constraint, Nij denotes the set of links within the neighborhood range from receiver j of link (i, j), and hij denotes total residual interference outside the neighborhood range experienced by active link (i, j), i.e., hij P −α σmn . = mn6∈Nij Pm lmj From Boltzmann’s law [8][13], the probabilistic model for a set of link-scheduling decisions at a time instance is, P (σ) = Zσ−1 · exp (−H(σ)) ,
(5)
where σ is a random field P (i.e., configuration) of all link activities σij ’s, and Zσ = σ exp (−H(σ)) is a normalizing constant. Note that both the number of active links and interference play a role: If there are multiple configurations that have the same number of active links, the system potential energy of each configuration is distinguished by the sum of the inverse SINR of active links. A desirable configuration corresponds to a small system potential energy, and a large Boltzmann probability [8]. Hence, an optimal decision σ ∗ is the one that maximizes the Boltzmann distribution, σ∗
= arg max P (σ). σ
(6)
Dependence Graph of random field σ
ψ12
ψ23
ψ34
ψ67
ψ78
σ12
σ23
σ34
σ67
σ78
Fig. 5. Factor Graph Representation of Dependence of Distributed ChannelAccess of Links in Figure 4.
Link-scheduling decisions are all dependent due to interference; thus the dependency graph is fully connected. Such a fully connected graph shows a less interesting case of a Markov random field, where the neighborhood of a node is the entire network. C. Optimal Configuration and Distributed Decisions A fully connected dependency graph impacts how distributed decisions are conducted using the probabilistic model. As node decisions are all dependent, optimal decisions require nodes to exchange information with all the other nodes in the network. Graphically, this means that an entire network is a neighborhood. That is, a node needs to obtain information on all other nodes. This, in reality, prohibits implementation of optimal distributed link-scheduling.
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VI. A PPROXIMATION
OF
P ROBABILISTIC M ODEL
We now derive approximations of the probabilistic model for distributed scheduling. Our approximation considers a (small) neighborhood range of each node where complete information is available to a node. Outside the neighborhood, instead of ignoring the residual interference, we characterize the residual interference as a random variable, i.e., the loss variable. The loss variable is then included in the probabilistic model. The resulting probabilistic model P l (σ) is an approximation of the exact model P (σ). A. Approximation by Mean Field We first characterize the residual interference by Mean Field (MF) approximation. The MF approximation obtains expected values of the loss variables through solving iteratively deterministic mean-field equations [15]. To derive the meanfield equations, we consider the system potential energy in (4). For simplicity, we assume that the SINR constraint is satisfied so that the system potential energy is reduced to X H(σ)
=
fij (σ)
(7)
ij
=
X
aij σij +
ij
ij
+
X X
X
aij,mn σij σmn
mn∈Nij
hij , Pm l−α mj
Nb , P l−α ij
where aij = −R0 + aij,mn = P l−α from (4), and i ij P hij = mn6∈Nij aij,mn σij σmn corresponds to the aggregated residual interference outside the neighborhood for link (i, j). Note that this system potential energy represents a secondorder Ising model in statistical physics, where all terms are dependent. Let h∗ij be a deterministic approximation of hij . An approximated system potential energy absorbs the impact of the aggregated residual interference into the coefficient aij ’s, X l =
β F˜
Q(σ) )) − ln(ZP ) P (σ)
=
EQ (ln(
=
EQ (ln(H(σ)) − ln(H l (σ))) − ln(ZQ )
=
− ln(ZQ ) +
X
(−hij µij ) +
ij
(11)
X X ij
aij,mn µij µmn ,
mn6∈Nij
where µij =E(σij )=P (σij = 1). Inserting Equation (10) in ∂ F˜ (11) and letting ∂h =0, the optimal value of hij is ij X aij,mn µmn . (12) h∗ij = mn6∈Nij
h∗ij is a weighted sum of expected values of link activities outside the neighborhood range, i.e., the mean field outside the neighborhood of link (i, j). As a special case when aij,mn = 0 for mn 6∈ Nij , h∗ij = 0. For (i, j) ∈ L, there are n such equations, where n is the total number of nodes in a network. Those equations are coupled. The coupling can be shown explicitly by rewriting 1−µ h∗ij as h∗ij = log µijij ηη12 , and then (12) in the following form (see Appendix I for derivations), X 1 − µij η1 log aij,mn µmn , (13) = µij η2 mn6∈Nij
ij
H (σ)
by minimizing the following objective function, i.e., variational free energy [15],
gij (σ)
(8)
where η1 and η2 are functions of h∗kl ’s for (k, l) 6= (i, j) (and in turn µkl ’s). Hence, the set of n equations are coupled through unknown values of h∗ij ’s (or µij ’s). The coupled mean-field equations can be solved iteratively and numerically starting from an initial condition [15]. The mean-field entity h∗ij can also be estimated when measurements are available. For example, the interference within the neighborhood range can be known with the information exchange. Thus, if we assume that the aggregated interference can be measured possibly with a measurement error, h∗ij can be estimated when interference measurements are available.
