On the Tree Construction of Multi hop Wireless Mesh Networks with Evolutionary Algorithms

Zahra Zarei & Shahram Jafari
International Journal of Computer Networks (IJCN), Volume (3) : Issue (1) : 2011 1
On the Tree Construction of Multi Hop Wireless Mesh Networks
With Evolutionary Algorithms
Zahra Zarei zahra.zarei@gmail.com
Department of Computer Science and engineering
Shiraz University , Shiraz, Iran
Shahram Jafari jafaris@shirazu.ac.ir
Department of Computer Science and engineering
Shiraz University , Shiraz, Iran
Abstract
In this paper, we study the structure of WiMAX mesh networks and the influence of tree structure
on the performance of the network. From a given network graph, we search for trees, which fulfill
some network, QoS requirements. Since the searching space is very huge, we use genetic
algorithm in order to find solution in acceptable time.
We use NetKey representation which is an unbiased representation with high locality, and due to
high locality we expect standard genetic operators like n-point cross over and mutation work
properly and there is no need for problem specific operators. This encoding belongs to class of
weighted encoding family. In contrast to other representation such as characteristics vector
encoding which can only indicate whether a link is established or not, weighted encodings use
weights for genotype and can thus encode the importance of links. Moreover, by using proper
fitness function we can search for any desired QoS constraint in the network.
Keywords: Wireless Mesh Networks, WiMAX, Network Planning, Multihop Networks.
1. INTRODUCTION
Wireless mesh networks have received much attention in recent years due to its low up-front
cost, easy network maintenance, robustness, and reliable service coverage [3-5]. Different from
traditional wireless networks, WMN is dynamically self-organized and self-configured. In other
words, the nodes in the mesh network automatically establish and maintain network connectivity.
In such networks, each mesh node plays both roles of a host and a router. Packets are forwarded
in a multihop fashion to and from the gateway (connected to the Internet). It has been shown that
the throughput and delay performances in wireless mesh networks are location dependent [6-9].
Mesh networks show great advantages such as good coverage, rapid and cost-efficient
deployment, and robustness. Mesh networks are built using various technologies, however the
most commonly used are WiFi (based on the IEEE 802.11 family) and WiMAX (based on IEEE
802.16). Using IEEE 802.11 for the wireless backbone leads to dense and suboptimal
deployments due to the short transmission range of the standard and, consequently, low
aggregate throughput capacity can be obtained. WiMAX, on the other hand, has a transmission
range of several kilometers with high data rates.
In WiMAX mesh networks, a MSS (Mesh Subscriber Station) can only have one path to the BS
(Base Station) which is through its immediate parent [1]. Thus, MSS nodes are organized in a
tree structure rooted at the BS. The tree topology is constructed through temporarily
disconnection of some links logically. From the other side, we know for a given network graph,
one can construct plenty of spanning trees (see the Kirchhoff’s Matrix Tree Theorem [2]). In each
resulted tree, nodes fan-out and the tree’s depth affected the performance of the WiMAX network
markedly. In this paper, we investigate this phenomenon and find desired tree topologies, which
satisfy intended delay and throughput trade-off. We first obtain per-node throughput and delay of
each node using the model proposed in [9]. In this model, we can obtain throughput and delay for

each node independent of tree structure. Using the obtained results, and, employing a proper
scheduling algorithm, we search for the best tree topologies. For this purpose, we use an
evolutionary algorithm in order to find trees, which satisfy some QoS requirements. The proposed
algorithm converges fast while it finds good enough answers. Due to huge searching space,
classical searching algorithms lead to unacceptable searching time and thus are not practical.
The rest of this paper is organized as follows: in section 2, we discuss the related work. In section
3, we define model assumptions. In section 4, we present the genetic algorithm for tree
construction. In section 5, we show the results. Finally, the paper is concluded in section 6.
2. RELATED WORKS
Designing mesh networks and specially WiMAX has been studied in some works.
In [4] the authors proposed an algorithm that selects a parent for each MSS, which maximize
throughput capacity. The object is to select links that have highest data rates among the set of all
possible paths between an MSS and the BS. In [9] the authors model and analyze the location-
dependent throughput and delay in WMN. They analyze the packet arrival and the packet
departure rate for the forwarding queues at each node, and based on the analytical model, they
proposed two network design strategies to provide fair resource sharing and minimize the end-to-
end delay in WMN. In [10] they investigate the throughput capacity of a WiMAX mesh tree, and
they try to balance the impact of the depth of the tree with its fan-out.The approaches for node
placements in WMN are depict in figure 1.
In this paper, we want to optimize WiMAX topology such that it meets some QoS constraints,
especially throughput and delay, so we assume that all nodes in the network are in the
transmission range and we do not consider the coverage provisioning in our work.
Using Genetic Algorithm (GA) in the field of network planning is widely investigated. Generally,
the application is to optimize some network performance like capacity, topology and routing. In
most of the works, the topology is graph, which has not the limitation of trees. Tree topologies
have limitations dealing with GA operators that we will address in this paper.
The representation that widely used for coding trees in GA is Prufer numbers or Prufer sequence
[11]. This encoding only represents trees, and each Prufer number represents exactly one tree.
These interesting characteristics make it a good option, but it also has an important problem, low
locality, which reduce the
FIGURE 1: Node placement approaches
GA performance dramatically. In [12] the authors investigate the properties of Prufer numbers
and show that it is a poor representation for trees.

