Flow Allocation in Multi-hop Wireless Networks:
A Cross-Layer Approach
Kun-Da Wu, Student Member, IEEE, and Wanjiun Liao, Senior Member, IEEE
Abstract— This paper addresses the flow allocation problem
in multi-hop wireless networks. We define and formulate a new interference model, referred to as the Node-based Interference Model, to better capture the behavior of medium access control protocols and the physical layer interference issues. Based on this model, we formulate the problem as a cross-layer network utility maximization problem that considers the coordination of the transport, MAC and physical layers, and avoid the maximum clique or independent set enumeration approach as adopted in most of the existing work. The objective of the problem is to maximize the aggregate network throughput while maintaining the fairness among flows. We then propose a gradient-based flow allocation algorithm by using the duality approach, and analyze the rate of convergence to the optimum for the proposed algo-rithm. The simulation results show that the proposed algorithm can rapidly converge to the optimum, and can also rapidly adapt to the changes in network topology and routing paths in different flow scenarios.
Index Terms— flow allocation, interference, multi-hop wireless
networks.
I. INTRODUCTION
R
ESEARCH in multi-hop wireless networks has received much attention in recent years. In such a network, packets are forwarded in a hop-by-hop manner without the assistance of a pre-deployed infrastructure. Each flow, in addition to contending for local resource at each intermediate node in its routing path, referred to as local interference, must also compete for the shared wireless medium with those flows located within its interference range, referred to as location-dependent interference. Local interference is characterized by the half-duplex property of the wireless transceiver, which means that it can either transmit or receive data at any time, but not both simultaneously; location-dependent interference is characterized by the property of the radio signal reception. These unique characteristics spawn many research challenges in resource management for end-to-end sessions in multi-hop wireless networks. Due to resource contention from different layers, traditional single layer design disciplines lead to inef-ficient performance. This calls for cross-layer design manner [1], [2] to coordinate among the transport, MAC and physical layers so that the resource can be efficiently utilized.Manuscript received July 17, 2006; revised January 8, 2007; accepted May 2 2007. The associate editor coordinating the review of this paper and approving it for publication was X. Zhang. This work was supported by the National Science Council (NSC), Taiwan, under a Center Excellence Grant NSC95-2752-E-002-006-PAE, and under Grant Number NSC95-2221-E-002-066.
The authors are with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan Univer-sity, Taipei, Taiwan (e-mail: wjliao@ntu.edu.tw).
Digital Object Identifier 10.1109/TWC.2008.060480.
The two models which are used widely for describing the location-dependent interference among packet transmissions in single channel wireless networks are the Protocol Model [13] and the Physical Model [13], [14]. Variations to the Protocol Model include the static interference model [15], [16] and the flow-dependent conflict graph [17]. Existing work [11], [20], [22] solve resource allocation by building on the concept of the conflict graph. In a conflict graph, each vertex corresponds to a wireless link and an edge between two vertices exists if the transmissions on the two wireless links contend with each other. A complete subgraph of the conflict graph is referred to as a clique. The maximal clique, representing a maximum set of mutually contending wireless links, is a clique that is not contained within any other cliques. The interference problem is then tackled by the enumeration of the maximal cliques in the network. However, clique computation is NP-complete [23], and worse, the clique constraints are insufficient to guarantee the optimality of link utilization [22]. Additional challenges arise when one attempts to implement these ideas in a distributed manner. Previous work [24]–[27] adopts the Physical Model for resource management. The idea is to optimize the network capacity while satisfying the power constraint of each node. Once the capacity of each link is determined, the flow allocation problem in wireless multi-hop networks can be treated in a similar way as that in wired networks. However, the calculation of the capacity region based on the Physical Model requires selecting the sets of concurrently active communication links. The determination of the sets of concurrently active links is time-consuming because each link in a set will interfere with the other links.
The relationship among the interference caused by wireless communications, the supportable data rate of a node, and the end-to-end flow rate control problem have not been explicitly addressed in all of the existing work. The MAC issues caused by the interference due to simultaneous transmissions have also not been addressed and characterized. This calls for new mechanisms which jointly consider these relations without complicated computation. The following two issues motivate our work: 1) avoiding the enumeration of maximum cliques or the sets of concurrently active links when considering location-dependent interference; 2) providing a general approach which accounts for the interference constraints in MAC protocol designs in arbitrary network topologies.