ij
=
X ij
bij σij +
X X ij
B. Approximation by Lyapunov Central Limit Theorem
aij,mn σij σmn ,
mn∈Nij
where bij = aij + h∗ij , and h∗ij is yet to be found. We denote the true free energy [15] with F , βF = − ln(ZP ),
(9)
where β is a large positive constant, and ZP = P exp(−H(σ)). Let variational free energy [15] be F˜ , σ β F˜ = − ln(ZQ ), (10) where β is a large positive constant, and ZQ = P l σ exp(−H (σ)). To find the simplest, i.e., the first-order Mean Field (MF) approximation, we make the following assumptions: (a) for link (i, j) 6= (m, n), the impacts only affect the first order terms; (b) for (m, n) 6∈ Nij , σij is independent of σmn ; and (c) for the approximated system potential energy, the −1 corresponding distribution is Q(σ) = ZQ exp(−H l (σ)). The approximated distribution is made close to the true distribution
Mean-filed approximation is deterministic that considers the expectation of the aggregated information outside a neighborhood. An alternative approach is to treat the aggregated residual interference as random and approximate its distribution. A motivation for such approximation is that the meanfield can be harder to estimate since measurements on the aggregative residual interference is not always available. This is because such measurements require nodes to transmit test signals on their timeslots during link-scheduling decisions, and in addition, the nodes need to be capable of measuring the aggregated interference. For example, in [21], packet reception ratio (PRR) versus SINR models need to be seeded by O(n) trials in an n-node network where each node transmits while receivers measure the channel conditions. Furthermore, it can be computationally intensive to solve n coupled mean-field equations. Now, let nij be a random variable that approximates the residual interference of an active link (i, j) . Our goal is to
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identify a probability distribution for nij . As residual interference aggregates interference signals from many interferers, one possibility is to approximate nij as a normal random variables. We first obtain an expression for nij to understand whether a normal approximation is feasible. Based on our assumptions, interferers have the same transmission power P0 . Locations of interferers outside the neighborhood range can be arbitrary/random but on the average follow the pattern of node positions within a neighborhood. Specifically, the following assumptions are posed on the location of interferers. • For an active link (i, j), all interferers are located uniformly within radial bands bounded by concentric circles from receiver j. The radius of the circles are rk = rk−1 +xk for k ≥ 1, where r0 =γf , and xk =x is a uniform random variable between rl and ru . Here, rl and ru are the minimum and maximum distance between any two active links which is the distance between the interferer source and the signal receiver, within the neighborhood of link (i, j). The circle with radius rk is called the k-th frontier. • On the k-th frontier for k ≥ 1, any two neighboring active links are separated by x0 . Random variables x and x0 are independent and identically distributed.
a product of marginal distributions, ∞ Y
P (ri | < ri−1 >),
where < ri−1 > = γf +(i−1)r c denotes the expected value of l u ri−1 . r c is an expected value of rc , i.e., r c = E(rc ) = r +r 2 , and < r0 >=γf . The proof can be found in Appendix III. Using such a meanfield distribution, we have the expected value E(nij )
=
Z
r1 ,···,r∞
Fig. 6.
