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Max-min Fair Scheduling in OFDMA-Based Multi-hop WiMAX Mesh Networks Shi Bai∗ , Weiyi Zhang∗ , Yang Liu∗ , Chonggang Wang†

∗ Department

†

of Computer Science, North Dakota State University, Fargo, ND 58105 InterDigital Communications, King of Prussia, PA 19406

Abstract— The emerging WiMAX technology (IEEE 802.16) is a fourth generation standard for low-cost, high-speed and longrange wireless communications for a large variety of civilian and military applications. IEEE 802.16j has introduced the concept of mesh network model and a special type of node called Relay Station (RS) for traffic relay for Subscriber Stations (SSs). A WiMAX mesh network is able to provide larger wireless coverage, higher network capacity and Non-Line-Of-Sight (NLOS) communications. This paper studies a Multi-hop FAir Scheduling for Throughput Optimization (MFASTO) problem in WiMAX mesh networks. The goal here is to maximize the minimum satisfaction ratio among all the SSs. In order to solve the MFASTO problem, an ILP formulation and an efficient heuristic algorithm are proposed in this work. Simulation results are presented to justify the performance and efficiency of our proposed solutions. Keywords: WiMAX mesh network; max-min fairness; multi-hop.

I. I NTRODUCTION The emerging WiMAX technology (IEEE 802.16 [16]) is the fourth generation (4G) standard for low-cost, high-speed and long-range wireless communications for a large variety of civilian and military applications. WiMAX uses large chunks of spectrum (10-20 MHz or more), and delivers high bandwidth (up to 75 Mbps). Despite the high bandwidth promised by WiMAX, there are several challenges to improving the network throughput. For the multi-hop WiMAX mesh networks, one of the important challenges is the multi-hop scheduling scheme for the network. The physical layer of WiMAX uses scalable-OFDMA (Orthogonal Frequency-Division Multiple Access) since OFDM has two-fold benefits in terms of robustness to multi-path fading, and ease of digital signal processing implementation.

Fig. 1.

OFDMA in frequency and time domain

An OFDMA system is defined as one in which each user occupies a subset of subcarriers (an OFDMA subchannel), and each subchannel is assigned exclusively to one user at any time. In OFDMA, users are not overlapped in frequency domain at The research developed in this paper is supported by NSF CNS-1022552, FAR-0016614, NSF ND EPSCoR under the Infrastructure Improvement Program FAR-0015846 and FAR-0017488.

any given time in one cell, which eliminates the co-channel interference in the same cell. Moreover, the frequency bands assigned to a particular user may change over time as shown in Fig. 1 (each type of shade represents resources allocated exclusively to a user). This paper is centered around the scheduling technique for the WiMAX mesh networks. OFDMA is typically used for WiMAX network scheduling. The heart of most scheduling problems in OFDMA relay networks is assigning transmission opportunities (subchannel, time slot) to each link in the network to maximize a certain objective function [1]. In relay networks, there are additional constraints due to synchronization in a multi-hop topology, use of a single transceiver at the relays, and flow conservation due to multi-hop relaying and fairness consideration among SSs. Another challenge is that the scheduling decisions in WiMAX networks have to be made in a timely fashion. Due to the typical order of magnitude of coherence time of the channel [12], the schedule is typically disseminated once every 5-10 ms. Thus, the problem of scheduling for fair-rate allocation in WiMAX relay networks poses several technical challenges. The objective of this paper is to provide a comprehensive WiMAX-based network resource scheduling and allocation. The rest of this paper is organized as follows. We discuss related work in Section II. The system model and problem definition are described in Section III. Proposed solutions are presented in Section IV, which is followed by simulation results in Section V. We conclude this paper in Section VI. II. R ELATED W ORK The network scheduling and resource allocation with relay stations received much attention in recent years in the wireless networks, including WiMAX mesh networks. In [15], the authors studied scheduling with a small number of relays in cellular wireless networks and proposed a centralized downlink scheduling scheme. In [11], the authors proposed a scheme termed as OFDM2 A that considers frequency-selectivity and provides significant gains over round-robin scheduling. The problem of scheduling in OFDMA-based IEEE 802.16j based WiMAX network was studied in [3]. The authors presented linear programming based heuristics for MAC scheduling in WiMAX relay networks in a fair manner while exploiting the multiuser diversity. [8] studied the capacity of the OFDMA relay networks. Two relay schemes, amplify-and-forward relay and decode-and-forward relay are analyzed. Relay node selection algorithm was presented to optimize the network capacity. [4] proposed a centralized scheduling algorithm for

