The effect of hexagonal grid topology on wireless communication ...

INTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMS Int. J. Commun. Syst. 2014; 27:1319–1337 Published online 25 March 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/dac.2782

The effect of hexagonal grid topology on wireless communication networks based on network coding Tao Shang* ,† , Fu-Hua Huang, Ke-Fei Mao and Jian-Wei Liu School of Electronic and Information Engineering, Beihang University, Beijing, China

SUMMARY Network coding has emerged as a promising technology that can provide significant improvements in the performance of wireless communication networks. Inspired by the recent advances in complex networks, we herein study the effect of topology structure on the performance benefit of network coding based on a highly structured wireless communication network with hexagonal grid topology. We propose a new concept called ‘network intensity’, namely  to characterize the property of hexagonal grid topology. We first derive a value of 12=7 for the upper bound on performance p benefit in a single regular hexagon network. This value holds only if the network intensity is in the case of 3 6  < 2. Based on the results in a single regular hexagon network, we then derive a value of 16=7 for the upper bound on performance benefit in a general hexagonal p grid network only if the network intensity is in the case of 1 6  < 3. Furthermore, a comparative analysis demonstrates that the network intensity affects the performance benefit of network coding in terms of the interference and the coding number. These findings will contribute to the design of network topology and the analysis of the bound on the performance benefit of network coding in wireless communication networks. Copyright © 2014 John Wiley & Sons, Ltd. Received 8 September 2013; Revised 18 February 2014; Accepted 25 February 2014 KEY WORDS:

network coding; complex networks; hexagonal grid; performance benefit; network intensity; upper bound

1. INTRODUCTION Network coding has been proved to be a promising technology for communication networks, and it can increase throughput, reduce delay, improve robustness and improve load balance [1–6]. Especially, the flooding and broadcasting peculiarity makes it more suitable for wireless communication networks. As a paradigm, the network coding scheme Complete OPportunity Encoding (COPE) proposed in [7], which is the first practical network coding protocol in multihop wireless networks, makes use of broadcast channel by overhearing native packets and broadcasting encoded packets to execute the network coding. Network coding has superb prospects in wireless communication networks, and thus its essential performance benefit to wireless communication networks is a strongly desirable to clarify. Research on complex networks helps achieve better understanding of communication networks. Complex networks can be characterized by large-scale topologies, distributed resource management, extreme heterogeneity of the constituent elements, high clustering coefficients, and so on, which are highly correlated to wireless communication networks such as mobile ad-hoc network, wireless sensor network, wireless mesh network or WiMAX network. Further analysis of wireless communication network paradigms in the context of complex networks will be very useful to network design. The key challenge is how to impose it on wireless communication networks. In the *Correspondence to: Tao Shang, School of Electronic and Information Engineering, Beihang University, Beijing, China. † E-mail: [email protected] Copyright © 2014 John Wiley & Sons, Ltd.

T. SHANG ET AL.

1320

past few years, many researchers [8] studied the structure and function of complex networks, and they have increasingly recognized that the characterization and the modeling of the structure of a network would lead to a better knowledge of its dynamical and functional behavior. Furthermore, the structural complexity of a network can be influenced from both node and connection diversity. Meanwhile, the network topology plays a crucial role in determining the emergence of collective dynamical behavior, such as synchronization, or in governing the main features of relevant processes that take place in complex networks, such as the spreading of information. Apparently, it remains a challenge to answer the fundamental question, that is, ‘How does topology structure affect wireless communication networks based on network coding? ’ or ‘Exactly how many performance benefits can be expected from network coding in the context of different topology structures of wireless communication networks?’. The studies in [9, 10] show that the performance benefit of network coding depends highly on the topology structure of multihop wireless networks. For instance, in Figure 1(a), a maximum of four packets can be encoded by an intermediate node within the ‘cross-style’ topology structure. By comparison, in Figure 1(b), a maximum of six packets can be encoded by an intermediate node within the ‘hexagon-style’ topology structure, which is three-halves times compared with the former case, and the details are described in Section 2.1. This phenomenon clearly shows that the topology, especially the basic unit of topology, significantly affects the performance benefit of network coding. By taking the topology constraints into consideration, Le et al. [11] proved that for a general wireless network structure, the performance benefit using the XOR coding scheme (COPE or COPE-like scheme) is upper bounded by 2n=.n C 1/ , where n denotes the maximum number of packets, which can be encoded within one single coding structure. This result illustrates that the topology can affect the performance benefit of network coding by changing the value of n. To seamlessly cover a full plane with regular polygons, there are three possible choices for the structural property of network topology, including regular triangle, square and regular hexagon. A network with hexagonal grid topology the basic unit of which is a regular hexagon is an optimal two-dimensional (2D) wireless communication network. This configuration can cover the geometrical range with the least number of nodes. In other words, if circles are used to cover a 2D plane with identical radius, their centers should coincide with the center of each hexagon in the hexagonal grid [12] in order to reach the largest geometrical range. A hexagonal grid network can be extensively applied in wireless communication networks. By means of network deployment and topology control, the hexagonal grid network facilitates the implementation of geometrical coverage of a certain area with the least number of nodes and provides a QoS approach for multiclass traffic wireless communication networks [13, 14]. Especially, the intrinsic characteristic of hexagonal grid topology makes it easy for the clustering management of nodes and hierarchical processing of a network. If the central node of each hexagon is treated to be a cluster header, intracluster session can be implemented in a hexagonal structure unit. If a hexagonal grid structure unit is located at the center of six surrounding structure units, the central node of the hexagonal grid structure unit can be treated