3. ANALYTICAL MODEL
The network that we consider consists of N Mesh Subscriber Station (MSS) nodes and one Base
Station (BS). Packets are forwarded in multihop fashion to or from the BS. We consider
unidirectional traffic, i.e., the traffic that only goes from the MSS to the BS.
Each MSSi has two queues, Qr for relay packets and Qs for packets originating from the node
itself. The forwarding rules at each node is described as follows: [9]
1- If Qr is empty, it sends one packet from Qs.( Qs always has packets to be transmitting)
2- If Qr is not empty, it sends a packet from Qr with a probability of qi or a packet from Qs
with a probability of 1-qi.
qi is forwarding probability for node i.
3.1 Throughput and Delay Analysis
We define per node throughput as the number of packets originating from node i successfully
received by the BS. It can be obtained by counting the packets, which are received successfully
by the BS without being blocked in any intermediate nodes. Thus, we need to calculate the
blocking probability at each intermediate MSS. Based on queuing theory analysis, it is given as:
Where ρi is the traffic intensity given above. Thus, we can state the throughput of node i, as
below:
Where |ni
u
| is number of nodes in the uplink path from node i to the BS. The average network
throughput is obtained as follows:
To drive the delay which a packet from node i encounter, we need to compute the waiting time of
a packet in each Qr. For this we first obtain the steady state queue size of Qr in node i as follows
(using M/M/1/K analysis):
Then we will have the following expression for the waiting time of a packet in Qr of node i:

Now, we can obtain D(i) by summing the waiting times spent in the intermediate nodes and the
transmission times(i.e. ), for traversing the i hopes:
In addition, the average delay of the network can be found by taking into account the delay of
packets, which have been successfully delivered:
4. PROPOSED GENETIC ALGORITHM
Given the number of nodes which exist in a WiMAX network; we intend to obtain a tree topology
that satisfies some QoS metrics. For this purpose, we use a Genetic Algorithm (GA) approach.
GA is one the most powerful approaches for optimizing complicated, multi-objective and large
scale problems. Our problem in this paper is also, large scale and constrained. By a slight
increasing in the number of nodes the search space grows dramatically and become too
complicated to be addressed by conventional problem solving strategies, e.g. Linear
Programming.
The proposed Genetic Algorithm is as followed:
Algorithm 1. Proposed GA
1. Create a specified number of random keys with length l=n(n-1)/2.
2. For each individual produced in step 1, generate it’s permutation sequence and it’s
corresponding spanning tree.
3. i=1
4. Do{
a. At iteration i, evaluate fitness of each individuals.
b. Select top 10 percent individuals from Pi and transfer them directly to population
Pi+1 , apply selection mechanism as defined in 4-2 to choose individuals.
c. Apply cross over to generate offsprings.
d. Apply mutation operator.
e. Generate the next population Pi+1.
f. Increment i.
5. } While a solution is not found or total number of generation is not reached or the function
tolerance is less than threshold.
Details of our algorithm are as follows:
4.1 Network Representation
We use NetKey representation which is simple and convenient to implement and characteristics
of this representation is studied in [13].
The NetKey encoding belongs to the class of weighted encodings. In contrast to other
representations such as CV encoding (Davis et al., 1993) which can only indicate whether a link
is established or not, these encodings use weights for the genotypes and can thus encode the

