CRITICAL TRANSMISSION RANGE FOR CONNECTIVITY IN AD-HOC NETWORKS
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1 CHAPTER CRITICAL TRASMISSIO RAGE FOR COECTIVITY I AD-HOC ETWORKS HOSSEI AJORLOO, S. HASHEM MADDAH HOSSEII, ASSER YAZDAI 2, AD ABOLFAZL LAKDASHTI 3 Iran Telecommunication Research Center, Tehran, Iran, {ajorloo, maddah}@itrc.ac.ir 2 Electrical and Computer Engineering Faculty, University of Tehran, Tehran, Iran,yazdani@ut.ac.ir 3 Rouzbahan Institute of Higher Education, Sari, Mazandaran, Iran lakdashti@rouzbahan.ac.ir Abstract: Keywords: One of the challenging problems in the ad hoc networks is how to determine the critical transmission range for each communicating node to achieve a connected network with minimum power consumption and communication interference. In this paper, an analytical approach is proposed to determine this parameter based on the number of nodes, physical dimensions of the network, and probability of connectivity. Our proposed approach resulted in Cumulative Distribution Functions (CDF) for the critical transmission range for various numbers of nodes Ad hoc networks, Cumulative distribution function, Critical transmission range. ITRODUCTIO One of the major challenging problems in ad hoc networks is the connectivity of the network. Reliability of connections depends on many factors, such as the transmission radius of each node, movement of nodes, environmental conditions, number of nodes, etc. In [] an analytical procedure is proposed for the computation of the node isolation probability in an ad hoc network in the presence of channel randomness, with applications to shadowing and fading phenomena. However, in a simplistic model some authors have tried to bind together the probability of connectivity, the number of nodes, size and shape of the area in which nodes are located, and the transmission radius of each node, given a certain distribution of nodes. H. Labiod and M. Badra (eds.), ew Technologies, Mobility and Security, Springer.
2 2 CHAPTER Santi and Blough [2] provided tight upper and lower bounds on the critical transmitting range for one-dimensional networks and non-tight bounds for two and three-dimensional networks. Gupta and Kummar [3] have shown that if n nodes are placed in a disc of unit area in R 2 and each node transmits at a power level so as to cover an area of r 2 = ( log n+c n ) /n, then the resulting network is asymptotically connected with probability one if and only if c n for n. This is a limit which does not help us to, e.g., determine the number of nodes required to have a connected network with a certain probability say 95%. Penrose [4] derived the distribution of the maximum of the edge lengths in a minimum spanning tree (MST), denoted M n constructed from n points distributed uniformly in the unit square and proved that () lim P n vm v n log n = n exp e R where v denotes the volume of the unit ball in v dimensions. As can be seen, here again we have an asymptotic relation for n which does not help us to determine n based on the size of the area, propagation radius of nodes and the probability of connectivity. Tang et al. [5] have proposed a model for the probability of connectivity in ad hoc networks considering various values for the propagation radius using Monte-Carlo simulations. In this paper, we propose another model in the opposite direction: finding the transmission radius considering the probability of connectivity, but with an analytical procedure. Some authors used a model with Poisson distribution of nodes (for example, see [2]). These models are appropriate for unlimited areas. But when we interest in finding models for limited areas, the distributions defined for limited areas such as the uniform distribution should be used instead. The aim of this paper is to find a model that binds together three quantities, namely, the number of nodes in an ad hoc network, the maximum distance over which two nodes can communicate, and the area over which the nodes are scattered, in such a way that the resulting network is connected with a high probability when the nodes are assumed to be spatially uniformly distributed. Designing power efficient protocols for ad hoc networks is a well documented topic in the literature (See [6 ] for some recently proposed solutions). In most of these protocols, it is required to determine the critical transmission power to achieve a connected network. One approach to determine this parameter is to use a message passing protocol such as one proposed in [2]. However, this approach suffers from the delay and communication load required for passing messages. On the other hand, if each node knows the approximate number of nodes in the network, it can determine the transmission power required to have a connected network with a certain probability using our proposed method. Although our method is not accurate