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

WiMAX mesh networks. Each node has one transceiver with multiple channels. The BS makes schedules intending to eliminate the secondary interference for reducing the length of scheduling. A resource allocation protocol that allocates subcarriers to cooperating subscriber and relay stations was proposed in [9]. [7] presented a centralized heuristic algorithm to allocate power and sub-carriers to user nodes and relays in a network where the node can establish a connection either through a direct connection or through the one relay but not in cooperative mode. But both work focused on maximizing the total network throughput rather than considering each user’s data requirement. In [2], the subchannel and relay station allocation problem was studied for the two-hop relay model. Each SS is allocated subchannels and RSs that are required to satisfy its minimum rate requirement. A 0/1 Integer Programming was formulated with QoS and synchronization constraints. Applying existing resource allocation algorithm to WiMAX networks is not trivial [10]. Han et al. [6] proposed a distributive non-cooperative game to perform subchannel assignment, adaptive modulation, and power control for multicell multi-user OFDMA networks. III. P ROBLEM S TATEMENT In this paper, IEEE 802.16j Mobile Multi-hop Relay-based (MMR) netowrk is used as the model for the network infrastructure. As suggested by the WiMAX standard [16], a tree rooted at the BS is usually constructed to support packet forwarding in a WiMAX mesh network. The BS is the root of the tree, the RSs are the intermediate nodes of the tree and the SSs are the leaf nodes of the tree. We focus primarily on the scheduling for SSs and RSs over time and frequency. We model only the uplink scenario, i.e. traffic flows from SSs to the base station. The extension to handle downlink resource allocation is along similar lines. The IEEE 802.16 series standards [16]–[18] include the PHY and MAC layer specifications but do not specify the scheduling algorithm or the routing protocol, which are the key components for mesh networking. In this paper, we investigate the scheduling problem in multi-hop relay WiMAX mesh networks with time-varying subchannels. The objective is to provide a fair and efficient complete schedule to ensure the minimum satisfaction ratio among all the SSs is maximized, which has not been addressed before. In a WiMAX network with a subscriber station set S = {s1 , . . . , sn } and relay station sets R = {r1 , . . . , rm }, SSs share a set of subchannel H = {hl |1 ≤ l ≤ Nh }. Using twohop relay cooperative AF protocol, the maximum achievable rate in (bits/sec/Hz) by a subscriber station si on subcarrier hl with the cooperation of rb is proved in [2], [8] to be: Ilib =

1 γ id |β b |2 |γ ib |2 |γ bd |2 log2 (1 + l + l b 2 l bd 2 l ) 2 N0 (|βl | |γl | )N0

In this paper, we study how to schedule and allocate subcarrier and time slots for each SS in a time frame. In other words, with the channel capacity of each SS given in each time slot, we need to allocate time slots and subchannel in a frame to each user to achieve max-min fairness. Definition 1 (Multi-hop FAir Scheduling for Throughput Optimization (MFASTO)). Based on WiMAX standard [16], a tree network G is given, with a base station BS as the root, a set of subscriber users S = {s1 , s2 , . . . , sn } as the leaf nodes, a set of relay stations R = {r1 , r2 , . . . , rm } as the intermediate nodes, the link capacity cit,h of each SS si at time slot t using subchannel h, and the package size requirement pi of each SS si in one frame, the MFASTO problem seeks a complete schedule for a scheduling frame. Specifically, we want to find subchannel-timeslot pair (denoted as STP in the following of this paper) allocation to each SS in a scheduling frame such that the minimum satisfaction ratio among all SSs is maximized with the following constraints: 1) There is no spatial reuse for any pair of links which interfere with each other 2) An RS has only single transceiver, and cannot transmit and receive at the same time 3) The total data sent by an RS to BS in a frame must equal to the data it receives from its children in the frame 2 IV. P ROPOSED SOLUTIONS A. Integer Linear Programming for MFASTO In [3], it was proved that scheduling with constant channel capacity is NP-hard. Therefore, our scheduling problem with time-varying channel will be NP-hard. To find an optimal solution, we provide an Integer Linear Programming (ILP) for the MFASTO problem. We denote S, R, H, and T as the set of SSs, RSs, subchannels, and timeslots, respectively.  The tree network is denoted as G. For each node i ∈ S R, pa(i) denotes the parent of i on the tree-topology. For each RS r ∈ R, we use cd(r) to represent the set of children (SSs or RSs) of r. For each node i in G, cit,h represents the link capacity (i, pa(i)) in timeslot i,j t using subchannel h. ft,h denotes that whether the link (i, j) is assigned with time slot t and subchannel h. We adopt the method in [13] and [14] to identify whether two links has interference or not. We denote the set of nodes which interfere with i as I(i). |H| |T |

