A Selection Region Based Routing Protocol for Random Mobile ad hoc Networks with Directional Antennas

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1 A Selection Region Based Routing Protocol for Rando Mobile ad hoc Networks with Directional Antennas arxiv: v1 cs.it] 8 Nov 010 Di Li, Changchuan Yin, and Changhai Chen Key Laboratory of Universal Wireless Counications, Ministry of Education Beijing University of Posts and Telecounications, China, E-ail: dean.lidi@gail.co, ccyin@bupt.edu.cn, chenchanghai@gail.co Abstract In this paper, we propose a selection region based ultihop routing protocol with directional antennas for wireless obile ad hoc networks, where the selection region is defined by two paraeters: a reference distance and the beawidth of the directional antenna. At each hop, we choose the nearest node to the transitter within the selection region as the next hop relay. By axiizing the expected density of progress, we present an upper bound for the optiu reference distance and derive the relationship between the optiu reference distance and the optiu transission probability. Copared with the results with routing strategy using onidirectional antennas in 1], we find interestingly that the optiu transission probability is a constant independent of the beawidth, the expected density of progress with the new routing strategy is increased significantly, and the coputational coplexity involved in the relay selection is also greatly reduced. 1 I. Introduction Deterining the capacity region of wireless networks has been an open proble for ore than half a decade. In the seinal work ], Gupta and Kuar proved that the transport capacity for wireless ad hoc networks, defined as the bit-eters puped every second over a unit area of the network, scales asθ n in an arbitrary network, where n is node density. In 3], Weber et al. derived the upper and lower bounds on the transission capacity of spread-spectru wireless ad hoc networks, where the transission capacity is defined as the product between the axiu density of successful transissions and the corresponding data rate, under a constraint on the outage probability. All these works address the single hop transission. Recently, Baccelli et al. 4] proposed a spatial reuse based ultihop routing protocol, and derived the optiu transission probability. In their protocol, at each hop the transitter selects the best relay to axiize the spatial density of progress. By assuing each transitter has a sufficient backlog of packets, Weber et al. in 5] proposed the longest-edge based routing protocol where each transitter selects a relay that akes the transission edge longest. In 6], Andrews et al. defined the rando access transport capacity with the axiu allowable total nuber of transissions per packet along ultiple hops. In 1], Li et al. proposed a 1 This work is partially supported by the NSFC grants , , , and , the National Great Science Specific Project under grants 009ZX and 010ZX selection region based ultihop routing protocol to guarantee the essage transitted towards the final destination, where the selection region is defined by two paraeters: a selection angle and a reference distance. By axiizing the expected density of progress, the author derived an upper bound on the optiu reference distance, and the relationship between the optiu reference distance and the optiu selection angle. The above literatures only considered the wireless networks with onidirectional antennas. In 7], Yi et al. investigated the wireless networks capacity using directional antennas, extending the network capacity with onidirectional antennas in ] to that with directional antennas. In 8], Spyropoulos et al. discussed the network capacity gain one can achieve by using directional antennas over that by using onidirectional antennas and how these bounds are affected by iportant antenna paraeters like gain and beawidth. In 9], Dai et al. cobined the ultiple channels and directional antennas together and shown that they iprove the network capacity due to the increased network connectivity and reduced interference. However, previous literature usually focuses on the scaling laws of the network capacity. In this paper, we extend the forer work in 1] with onidirectional antennas to that with directional antennas, and derive the close-fored expected density of progress of the network, which is defined as the nuber of packets progress toward their destinations in a unit area of the network. Copared with the routing strategy using onidirectional antennas in 1], due to the directional antennas, the selection region based routing is ipleented uch easier and the calculation burden for nodes to select the relay is also decreased. The rest of the paper is organized as follows. Directional antenna odel, network odel and routing strategy are described in Section II. The optiization for selection region and transission probability are presented in Section III. Nuerical results and interpretations are given in Section IV. Finally, Section V suarizes our conclusions. II. Network Model and Routing Protocol In this section, we first present the siplified directional antenna odel, then define the network odel and the selection region based routing protocol using directional antennas.