In this paper, we study the flow allocation problem in wireless multi-hop networks. The objective of this study is to optimize global resource allocation by maximizing the aggregate utilization of wireless resource with coordination between the transport, MAC and physical layers. To achieve 1536-1276/08$25.00 c 2008 IEEE
this objective, we define and formulate a new interference model, referred to as the Node-based Interference Model, which accounts for MAC protocols and captures the behavior of local interference and location-dependent interference for multi-hop wireless networks. This model enables each node to locally identify the interference that occurred at the phys-ical layer and contention behavior at the MAC layer only through signal power measurement. Therefore, the complexity of mutual interference and contentions among neighboring nodes can be reduced while the key factors of physical and MAC layers can be characterized. Compared with the Protocol Model and Physical Model, this model can simplify the cross-layer design for network planning and wireless resource management, and can characterize the relationship among interference, data rate, and medium access contentions. Based on the Node-based Interference Model, we formulate the optimal flow allocation problem as a convex optimization problem such that 1) the behavior of the interference and the supportable data rate at the physical layer, medium contention at the MAC layer, and end-to-end flow issues at the transport layer can be jointly considered, and 2) the clique or the independent set computation can be eliminated with the node-based interference constraints. We then propose a gradient-based flow allocation algorithm with the duality approach, which can be easily extended to a distributed algorithm in a way similar to [3], [4]. The proposed algorithm is shown to be primal-dual optimal and can converge to the optimum within a limited number of iterations. The performance of the proposed algorithm is evaluated via numerical study in different flow scenarios.
The rest of the paper is organized as follows. In Sec. II, the new interference model with respect to local interference and location-dependent interference in multi-hop wireless net-works is characterized and formulated. In Sec. III, the flow allocation problem is formulated and solved by the duality approach. In Sec. IV, an optimal flow allocation algorithm based on the gradient projection method is proposed. In Sec. V, the simulation results are provided. Finally, the paper is concluded in Sec. VI.
II. SYSTEMMODEL A. Interference in Wireless Networks
We consider a multi-hop wireless network G=(V ,E), where V is the set of nodes and E is the set of links in the network. Let Pt(i) denote the transmission power of node i; di,j be the distance between nodes i and j; L(.) be the path gain function, and σ be the thermal background noise. Consider a pair of nodes i, j ∈ V , the received power at node j, i.e., Pr(j) = Pt(i)L(di,j), must exceed a threshold to correctly receive a data unit from transmitter i. Hence, we have SN Ri,j ≥ θ, where SN Ri,j= Pt(i)L(di,j)/σ is the Signal to Noise Ratio (SNR) of the wireless link(i, j), σ is a constant, and θ is the SNR threshold for a node to correctly decode the signal. The transmission range ri of transmitter i is the largest distance from i that node i’s data packets can be correctly decoded. It is determined once the transmission power Pt(i) of node i and θ are given.
For a multi-hop wireless network, multiple pairs of nodes may transmit data units simultaneously. In addition to the
thermal noise, the transmission from node i to node j may be interfered by other concurrent transmitters. Let K denote the set of concurrent transmitters. The Signal to Interference Ratio (SIR) for link(i, j) is defined in [13], [14] by
SIRi,j= Pt(i)L(di,j)
k∈KPt(k)L(dk,j) + σ. (1) For node j to receive a data unit from node i correctly, the SIRi,j of link (i, j) must exceed the threshold β. The value of threshold β is determined by the settings of wireless physical layer (PHY). In this paper, we adopt the default setting of IEEE 802.11 PHY, i.e., the collision threshold (CP T hreshold) of IEEE 802.11 PHY is defined as the signal to interference ratio. Thus, we have CP T hreshold= β×k∈KPt(k)L(dk,j).
B. Node-based Interference Model
We assume that each node i ∈ V is characterized by an SNR threshold θi to receive one data unit from a transmitter. The SIR threshold βi, provided that θi > βi, is also given so as to guarantee correct signal decoding when there are concurrent transmissions contending for the resource. Based on the Shannon theorem, the supportable data rate of any communication link incident to node i is at least Ri = W×log2(1+βi), where W is the frequency bandwidth of the communication channel. Only when the SIR of the received signal is smaller than βi can the supportable data rate of this node be assumed to be zero, and thus the transmission be prohibited from accessing this wireless link.
Let Pmax(i) denote the maximum transmission power of node i. Suppose each node i can adjust its transmission power Pt(i), 0 ≤ Pt(i) ≤ Pmax(i) such that the signal power of the receiver node j is slightly larger than θj× σ. Then, the maximum supportable data rate of a wireless link connecting node j is given by Rj,max = W × log2(1 + θj) if there is no interference contributed by the neighboring nodes. The maximum interference budget Bj, which denoted that node j can sustain to correctly decode the signal from a transmitter, is given by Bj = (θj × σ/βj) − σ. For a particular node k, the ratio of the interference contributed by the concurrent transmission from node i to node j, denoted by ωi,k,j, can be expressed by
ωi,k,j= Pt(i) × L(di,k)
Bk =
L(di,k)θkβk
L(di,j)(θk− βk). (2)
The set of nodes which renders the interference ratio ωi,k,j≥ 1 is called the set of contending nodes for node k. The occurrence of any communication at each contending node will cause the supportable data rate of node k to drop to zero, and therefore, prohibit node k from accessing the wireless medium. Let ςi,k,j denote the interference indicator for the communications performed at the set of contending nodes of node k. ςi,k,j = 1 if node k contends with the transmission from node i to node j; otherwise, ςi,k,j = 0.