2πrk · x0
(16)
where P (r1 | < r0 >)=P (r1 |γf ). Closed-form expressions can be derived as follows for the individual expectations in the sum. Lemma 3: Let uk be the term in the summation for nij in (14), where uk = 2πP0 rk1−α /γf = 2πP0 (rk−1 + xk )1−α /γf . Denote the mean and the variance of uk with E(uk ) and V (uk ). For α > 2, 2πP0 (D12−α − D22−α ) , (2 − α)(ru − rl ) E(u2k ) − E(uk )2 ,
E(uk ) =
k) and V (vk ) = Vr(u u rl . The proof is given in Appendix IV. Using the results from Lemmas 1-3, we obtain the mean and the variance of the residual interferences below.
Lemma 1: With the above two assumptions, the residual interference hij for active link (i, j) can be modeled as nij , ∞ X
(P0 rk−α ) ·
x0 k=1
E(uk ) ln(r u /r l ) , r u −r l
rc
Configuration of Active Links outside Neighborhood range.
nij =
∞ X
where D1 = γf + (k − 1)rc + ru , D2 = γf + (k − 1)r c + rl , and E(u2k ) = 4π 2 P02 (D13−2α − D23−2α )/((3 − 2α)(ru − rl )). Let the k-th term of nij be vk = uxk0 . Then, E(vk ) =
γ f + (k − 1)rc
γf
Z
P (rk | < rk−1 >)P (x0 )dr1 · · · dr∞ dx0 ,
V (uk ) =
θ
(15)
i=1
(P0 rk−α ) · (2πrk /x0 ),
(14)
k=1
where rk = rk−1 + xk with r0 =γf as a constant, and xk = x is a random variable for k ≥ 1. x and x0 both follow a uniform distribution between rl and ru . xk and xm are independent for k 6= m. The proof can be found in Appendix II. This lemma means that random positions of interferers outside a neighborhood can be predicted from those within the neighborhood. Next, we derive the mean and the variance of nij ’s. For analytical feasibility of obtaining the mean value, we approximate the joint probability density P (r1 , · · · , r∞ ) with a product form based on the Mean-field approximation [19]. Lemma 2: Consider {r1 , · · · , r∞ } as a random field. Based on the first-order Mean-field approximation, the joint probability density function P (r1 , · · · , r∞ ) can be approximated as
Theorem 1: For a link (i, j), residual interference nij is a finite sum of independent random variables with different probability distributions. Thus, the probability distribution of nij is approximately normal P with mean E(nij ) ∞ and variance V (n ), where E(n ) = ij ij k=1 E(vk ), and P∞ V (nij ) = k=1 V (vk ). The proof is provided in Appendix V. How accurate is the normal approximation? We check the modeling error using Lypunov condition. For nij to converge to a normal distribution, the Lyapunov condition on nij = P ∞ k=1 vk should satisfy P∞ 3 1/3 k=1
E(|vk − E(vk )| )
P∞
k=1
V (vk )
1/2
→ 0.
(17)
Such condition is valid for an infinite sum of independent random variables. As the interference from far apart interferers is diminishing to zero, nij is effectively a sum of a finite number of independent random variables. As a result, the Lyapunov condition converges to a small constant instead of zero. A numerical analysis on the normal distribution modeling is conducted on a network with infinite nodes and with the increase of neighborhood size to k-th frontier neighbors, and channel attenuation parameter α = 4. Figure
8
7 shows that the Lyapunov condition converges to a small constant around 0.97.