Figure 1. Difference of topology structures. Copyright © 2014 John Wiley & Sons, Ltd.

Int. J. Commun. Syst. 2014; 27:1319–1337 DOI: 10.1002/dac

THE EFFECT OF HEXAGONAL GRID TOPOLOGY ON NETWORK CODING

1321

to be a higher level cluster header, and the total nodes of the hexagonal grid structure unit can be treated to be relay nodes so as to implement intercluster communications. Furthermore, for the current construction of wireless metropolitan, based on current wireless infrastructure, newly-built wired and wireless base stations can be arranged according to hexagonal grid topology so that it can be deployed to optimize wireless coverage and provide the maximum-throughput transmission performance with the economical minimum-node infrastructure. Until now, there is still no explicit conclusion about the performance benefit of network coding in a network with hexagonal grid topology. We herein attempt to derive the upper bound on the performance benefit of network coding in hexagonal grid topology and to generalize the results to the other types of topology. In addition, the highly structured geometrical shape of hexagonal grid is also helpful to analyze the upper bound and provide the ground for the quantification of the relationship between the topology structure and the performance benefit of network coding. In order to analyze the effect of topology structure on the performance benefit of network coding, we herein first propose a quantitative network parameter called ‘network intensity’, namely  to characterize the connectivity between nodes or the density of a network, we will discuss ‘network intensity’ in details in Section 3. On one hand, we quantify the exact relationship between the network intensity  and the performance benefit of network coding, that is, we study how exactly the topology structure affects the performance benefit of network coding. On the other hand, we try to obtain a tightest upper bound on the performance benefit of network coding in the multihop wireless network of hexagonal grid topology on the basis of Le’s work. As the main contributions of our work, (1) we introduced the network intensity  to characterize the property of topology structure and to quantify the exact relationship between the network intensity  and the upper bound on the performance benefit of network coding. (2) We derived the tightest upper bound on the performance benefit of network coding in a single regular hexagon network for now, and we found that the exact upper bound is 12=7 only if p 3 6  < 2 is satisfied. (3) Furthermore, we derived the tightest upper bound on the performance benefit of network coding in a general p hexagonal grid network, and we found that the exact upper bound is 16=7 only if 1 6  < 3 is satisfied. (4) We verified that the topology structure described in this way can certainly affect the performance benefit of network coding, and we concluded that the compensative effect of the interference and the coding number results in the improvement of performance benefit in hexagonal grid topology. This paper is structured as follows. Section 2 presents the main works related to this paper and highlights the problems encountered in these works. Section 3 describes the network model in details and the definition of network intensity. Section 4 discusses the relationship between network intensity and performance benefit and derives the upper bounds on performance benefit in a single regular hexagon network under different cases. Section 5 extends the upper bounds from a single regular hexagon network to a general hexagonal grid network. Finally, Section 6 concludes the paper.