The concept of the Node-based Interference Model is il-lustrated in Fig. 1. Fig. 1(a) shows the relationship between the location-dependent interference contributed by the set of concurrent transmitting nodes and the capacity shared by a
Time Time (3) (3) 0 0 Rmax
The capacity shared by input traffic
The capacity shared by input traffic
The capacity shared by output traffic The capacity shared by contention transmissions
Data rate
(1) (1)
T
The interference contributed by concurrent transmissions Interference budg et (2) (4) (4) (1) (1) T R B R (a) (b)
Fig. 1. Node-based interference model: (a) location-dependent interference; (b) local interference.
particular node. During each fixed time period T , the capacity region 1 can be classified into two types of regions: 1)
the region which can be shared with the set of concurrent transmitters (i.e., area (1)+(2)), and 2) the entire region which is consumed by the set of contending nodes (i.e., area (3)). During the portion of time that the interference caused by the set of concurrent transmitters (i.e., area (2) in Fig. 1(a)) does not exceed the budget B, the node is allowed to receive input traffic at least data rate R (i.e., area (1) in Fig. 1(a)). During the portion of time for area (3) in Fig. 1(a), the interference contributed by the set of contending nodes is larger than B, and thus the data reception at this node is prohibited.
Consider a set of end-to-end flows, denoted byΓ, in a multi-hop wireless network. Each end-to-end flow, denoted by f = {s, d}, traverses the system from the source node s through multiple hops to the destination node d. Let tfi,j denote the portion of time shared by flow f traversing from node i to node j. Based on the location-dependent interference from both sets of concurrent transmitters and contending nodes, we have f∈Γ j∈V tfj,i+ f∈Γ j∈V k∈V (i,j) ςj,i,k× tfj,k≤ T. (3) f∈Γ j∈V k∈V (i,j) ωj,i,k(1 − ςj,i,k)tfj,k− f∈Γ j∈V tfj,i≤ 0. (4) Fig. 1(b) illustrates the impact of local interference of a node on the channel capacity sharing. The capacity can be shared by the input traffic and the output traffic. This kind of interference is imposed by most MAC protocols, giving rise to the name the MAC constraint, and expressed by
f∈Γ j∈V tfj,i+ f∈Γ j∈V tfi,j ≤ T. (5)
III. FLOWALLOCATION INMULTI-HOPWIRELESS NETWORKS
In this section, we formulate the problem based on the previous discussion with respect to the interference and MAC constraints in multi-hop wireless networks. Note that the
1Each capacity region is given by a data rate times a period of time.
formulation below is based on single-path routing. It can also be extended to the multi-path flow allocation problem in a straighforward manner.
A. Problem Formulation
Each end-to-end flow f ∈ Γ is associated with a utility function Uf(xf), which indicates the degree of satisfaction of its end-user. Let Ci = T × Ri denote the total capacity of node i ∈ V . Assume that the utility function Uf(.) is an increasing, strictly concave, twice continuously differentiable function of xf over the interval 0 < xf < max{Ci|i ∈ V }. The traffic satisfying such a utility function is described as elastic [30]. We further assume that the utilities are additive such that the aggregate utility of flows can be regarded as the network utility. If link (i, j) carries the traffic of flow
f , then rfi,j = 1; otherwise, rfi,j = 0. Thus, we have xf =
f∈V tfs,j×Rj. The objective of this problem is to maximize the total network utility over x = (xf, f ∈ Γ) subject to the local and location-dependent constraints along with the medium contention consideration. Such an objective function can achieve the optimal resource utilization and realize the fairness models such as max-min or proportional fairness [30]. Substituting the definition of a flow into (3) and (5), we can formulate the problem of flow allocation in multi-hop wireless networks as a convex optimization problem as follows.