1.16 1.14 1.12
where n is a random vector including all nij ’s. This approximation is a random-bond Markov Random Field [13], where nij ’s are random coefficients (bonds) for σij ’s. As distributed decisions only need to determine the values of σij ’s, the marginal probability model for σij ’s is Z P l (σ) = P l (σ, n)dn. (22)
Lyapunov Condition
1.1
VII. M ESSAGE -PASSING
1.08
1.04 1.02 1 0.98 0.96
Fig. 7.
0
50
100 K−th Frontier
150
200
Lyapunov Condition of the residual interference nij
C. Approximated Potential Energy and Probabilistic Model We now summarize the approximated network potential energy and the resulting probabilistic model from the two approximations. First, from the mean-field approximation in Section VI-A, the residual interference is approximated as a (deterministic) mean-field outside the neighborhood. The approximated system potential energy is X gij (σ) (18) H l (σ) = ij
=
FACTOR G RAPH
σ ˆ = arg max P l (σ). (23) σ Randomized and distributed decisions implement such optimization through message passing over localized factor graphs. The general message passing algorithm is provided in [18] and the references therein. The novelty here is to couple message passing with approximation of the residual interference. Appropriate functional nodes and variable nodes need to be chosen accordingly for such coupling and distributed scheduling. The resulting probabilistic dependency graph is shown in Figure 8. l P(Ψ12 | σNI ) 12
l ψ12
ψl23
ψl34
ψl67
ψl78
σ34
σ67
σ78
P(σ23 | σ N I ) 23
l , σNI ) P(σ12 | Ψ12 12
X
(aij + h∗ij )σij +
X X
aij,mn σij σmn ,
ij mn∈Nij
ij
σ12
h∗ij
where is deterministic and obtained from the meanfield equation. The corresponding approximated Boltzmann distribution is (19) P l (σ) = Zσ−1 exp −H l (σ) ,
which is a second-order Markov Random Field with deterministic coefficients aij ’s, h∗ij ’s and aij,mn ’s. The resulting probabilistic dependency graph is shown in Figure 8. Second, from the normal approximation, the residual interference outside the neighborhood is approximated as a random variable that aggregates contributions from unseen interferers. The approximated system potential energy is, for active link (i, j), X Nb l H (σ, n)
OVER
A simplified probabilistic model P l (σ) from either the mean-field approximation or the normal approximation can be used to determine link activities, such as,
1.06
(−R0 +
=
ij
+
−α Pi lij
)σij
−α X X Pm lmj ij
+β
mn∈Nij
X
−α Pi lij
σij σmn +
Fig. 8. Factor Graph Representation of the Dependence of the Distributed Channel-Access of Links with Local Information.
A. Message Passing with Normal Residual Interference The local potential function in message passing, for exl ample, P (ψij |σNijI ) for link (i, j), exhibits a special form from approximating residual interference as a normal random variable, i.e., l l l P (ψij |σN I ) = N (E(ψij ), V (ψij )),
(20)
X nij ij
−α Pi lij
σij
U (Rlij − Rth )σij .