2. RELATED WORKS 2.1. Complete opportunity encoding scheme Here, we describe the working mechanism of COPE and take the typical scenario in Figure 2 as an example. COPE incorporates three main techniques as follows [7]: (a) Opportunistic listening: Wireless is a broadcast medium, creating many opportunities for nodes to overhear packets when they are equipped with omnidirectional antenna. COPE sets the nodes in promiscuous mode, makes them snoop on all communications over the wireless medium and store the overheard packets for a limited period. Copyright © 2014 John Wiley & Sons, Ltd.

Int. J. Commun. Syst. 2014; 27:1319–1337 DOI: 10.1002/dac

T. SHANG ET AL.

1322

Figure 2. Typical topology of wireless network coding.

In addition, each node broadcasts reception reports to tell its neighbors which packets it has stored. Reception reports are sent by annotating the packets the node transmits. A node that has no packets to transmit periodically sends the reception reports in special control packets. In Figure 2, let us suppose that the node S1 has a packet a to transmit to the node D1 via the intermediate node C and that the node S2 has a packet b to transmit to the node D2 via the node C . First, the node S1 .S2 / transmits the packet a.b/ to C . Using the opportunistic listening, D2 .D1 / overhears the packet a.b/. And then D2 .D1 / broadcasts reception reports to tell C that it has stored the packet a.b/. (b) Opportunistic coding: Packets from multiple unicast flows may have encoded together at some intermediate hop. The nodes that perform encoding should aim to maximize the number of native packets delivered in a single transmission, while ensuring that each intended nexthop has enough information to decode its native packet. This can be achieved using the following simple rule: to transmit n packets, p1 , : : : , pn , to n nexthops, r1 , : : : , rn , a node can XOR the n packets together only if each nexthop ri has all n  1 packets pj for j ¤ i. In Figure 2, C knows that D2 .D1 / has stored the packet a.b/ via D2 .D1 /’s reception reports; therefore, the rule is satisfied. Then C XORs a and b, and broadcasts the encoded packet a˚b to D1 and D2 . After receiving a˚b, D2 .D1 / makes use of the overheard packet a.b/ to decode the packet a˚b. The decoding processes in D1 and D2 are executed as a D a˚b˚b and b D a˚b˚a, then D1 .D2 / receive the packet a.b/. (c) Learning neighbor state: Besides the reception report, COPE provides another method for nodes to know what packets its neighbors have. This method is that COPE estimates the probability that a particular neighbor has a packet as the delivery probability of the link between the packet’s previous hop and the neighbor. In Figure 2, C may guess that D2 .D1 / has stored the packet a˚b for the delivery probability of the link between S1 .S2 / and D2 .D1 / is high enough. Now, let us explain the situation in Figure 1. In Figure 1(a), by using COPE, the intermediate node can encode a maximum of four packets when there are four data flows crossing it at the same time. Similarly, in Figure 1(b), by means of COPE, the intermediate node can encode a maximum of six packets when there are six data flows crossing it at the same time. In conclusion, COPE is executed in a multihop wireless network by overhearing and broadcasting packets. The characteristic of overhearing generates coding opportunities that can benefit both encoding and decoding, and reduce the times of packets transmission, thus improving network throughput. In the example, S1 and S2 transmitting the packets a and b to D1 and D2 need three transmissions with network coding compared with four transmissions without network coding (for C should send a and b separately). The times of packet transmissions decrease when using network coding, thus improving the network throughput. 2.2. Performance benefit In general, performance benefit is defined as the ratio of performance with and without network coding. It evaluates the performance improvement for the employment of network coding. Liu et al. [15] were first to study the upper bound on the performance benefit of network coding in a multihop wireless network. They found that forparbitrary network coding in 2D random topology, the upper bound on performance benefit is 2c .1 C /= for large n, where  > 0 is a Copyright © 2014 John Wiley & Sons, Ltd.