P:M aximize f(x) = f∈Γ Uf(xf), (6) subject to f∈Γ j∈V rfj,ixf+ f∈Γ j∈V rfi,jxfRi Rj ≤ Ci, (7) f∈Γ j∈V rfj,ixf+ f∈Γ j∈V k∈V (i,j) ςj,i,krfj,kxfRi Rj ≤ Ci. (8)
The objective function in (6) is to maximize the aggregate utility over all flows. By optimizing this objective function, both optimal and fair flow allocation can be achieved. The feasible region of the optimization problem jointly formed by constraints in (7) and (8) is a convex and compact set. B. Duality
With the assumptions on the utility function, the objective function of the primal problem P in (6) is differentiable and concave. In addition, the feasible region of the optimization problem in (7)(8) is convex and compact. Based on the non-linear optimization theory, there exists an optimal value of
x∗ for the primal problem P. The Lagrangian form of the optimization problem P can be expressed as follows.
L(x, λ, μ) = |Γ| f=1 U(xf) + |V | i=1 λi[Ci− |Γ| f=1 aifxf]+ |V | i=1 μi[Ci− |Γ| f=1 bifxf], (9)
where aif = |V |j=1rj,if + |V |j=1rfi,jRi Rj, and bif = |V | j=1rfj,i+ |V | j=1 |V | k=1;k=i,jςj,i,krj,kf RRki.
In (9), λi,, and μi, i ∈ V , are Lagrange multipliers as-sociated with a local interference constraint and a location-dependent interference constraint on node i, respectively. The addition of total network utility and the linearity of constraints lead to a Lagrangian dual decomposition into each individual flow f as follows. L(x, λ, μ) = |Γ| f=1 Lf(xf, λf, μf) + |V | i=1 Ci(λi+ μi), (10)
where λf =|V |i=1λiaif and μf =|V |i=1μibif.
For each flow f ∈ Γ, Lf(xf, λf, μf) = Uf(xf) − (λf + μf)xf and its value is determined by xf and flow prices λf and μf. Considering the expression λf+ μf, we obtain
λf+μf= |V | i=1,i=j |V | j=1 rj,if (λi+λj×Rj Ri+μi+μj+ηj,i), (11) where ηj,i = |V |k=1μkςj,k,i represents the price of link (j, i) that is the aggregate interference price from the neigh-borhood of link(j, i).
To determine the Lagrange multipliers, we introduce the dual problem g of the optimization problem P, which can be formulated as follows. g: minλ≥0,μ≥0 g(λ, μ), (12) where g(λ, μ) = maxxL(x, λ, μ) = |Γ|f=1Sf(λ, μ) + V(λ, μ)), and Sf = maxx f (Uf(xf)− |V | i=1,i=j |V | j=1 rf j,i(λi+ λRjRj i +μi+μj+κj,i)xf), V (λ, μ) = maxcCi( |V | i=1 (λi+ μi)).
The dual approach decomposes the original problem into the rate control problem Sf and the scheduling problem V(λ, μ) given the Lagrange multipliers λ and μ . In (10)-(11), the Lagrange multipliers λican be interpreted as the implied cost of a unit flow accessing node i, and the Lagrange multiplier μi can be interpreted as the implied cost of a unit flow contributing interference to node i. From these equations, we observe that each flow f incurs a cost to each node that it traverses and to each node to which it contributes interference. IV. GRADIENT-BASEDFLOWALLOCATIONALGORITHM
To solve the optimization problem presented in the previous section, we propose a de-centralized algorithm based on the node-based pricing framework. The objective is to achieve optimal flow allocation in multi-hop wireless networks. We first design an algorithm to determine the per-node price and to obtain the flow allocation schedule by using the gradient approach. Then, we analyze the properties of the algorithm.
TABLE I
GRADIENT-BASEDFLOWALLOCATIONALGORITHM
Input: A set of nodesV , a set of source-destination pairs Γ, and
the routing path of each flow.
Output: Flow assignmentxf for each flowf ∈ Γ.
1: Initialize flowxf(0) 0, ∀f ∈ Γ, and node prices λi 0, μi 0, ∀i ∈ V .
2: Update the price at each nodei ∈ V .
λi(t+1) = [λi(t)−α(Ci− |Γ|f=1( |V |j=1rfj,i+ri,jf RRij)xf)]+. μi(t + 1) = [μi(t) − α(Ci− |Γ|f=1( |V |j=1rfj,i+ |V | j=1 |V | k=1 rfj,kςj,i,kRRi k)xf)] +.
3: For each nodei ∈ V , send the prices λi(t + 1) and μi(t + 1) to the sender of the flowf ∈ Γ, for which ri,jf = 1 or rfj,i= 1 or
rfj,kςj,i,k= 1.
4: For each flow originator, after receiving node pricesλi(t + 1) andμi(t + 1) from each node i ∈ V , calculate the gradient by
ζf(t + 1) = |V |i=1,i=j |V |j=1rfj,i[λi(t + 1) + μi(t + 1)RRji+
λj(t + 1) + μj(t + 1) + |V |k=1μk(t + 1)ςj,k,i].