σ23
ij
l l l l where ψij =Rij , and N (E(ψij ), V (ψij )) denotes a normal distribution with a mean P −α l E(ψij )=
|Xm −Xj |≤γf
P0 lmj σmn + Nb
−α P0 lij
+
ij
The corresponding approximated Boltzmann distribution is P l (σ, n) = Zσ−1 exp −H l (σ, n) , (21)
(24)
1 −α E(nij ), P0 lij (25)
and from (2), a variance l V (ψij )=(
1 2 −α ) V (nij ). P0 lij
(26)
9
l Message from a function node ψij to a variable node σij is l l P (σij = 1|ψij , σN I ) = U (Rth − ψij ), ij
B. Simulation Results
(27)
40
l Message from a variable node σij to a function node ψmn
ij
1 2
+ Q( l
l Rth −E(ψij ) l ) V (ψij
Q( E(ψij )−Rth ), V (ψ l )
),
l E(ψij ) ≤ Rth
otherwise,
ij
Rx
2
(28)
20
−t √1 e 2 0 2π
B. Message Passing with Mean-Field Residual Interference The local function with MF approximation of the residual interference, with estimation of h∗ij in (8), is l l P (ψij = a0 |σN I , h∗ij ) = δ(ψij − a0 ), ij
(29)
l l where ψij =Rij in (2) with Resij =h∗ij in (8), a0 is a constant, l l δ(ψij − a0 )=1, if ψij = a0 ; 0, otherwise. l Message from a function node ψij to a variable node σij is l l ), P (σij = 1|ψij , σN I , h∗ij ) = U (Rth − ψij ij
(30)
where U (x) is a unit step function. l Message from a variable node σij to a function node ψmn is l P (σij = 1|σN I , h∗ij ) = U (Rth − ψij ). ij
(31)
The overall algorithm includes two steps: (a) estimating the mean-field parameter h∗ij ’s, and (b) obtaining decisions σij ’s through message passing. These two steps alternate until convergence. VIII. S IMULATION We conduct simulations for the shared channel access on a multi-hop wireless network. Our goal is to assess the performance of the two approximated distributed algorithms for link-scheduling. A. Simulation Setup Network nodes are positioned uniformly in a square area of 100 square meters, and composed of L = 200 links. We consider 2 ≤ α ≤ 6 for the channel attenuation factor, and 10 ≤ SINRth ≤ 100. We consider the maximum spatial channel-reuse at a time instance.
0
10
where Q(x)= dt. This equation is obtained from R l an equality of P (σij = 1|σNijI ) = P (σij = 1|ψij , σNijI ) l l P (ψij |σNijI )dψij . For example, in Figure 8, since node 2 determines the scheduling of link (2, 3), node 2 maintains only the local factor l graph that is related to link (2, 3), which are ψ23 , σ12 ,σ23 and σ34 . The overall algorithm includes two steps: (a) estimating the parameters for the mean and variance of the residual interference, (b) using the estimated parameters in the local functions for message passing.
30
P (σij = 1|σN I ) =
Percentage
is
50
l l where U (Rth − Rij )=1, if Rth > Rij ; 0, otherwise.
1
2
3
4
5
6
7
Number of Iterations until Convergence
Fig. 9.
Convergence of Distributed Link-Scheduling Decisions
Using the above experimental setting, we first study through simulation whether distributed decisions converge sufficiently fast. Figure 9 shows that the distributed link-scheduling decisions converge through interactions among neighbors with a few iterations. We now compare the performance of two algorithms from the mean-field and the normal approximation with the conventional distributed link-scheduling algorithms [1][7][13] that simply ignore the residual interference outside neighborhood range. The link outage probability is used as a performance measure. Such a performance measure assesses the probability that a link cannot satisfy the SINR requirement, where the link outage is mainly caused by the limited neighborhood size. There, improved performance signifies the importance of proper approximations of information outside a neighborhood. Figure 10 plots the outage probability as a function of neighborhood size γf assuming all nodes have the same neighborhood size. This figure shows that our two approaches have a small link outage probability across different neighborhood range including small γf . The conventional scheme, however, has a large link-outage probability for small γf . This shows that for a small neighborhood γf , learning the residual interference with a loss variable provides a big gain. The performance with mean-field approximation varies with respect to the measurement error of residual interference, which is characterized with zero mean and non-zero variance. When the measurement error is small, the mean-field approximation results in a small link-outage probability across different neighborhood size. Overall, the two approximation approaches outperform significantly the method that ignores the residual interference, showing the importance of approximating the information outside a neighborhood. Figure 11 shows how the link-outage probability varies with respect to channel attenuation factor α. The link outage probability for the two approximation schemes remains small for
0.30
10
0.20 0.15 0.10 0.00
0.05
Link Outage Probability
0.25
Without Loss Variable With Lyapuanov CLT With MF (Measure Error Var=0.001) With MF (Measure Error Var=0.01)
3
4
5
6
7
8
9
Neighborhood range γf
MF approximation for large measurement error gets worse as the SINR threshold increases. This is because the SINR constraint is more easily violated due to the measurement error as the SINR threshold increases. For the proposed scheme with Lyapunov central limit theorem (CLT), once γf is moderately large, the proposed scheme copes well with a large SINR threshold. When neighborhood size γf is not sufficiently large, for a moderate SINR threshold, conventional schemes in the literature often fail to satisfy the SINR requirement, resulting in a high link outage probability. On the contrary, across a wide range of SINR threshold values, the proposed scheme with Lyapunov CLT perform efficiently and improves the link outage probability significantly. In particular, if γf is moderate (e.g., γf = 4 in Figure 13), the link outage probability quickly approaches zero. Therefore, with both approximation schemes, resource utilization is significantly improved overall.