Int. J. Commun. Syst. 2014; 27:1319–1337 DOI: 10.1002/dac

THE EFFECT OF HEXAGONAL GRID TOPOLOGY ON NETWORK CODING

1323

parameter of the wireless medium that characterizes the p intensity of interference between nodes, n is the number of nodes in the network, and c D max¹2, 2 C 2º. This conclusion contributes to the analysis of relationship between performance benefit and network intensity. However, the derived upper bound is not tight enough to characterize a practical performance benefit. In addition, the effect of topology structure on performance benefit is not considered as well. If the topology structure was to be considered, a tighter bound would have been achieved. Goseling et al. [9] studied the bound on the performance benefit of network coding from the viewpoint of energy consumption. They found that if coding and noncodingp solutions use the same transmission range, the benefit in d -dimensional network is at least 2d=b d c. Moreover, if the transmission range can be optimized for coding and noncoding individually, the benefit in 2D networks is at least 3. Meanwhile, a constructive coding scheme is established in a hexagonal grid network and quadrilateral grid network, respectively, but the coding scheme is too theoretical to represent general practical cases. Koutsonikolas’s empirical study [10] shows that the performance benefit of network coding in a moderate-size multihop wireless network for general traffic patterns is extremely limited. The performance of a noncoding scheme even outperforms the performance of a coding scheme. Empirical experiments were conducted in a sparse network with quadrilateral grid topology and one-hop transmission range, as shown in Figure 3. In fact, such configuration restricted the performance benefit of network coding. Considering the scenario in Figure 3, the coding opportunities emerge only when there are two opposite flows traversing the same node simultaneously, and even under this condition, the number of encoded packets is just 2, which is the least in all feasible coding schemes. This is different from the original results reported in [7], where in a random network and with a random traffic pattern, the original COPE protocol offers a three to four times throughput improvement over a noncoding scheme. The significant gain in [7] is due to the small (20 nodes) and dense network it used. Such a configuration offers high overhearing probabilities, giving nodes chances to XOR more than two packets together. In contrast, the evaluation in [10] was conducted in a larger and sparser network where the overhearing cannot bring any benefit. Meanwhile, the parameter  in [15] also implies the influence of the network intensity on performance benefit to a certain extent. The former results show that the network intensity greatly affects the performance benefit of network coding. Thus the question of how the network intensity exactly affects the performance benefit of network coding is still open. Le et al. [16] proposed the concept of ‘coding number’, which is equal to the number of packets that can be encoded in a single ‘coding structure’. Le pointed out that all the coding opportunities are within one single coding structure and that the performance benefit in one coding structure is associated with the coding number. The upper bound on the coding number in any possible coding structure is O..r=ı/2 / in a 2D random wireless network, where r is the transmission range, and ı is the transmission gap. This result is contradictive with the result in [7], which indicates that the coding number can be infinite. In fact, as described in [10], there exist geometrical constraints within a multihop wireless network, that is, the network intensity can affect the performance benefit

Figure 3. Quadrilateral grid topology. Copyright © 2014 John Wiley & Sons, Ltd.

Int. J. Commun. Syst. 2014; 27:1319–1337 DOI: 10.1002/dac

T. SHANG ET AL.

1324

of network coding. By taking the geometrical constraints into consideration, Le et al. [11] proved that for a general wireless network, the upper bound on the performance benefit by using the XORstyle coding scheme is 2n=.nC1/ , where n denotes the maximum coding number within one single coding structure. This upper bound is tighter compared with the one in [15], but the drawback is that this upper bound is the same with the one in a general network with the interference of uncorrelated nodes, which implies that an actual upper bound could be tighter. 3. NETWORK MODEL 3.1. Description of network model As shown in Figure 4, the topology with a hexagonal grid is optimal in a 2D plane. It can cover the geometrical range with the least number of nodes. Our study on the performance benefit of network coding in a multihop wireless network is based on hexagonal grid topology. The basic structure unit of hexagonal grid topology shown in Figure 5 is a single regular hexagon network, which consists of six vertices and one central node. As shown in Figure 5, we studied the multiple unicast traffic pattern as follows: each source node, denoted as Si .i D 1, : : : , 6/, has packets to deliver to its opposite destination node, which is Di .i D 1, : : : , 6/. C denotes the intermediate node. We schedule the source packets transmission in order to maximize the throughput with or without network coding. We assume that the nodes Si and Di cannot communicate with each other directly, and they have to use the intermediate node C to relay the communication. Provided that j  j denotes the Euclid distance and jSDj denotes the distance between the node S and the node D, we assume that the node D successfully receives

Figure 4. Hexagonal grid topology.

Figure 5. Single regular hexagon network. Copyright © 2014 John Wiley & Sons, Ltd.