5: The flow allocation is adjusted by
xf(t + 1) = xf(ζf(t + 1)).
A. Gradient-Based Algorithm
By applying the gradient-based approach to the dual prob-lem g, we propose an algorithm to calculate the multipliers of each node iteratively. The net benefit of the flow f is defined as follows.
φf(xf) = Uf(xf) − ζfxf. (13) In (17), ζf = λf + μf is the shadow price of flow f , and φf(xf)is the net benefit of flow f corresponding to the difference between its utility and its cost. Since Uf(.) is an increasing, strictly concave, twice continuously differentiable function of xf, the maximizer of φf(xf) exists when
dφf(xf)
dxf = U
f− (λf + μf) = 0, (14) where the maximizer is defined by
xf(ζf) = arg maxxf∈R φf(xf). (15) We apply the iterative gradient projection method to solve the dual problem g. Let = [λ, μ]T , where λ= (λi, i∈ V ) and μ = (μi, i∈ V ) are Lagrange multiplier vectors. In this method, is adjusted in the opposite direction to the gradient g( ) as follows. i(t + 1) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ i(t) − α(Ci−|Γ|f=1(|V |j=1(rj,if + rfi,jRi Rj))xf) + , if i= λi i(t) − α(Ci−|Γ|f=1(|V |j=1rfj,i+ |V | j=1 |V | k=1rfj,kςj,i,kRRkixf)) + , if i= μi. (16) The iterative algorithm of computing an optimal flow allo-cation is summarized in Table I.
B. Convergence Analysis
In this section, we analyze the convergence behavior of the algorithm in Table I and characterize the property of its limit points. The result shows that the algorithm can converge to a unique flow allocation schedule such that the summation of all users’ utilities is maximized.
To show the properties of the iterative algorithm, we define Y(f) = 2|V |i=1 Aif, which gives the definition of ¯Y = maxf∈ΓY(f) as, intuitively speaking, the ”length” of the ”longest” path. We further define W(n) = |Γ|f=1Anf, n = 1, 2, ..., i, ..., 2i, and ¯W = maxn∈1,...,2iW(n), which gives the number of flows at the most ”congested” node. Let¯κ = maxf∈Γκf, where κf is the upper bound of Uf(.) within the range[0, max Ci|i ∈ V ].
Since = [λ, μ]T, we show that g is the Lipschitz continuity [29] ong . For any , ϕ( ) is defined by
ϕ( ) = 1 −U f(xf()), U f(Mf) ≤ ζf ≤ Uf(mf), 0, otherwise (17)
where xf( ) = xf(ζf), Mf = max{Ci|i ∈ V }, and mf = 0. If Uf is bounded away from zero on I: −Uf(xf) ≥ 1/κ > 0, we have 0 < ϕ( ) < (¯κ). We note that the above expression is of the same form as that used in [18]. The sequence(x(t), (t)) generated by the iterative algorithm in TABLE I is primal-dual optimal assuming that ≤ α ≤ (2 − )/¯κ ¯W ¯Y , where is a fixed positive scalar.
V. NUMERICALSTUDIES
In this section, we evaluate the proposed algorithm by nu-merical studies. We assume that message updates are synchro-nized and communication delays are bounded. The thermal noise σ of each node is assumed to be −90dBmW . The maximum transmission power is0.2mW . The SNR threshold and SIR threshold of each node are set to 1.8 and 1.5, respectively. We use a simplified path gain function defined as L(di,j) = 1/d4i,j, where di,j is the distance between the transmitter and the receiver. If the frequency bandwidth is assumed to be 1.6MHz, then the wireless channel capacity can be derived as 2Mbps. The routing path of a flow is determined by the shortest path routing algorithm. The utility function used in our simulation is defined by Uf(xf) = ln(xf), because it has been shown in [20] that the proportional fairness can be achieved and the optimal condition can be satisfied with this utility function. We evaluate the convergence rate, the aggregate utility, and the flow throughput in a tandem network in different flow allocation scenarios as shown in Fig. 2. The distance between two adjacent nodes is set to 100 meters. Then, we evaluate the impact of routing configuration on our proposed algorithm in a random network.