0.6 0.4 0.2
Link Outage Probability
Without Loss Variable With Lyapuanov CLT With MF (Measure Error Var=0.001) With MF (Measure Error Var=0.01)
0.0
different α. As α increases, the residual interference outside the neighborhood decrease, and the link outage probability of the two approximation schemes approaches zero. Overall, the two proposed schemes learn the ignored long-range dependence outside a neighborhood better when the long-range dependence decreases rapidly. For small α, the two approaches significantly outperform the conventional approach, showing the importance and the ability to approximate the aggregated residual interference outside a neighborhood.
0.8
1.0
Fig. 10. Link Outage probability with SINRth =10. Dotted line: Conventional scheme. Other curves: Our approximation approaches
20
40
60
80
100
SINR Threshold
0.4 0.3
Fig. 12. Link Outage probability with γf =3. Dotted line: Conventional scheme. Other curves: Approximation schemes.
0.2
IX. C ONCLUSION
0.0
0.1
Link Outage Probability
0.5
Without Loss Variable With Lyapuanov CLT With MF (Measure Error Var=0.001) With MF (Measure Error Var=0.01)
3.0
3.5
4.0
4.5
5.0
5.5
6.0
Power Attenuation Factor α
Fig. 11. Link Outage probability with γf =4 and SINRth =10. Dotted line: The conventional scheme. Other curves: The two approximation schemes.
Figure 12 and 13 show the performance as a function of SINR. As SINR threshold increases, the link outage probability increases proportionally except for the mean-field approximation with small measurement error. The performance of the
A main theme we have studied in this work is distributed scheduling decisions where each wireless node uses complete information within a neighborhood and approximated information outside. Our study is motivated by the fact that strong interference results in long-range dependence among network nodes. We have characterized the complex spatial dependence of distributed link-scheduling decisions with a factor graph. We then simplify the factor graph to a localized factor graph that exhibits only dependencies among neighboring nodes. Next, instead of simply ignoring the long-range dependence in residual interference, we characterize the aggregated dependence outside a neighborhood with two approximations. One is deterministic Mean Field approximation. The other is random, i.e., normal approximation with Lyapunov central limit theorem. The approximated dependencies are incorporated into the localized factor graph.
11
Without Loss Variable With Lyapuanov CLT With MF (Measure Error Var=0.001) With MF (Measure Error Var=0.01)
0.6
II. P ROOF
OF
L EMMA 1
0.4
On the k-th circle for k ≥ 1, the total number of active k links on the circle is 2πr x0 , which is random. Each active link on the k-th circle results in as much interference as P0 rk−α . As a result, total interference from the active the k-th P∞ links on −α 2πrk k (P r ; and, thus n = circle is (P0 rk−α )· 2πr 0 ij k=1 k )· x0 . x0
0.2
Link Outage Probability
0.8
1.0
Note that η1 and η2 are functions of h∗mn ’s (µij s) for (m, n) 6= (i, j). µij is also a function of h∗mn ’s. Hence, the equation is coupled. In addition, there are multiple equations that contain common h∗ij ’s. Hence, for all (i, j)’s, the meanfield equations are coupled, and can be solved iteratively.
0.0
III. P ROOF 20
40
60
80
100
SINR Threshold
Fig. 13. Link Outage probability with γf =4. Dotted line: Conventional scheme. Other curves: Approximation schemes.