3 CRITICAL TRASMISSIO RAGE 3 as message passing approaches, it is faster and does not pose any communicating burden on the network. The remainder sections of this paper are organized as follows: In section 2 we have discussed our analytical modeling. In section 3 the experimental results are presented. Finally, in section 4 we conclude the paper. 2. FIDIG THE CRITICAL TRASMISSIO RAGE FOR COECTIVITY In this section, we propose an analytical model for determining the critical transmission range given the probability of connectivity of ad hoc networks and the number of nodes. For the sake of simplicity, we consider these assumptions: The propagation radius is equal for all nodes. n nodes are distributed uniformly in the unit square. The x and y coordinates of the nodes are independent. The coordinates of nodes are independent. Moreover, we have not assumed the mobility of nodes in our analytical approach. However, one can use the results for an instance, when assuming mobility in the network. By considering any mobility model, one can determine the critical transmission range to have a connected network for a given probability of connectivity in a certain time interval. To begin, an important result given in [3,4] is presented: Theorem : The critical transmission range for connectivity R crit is equal to the longest link distance in the minimum spanning tree of the nodes. From several algorithms proposed for finding the MST, we used the Prim algorithm [5]: starting with any single node, new nodes are added to the tree one by one, so that at each step the node closest to the nodes included so far is added [4]. One realization with nodes as well as their MST is depicted in Fig.. For finding the probability density function (PDF) of the maximum edge of the MST, we begin by the following equation: (2) f d d 2 d d d 2 d = f d d f d2 d 2 f d d where the left side denotes the joint PDF on distances, and f di d i is the marginal PDF of the distance d i. This means that the joint PDF of distances between nodes distributed independently in a square area equals to the product of the marginal PDFs of each of them. In the next subsection, we derive the PDF and CDF of the distance between two uniformly distributed nodes.
4 4 CHAPTER Figure. A sample set of nodes and their MST. The longest link is shown with a dark line 2.. PDF and CDF of the Distance Between Two Uniformly Distributed odes We proceed by finding the PDF of the distance between two uniformly distributed nodes, d, in the unit square. In Appendix 4 we proved that this parameter has the following PDF (3) 2d 3 8d d d< f D d = 8d d 2 + 8d sin 2d 4d d 2d3 d 2 The probability distribution and density functions of d is shown in Fig PDFs in the Prim Algorithm In the Prim algorithm, at the first step, one node is chosen randomly. Then, the nearest node to this node is selected. In other steps, the nearest node to the set of selected nodes is chosen. We denote the edge chosen at the ith step by i. Finally, R crit = max 2. If we denote the node chosen at step i by number i, then the PDFs of the random variables to is
5 CRITICAL TRASMISSIO RAGE d (Distance) (a) d (Distance) (b) Figure 2. (a) Cumulative distribution and (b) Probability density functions of the distance between two uniformly distributed nodes in the unit square ( f = P min ( f 2 2 = P j=2 d j min i= 2 3 j ) d ij d 2 = min j=2 d j ) ( f 3 3 = P min d ij d 2 = min d j d 3 = min j=2 i= 2 i 3 4 j 3 j ) d ij (4) f = P ( min d i i= d 2 = min d j j=2 d 3 = min i= 2 3 j d ij d 2 = min i 2 j= ) d ij We should first determine the PDF of f. Because of Theorem, the random variables d ij are independent. Therefore, we should determine the joint PDF of
6 6 CHAPTER independent, identically distributed (i.i.d.) random variables. This is done in the next subsection Joint PDF of the Minimum of Independent Random Variables We first find the joint pdf of the minimum of two i.i.d. random variables x and y. Defining z = min x y, we proved in Appendix 4 that the PDF of z is f Z z = 2f X z ( F X z ) (5) For determining the joint PDF of i.i.d. random variables, we use the following recursive formula min x i = min x (6) min x i i= i= f min x i = f (7) min x i f x i= i= ote that there does not exists any formula in an enclosed form for the joint PDF of the minimum of i.i.d. random variables, and hence, we should use the above equations PDF of R crit As mentioned earlier, R crit is the largest edge of MST of nodes, i.e., (8) R crit = max i i= ow, similar to the previous section we find the joint PDF of the maximum of independent random variables. Suppose that two random variables x and y are independent. Then, if z = max x y, according to Fig. 7 b the CDF of z is (9) () F Z z = P max x y z = z = f X z f XY z y dy + z z f y y dy + f Y z = f X z F Y z + f Y z F X z f XY x z dx z f X x dx Moreover, the following recursive formula is valid () (2) max x i = max x max x i i= i= f max x i = f max x i f x i= i=