A. Tandem Networks
We vary the tandem network size from 6 to 20 nodes and consider four scenarios for flow allocation: 1) a 3-flow scenario, 2) a mutual flow scenario, 3) an aggregate flow scenario, and 4) a reverse flow scenario. Fig. 2 (a) gives the 3-flow scenario, in which flow 1 goes through all hops of the
(a) 3-flow scenario Flow 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 1 1 1 Flow 3 Flow 3 Flow 4 Flow 2 Flow 1 Flow 3 Flow 1 Flow 2 (c) Agregate flow scenario
Flow 3 Flow 4
Flow 4
Flow 5 Flow 6 Flow 2
(b) Mutual flow scenario
(d) Reverse flow scenario Flow 1 Flow 2
Fig. 2. A 6-node tandem network with three end-to-end flow scenarios. TABLE II
THESTEPSIZE FORCONVERGENCE
3-flow Mutual flow Aggregate flow Reverse flow
6-node 3.65 6.08 7.76 7.85
10-node 3.50 5.99 10.6 2.89
15-node 3.28 5.25 16.5 1.39
20-node 3.28 5.00 23.3 0.9
network and flows 2 and 3 each traverse only one hop, i.e., the first hop and last hop, respectively. In this example, flow 1 interferes with both flows 2 and 3, and flows 2 and 3 are forwarded simultaneously. Fig. 2 (b) gives the mutual flow scenario, in which one flow (i.e., flow 1) traverse all hops of the network, and the other five flows are non-overlapped single-hop flows. Fig. 2 (c) gives the aggregate flow scenario with five flows in the network, each with a different source node but destined for the same node, i.e., node 6. This scenario can be used to investigate the optimal flow allocation in wireless mesh networks. In Fig. 2(d), the reverse flow scenario with four flows is given. In this scenario, flows 1 and 2 traverse all hops in the network but in opposite directions. Flows 3 and 4 are both sent from node 3 but each traverses along the path in a different direction.
We evaluate the convergence rate of the proposed gradient-based flow allocation algorithm by properly adjusting step sizes. In all simulations, the initial values of all flows are fixed at 0 Kbps and the initial shadow prices are set to 1. Table II and Table III give the relationship between the step size and the number of iterations, respectively, in these four flow scenarios. Given the number of nodes and the flow scenario, we adjust the step size such that convergence to the optimum can be achieved at the least number of iterations. The optimum is achieved if|xf(t)−x∗f| < , |λi(t)−λ∗i| < , and |μi(t) − μ∗i| < are satisfied for all f ∈ Γ and i ∈ V , where = 10−4.
From these results, we observe that when the network size grows, smaller step sizes and more number of iterations are required for convergence in the 3-flow, the mutual flow, and the reverse flow scenarios. Among these three scenarios, the reverse flow scenario gives the largest step size variance and the most number of iterations to converge. This is because flows1 and 2 traverse the same set of nodes but in the opposite directions. Since the neighboring links along the path of a flow will interfere with each other, and the communication in the reverse direction will also contend for the node’s bandwidth, a small change in price at each node will result in a higher
TABLE III
NUMBER OFITERATIONS FORCONVERGENCE
3-flow Mutual flow Aggregate flow Reverse flow
6-node 24 17 19 21 10-node 30 40 23 37 15-node 31 63 27 52 20-node 31 75 37 70 0 100 200 300 400 500 -10 -5 0 5 10 Number of iterations
Number of iterations Number of iterations
Number of iterations
6 node 10 node 15 node 20 node
(a) Convergence of the objective value vs. the number of iterations under the 3-flow scenario
(c) Convergence of the objective value vs. the number of iterations under the aggregate flow scenario
(d) Convergence of the objective value vs. the number of iterations under the reverse flow scenario (b) Convergence of the objective value vs. the number of iterations under the mutual flow scenario
Aggregated network utilization
Aggregated network utilization Aggregated network utilization
Aggregated network utilization
0 100 200 300 400 500 -40 -20 0 20 40 0 100 200 300 400 500 -20 -15 -10 -5 0 5 0 100 200 300 400 500 -120 -100 -80 -60 -40 -20 0 20 6 node 10 node 15 node 20 node 6 node 10 node 15 node 20 node 6 node 10 node 15 node 20 node
Fig. 3. The convergence of objective value under different flow scenarios.
cost to the neighboring nodes and this increased cost will be propagated through the network. Thus, when the network size increases, it needs smaller step sizes and more number of iterations to achieve the optimum. In addition, the variance of step sizes and the number of iterations in the mutual-flow scenario is larger than that in the 3-flow scenario in the steady state for a network with larger than 10 nodes. While there is a long-path flow traversing all nodes in the network, all intermediate nodes of this flow are only interfered by the two neighboring wireless links two hops away. Due to this regularity among all intermediate nodes, the pricing updates at these nodes are consistent and hence the flow converges once other non-mutual interfering flows is determined. In the mutual-flow scenario, flows interfere with each other mutually, and thus more flows are affected when prices are updated at a particular node in the network, compared with the 3-flow scenario, and this may require further flow adjustments. Any change in flow rates may result in further price coordination to approach the optimum. Hence, a higher degree of mutual interference among flows leads to more iterations and a smaller step size for convergence to the optimum. In the aggregate flow scenario, the number of iterations and the step sizes increase with the network size. This is because the degree of mutual interference among flows at a particular node and the number of flows going through the node are un-balanced compared with the mutual flow scenario.