Int. J. Commun. Syst. 2014; 27:1319–1337 DOI: 10.1002/dac

THE EFFECT OF HEXAGONAL GRID TOPOLOGY ON NETWORK CODING

1325

the packets originating from the node S with the probability of 100% if r > jSDj and with the probability of 0% if r < jSDj. We consider a stationary wireless network, because dynamic node will destroy the validity of opportunistic coding of COPE. The static hexagonal grid can offer good characteristics to make full use of the performance of COPE. We also assume all nodes in the network have identical transmission range. This configuration can cover the geometrical range with the least number of nodes. Practically, there are many types of multihop wireless network the nodes of which are stationary and have identical transmission range, such as the mesh router of wireless mesh networks and the sensor node of wireless sensor networks. The essential limit to network throughput is the shared physical channel within a single regular hexagon network. All seven participant nodes are within one shared wireless channel. We assume that the capacity of the shared channel is 1, and the bandwidth resource is time-shared such as distributed coordination function in 802.11. Only one node is allowed to send data in a time slot. Consequently, there exists no transmission collision in a neighbor scope, that is, there are no lost packets in a neighbor scope. Thus such assumption is helpful to the analysis of the upper bound on performance benefit. Meanwhile, because the half-duplex mode is easier to calculate performance benefit than the full-duplex mode, we assume that nodes operate at halfduplex mode. We use COPE as a representative protocol and expect to generalize the results of COPE to other similar protocols [17–23]. We show how COPE or COPE-like coding scheme works under the traffic pattern described previously in a hexagonal grid network. When each source node Si sends a packet ai , all other nodes except for opposite node Di should be able to overhear this packet. When C receives all six packets from Si .i D 1, : : : , 6/, it then XORs these packets and broadcasts them as a single packet, denoted as the following: a1 ˚ a2 ˚ a3 ˚ a4 ˚ a5 ˚ a6 . All the six nodes Di .i D 1, : : : , 6/ receive the broadcast packet and decode it to obtain the target packet by means of the overheard packets, that is, ai D .a1 ˚a2 ˚a3 ˚a4 ˚a5 ˚a6 /˚.a1 ˚: : :˚ai 1 ˚ai C1 ˚: : :˚a6 /. So far, the multiple unicast traffic has been fulfilled by the coding scheme. Thus, in our network model, we give the following assumptions: (1) (2) (3) (4) (5)

The hexagonal grid is static. All nodes in the network have identical transmission range. The communication mode between nodes is half-duplex. The bandwidth resource is time-shared. The COPE or COPE-like scheme is used as typical coding scheme.

On the other hand, we define the coding number nc as the number of packets, which can be encoded by the central node C at one time. With COPE or COPE-like coding scheme, the coding number in a single regular hexagon network achieves a maximum value 6 when all six source nodes send packets to its opposite destination nodes. Let Ti .i D 1, : : : , 6/ denote the end-to-end throughput of the data flow from the source node Si .i D 1, : : : , 6/ to the destination node Di .i D 1, : : : , 6/, which is equal to the incoming data of Di .i D 1, : : : , 6/. Then we define the throughput of a network as the sum of Ti .i D 1, : : : , 6/, that P6 is, T D i D1 Ti . Let Tc and Tnc denote the throughput with coding scheme and noncoding scheme under same  topology structure and network intensity, respectively. Let Tc and Tnc denote the maximum of Tc  and Tnc . Then we define the performance benefit of network coding as Tb D Tc =Tnc to evaluate the throughput improved by network coding. Thus the upper bound on performance benefit is defined  as Tb D Tc =Tnc . 3.2. Definition of network intensity As we know, network topology can affect the performance benefit of network coding. Here, we propose a new concept called ‘network intensity’ to quantify the effect of hexagonal grid topology on the performance benefit of network coding. Copyright © 2014 John Wiley & Sons, Ltd.