2 A. Directional Antenna Model In the study of wireless networks, the antenna odel is often grouped into onidirectional and directional. Onidirectional antenna radiates signals equally well in all directions, while directional antenna has gain in the direction of the ain lobe at which it is pointing. Thus, with directional antennas the interference can be decreased and nodes located in each others neighborhood ay transit siultaneously, which increase spatial reuse of the channel. To siplify the analysis, we odel the power pattern of the directional antenna as a circular sector with angleϕ, whereϕis the beawidth of the antenna, see Fig. 1. In the following analysis, we assue the transitters with directional antenna and the receivers with onidirectional antenna, which is called Directional Transission and Onidirectional Reception DTOR as entioned in 7]. B. Network Model Assue nodes in the network follow a hoogenous Poisson Point Process PPP with density λ, and slotted ALOHA as the ediu access control MAC protocol. During each tie slot a node chooses to transit data with probability p, and to receive with probability 1 p. Therefore, at a certain tie instant, the transitters follow a hoogeneous PPP Π T x with density pλ, while the receivers follow another hoogenous PPP Π Rx with density 1 pλ. At each hop in the ultihop transissions, a transitter tries to find a receiver inπ Rx as relay. We consider the nodes are obile, to eliinate the spatial correlation, which is also discussed in 4]. We also assue that all transitters use a fixed transission power ρ and the wireless channel cobines the large-scale pathloss and sall-scale Rayleigh fading. The noralized channel power gain is given by Gd= γ dα, 1 where γ denotes the sall-scale fading, drawn fro an exponential distribution of ean 1 µ with probability density function PDF f γ x=µ exp µx, andα> is the path-loss exponent. For the transission fro transitter i to receiver j, an attepted transission is successful if the received signal-tointerference-plus-noise ratio SINR at the receiver j is above a threshold β. Thus the successful transission probability over this hop with distance d i j is given by ργ 0 d α i j P s = Pr k Π T X \{i}ργ i d α k j +η >β, where i Π T x, j Π Rx. Since we use directional antenna with beawidth ϕ for transission, the interfers seen by a specific receiver follow a hoogeneous PPP Π T x with density p ϕ π λ. Thus k Π T x \{i}ργ i d k j α is the su interference seen at the receiver j, where d k j is the distance fro interferer k to receiver j, and η is the average power of abient theral noise. In the sequel we approxiateη=0, which is reasonable in interference-liited ad hoc networks. Fro 4], the successful transission probability fro transitter i Fig. 1. to receiver j is derived as where! r d Selection region with the directional antenna P s = exp p ϕ π λtd i j, 3 t= π /α sinπ/α β/α. C. Routing Strategy with Directional Antennas 3a Considering a typical ultihop transission scenario, where a data source sends inforation to its final destination that is located far away, and it is ipossible to coplete this operation over a single hop. Thus a ultihop transission is needed. In ultihop wireless networks, if we assue positiondeterined relays exist to ensure each hop shares the sae distance that aggregates to for the path fro the data source to its final destination, the optiu transission distance at each hop is derived in 6]. In this case, the transission distance is used to deterine the location of a relay. In a practical case, nodes are usually randoly distributed, thus relays ay not be located just over the optiu transission distance. To guarantee a relay existing at a proper position, we use the selection region based ultihop routing protocol with directional antennas. For each transitter along the route to the final destination, the selection region includes two paraeters: the beawidthϕand a reference distance r, as shown in Fig. 1, where the selection region is defined as the region that is located within angle ϕ and outside of the distance r. Here, the transitter is located in the circle center O, BOC = AOC = ϕ/, and OC points to the direction of the final destination. At each hop, the transitter selects the nearest receiver located in the selection region as the relay. Copared with the routing strategy using onidirectional antennas in 1], the selection region based routing with directional antennas can be ipleented uch easier. Since the transitters are equipped with directional antennas, only the receivers within the angle ϕ can receive the radiated signals fro the transitters, thus the process to deliit the potential receivers within the angle ϕ do not need any calculation. However, for the routing protocol with onidirectional antennas, to deterine if a receiver is located within the angle ϕ or not needs a coplicated calculation, e.g., calculating the angle φ between the line fro the transitter towards the final destination and the edge fro the transitter and the potential receivers, and aking coparisons between these φ with ϕ to

