# Optimisation of spatial CSMA using a simple stochastic geometry model for 1D and 2D networks

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1 Optimisation of spatial CMA using a simple stochastic geometry model for D and 2D networks adjib Achir, Younes Bouchaala, Paul Muhlethaler, Oyunchimeg hagdar To cite this version: adjib Achir, Younes Bouchaala, Paul Muhlethaler, Oyunchimeg hagdar. Optimisation of spatial CMA using a simple stochastic geometry model for D and 2D networks. IWCMC 26-2th International Wireless Communications & Mobile Computing Conference, ep 26, Paphos, Cyprus. pp , 26, Proceedings of the 2th International Wireless Communications & Mobile Computing Conference. <.9/IWCMC >. <hal > HAL Id: hal ubmitted on 2 Oct 26 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Copyright

4 This result is very simple. Let Fj be the fading between the node at the origin X i and node X j This is just the application of livnyak s theorem and Campbell s formula, see [2], [9] = E [ ] I(F j l( X j X i P cs X j φ = λ P {F P cs l( x }dx A straightforward computation provides the explicit value of in the D and 2D cases. Proposition III.2. The probability p that a given node X transmits i.e. e = is: p = E [e ] = e. Proof: The proof is obtained by computing the probability that a given node X at the origin with a mark m = t is allowed to transmit. The result is then obtained by deconditioning on t. The details of the proof can be found in [9]. Thus p measures the probability of transmission in a CMA network. If p is close to this means that the carrier sense does not restrain transmissions. In contrast, if p is small, this means that the carrier sense imposes a severe restriction on transmissions. Proposition III.3. The probability that X transmits given that there is another node X j Φ at distance r is p r with p r = p e Pcsµl(r( e 2 e Proof: The proof is the same as that of Proposition III.2. Proposition III.4. Let us suppose that X and X 2 are two points in Φ such that X X 2 = r. We suppose that node X 2 is retained by the selection process. The probability that X is also retained is: with h(r = 2 b(r ( e e e b(r b(r ( e Pcsµl(r e Pcsµl(r( e 2 e b(r = 2 λ e Pcsµ(l( x +l( r x dx. In a 2D network, we have: b(r = 2 λ 2π In a D network, we have: b(r = 2 λ e Pcsµ(l(τ+l( τ 2 +r 2 2rτcos(θ dτdθ. e Pcsµ(l(τ+l( r τ dτ Proof: The proof can be found in [9] Proposition III.5. Given the transmission of a packet, we denote by p c (r, P cs the probability of successfully receiving this packet at distance r in a CMA system (modeled by a Matern selection process with a carrier sense threshold P cs and with a capture threshold T. We have: ( p c (r, P cs exp λ In a 2D network, we have: ( p c (r, P cs exp λ 2π In a D network, we have: ( p c (r, P cs exp λ h( x + l( x r T l(r τh(τ dx + l( τ 2 +r 2 2rτcos(θ T l(r h(τ + l( r τ T l(r dτ dτdθ Proof: The idea of the proof is to consider a transmitter at the origin and to compute the probability of successful reception by a receiver at distance r. To do so, we condition by the presence of another transmitting node at distance τ. According to proposition III.4, the density of such nodes is λh(τ. We approximate the interference by the interference of a Poisson Process of density λh(τ. The result is obtained by integrating on τ. The details of the proof can be found in [9]. It is easy to add a thermal noise W to the model. The expression of p c (r, P cs must then be multiplied by L W (µt l(r where L W (. is the Laplace Transform of the noise. In a more advanced model using hannon s law, we have the average transmission rate for X E (log(+ir e = = = P (log(+ir > t e = dt p c (r, P cs, e t dt with p c (r, P cs, x = p c (r, P cs where T is substituted by x, see [9]. Although more complicated, such an approach seems computationally achievable, and will form the subject of a more extensive study of spatial CMA. We resume with the capture model in the IR model. Proposition III.6. The spatial is: λpp c (r, P cs This spatial density has a D and a 2D version and the values of p and p c (r, P cs are chosen accordingly. Proof: Proposition III.6 is just the exploitation of propositions III.2 and III.5. IV. REULT OF THE MODEL In this section, the model is used to analyze the network performance and the influence of the model s parameters. We study the transmissions for pairs of source-destination nodes at distance r. r is set at / λ or /λ for 2D and D networks respectively. r can be seen as a typical distance in these