Int. J. Commun. Syst. 2014; 27:1319–1337 DOI: 10.1002/dac

T. SHANG ET AL.

1326

Definition 1 ‘Network intensity’ is defined by  D r=R , where R denotes the distance between two neighbor nodes in a hexagonal grid, and r denotes the transmission range of a node, which considers the effect of both the transmission range and the density of nodes, as shown in Figure 6. Network density generally denotes the number of nodes in a certain transmission range. Because it does not consider the topology structure and opportunistic coding, the concept of network density is insufficient to characterize the hexagonal grid topology. Instead of network density, network intensity combines the factor of distance between two neighbor nodes with the factor of transmission range. Furthermore, the factor of the distance between two neighbor nodes can be related to opportunistic coding. Thus network intensity can describe the relationship of acquiring encoding opportunity and evaluate how network topology affects the performance benefit of network coding. Specifically, the performance benefit depends on the coding number nc , and the coding number further depends on network intensity. For hexagonal grid topology, R is isometric. When r is different, the coding number is also different due to the opportunistic listening ability. The smaller the value of , the smaller the coding number, just like the strength of wireless signal. Such phenomenon leads to the concept of network intensity. Note that the network intensity is a general concept, and it can be expanded to general wireless networks. For a wireless network with nodes randomly distributed, if the network intensity is larger, the coverage is relatively larger. That mean that there are more neighbor nodes receiving the transmitted packets, and the coding number is also larger. According to this definition, the network intensity actually describe the relative relationship between R and r. Virtually, it is a precise measurement of the coding number, and it refines the essential characteristics of network topology. The network intensity in different ranges will lead to different coding number, which is an important property of the network intensity. Theorem 1 describes this property in details. Theorem 1 p The coding number in a single regular hexagon network is nc D 6 and nc D 2 when 3 6  < 2 p and 1 6  < 3 are satisfied, respectively. Proof In a single regular hexagon network with COPE or COPE-like coding scheme, there exist geometrical constraints for the coding scheme to achieve a maximum throughput. After the central node C broadcasts the encoded packet, all six vertices Di .i D 1, : : : , 6/ should be able to decode the encoded packet, which means that each transmission of the source node Si .i D 1, : : : , 6/ should be overheard by the vertices Dj .j D 1, : : : , 6, j ¤ i/, but cannot be overheard by the vertex Di ,

Figure 6. Network intensity  D r=R. Copyright © 2014 John Wiley & Sons, Ltd.

Int. J. Commun. Syst. 2014; 27:1319–1337 DOI: 10.1002/dac

THE EFFECT OF HEXAGONAL GRID TOPOLOGY ON NETWORK CODING

1327

Figure 7. The effect of  on nc .

Figure 8. Two types of topology.

just as shown in Figure 7. Each transmission of the source node p Si .i D 1, : : : , 6/ will be overheard by the destination node Dj .j D 1, : : : , 6, j ¤ i/ if r > 3R. Moreover, each transmission of p Si .i D 1, : : : , 6/ will not be overheard by the opposite node Di if r < 2R. In summary, if 3 6  < 2, then the central node C in COPE or COPE-like coding scheme will be able to encode all six packets from Si .i D 1, : : : , 6/ and broadcast the encoded packet p to Dj .j D 1, : : : , 6/, that is, nc D 6. On the other hand, under the condition of 1 6  < 3, because of lack of overhearing opportunities, there only exist two kinds of feasible topology structure for coding in the single regular hexagon network: (1) ‘X-style’ topology, and (2) topology with two opposite flows. In Figure 8(a), both the node Si and the node Di send packets to the node C and store them in their buffers, the node C XORs the received packets and broadcasts the encoded ones; after receiving the encoded packet, Si and Di use the packet they store in the buffer to decode and obtain the target packets. Obviously, the number of the packets that C can encode is 2. The other case in Figure 8(b) has the same coding number p as the scenario in Figure 8(a), which is also 2. So the coding number is nc D 2 when 1 6  < 3. Besides the constraints stated in Theorem 1, it is evident that there will be no communication when  < 1 and communication will be excessive in the single regular hexagon network where it is not necessary for the node C to be an intermediate node when  > 2. In conclusion, the property of the network intensity is summarized as follows: the codingpnumber p increases from nc D 2 to nc D 6 as the network intensity increases from 1 6  < 3 to 3 6  < 2, nc D 0 when  < 1, and nc is undefined when  > 2.  4. PERFORMANCE BENEFIT IN A SINGLE REGULAR HEXAGON NETWORK As the basic unit of a hexagonal grid network, the single regular hexagon network is the place where the coding process of COPE or COPE-like protocols happens. In this section, we discuss the performance benefit of network coding within a single regular hexagon network. The main mathematical symbols used in this section are defined in Table I. The work in [11] illustrates that the coding number will affect the throughput of network coding  and provides two lemmas to calculate the values of Tc and Tnc in a circle topology with a central Copyright © 2014 John Wiley & Sons, Ltd.