3 decide which nodes are located within the selection region upper bound siulation III. Reference Distance and Transission Probability Optiization In this section, we derive the optiu values of the transission probability p and the reference distance r for different beawidth ϕ by axiizing the expected density of progress. A. Upper Bound for Optiu Reference Distance As in 4], the density of progress is defined as D= pλ P s d cosφ, where P s is the successful transission probability defined in, d cosφ is the projection of the transission distance d along the line connecting the transitter and the final destination. Since the receivers follow a hoogeneous PPP with density λ1 p, the cuulative distribution function of the transission distance d is given as Prd r=1 exp λ1 p ϕ ] r r, r r<. 4 Since φ is uniforly distributed over ϕ/, ϕ/], which is independent of d, the expected density of progress is given by ED]= pλ ϕ r ϕ =λ p1 p e pϕ π λtx x cosφf d xdφdx, kr k 3/ exp λ1 p ϕ ϕ r sin, where f d x is the probability density function of d obtained fro 4, k = λϕ pt π + 1 p, t is defined in 3a, and 3, kr = e t x 3 1 dx is the incoplete Gaa function. kr To optiize the objective function in 5 with respect to the beawidth ϕ, let us first assue that p is a constant, and try to derive the optiu value of r. For brevity, in the following discussion, we write the objective function as E. Setting the derivative with respect to r as 0, after soe calculations we have de = exp λ1 p ϕ dr r 3, kr λ1 pϕr + d 3, kr dr = 0, 6 where 3, kr is calculated as 3 3, kr = + kr exp π kr er f kr. 7 5 Optiu reference distance Beawidth unit:π Fig.. Nuerical results and the analytical upper bound for the optiu reference distance Therefore, d 3, kr dr = k exp kr 1 kr k exp kr = k 3/ r exp kr. 8 Applying 8 to 6, we obtain, kr λ 1 pϕr k 3/ r exp kr = 0. 9 Since it is difficult to analytically derive the exact solution for r fro 9, here we present an upper bound for r. Since, kr > 1 ] 1, kr +, kr using 10 in 9, we have Therefore, = 1 exp kr +kr, 10 λ1 pkϕr 4k 3/ r +λ1 pϕ>0. 11 r < k3/ 4k 3 kλ1 pϕ]. 1 kλ1 pϕ In Fig., we copare the upper bound of the optiu r with the nuerical results for different beawidth ϕ, when p=0.1. B. Jointly Optiizing the Reference Distance and the Transission Probability Now, let us axiize the objective function by jointly optiizing r and p, for different beawidthϕ. Rewrite 5 as E=λ p1 p exp ϕ π λptr, kr exp ϕ kr k 3/ sin. For brevity, we denote expkr as e and 3, kr as.

4 With partial derivatives, we have E ϕ =λ p1 p sin k 3/ exp ϕ r π λptr ϕ π λptr e+ e + e ] = 0, 13 r r This holds only if e r + e r = ϕ π λptr e. 14 Transission probability Since k= ϕλt π π p+ ϕλ k, there is p = ϕλt π π. To siplify things, we can then calculate the derivative with respect to k instead of p as E ϕ Fig. 3. sin k =λ { tr t π 3 k p ] π e+ e } p1 p ϕλt π k + e = 0. k 15 By using the relationship 1 r e r 14, we have e k + e k = r k = k e k, 1 r r = k k, and e + e = ϕ r r πk λptr e. 16 Applying 16 to 15, the following holds: tr t π 3 k p ] π e+ ϕ p1 p ϕλt π kπ λptr e=0. 17 After soe calculation, we have ϕt t π λr 3 + p+π/t π]1 p = p1 p Given beawidthϕ, we useϕand the optiu p to express the optiu r as r = 1 pt π+π]1 p ϕλ p1 pt 3t π. 19 t Interestingly, by jointly optiizing r and p, we find that the optiu transission probability p in 19 is a constant independent of beawidthϕ, which is different fro the result we had in 1]. The proof is shown in Appendix. Since in 19 p is an constant, r is only related to the beawidth ϕ. Thus for a given beawidth ϕ, there is an associated optiu reference distance r for the transitters to select next hop relay node, as shown in Fig. 4. Also note that in 19 r scales asλ 1/, which intuitively akes sense. This is because as the node density increases, the interferers relative distance to the receiver decreases as λ, it requires a shorter transission distance by the sae aount to keep the required SINR. By applying 19 in 5, we observe that 5 becoes N λ, where N is a constant independent ofλ. This eans that the axiu expected density of progress scales asθ λ, which confors to the results in ] and 6]. Optiu reference distance Beawidth unit:π The optiu transission probability p vs. the beawidth ϕ Fig. 4. analytical solution siulation Beawidth unit:π The optiu reference distance vs. the beawidth ϕ IV. Nuerical Results and Interpretations In this section, we present soe nuerical results based on the analysis in Section III. We choose the path-loss exponent α as 3, the node densityλas 1, and the outage thresholdβas 10 db. In Fig. 3, we show the optiu transission probability obtained nuerically vs. the beawidth ϕ. As shown in the figure, the optiu transission probability p is a constant, which does not change with the beawidth ϕ, this confirs our proof in Appendix. Thus, with the selection region based routing with directional antennas, no atter how uch the directional antenna s beawidth we select, the optiu transission probability always keeps the sae as p=0.1. However, when we use the routing strategy with onidirectional antenna, the optiu transission probability p changes with the selection angleϕsee Fig. 5 in 1]. In Fig. 4, we copare the optiu reference distance r obtained nuerically with that derived in 19, where p is chosen optially as the constant shown in Fig. 3. We see that increent of the beawidthϕleads to the decease of the optiu reference distance. This can be explained as follows: Increent of the beawidth ϕ eans ore interference seen