5 networks since it is the average distance between a node and its closest neighbor. Thus the transmitters are in the Poisson Point Process and for each transmitter, we create a random receiver at distance r. A. Optimizing the with the carrier sense threshold P cs We consider that the parameters of the model λ, T and µ are constant and we vary P cs to maximize the density of successful transmissions. It is easy to show that p is an increasing function of P cs. When the carrier threshold increases, the probability of transmission in CMA increases. VAET As the carrier threshold increases, transmission becomes easier and thus p increases. This can be verified using the equation of proposition III.2. In contrast, when P cs increases then p c (r, P cs decreases. This can been shown using the equation of proposition III.4. When P cs increases, h(τ increases and thus p c (r, P cs decreases. ince the density of successful transmissions is upper-bounded by λ we know that there is an optimal value of P cs which optimizes the density of successful transmissions. tudying a few examples, we have seen that the always has the same behavior, as shown in Figure IV.. For small value of P cs and when we increase P cs, p increases faster than p c (r, P cs decreases, and thus the is an increasing function of P cs. This density reaches a maximum for a given value of P cs and then becomes a decreasing function of P cs. We assume that this is always the case although it seems difficult to show it mathematically. We use Maple to numerically compute this optimum of the density of successful transmissions carrier sense level Fig. IV.. Density of successful transmissions versus carrier threshold (T =, µ =, β = 4. In Figures IV.2 and IV.3, we present the carrier sense threshold versus the when the density of successful transmissions is optimized. B. Effect of the fading rate µ We note that in the probability of transmission p found in Proposition III.2, we can isolate µp cs. It is the same for p c (r, P cs. This means that if we multiply µ by, exactly the same performance can be obtained with P cs divided by. Thus there is no influence of µ on the global performance of the system; the optimum, the probability of capture p c (r, P cs and the probability of transmission p at the optimum value of P cs. This remark regarding µ is valid for both D and 2D networks. In the following we use µ =. carrier sense threshold carrier sense threshold for optimized CMA Fig. IV.2. Optimized carrier threshold versus density, 2D network (T =, µ =, β = 4. carrier sense threshold carrier sense threshold for optimized CMA Fig. IV.3. Optimized carrier threshold versus density, 2D network (T =, µ =, β = 4. C. Effect of the λ We compute the optimum when P cs is optimized versus λ the in the network. We use the following parameters T =, µ = and β = 4. The results of these computations are shown in Figure IV.4 for 2D networks and in Figure IV.5 for D networks. Our numerical study shows that the density of successful transmissions is linear in λ. This means that the maximum of the product of pp c (r, P cs does not depend on λ. This is an interesting result and one which is not easily apparent in the analytical formulas of pp c (r, P cs. Figures IV.4 and IV.5 also show the density of successful transmissions when the carrier sense threshold is constant and taken as the optimal value for λ =. The loss is significant for small values of λ: 26% for λ =. and much more significant for large values of λ : 8% for λ = in 2D networks and 85% for λ = in D networks. For instance, this means that, in a VAET, the channel cannot be used efficiently if the carrier sense threshold is not properly optimized according to the density of vehicles. When λ =. and if we use the optimization for λ = we do not have any restriction on the transmission rights and we actually have an excess of transmission rights. The problem comes from the probability of success for a given transmission. When λ = and if we use the optimization for λ = we have a stringent restriction 3 on the transmission rights, whereas a given transmission is very well protected by the CMA scheme and thus every transmission is nearly always successful. The model shows 3 the access right of CMA (excess or stringent restriction is determined by the equation given in Proposition III.2.

6 that the problem concerning the access right is much more detrimental to the global throughput than collisions would have been if the had been overestimated. We have studied the probability of capture when the throughput is optimized. We observed that the optimum throughput is not obtained when most of the transmissions are successful but rather when the success rate is around 55% in 2D networks and around 7% in D networks. The numerical results we obtained show that, at the optimum, p c (r, P cs does not depend on λ and we also deduce that p does not depend on λ. This is an interesting result which is not brought to light using the analytical formulas (optimized CMA (CMA Fig. IV.4. Density of successful transmissions versus (T =, µ =, β = 4. patial network (2 D (optimzed CMA (CMA Fig. IV.5. Density of successful transmissions versus (T =, µ =, β = 4. Linear network ( D D. Evaluation of exclusion area when the system is optimized We computed the optimum value of P cs : P cs (opt in the previous section. CMA is optimum when a transmission at a given point forbids a transmission where the signal power exceeds P cs (opt. This means that on average, around a transmitter, any transmission at distance R cs is forbidden such that µl(r = P cs cs(opt. We study the ratio of R cs by the distance between the transmitter and its receiver. We recall that this distance is / λ in 2D networks and /λ in D networks. In Figure IV.6, we show the ratio of R cs by the distance between the transmitter and its receiver for 2D networks versus the λ. We see that the average exclusion area around a given transmitter ranges, on average, from.92 to.47 times the distance between the source and destination nodes for 2D networks T =, µ = and β = 4. For D networks as there is only one degree of freedom, the nodes are more grouped and the average exclusion area around a given transmitter is larger; it ranges from.47 to.63 times the distance between the source and destination nodes depending on the density of the nodes. carrier sense range divided by the average transmission range Fig. IV.6. Ratio of carrier sense range and transmission range versus λ for T =, µ =, β = 4 and a 2D network. carrier sense range divided by the average transmission range Fig. IV.7. Ratio of carrier sense range and transmission range versus λ for T =, µ =, β = 4 and a D network. E. Effect of the capture threshold T We study the effect of the capture threshold on the maximum. In Figure IV.8 and Figure IV.9 we plot the maximum density of successful transmissions for T varying from. to respectively for 2D and D networks. We observe that dividing the capture threshold by leads to multiplying the density of successful transmissions by 5.6 and.9 for 2D and D networks. This means that a small capture threshold is much more beneficial in 2D networks. The study of the analytical model does not show any obvious scaling of the with the capture threshold T. F. Effect of the transmission decay β In Figures IV. and IV., we plot the maximum density of successful transmissions for β varying from 2 to 6 respectively for 2D and D networks. In 2D networks, we observe that the maximum is multiplied by.9 when β varies from 2.5 to 6. For linear networks (D the maximum is multiplied by.32 when β varies from 2.5 to 6. As for the capture threshold, the effect of a large transmission decay is less beneficial for D networks than for 2D networks. The study does not show any apparent scaling of the with the capture threshold β.