Maximize

1  i i,pa(i) ct,h · ft,h pi t=1

min i∈S

(4.1)

h=1

subject to: i ,j  i,j max ft,h + ft,h ≤ 1, ∀h ∈ H, ∀t ∈ T, ∀i ∈ S ∪ R(4.2)  i ∈I(i)

(3.1)

where |γlid |2 , |γlib |2 , |γlbd |2 , respectively, are the lth subcarrier SNR from si to d, si to rb and rb to d. β b is the relay rb ’s amplifying gain. This gives us the channel capacity of an SS si with RS rj using subchannel hl .

r,pa(r)

max ft,h h∈H

|H| |T |   

+

v,r ft,h

v∈cd(r) h=1 t=1 i,j ft,h

max

h∈H,v∈cd(r)

·

cvt,h

≤

v,r ft,h ≤ 1, ∀t ∈ T, ∀r ∈ R(4.3)

|H| |T |  

r,pa(r)

ft,h

· crt,h , ∀r ∈ R(4.4)

h=1 t=1

= {0, 1},

∀(i, j) ∈ G, ∀t ∈ T, ∀h ∈ H (4.5)

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

In the ILP formulation, Constraint (4.2), which is the Spatial Reuse Constraint, states that a particular STP can be used no more than once in each pair of interference links; Constraint (4.3) is the Single Transceiver Constraint which states that an RS can not transmit and receive package concurrently due to that each RS just has one single transceiver; Constraint (4.4) is the Flow Constraint that all the data an RS receives in a frame must be sent out in the same frame. B. Heuristic Algorithm Though the ILP solution can be used to obtain optimal solutions for small sized problem, it has high time and space consumption for large-sized networks. Therefore, in practice, heuristics algorithms are needed for better running time and scalability performance. For the set of SSs S = {s1 , s2 , ..., sn }, we are given a corresponding set P of data package demands. Given C = {c0 , . . . , cN } which is the set of capacities of all the nodes on G, where N is the total number of nodes in G, a collection I = {I(vi )|1 ≤ i ≤ N } can be pre-determined, where I(vi ) is the set of nodes which interfere with vi on G. We use Algorithm 1 to allocate STPs for each node in the network. Our heuristic algorithm has the following main steps: 1) Timeslot allocation for hops (Algorithm 2): The first step is to assign timeslots for each hop in the network. This step can guarantee that each node will not transmit and receive data package concurrently. 2) STP allocation for nodes (Algorithm 3): The second step is to allocate STPs for each node in the network. In this step, we assume that there is no spatial reuse in the whole network. This assumption can guarantee us to obtain a resource allocation without any interference. 3) Maximum Flow Improvement (Algorithm 4): After allocating STPs for each node, we use a maximum flow based algorithm to improve the network throughput.

Algorithm 2 Hops Allocation (T , LD , LA ) 1: γ ←

|T | K lD k=0 k

;

2: for all k ∈ {1, 2, . . . , K} do 3: lkA ← γlkD ; γk ← lkA /lkD ; γk ← (lkA − 1)/lkD . (γk ← 1 and

γk ← 1 if lkD = 0); end for  A Lover ← K k=1 lk − |T |; while Lover > 0 do Choose a hop j with the greatest γ  ; ljA ← ljA − 1; γj ← ljA /ljD ; γj ← (ljA − 1)/ljD ; Lover ← Lover − 1; end while for all k ∈ {1, 2, . . . , K} do Construct set Tk of timeslots; Choose lkA elements from set T and add them to Tk ; T ← T \ Tk ; 13: end for 14: return T= {Tk |1 ≤ k ≤ K}; 4: 5: 6: 7: 8: 9: 10: 11: 12:

Algorithm 3 Nodes Allocation (τ , Λ, Δ, Q, X) 

1: θ ← τ / v ∈G qi (θ ← 1 if θ > 1); i 2: for all SS node vi ∈ G do 3: δi ← θqi ; λi ← δi · ci ; (λi ← pi and δi ←  pci  if λi > pi ) i

4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19:

i si ← λpii ; si ← (δi −1)c ; {si is the satisfaction ratio if subtract pi 1 STP from δi .} for all RS vj on the path from vi to BS do λ λj ← λj + λi ; δj ←  cjj ; end for end for  while ∃k, 1 ≤ k ≤ K and |Xk | < vi ∈Vk δi (Vk is the set of nodes which has k hops from BS) do Choose an SS vi with the greatest s ; i δi ← δi − 1; λi ← δi · ci ; si ← λpii ; si ← (δi −1)c ; pi for all RS vj on the path from vi to BS do λ λj ← λj − ci ; δj ←  cjj ; end for end while for all link (i, pa(i)) ∈ G do Choose δi STPs from Xki and add them to Ui ; {ki is the number of hops from BS to vi .} end for return U= {Ui |vi ∈ G};

Algorithm 1 Schedule (G, P , C, T , H, I) 1: Construct a set Q= {q0 , . . . , qN } which is the set of STP

demands of all the nodes on G;

2: Construct LD = {lkD |1 ≤ k ≤ K} and LA = {lkA |1 ≤ k ≤ K}, 3:

4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18:

where lkD is the timeslot demand of Hop k, and lkA is number of timeslots allocated to Hop k; Initialize each lkD and lkA to 0; Construct a set Λ = {λi |0 ≤ i ≤ N }, where λi is the size of package node vi have to send to its parent; Construct a set Δ = {δi |0 ≤ i ≤ N }, where δi is the number of STPs node vi needs; Initialize each λi and δi to 0; for all each  node vi on G do pi ← vj ∈Gi pj ; qi ←  pcii ; {Gi is subtree rooted at vi } end for for all k ∈{1, 2, . . . , K} do v ∈V qi lkD ←  i|H|k ; {Vk is the set of nodes k hops from BS} end for T ← Hops Allocation (T , LD , LA ); Construct a collection X= {Xk |1 ≤ k ≤ K}; for all k ∈ {1, 2, . . . , K} do Construct a set of STPs Xk ← H × Tk ; end for Total number of STPs τ ← |H| · |T |; U ←Nodes Allocation (τ , Λ, Δ, Q, X); U ←Max Flow Improvement (Λ, Δ, I, U); return U ;

Let us use an example to illustrate our algorithm in Fig. 2. The network includes 1 BS, 2 RSs (R1 and R2 ) and 4 SSs (S1 , S2 , S3 and S4 ). The package requirements of these SSs are 2, 3, 2 and 1, respectively. The capacity of each link is set to be 1. The number of timeslots and number of subchannels in a frame are 3 and 2. The interference node sets are: I(R1 ) = {R2 , S1 , S2 }, I(R2 ) = {R1 , R3 , R4 }, I(S1 ) = {R1 , S2 }, I(S2 ) = {R1 , S1 , S3 }, I(S3 ) = {R2 , S2 , S4 } and I(S4 ) = {R2 , S3 }. We first calculate the timeslot demand for each hop (Lines 1 - 6 in Algorithm 1). Consequently, the timeslot demands are 8 for both Hop 1 and Hop 2. Then, we call Algorithm 2 (Line 10) to allocate timeslots for each hop. In Algorithm 3 in this case (Line 1). 2, the ratio γ is calculated, which is 16 After that, we pre-allocate timeslots for each hop based on γ (Lines 2-4). The corresponding results l1A = 2, γ1 = 14 , γ1 = 18 and l2A = 2, γ2 = 14 , γ2 = 18 are obtained. Consequently, the over-allocated timeslots Lover = 1 (Line 5). Then, we choose one hop with the maximum γ  and subtract 1 timeslot from

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Algorithm 4 Max Flow Improvement (Λ, Δ, I, U) 1: Construct a directed graph GA (V, E) and a virtual node s; 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29:

V ← V {s}; Set COUNT = 0; while COUNT < |V | − 2 (no allocations for BS and s) do Choose node vmin with the smallest satisfaction ratio smin ; if (U \ Imin ) = ∅ then  Choose (h, t) ∈ (U \ Imin ); Imin ← Imin {(h, t)};  if t is not used by pa(vmin ) or any v ∈ cd(vmin ) then ·cmin λmin ← λmin + 1; smin ← λmin ; pmin    Umin ← Umin {(h, t)}; end if else COUNT ← COUNT + 1; end if end while for all link (i, pa(i)) in E do Set the capacity of (i, pa(i)) to be λi · ci ; end for for all leaf node l in GA (V, E) do  Construct a link (s, l) with capacity +∞; E ← E {(s, l)}; end for Find the maximum flow from s to BS and the link flows; for all link (i, pa(i)) ∈ G do if vi is an SS then λi ←  fcii , where fi is the flow value of (i, pa(i)); else λi ←  fcii ; end if Keep λi elements in Ui and remove the rest; end for return U’= {Ui |vi ∈ G};

(a) Network topology

(b) Scheduling assuming no spatial reuse in network

(c) Scheduling with spatial (d) Maximum flow (e) Result: throughput = 4, reuse if no interference smin = 13 Fig. 2.

Illustration of the scheduling algorithm

this hop. We repeat the same procedure until Lover becomes 0 (Lines 6 - 10). The allocation results are shown in Fig. 2(a). Back to Algorithm 1, we allocate STPs for each node (Lines 11-16). We first constructed set of STPs for each hop. More specifically, X1 = {(h0, t2), (h1, t2)} and X2 = {(h0, t0), (h1, t0), (h0, t1), (h1, t1)}. At Line 15, the total number of available STPs is calculated, which is τ = 2×3 = 6 in this case. Then, we call Algorithm 3 to allocate resources for all the nodes in this network (Line 16) shown in Fig. 2(b). In Algorithm 3, we use the similar idea of Algorithm 2. We 6 = 38 (Line 1). Then, first calculate a ratio θ = 2+3+2+1+5+3 we allocate STPs using this ratio (Lines 2 - 8). After allocating resources for all the nodes, we subtract the over-allocated STPs from some nodes. We choose the SS with greatest s in the

network, and subtract 1 STP from this node (Lines 10 - 11). Then, we update the allocated STPs for the nodes on the path from BS to it (Lines 12 - 14). The corresponding allocation results, shown in Fig. 2(b), are returned to Algorithm 1. Then, Algorithm 4 is called to allocate resources allowing spatial reuse if no interference (Line 17 in Algorithm 1). In Algorithm 4, we choose an SS with smallest satisfaction ratio (Line 5 in Algorithm 4), and check whether there is any STP can be used for it (Lines 6 - 12). If possible, we allocate 1 STP for this node (Lines 6 - 9). This procedure is repeated until there is no more available STP for any SS. As the result, we allocate 1 STP for S1 , S4 , R1 and R2 , shown in Fig. 2(c). Then, we assign the capacity for each link (i, pa(i)) (Lines 14 - 16). The link capacities are shown in Fig. 2(d). After that, we calculate a maximum flow in this graph, and output the corresponding STP allocation (Lines 20 - 29). The final resource allocation results are shown in Fig. 2(e). V. N UMERICAL R ESULTS In this section, we presented numerical results to evaluate the performances of our solutions. We implemented the ILP solution and our proposed heuristic algorithm. To evaluate our heuristic algorithm, we divided it into two sub-solutions, the algorithm with Maximum Flow improvement and the one without Maximum Flow improvement, which were denoted as MaxFlow and NoResue in the figures. All our simulation runs were performed on a 2.8 GHz Linux PC with 2G bytes of memory. The transmission range and interference range of each SS were set to be 500 and 1000, respectively. For RS and BS, the transmission range and interference range were 1000 and 2000. One base station was deployed at the center of the field. Multi-hop shortest path routing was adopted to obtain the network topology. The SSs were distributed randomly and uniformly in the playing fields. The data package requirement of each SS and the link capacity between it and its parent were randomly distributed in [2, 8] and [5, 10], respectively. First, we compared the minimum satisfaction ratios and the running times obtained by the ILP formulation and our heuristic algorithm in a 1500 × 1500 sq. units playing field with 4 RSs. We used Gurobi Optimizer [5] to solve the ILP formulation. Due to the limitation of memory space, we set the number of timeslots and number of subchannels in a frame to 12 and 5, respectively. The corresponding results were shown in Fig. 3. In Fig. 3(a), we noticed that when the number of SSs was more than 15, the ILP formulation cannot provide solution due to the memory limitation. On the other hand, a good performance can be obtained from our heuristic algorithm. In Fig. 3(b), we tested the running time performances of ILP formulation and our heuristic. Fig. 3(b) showed that our heuristic algorithm is much faster than the ILP solution. Then we tested the performances of NoReuse and MaxFlow in terms of minimum satisfaction ratio, average satisfaction ratio and network throughput in larger playing field. We set the number of timeslots and number of subchannels in a frame to 48 and 5. In Fig. 4(a), as the number of SSs increased, the total package requirements also increased. Consequently, the