This results in an extended message-passing algorithm over factor graph with the two phases. First, the approximations are obtained iteratively from the available information at neighboring nodes. Second, the approximations are combined with messages from neighboring nodes and allow each node to make its link-scheduling decisions. The algorithm alternates between the two phases until convergence. We have shown through simulations that the link outage probability of the resulting link-scheduling decisions is significantly improved. The improvement is especially profound for distributed decisions with a small number of neighbors. Open issues that have not been considered in this work include possible dependencies among approximated normal random variables outside neighborhoods, and further comparison of the two approximations. A PPENDIX I. D ERIVATION
OF
E QUATION (13)
µij = P (σij = 1), where the probability is evaluated using the simplified Boltzmann distribution Q(σ) from the meanfield approximation. Expanding the expression of µij , we have exp(−h∗ij )η1 , (32) exp(−h∗ij )η1 + η2 P where η1 = exp(−aij ) (m,n)6=(i,j) exp(−H(σ \ σij )) with σ P\ σij being all σmn ’s except for (m, n) = (i, j). η2 = (m,n)6=(i,j) exp(−H(σ|σij = 0)). Representing h∗ij from the above expression, we have µij =
h∗ij = log
1 − µij η1 . µij η2
Replacing this expression in (13), we have X 1 − µij η1 aij,mn µmn . = log µij η2 mn6∈Nij
(33)
(34)
OF
L EMMA 2
Mean field approximation shows that the joint distribution of a random field (y1 , · · · , yn ) can be approximated Q by a product of marginal distributions. Thus, P (y1 , · · · , yn ) = ni=1 P (yi | < yNi >) with a good approximation, where yNi is the set of neighboring random variables of yi . From the assumptions, rk =rk−1 + xk . Thus, P (rk | < rNk >)=P (rk | < rk−1 >). IV. P ROOF
OF
L EMMA 3
The average of uk , Rdenoted with E(uk ), can be derived from an integration of 2πP0 rk1−α P (rk | < rk−1 >)drk . The average of u2k can be derived in a similar way. V. P ROOF
OF
T HEOREM 1
Consider the configuration in Figure 6. For an active link (i, j), the residual interference of this configuration is denoted with a random variable nij , which is a function of random variables rk for k ≥ 1, where rk is the radius of the k-th frontier. Note that rk ’s are independent random variables with different uniform distributions. Lyapunov’s central limit theorem [27] shows that the summation of a large number of independent random variables (even with different distribution) results in a normal distribution. nij is a summation of hundreds and thousands of independent random variables (i.e., rk ’s), thus nij can be approximated by a normal distribution. Furthermore, a normal distribution can be completely characterized by mean and variance. The mean and variance of nij is P denoted with E(nij ) and V (n Pij∞), respectively, i.e., E(nij ) ∞ = k=1 E(vk ), and V (nij ) = k=1 V (vk ). R EFERENCES [1] P. Bjorklund, P. Varbrand, and D. Yuan, “Resource Optimization of Spatial TDMA in Ad Hoc Radio Networks: A Column Generation Approach,” In Proc. Of IEEE Infocom, April 2003. [2] V. Chandrasekhar, J. Andrews, A. Gatherer, “Femtocell Networks: A Survey,” IEEE Comm. Magazine, pp.56-67, Sept. 2008. [3] M. Chiang, “Distributed Network Control Through Sum Product Algorithm on Graphs,” In Proc. of IEEE Globecom, vol. 3, pp. 2395-2399, Nov. 2002. [4] P. Djukic and S. Valaee, “Link Scheduling for Minimum Delay in Spatial Re-Use TDMA,” In Proc. Of IEEE Infocom, April 2007. [5] T. Elbatt, and A. Ephremides, “Joint Scheduling and Power Control for Wireless Ad Hoc Networks,” IEEE Trans. Wireless Communications, vol. 3, pp.74-85, Jan. 2004.
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