importance of the links Also, an additional construction algorithm is necessary which constructs a
valid tree from the genotypic weights.
For coding a tree, we first generate a random key sequence with length l=n(n-1)/2. Each element
of this sequence which is between zero and one, describes the importance of the corresponding
link (link ordering is like CV encoding).
The permutation r
s
corresponding to a key sequence r of length l is the permutation δ such that δr
is decreasing. These definitions say that the positions of the keys in the key sequence r, are
ordered according to the values of the keys in descending order.
After generating a random key sequence and a permutation, we should construct a valid tree
from that permutation. When constructing the tree, the positions of the key sequence r, are
interpreted in the same way as for the CV.
The positions are labeled and each position represents one possible link in the tree. From a Key
sequence r of length l, we have a permutation rs
,of l numbers. Then the tree is constructed from
the rs
as follows:
Algorithm 2. Tree Construction
1- Let i=0 , T be an empty tree with n nodes, and rs
the permutation of length l=n(n-1)/2 that
can be constructed from the key sequence r. All possible links of T are numbered from 1
to l.
2- Let j be the number at the ith position of the r
s
.
3- If the insertion of the link with number j in T would not create a cycle, then insert the link
in the T.
4- Stop, if there are n-1 links in T.
5- Increment I and go to step 2.
The construction rule is based on Kruskal’s algorithm (Kruskal 1956) and only consider the
weights of the Random Keys vector for building the tree. With this rule, we can construct a
unique, valid tree from every possible Random key sequence.
4.2 Selection and Reproduction
Based on trial and error we employ elitism to directly transfer top 10 percent individuals from
current population to the next one. The rest of the population is created by crossing and mutating
the genes. We use tournament selection with tournament size of four, for selecting the parents for
mating pool. The reason why we employ this selection mechanism is to keep a balance between
randomness and proportionality in search process.
4.3 Cross Over and Mutation Operator
Since the NetKey encoding have high locality, the standard cross over, mutation operators will
work properly, and there is no need for problem specific operators. Because the operators will
change the Random key sequence there will be no infeasible solution.
4.4 Fitness Function
With fitness function, we can evaluate the goodness degree of each individual in the population.
For any desired QoS, we can have different fitness functions.
A well-known problem in wireless mesh networks is the fairness problem [5-7], i.e., the nodes
farther away from the gateway may have a lower throughput than the nodes closer to the
gateway. We address this problem here by searching for trees, which have smoothest throughput
among its nodes. So in our fist scenario we want to find a tree t such that function f(t)=std(T) is
minimized.
In the second scenario, we study the delay-minimization problem based on the analytical model
described in 3. Here we want to find a QoS tree, which minimize the average delay in whole
network. The fitness function here is f(t)=min(D).

5 NUMERICAL STUDY AND RESULTS
The network we choose for our numerical study is consisting of 18 nodes and one gateway.
Based on Cayley(1889), there is exactly nn-2
possible tree for a graph with n nodes. This huge
search space dare us to use GA, otherwise it will be too complex.
For simplicity we do not consider spatial reuse in the simulation, thus only one node is allowed to
send within one time slot. The channel capacity is 75Mbs and the packet size is 1500B, so the
length of time slot is set to 1500B/75Mbs=0.16 ms. The buffer size of Qr is set to 30 for each
node. In addition, for each node the channel access probability is assumed to be proportional to
its sub tree and, probability for relaying packets is q=0.7 for all nodes. Population size is 50;
crossover and mutation rates are adjusted to 0.8 and 0.02 respectively. Total number of GA
iterations is set to 200.
The resulted tree topologies for fairness throughput and minimum delay are show in figure 2, 3
and the corresponding per-node throughput and delay for each node is depicted in figure 4(a), (b)
respectively. As we can see, in the first scenario, the throughput among nodes is smooth and
close to each other as we expected and the resulted tree is also almost balance. We know that
for having a fair throughput among nodes especially for those that are far away from gateway we
should consider tuning the forwarding probability for nodes,in the leaf nodes q=0 because there is
no data to be forwarded but in intermediate nodes the q should be set to a value such that the
node forward the childs traffic and also it’s packets.
1
2
3
4
5
6 7
8
9
10
11
1213
14
15
16 17
18
FIGURE 2: Resulted tree topology for fair Throughput
1
2 3
4
5
6 7
8 9
1011
1213 1415
16
17 18
FIGURE 3: Resulted tree topology for minimum delay
In the second scenario the delay is minimized in the cost of decrease in throughput. As we can
see in the resulted tree the depth is decreased and the node fan-out increased. Tuning
forwarding probability (q) in this scenario is also important, if we set the q so high, the

intermediate nodes will always be busy by forwarding the leaf nodes data and their delay may
increase or even they may starve.
We also change our scenario a bit to compare our proposed algorithm with [14]. We increase the
number of nodes from 5 to 120 with step of 5 like [14] and change our fitness function to gain
maximum throughput for each network.
For each network with n nodes we set the population size to n
1.5
based on the [13] and increase
the number of generations based on the size of the problem from 100 to 700.
Figure 5 show the results from [14] and figure 6 show our result. We can see that our algorithm
can find a topology with better overall throughput for networks from 5 to 65 nodes and the remain
is almost the same.
2 4 6 8 10 12 14 16 18
115
120
125
130
135
140
145
150
155
160
Node Numbers
Throughputpkt/sec
Smooth Throughput
Minimun Delay
(a)
2 4 6 8 10 12 14 16 18
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Node Numbers
Delay(second)
Smooth Throughput
Minimun Delay
(b)
FIGURE 4: Per-node delay and throughput of output tree, (a) per-node throughput, (b) per-node
delay
FIGURE 5: Resulted throughput from [14]