7 CRITICAL TRASMISSIO RAGE 7 Again, there does not exists any formula in an enclosed form for the joint PDF of the maximum of random variables. 3. EXPERIMETAL RESULTS In this section, we present our experimental results. We implemented Eqs. (5) (2) using numerical methods. More precisely, we partitioned the range d 2 into infinitesimal sections of equal length, and for various numbers of nodes, we calculated the CDF of R crit in a discrete form. The resultant CDFs for some selected values of n, the number of nodes, is shown in Fig. 3. In this figure, a horizontal line indicating 95 percentile line is sketched. From the intersection of this line by each of the curves, one can determine the required critical transmission range (and consequently the required power) for a known number of nodes to have a network that is connected with the probability of 95%. For example, if we have nodes distributed uniformly in a unit square, the critical transmission range for all nodes required to have a connected network with the probability of 95% is about 2. CDF n=25 95%.8 n=.6 n=25.4 n= R crit Figure 3. Computed CDFs for various numbers of nodes
8 8 CHAPTER Figure 4 shows the critical transmission range as a function of n for various values of percentiles. Using this figure, one can determine the number of nodes given the critical transmission range, required to have a connected network with a known probability. For example, if the critical transmission range (resulted from a known power) for all nodes distributed uniformly in a unit square equals to 3, then the number of nodes required to have a connected network with the probability of 99% should be 63 nodes and 45 nodes for the probability of 95%. ow, we are going to describe how the results can be used in potential applications toward more efficient networks. For example, consider one application in which scientists scatter some sensors in an area (such as around a volcano) which can communicate each other to convey their registered information. They may fix the transmission power for the nodes and desire in the number of nodes required to scatter in the area to have connected network with the probability of, say 95%. They can generate a plot similar to Fig. 4 using our proposed method and from which determine the number of nodes. The reverse situation may also be arisen: The number of nodes is fixed and the transmission power is of interest given a certain probability for connectivity.. R crit % 95% 75% 5% umber of odes Figure 4. R crit vs. number of nodes for various values of percentiles for connectivity
9 CRITICAL TRASMISSIO RAGE 9 In design of power efficient routing protocols for ad hoc networks, the plots of Fig. 4 can be used, for example in the form of look-up tables, to adjust the power levels for the nodes to have a connected network with a certain probability and with least communication interferences. Using this approach, the nodes in the network only require to know the approximate value of the number of nodes in a certain area. 4. COCLUSIO In this paper, an analytical approach is proposed to compute the CDF of the critical transmission range required to have a connected ad hoc network with a given probability as a function of the number of nodes and the physical dimensions. We obtained two recursive formulas to determine the mentioned CDF and used numerical methods to implement them. Our experiments resulted in curves of CDFs for a given number of nodes and critical transmission ranges as a function of the number of nodes. One can use these curves to determine the number of nodes required to have a connected network with a given radius of propagation for each node, or conversely, determine the required power for nodes to have a connected network with a given number of nodes in a square of known size and a certain probability of connectivity. REFERECES. Miorandi, D., Altman, E.: Coverage and connectivity of ad hoc networks in presence of channel randomness. Proceedings of IEEE IFOCOM th Annual Joint Conference of the IEEE Computer and Communications Societies, Vol March (25) Santi, P., Blough, D.M.: The critical transmitting range for connectivity in sparse wireless ad hoc networks. IEEE Transactions on Mobile Computing, Vol. 2(). Jan March (23) Gupta, P., Kumar, P.R.: Critical power for asymptotic connectivity in wireless networks. Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming, W.M. McEneaney, G. Yin, and Q. Zhang (Eds.), Birkhauser, Boston (998) Penrose, M.D.: The longest edge of the random minimal spanning tree. The Annals of Applied Probability, Vol. 7(2). (997) Tang, A., Florens, C., Low, S.H.: An empirical study on the connectivity of ad hoc networks. Proceedings of IEEE Aerospace Conference, Vol. 3. March (23) Zhang, J., Zhang, Q., Li, B., Luo, X., Zhu, W.: Energy-efficient routing in mobile ad hoc networks: mobility-assisted case. IEEE Transactions on Vehicular Technology, Vol. 55(). January (26) Li, D., Jia, X., Liu, H.: Energy efficient broadcast routing in static ad hoc wireless networks. IEEE Transactions on Mobile Computing, Vol. 3(2). April June (24) Chen, K., Qin, Y., Jiang, F., Tang, Z.: A Probabilistic Energy-Efficient Routing (PEER) Scheme for Ad-hoc Sensor etworks. 3rd Annual IEEE Communications Society on Sensor and Ad Hoc Communications and etworks, 26. SECO 6, Vol. 3. (26) Ping, Y., Yu, B., Hao, W.: A Multipath Energy-Efficient Routing Protocol for Ad hoc etworks. 26 International Conference on Communications, Circuits and Systems Proceedings, Vol. 3. June (26) El-Hajj, W., Kountanis, D., Al-Fuqaha, A., Guizani, M.: A Fuzzy-Based Hierarchical Energy Efficient Routing Protocol for Large Scale Mobile Ad Hoc etworks (FEER). 26 IEEE International Conference on Communications, Vol. 8. June (26)
10 CHAPTER. Li, F., Wu, K., Lippman, A.: Energy-efficient cooperative routing in multi-hop wireless ad hoc networks. 25th IEEE International Performance, Computing, and Communications Conference, 26. IPCCC 26, Vol. 8. April (26) 2 2. Ovalle-Martinez, F. J., Stojmenovic, I., ocetti, F. G., Solano-Gonzalez, J.: Finding minimum transmission radii for preserving connectivity and constructing minimal spanning trees in ad hoc and sensor networks. Journal of Parallel and Distributed Computing, Vol. 65(2). February (25) Sanchez, M., Manzoni, P., Haas, Z. J.: Determination of critical transmission range in Ad-Hoc etworks. Proceedings of Multiaccess Mobility and Teletraffic for Wireless Communications 999 Workshop (MMT 99). October (999) 4. Koskinen, H.: Connectivity and Reliability in Ad Hoc etworks. Master thesis, Helsinki University of Technology, Department of Electrical and Communications Engineering, February (23) 5. Prim, R.C.: Shortest connection networks and some generalizations. Bell Systems Technology Journal, Vol. 36. (957) APPEDIX I: FIDIG THE DISTRIBUTIO OF THE DISTACE BETWEE TWO UIFORMLY DISTRIBUTED ODES Define z as z = x y where x and y have uniform distribution in the range and are independent (Fig. 5 (a)). Considering Fig. 5 (b), the CDF of z is (3) F Z z = P Z z = z 2 = 2z z 2 z Taking the derivative of F Z z with respect to z, weget (4) f Z z = 2 z z Defining two new variables z = x x 2 and z 2 = y y 2 with the PDF of (4), we observe that these variables are independent. Since, d = ( x x y y 2 2) /2, the CDF of d is (5) F D d = P D d = P z 2 + z2 2 d (a) (b) Figure 5. (a) The distance between two nodes having uniform distributions; (b) Computation of the distribution function of z = x y
11 CRITICAL TRASMISSIO RAGE (a) 2 d (b) Figure 6. Computation of the CDF of d for (a) d<; (b) d 2 To obtain this distribution, we should consider two cases: Case : d<, Considering Fig. 6 (a), we obtain (6) F D d = d d 2 z 2 4 z z 2 dz dz 2 Case 2: d 2 (Fig. 6 (b)), F D d = d 2 4 z z 2 dz dz 2 d 2 z 2 (7) + 4 z z 2 dz 2 dz d 2 Combining these two cases, we get 2 d4 8 3 d3 + d 2 d< (8) F D d = 4 3 d2 + 8 d2 d d 2 sin d d 2 2d 2 2 d4 d 2 3 Taking the derivative of (8), we obtain (3). APPEDIX II: DETERMIIG THE JOIT PDF OF THE MIIMUM OF TWO I.I.D. RADOM VARIABLES Defining z = min x y, according to Fig. 7 (a) the CDF of z equals to F Z z = P min x y z (9) = z x + z y f XY x y dydx + f XY x y dxdy + 2 z z 2 z z f XY x y dydx f XY x y dxdy
12 2 CHAPTER 2 2 (a) 2 (b) 2 Figure 7. Computation of the CDF of (a) z = min x y ; (b) z = max x y According to Leibnitz theorem, if F Z z = b z f x z dx, then a z (2) f Z z = db z dz f b z z da z b z dz f a z z + a z f x z dx z In our example, z = min x y and x and y are i.i.d. random variables. Therefore, (2) f Z z = 2 z = f Y z = 2f X z f XY x z dx + 2 z 2 z 2 f X x dx + f X z f X x dx z f XY z y dy 2 z f Y y dy Determining (5) from (2) is straightforward.
Computing functions over wireless networks
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License. Based on a work at decision.csl.illinois.edu See last page and http://creativecommons.org/licenses/by-nc-nd/3.0/
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