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Min Ratio

∞

ILP MaxFlow No Reuse

0.969 0.949 0.926

1 1 1

NA 0.843 0.809

Running time

1 1 1

1 0.8

NA 0.553 0.515

0.6 0.4

5.787

6

4

2

0.2 0

5

10

15

20

1.16 0.01 0.01

0.265 0.01 0.01

0

25

5

10

(a) Minimum satisfaction ratio

Max Flow No Reuse

1

0.932 0.734

0.715 0.6

0.583

0.573 0.4

Min Ratio

Min Ratio

25

0.422

0.412

50

60

70

0.6

0.483 0.45

0.4

0.2

0.115

0.112 40

0.706

0.684

0.256

0.256

0.2

0.852

0.835

80

90

0.109 0 10

100

0.07 20

30

40

50

# of SSs

60

70

80

90

100

# of SSs

(a) Network in a 3000 × 3000 playing field with 16 RSs Fig. 4.

(b) Network in a 4000 × 4000 playing field with 36 RSs

Minimum satisfaction ratio

minimum satisfaction ratio decreased. From Fig. 4(a), we also observed that the minimum satisfaction ratios obtained from NoReuse and MaxFlow were similar in this network topology. In Fig. 4(b), the minimum satisfaction ratio decreased more sharply due to the increased number of hops and RSs. As shown in Fig. 5(a), as the number of SSs increased, the average satisfaction ratios decreased. Fig. 5(a) also showed the average satisfaction ratio from MaxFlow were better than the one from NoReuse. In Fig. 5(b), the difference between the performances of MaxFlow and NoReuse were much greater than the one in Fig. 5(a). This is because, after NoReuse, the number of potential available resources in large network was more than the one in a relative small network. In Fig. 6(a), the performance of network throughput of MaxFlow was better than the one obtained from NoReuse. 1

1

0.874

0.832

0.8

0.785

0.739

0.7

0.661

0.688 0.6

0.609

0.4 10

0.8

0.879

0.808

0.7

0.685 0.63

0.6 0.5 0.4

0.512

0.5

0.3

0.458 20

30

40

50

60

70

80

90

0.2 10

100

Max Flow No Reuse

0.972 0.939

0.9

0.91

Average Ratio

Average Ratio

Max Flow No Reuse

0.972 0.951

0.9

20

30

40

# of SSs

0.567 0.508 0.495 0.441 0.441 0.384 0.374 0.321

50

60

70

80

90

100

# of SSs

(a) Network in a 3000 × 3000 playing field with 16 RSs Fig. 5.

(b) Network in a 4000 × 4000 playing field with 36 RSs

Average satisfaction ratio

500

Max Flow No Reuse

450

300

Throughput

Throughput

Max Flow No Reuse

350

400 350 300 250 200

250

200

150

150 100

100 10

20

30

40

50

60

70

80

90

100

10

20

30

# of SSs

40

50

60

70

80

90

100

# of SSs

(a) Network in a 3000 × 3000 playing field with 16 RSs Fig. 6.