Int. J. Commun. Syst. 2014; 27:1319–1337 DOI: 10.1002/dac

T. SHANG ET AL.

1328

Table I. The definition of mathematical symbols. Symbol  nc Tb Tb ˛i .i D 1, : : : , 6/ 

Definition Network intensity Coding number The performance benefit of network coding The upper bound on the performance benefit The draining rate of the source node Si .i D 1, : : : , 6/ The draining rate of the central node C

node and n nodes homogeneously distributed on the circumference. We extend the related conclu sions to deduce the value of Tc and Tnc . As proved in Theorem 1, in a single regular hexagon network, the coding number strongly depends on the network intensity, so we discuss the performance benefit of network coding in a single regular hexagon network under two different conditions: p p 3 6  < 2 and 1 6  < 3. p 4.1. Upper bound on performance benefit under the condition of 3 6  < 2 Lemma 1 p  In a noncoding scheme, when 3 6  < 2, the maximum throughput Tnc is equal to 1=2 under the condition that the sum of bandwidth allocated to six vertices Si .i D 1, : : : , 6/ should be the same as the bandwidth allocated to the central node C . Proof Let ˛i .i D 1, : : : , 6/ denote the draining rate of vertex Si .i D 1, : : : , 6/, and let  denote the draining rate of the central node C . Because C is the bottleneck of the network, the total throughput P is equal to . Obviously we get  6 6iD1 ˛i . Because all seven nodes share the same channel, P  we get 6iD1 ˛i C  6 1. Therefore, the throughput reaches the maximum Tnc D 1=2 when P6   D i D1 ˛i . Lemma 2 p In a coding scheme, when 3 6  < 2, the maximum throughput Tc is equal to 6=7 under the condition that the transmission schedule follows some cyclic pattern such as S1 , S2 , : : : , S6 , C , and equal bandwidth is allocated to each node. Proof The proof is almost the same as that in Lemma 1, except that the total throughput is now equal P6 to the node C ’s draining rate nc . Similar to the proof of Lemma 1, we have nc  6 i D1 ˛i P6 and i D1 ˛i C  6 1. According to these two formulae, we obtain nc  6 nc =.1 C nc /. When p 3 6  < 2, the coding number is nc D 6. So the throughput reaches the maximum Tc D 6=7 on P 6 the condition that  D  i D1 ˛i =6. Theorem 2 p In a single regular hexagon network, when 3 6  < 2.nc D 6/, the upper bound on the performance benefit of COPE or COPE-like coding scheme is equal to Tb D 12=7. Proof p From Lemma 1, Lemma 2 and the definition of performance benefit, we derive that when 3 6  <  2, Tb D Tc =Tnc D 12=7.  Example 1 In Figure 5, we assume that Si .i D 1, : : : , 6/ wants to send packet pi .i D 1, : : : , 6/ to its opposite vertex Di .i D 1, : : : , 6/. The size of packet is one unit. Figure 9 shows how packets are Copyright © 2014 John Wiley & Sons, Ltd.

Int. J. Commun. Syst. 2014; 27:1319–1337 DOI: 10.1002/dac

THE EFFECT OF HEXAGONAL GRID TOPOLOGY ON NETWORK CODING

1329

Figure 9. An example without coding.

Figure 10. An example with coding.

transmitted to achieve the maximum throughput without coding in one cycle. In this scenario, six packets are transmitted in 12 time slots, thus the maximum throughput is 1=2. On the other hand, Figure 10 shows how packets are transmitted to achieve p the maximum throughput with coding in one cycle. In this scenario, under the condition of 3 6  < 2, each node can overhear every other node. Si .i D 1, : : : , 6/ first transmits pi .i D 1, : : : , 6/ to C , then C encoded six packets as pal l D p1 ˚ p2 ˚ p3 ˚ p4 ˚ p5 ˚ p6 and broadcasts pal l to all six nodes Sal l , which includes Si .i D 1, : : : , 6/. In this scenario, six packets are transmitted in seven time slots; thus the maximum throughput is 6=7.

4.2. Upper bound on performance benefit under the condition of 1 6