7 Capture threshold T power decay β Fig. IV.8. Density of successful transmissions versus capture threshold T for 2D networks (λ=, µ =, β = 4. Fig. IV.. Density of successful transmissions versus the decay factor β for 2D networks (λ=, µ =, T = Capture threshold T power decay β Fig. IV.9. Density of successful transmission versus the capture threshold T (λ=, µ =, T =. Linear network ( D Fig. IV.. Density of successful transmissions versus the decay factor β (λ=, µ =, T =. Linear network ( D V. COCLUIO In this paper, we present a simple model of CMA and we show the importance of optimizing it according to the. We have shown that the optimized density of successful transmissions scales linearly with the density of nodes. We have observed that using a constant carrier threshold leads to a very significant loss in the network s global throughput. This effect is much more penalizing when the in the network is underestimated than when it is overestimated. The numerical computations we have carried out show that the best performance of the network is not reached when transmissions are nearly always successful but when there is a success rate of around.6. We have also studied the influence of the model s parameters : µ, T and β. The rate of fading does not influence the performance of the network if it is optimized. We show that T and β have a greater impact on 2D networks than on D networks. The results of this study have yet to be compared with simulation results, preliminary tests show a good matching between the results of both approaches. This will form the subject of our future work. In addition the approximation of CMA induced by the Matern selection process should be further investigated. REFERECE [] L. Kleinrock and F. Tobagi, Packet switching in radio channels: Part I carrier sense multiple-access modes and their throughput-delay characteristics, IEEE Transactions on Communications, vol. COM- 23, no. 2, pp. 4 46, December 975, (Also, Multiple Access Communications, Foundations for Emerging Technologies, orman Abramson (Ed, IEEE Press, 992, pp [2] G. Bianchi, Performance Analysis of the IEEE 82. Distributed Coordination Function, IEEE Journal of elected Areas in Communications., vol. 8, no. 3, pp , March 2. [Online]. Available: [3]. Ghez,. Verdu, and. chartz, tability properties of slotted Aloha with multipacket reception capability, IEEE Trans. Automat. Contr., vol 7, pp , 988. [4] F. Baccelli, B. Blaszczyszyn, and P. Muhlethaler, An aloha protocol for multihop mobile wireless networks, Information Theory, IEEE Transactions on, vol. 52, no. 2, pp , Feb 26. [5] P. Jacquet and P. Muhlethaler, Mean number of transmissions with csma in a linear network, in 2 IEEE 72nd Vehicular Technology Conference: VTC2-Fall, 69 eptember 2, Ottawa, Canada 2. [6] D. toyan, W.. Kendall, and J. Mecke, tochastic geometry and its applications. 2nd edition. Wiley, 995. [7] P. Muhlethaler and A. ajid, Throughput optimization in multihop csma mobile ad hoc networks, in EW 24. The 5th European Wireless Conference, February Barcelona 24. [8] A. Busson and G. Chelius, Point processes for interference modeling in csma/ca ad-hoc networks, in Conference: Proceedings of the 6th ACM International Workshop on Performance Evaluation of Wireless Ad Hoc, ensor, and Ubiquitous etworks, PE-WAU 29, October , Tenerife, Canary Islands, pain, 29. [9] F. Baccelli and B. Błaszczyszyn, tochastic Geometry and Wireless etworks, Volume II Applications, ser. Foundations and Trends in etworking. ow Publishers, 29, vol. 4, o 2. [] I. Ramachandran and. Roy, Analysis of throughput and energy efficiency of p-persistent csma with imperfect carrier sensing, in GLOBECOM 5. IEEE, 2-4 Dec 25, t Louis, MO UA 25. [], On the impact of clear channel assessment on mac performance, in GLOBECOM 6. IEEE, 27 ov Dec 26, an Francisco, California UA 26. [2] F. Baccelli and B. Błaszczyszyn, tochastic Geometry and Wireless etworks, Volume I Theory, ser. Foundations and Trends in etworking. ow Publishers, 29, vol. 3, o 3 4.

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