Fig. 3 gives the trajectories of the aggregate network
utiliza-Throughput of each f
low (x 10 )
(d) Throughput comparison for each flow under reverse flow scenario (c) Throughput comparison for each flow under
the aggregate flow scenario (a) Flow through comparison under the 3 flow scenario
(b) Throughput comparison for each flow under the mutual flow scenario
Number of nodes
Number of nodes Number of nodes
Number of nodes 5 Throughput of each f low (x 10 ) 5 Throughput of each f low (x 10 ) 5 Throughput of each f low (x 10 ) 5
Fig. 4. Throughput comparison under different flow scenarios.
tion with different network scales in different flow scenarios, and Fig. 4 gives the throughput of each flow in each scenario. Consider Figs. 3 (a) and 4 (a) first, which are the results in the 3-flow scenario. The network utilization remains unchanged even though the network size is larger and the longest flow traverses over more number of nodes. This is because all intermediate nodes for the longest flow are interfered in the same way. Once the longest path flow is determined, the flows traversing the first hop and last hop can use the remaining resource of the destination node optimally since they will not interfere with one another. However, in the mutual-flow scenario as shown in Fig. 3(b), the aggregate network utiliza-tion increases with the network size. Due to the spatial reuse property of the wireless channel, the flows not contending mutually may share a node’s capacity simultaneously. Fig. 4(b) gives the throughput of each flow in this scenario. The throughput of flow 1 may decrease due to traversing more intermediate nodes and thus incurring more contentions with the other flows.
Next, we move to the aggregate flow scenario. Fig. 3(c) shows that the overall network utilization is decreased as the network size increases. This is because the last hop suffers from the resource competition from all flows, and experiences the highest degree of mutual interference among flows. There-fore, this hop can be regarded as the bottleneck in the network and all other flows going through it will be affected. Fig. 4(c) shows that the two flows with their source nodes closer to the destination node gain more throughputs than the other flows. This is because these two flows experience less wireless resource contentions, and thus are allocated more resources. Finally, we observe the results of the reverse flow scenario. Fig. 3 (d) shows that as the network size increases, the total network utilization stays unchanged but the convergence time is increased. Fig. 4 (d) shows that each flow is allocated the same amount of resources under different network sizes. These results show that the flow change is sensitive to the
11 16 14 13 10 5 20 1 7 6 15 4 3 19 18 9 8 12 17 2 flow 1 flow 2 flow 5 flow 4 flow 3 flow 6 11 16 14 13 10 5 20 1 7 6 15 4 3 19 18 9 8 12 17 2 flow 1 flow 2 flow 5 flow 4 flow 3 flow 6 11 16 14 13 10 5 20 1 7 6 15 4 3 19 18 9 8 12 17 2 flow 1 flow 2 flow 5 flow 4 flow 3 flow 6
(a) Routing paths in the initial network topology (b) Routing paths after node 16 becomes unavailable. (c) Routing paths after node 18 becomes unavailabl Fig. 5. A 20-node wireless network topology with 6 flows.
network scale, and thus the convergence rate degrades as the network size increase. However, the aggregate network utilization is not sensitive to the network size. When looking into the throughput of each flow in Fig. 4(d), we find that the same throughput distribution can be obtained under different network sizes. This is because the set of nodes that will be interfered by a particular link is limited, and the nodes outside of the interference range of a particular link can transfer data simultaneously thanks to the spatial reuse factor. Thus, irrespective of the increase in the network size, only a finite set of links contend for a node’s resource.
B. Random Networks
In this section, we study the impact of sudden changes in the routing path on the convergence rate of our proposed algorithm in a randomly generated network as shown in Fig. 5(a). This network consists of 20 nodes distributed in a 500 × 500m2 region. In this simulation, 6 flows between 6
pairs of source and destination nodes start at the same time instant. The simulation is performed for 1000 iterations. At the 250thiteration, node16 is assumed to be down, and the set of links incident to it becomes disabled immediately. The routing paths for flows 3 and 4 are changed as shown in Fig. 5(b). Similarly, node18 becomes unavailable at the 500th iteration and the routing paths for flows 1 and 2 are also changed as shown in Fig. 5(c).