(b) Network in a 4000 × 4000 playing field with 36 RSs

Network throughput

Also, when the number of SSs was no greater than 70, the network throughput performances of NoReuse and MaxFlow increased as the the network size increased. However, the network throughputs decreased when the number of SSs was greater than 70. Because of the fixed number of resources, to obtain a higher minimum satisfaction ratio, more resources have to be allocated. Similar trends were found in Fig. 6(b). VI. C ONCLUSIONS

Max Flow No Reuse

1

0.8

30

0.01 0.01

20

(b) Running time

0.8

20

0.01 0.01

15

Network in a 1500 × 1500 playing field with 4 RSs

Fig. 3.

0 10

0.01 0.01

# of SSs

# of SSs

0.919

∞

∞

ILP(second) MaxFlow(millisecond) No Reuse(millisecond)

In this work, we studied the Multi-hop FAir Scheduling for Throughput Optimization (MFASTO) problem, which seeks the maximized minimum satisfaction ratio scheduling in OFDMA-based multi-hop WiMAX mesh networks. For the MFASTO problem, we presented an Integer Linear Programming (ILP) formulation providing optimal solutions and a heuristic algorithm with better running time and scalability. Simulation results have been shown to justify the performance and efficiency of the solutions. R EFERENCES [1] M. Andrews and L. Zhang, Scheduling algorithms for multi-carrier wireless data systems, ACM Mobicom’2007, September 2007. [2] M. Awad and X. Shen, OFDMA Based Two-Hop Cooperative Relay Network Resources Allocation, IEEE ICC, 2008. [3] S. Deb, V. Mhatre, V. Ramaiyan, WiMAX relay networks: opportunistic scheduling to exploit multiuser diversity and frequency selectivity, ACM MobiCom 2008, pp. 163-174. [4] P. Du, W. Jia, L. Huang, W. Lu, Centralized Scheduling and Channel Assignment in Multi-Channel Single-Transceiver WiMax Mesh Network, IEEE WCNC’2007, pp. 1734-1739. [5] Gurobi Optimizer, http://www.gurobi.com/. [6] Z. Han, Z. Ji, K. J. R. Liu, Non-Cooperative Resource Competition Game by Virtual Referee in Multi-Cell OFDMA Networks, IEEE Journal on Selected Areas in Communications, Vol. 25, Issue 6, pp. 1079-1090. [7] L. Huang, M. Rong, L. Wang, Y. Xue, and E. Schulz, Resource allocation for OFDMA based relay enhanced cellular networks, IEEE VTC-Spring, 2007, pp. 3160-3164. [8] G. Li and H. Liu, On the Capacity of the Broadband Relay Networks, Thirty-eighth asilomar conference on signals, systems and computers, Vol. 2, 2004, pp. 1318-1322. [9] G. Li and H. Liu, Resource allocation for OFDMA relay networks with fairness constraints, IEEE Journal on Selected Areas in Communications, vol. 24, no. 11, pp. 2061-2069, 2006. [10] G. Li and H. Liu, Downlink radio resource allocation for multicell OFDMA system, IEEE Transactions on Wireless Communications, Vol. 5, Issue 12, pp. 3451-3459. [11] O. Oyman, OFDMA2 A: A centralized resource allocation policy for cellular multi-hop networks, IEEE Asilomar Conference on Signals, Systems and Computers, Nov 2006. [12] T. S. Rappaport, Wireless Communications: Principles and Practice, Prentice Hall, 2001. [13] K. Sundaresan, W. Wang and S. Eidenbenz, Algorithmic aspects of communication in ad hoc networks with smart antennas, Proceedings of ACM MobiHoc’2006, pp. 299–309. [14] Y. Xu, S. Wan, J. Tang and R. S. Wolff, Interference Aware Routing and Scheduling in Wireless Backhaul Networks with Smart Antennas, IEEE SECON’2009. [15] H. Viswanathan, S. Mukherjee, Performance of cellular networks with relays and centralized scheduling, IEEE Transactions on Wireless Communications, Vol. 4, Issue 5, pp. 2318-2328. [16] IEEE 802.16 Working Group, Part 16: Air interface for fixed broadband wireless access systems, IEEE Standard, 2004. [17] IEEE 802.16 Working Group, Part 16: Air Interface for fixed and mobile broadband wireless access systems amendment 2: physical and Medium Access Control layers for combined fixed and mobile operation in licensed bands, IEEE Standard, 2005. [18] IEEE 802.16 Working Group, Part 16: Air Interface for fixed and mobile broadband wireless access systems–multihop relay specification, IEEE Draft Standard, 2007.