FIGURE 6: Maximum Throughput for different networks
6 CONCLUSION
In this paper, we propose a GA algorithm for finding a QoS tree for WiMAX mesh networks.
Buffer size for each MSS node, transmission opportunity, lost packets, traffic load per node
and delay in each node is considered. Our proposed GA algorithm is able to find QoS tree
topology for given network. Any change in network parameter or nodes or application of
network will not affect the algorithm at all. With defining proper fitness function this approach
can always find the QoS tree with any desired delay and throughput. The obtained results
show that different delay and throughput will lead to different tree’s depth and fan out. For
future works, node movements can be considered to overcome the limitation of the proposed
algorithm, which are for the fixed nodes.
7 REFRENCES
1. IEEE 802.16d-2004,”Draft IEEE Standard for Local and Metropolitan area networks”, May
2004.
2. S. Skiena, "Implementing Discrete Mathematics: Combinatorics and Graph Theory with
Mathematica",Addison-Wesley,1990.
3. M. J. Lee, J. Zheng, Y.-B. Ko, and D. M. Shrestha, “Emerging standards for wireless mesh
technology,” Wireless Commun”, 13( 2): 56–63, 2006.
4. Salim Nahle, Luigi Iannone, Benoit Donnet and Naceur Malouch, “On the Construction of a
wimax mesh tree”, IEEE COMMUNICATIONS LETTERS, 11(12): 967 – 969, 2007
5. I. F. Akyildiz, X. Wang, and W. Wang, “Wireless mesh networks: A survey,” Comput. Netw.,
47(4):445–487, 2005.
6. V. Gambiroza, B. Sadeghi, and E. W. Knightly, “End-to-end performance and fairness in
multihop wireless backhaul networks,” in Proc. ACM MOBICOM, Sep. 2004.
7. J. Jun and M. L. Sichitiu, “Fairness and QoS in multihop wireless networks,” in Proc. IEEE
VTC, Oct. 2003.
8. J.-F. Lee, W. Liao, and M.-C. Chen, “An incentive-based fairness mechanism for multi-hop
wireless backhaul networks with selfish nodes,” IEEE Trans. Wireless Commun., 7(2): 697 -
704,2008.
9. T.Liu, W.Lio , “Location Dependent Throughput and Delay in Wireless Mesh Networks” ,
IEEE Transactions on Vehicular Technology, 57(2):1188-1198, 2008.

10. S.Nahle, L.Iannone, B.Donnet, and T. Friedman, “Investigating depth fanout trade-off in
WiMAX mesh networks”, in Proc. 1st
WEIRD workshop, May 2007.
11. H. Prufer. “Neuer Beweis eines Satzes uber Permutationen” Arch.Math. Phys.27(1) : 742-
744,1918.
12. J. Gottlieb, G. R. Raidl, B.A. Julstrom, F. Rothlauf ,” Pr¨ufer Numbers: A Poor Representation
of Spanning Trees for Evolutionary Search” IEEE Transactions on Evolutionary Computation,
4(2):125–134, 2002.
13. F.Rothluf , D.E. Goldberg and A.Heinzel, “Network Random Keys—A Tree Representation
Scheme for Genetic and Evolutionary Algorithms” MIT press journals , 10(1): 75-97,2002.
14. Ali Al-Hemyari, Nor Kamariah Noordin, Alyani Ismail, Sabira Khatun, Yaseen H. Tahir, and
Y.A. Qassem “Centralized Scheduling, Routing Tree in WiMAX Mesh Networks”, Innovations
in Information Technology, 2008. IIT 2008, Al Ain UAE.

On the Tree Construction of Multi hop Wireless Mesh Networks with Evolutionary Algorithms

More Related Content

What's hot (17)

Viewers also liked (8)

Similar to On the Tree Construction of Multi hop Wireless Mesh Networks with Evolutionary Algorithms (20)

Recently uploaded (20)

On the Tree Construction of Multi hop Wireless Mesh Networks with Evolutionary Algorithms