The actual throughput of each flow is plotted in Fig. 6. Initially, each flow is assumed to transmit at 0Mbps. After about 50 price updates, all flow allocations converge to500Kbps which is a proportional fair value. At the 250th iteration, the routing paths for flows3 and 4 are changed due to the unavailability of node16. The sudden change to routing paths causes nodes 1 and 20 to be overloaded. Since nodes 1 and 20 are the bottlenecks for flows 3, 4, and 5, the actual allocated throughput for each of these flows is thus decreased rapidly. Flow3 suffers more throughput degradation since the number of hops it traverses is more than flows 4 and 5. In addition, the effects of interference and MAC contention make flows1,2 and 6 contending for the resources of nodes 1 and 20, resulting in a slight throughput degradation for each of these flows even though their routing paths are not affected.
When the routing paths of flows1 and 2 are changed due to the unavailability of node18 at the 500 iteration, the rate of
0 100 200 300 400 500 600 700 800 900 1000 0 0.1 0.2 0.3 0.4 Number of iterations T h ro ug hpu t( M b ps) flow 1 flow 2 flow 3 flow 4 flow 5 flow 6 flow 1,2 flow 4, 5, 6 flow 1 ~ 6 flow 1,6 flow 2,3,4,5 flow 3
Fig. 6. Throughput trajectory of each flow when the network topology is changed.
each flow still converges within 50 iterations. This time, the flows converge to two groups of rates. The group of the higher rate includes flows1 and 6. The other flows obtain a lower rate when flow allocation converges. At this stage, flows1 and 6 obtain a higher throughput since they have lower interference caused by other concurrent flows. We can observe from tracing the trajectory of each flow in Fig. 5 that the throughput of flow 6 decreases when node 16 is down but increases after node 18 becomes unavailable. Therefore, we find that the impact of topology and routing change is not always negative for a particular flow. The flow contending with less number of flows in the routing path and suffering less interference from other concurrent flows will gain higher throughput.
VI. CONCLUSION
In this paper, we developed the Node-based Interference Model and a flexible theoretical framework to consider inter-ference, data rate, and signal reception power at the physical layer, the contention behavior at the MAC layer, and end-to-end flows at the transport layer for multi-hop wireless net-works. Based on this framework, we formulate an interference-aware optimal flow allocation problem without clique or independent set enumeration. The objective of the problem is to maximize network utilization and maintains fairness among flows. We then propose a gradient-based flow allocation algorithm by using the duality approach. The convergence of the gradient-based flow allocation algorithm is analyzed. The numerical results show that our proposed algorithm can achieve the optimum within a small number of iterations
and can allocate resource to the end-to-end multi-hop flows to maximize optimal network utilization while maintaining fairness among flows. The simulation results show that the proposed algorithm can rapidly adapt to changes in network topology and routing paths. To the best of our knowledge, this is the first work which formulates the interference constraints for the flow allocation problem without any global information in multi-hop wireless networks. The results demonstrate that the proposed solution can also be used in emerging wireless mesh networks.
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Kun-Da Wu received his B.S. degree from the department of Computer Science and Information Engineering in National Chiao-Tung University, Tai-wan, ROC in 1996, and M.S. degree from the department of Computer Science and Engineering in National Sun Yat-Sen University in 1998. He is currently a PhD candidate in the department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, ROC. His research interests include wireless ad hoc networks, resource allocation, and network optimization.
Wanjiun Liao received the BS and MS degrees from National Chiao Tung University, Taiwan, in 1990 and 1992, respectively, and the Ph.D. degree in Electrical Engineering from the University of Southern California, Los Angeles, California, USA, in 1997. She joined the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, as an Assistant Professor in 1997. Since August 2005, she has been a full professor in the EE department and the Graduate Institute of Communication Engineering at NTU. Her research interests include wireless networks, multimedia networks, and broadband access networks.
Dr. Liao is currently an Associate Editor of IEEE Transactions on Wireless
Communications and IEEE Transactions on Multimedia. She served as the
Technical Program Committee (TPC) chairs/co-chairs of many international conferences, including the Tutorial Co-Chair of IEEE INFOCOM 2004, the Technical Program Vice Chair of IEEE Globecom 2005 Symposium on Au-tonomous Networks, and the Technical Program Co-Chair of IEEE Globecom 2007 General Symposium. Dr. Liao has received many research awards. Papers she co-authored with her students received the Best Student Paper Award at the First IEEE International Conferences on Multimedia and Expo (ICME) in 2000, and the Best Paper Award at the First IEEE International Conferences on Communications, Circuits and Systems (ICCCAS) in 2002. Dr. Liao was the recipient of K. T. Li Young Researcher Award honored by ACM in 2003, and the recipient of Distinguished Research Award from National Science Council in Taiwan in 2006. She is a Senior member of IEEE.