5 Expected density of progress onidirectional antenna directional antenna Transission probability p Fig. 5. Coparison of the expected density of progress for the routing protocol using directional antennas with that using onidirectional antennas by a receiver; therefore, the optiu reference transission distance should be decreased to guarantee the quality of the received signal and the transission successful probability. In Fig. 5, we copare the expected density of progress of the routing protocol using directional antennas with that using onidirectional antennas in 1]. Fro the figure, we see that the expected density of progress for selection region based routing with directional antennas owns a great advantage to that of the routing strategy with onidirectional antennas. This is because directional antennas can bring benefits such as reduced interference and increased spatial reuse copared with onidirectional antennas. V. Conclusions We propose a selection region based ultihop routing protocol for wireless ad hoc networks with directional antennas, where the selection region is defined by the beawidth and the reference distance. By axiizing the expected density of progress, we present soe analytical results on how to refine the selection region and the transission probability. Copared with the routing strategy with onidirectional antennas in 1], the routing protocol in this paper owns two ain advantages: higher expected density of progress and less coputational coplexity when choosing the next hop relay. Appendix Here, we prove that when jointly optiizing the reference distance r and the transission probability p, the optiu transission probability p is a constant independent of beawidth ϕ. Step 1: We consider the partial derivitive to the reference distance r, which is shown in 9, here rewrite as, kr λ 1 pϕr k 3/ r exp kr = 0, 0 pt where k= λϕ π + 1 p. Applying k to 0, we have pt ], kr 1 p π + 1 p kr exp kr = 0. 1 Step : Now let us focus on the partial derivitive to the transission probability p. As entioned in Section III-B, we calculate the derivitive with respect to k instead, here rewrite 15 as E k =λ sin ϕ { tr t π 3 k p ] π e+ e } p1 p ϕλt π k + e = 0. k By using the sae notation, defined in Section III-B, i.e., e=exp kr and= 3, kr, we have e k = expkr r. 3 k = k 1 r 3 expkr. 4 Applying 3 and 4 to, we have tkr t π p ] pt+π1 p e p1 p t π + exp kr kr kr 3 = 0. 5 Step 3: Applying 19 to 1 and 5, we see that in kr, ϕ can be cancelled, thus 1 and 5 becoe two equations that are independent of ϕ. Therefore, when jointly optiizing r and p, the optiu p is a constant independent of the beawidthϕ, only related to t which is defined in 3a. And the nuerical value for the optiu p is given in Fig. 3. References 1] D. Li, C. Yin, C. Chen, and S. Cui, A Selection Region Based Routing Protocol for Rando Mobile ad hoc Networks, IEEE Globeco010 Workshop on Heterogeneous, Multi-Hop, Wireless and Mobile Networks, Dec. 010, accepted. Online] Available: ] P. Gupta and P. R. Kuar, The capacity of wireless networks, IEEE Transactions on Inforation Theory, vol. 46, no., pp , Mar ] S. Weber, X. Yang, J. G. Andrews, and G. de Veciana, Transission capacity of wireless ad hoc networks with outage constraints, IEEE Transactions on Inforation Theory, vol. 51, no. 1, pp , Dec ] F. Baccelli, B. Blaszczyszyn, and P. Muhlethaler, An Aloha protocol for ultihop obile wireless networks, IEEE Transactions on Inforation Theory, vol. 5, no., pp , Feb ] S. Weber, N. Jindal, R.K. Ganti, and M. Haenggi, Longest Edge Routing on the Spatial Aloha Graph, Proceedings of the IEEE GLOBECOM, pp. 1-5, Nov ] J. G. Andrews, S. Weber, M. Kountouris and M. Haenggi, Rando Access Transport Capacity, IEEE Transactions On Wireless Counications, subitted. Online] Available: cache/arxiv/pdf/0909/ v1.pdf. 7] S. Yi and Y. Pei and S. Kalyanaraan, On the capacity iproveent of ad hoc wireless networks using directional antennas, Proceedings of Mobihoc,003. 8] A. Spyropoulos, and C.S. Raghavendra, Capacity bounds for ad-hoc networks using directional antennas, Proceedings of IEEE ICC, vol. 1, pp , May ] H. Dai, K. Ng, R. Wong, and M. Wu, On the Capacity of Multi- Channel Wireless Networks Using Directional Antennas, Proceedings of the IEEE INFOCOM, pp , Apr. 008.