An Evaluation of Connectivity in Mobile Wireless Ad Hoc Networks

Transcription

1 An Evauation of Connectivity in Mobie Wireess Ad Hoc Networks Paoo Santi Istituto di Informatica e Teematica Area dea Ricerca de CNR Via G.Moruzzi, 5624 Pisa Itay santi@iit.cnr.it Dougas M. Bough Schoo of Eec. and Comp. Eng. Georgia Institute of Technoogy Atanta, GA USA doug.bough@ece.gatech.edu Abstract We consider the foowing probem for wireess ad hoc networks: assume n nodes, each capabe of communicating with nodes within a radius of r, are distributed in a d- dimensiona region of side ; how arge must the transmitting range r be to ensure that the resuting network is connected? We aso consider the mobie version of the probem, in which nodes are aowed to move during a time interva and the vaue of r ensuring connectedness for a given fraction of the interva must be determined. For the stationary case, we give tight bounds on the reative magnitude of r, n and yieding a connected graph with high probabiity in -dimensiona networks, thus soving an open probem. The mobie version of the probem when d=2 is investigated through extensive simuations, which give insight on how mobiity affect connectivity and revea a usefu trade-off between communication capabiity and energy consumption. Introduction Wireess ad hoc networks are networks where mutipe nodes, each possessing a wireess transceiver, form a network amongst themseves via peer-to-peer communication. An ad hoc network can be used to exchange information between the nodes and to aow nodes to communicate with remote sites that they otherwise woud not have the capabiity to reach. Wireess ad hoc networks are sometimes referred to as wireess muti-hop networks because, as opposed to wireess LAN environments, messages typicay require mutipe hops before reaching a gateway into the wired network infrastructure. Sensor networks [8] are a particuar cass of wireess ad hoc networks in which there are many nodes, each containing appication-specific sensors, a wireess transceiver, and a simpe processor. Potentia appications of sensor networks abound, e.g. monitoring of ocean This research was supported in part by the Nationa Science Foundation under Grant CCR temperature to enabe more accurate weather prediction, detection of forest fires occurring in remote areas, and rapid propagation of traffic information from vehice to vehice, just to name a few. Whie the resuts in this paper appy to wireess ad hoc networks in genera, certain aspects of the formuation are specificay targeted to sensor networks. For exampe, we assume nodes are randomy paced, which coud resut when sensors are distributed over a region from a moving vehice such as an airpane. We are aso concerned, in part, with minimizing energy consumption, which, athough being an important issue in wireess ad hoc networks in genera, is vita in sensor networks. Sensor nodes are typicay batterypowered and, because repacing or recharging batteries is often very difficut or impossibe, reducing energy consumption is the ony way to extend network ifetime. In many appications of wireess ad hoc networks, the nodes are mobie. This compicates anaysis of network characteristics because the network topoogy is constanty changing in this situation. In this work, we consider networks both with and without mobiity. We present anaytica resuts that appy to networks without mobiity and confine ourseves to simuation resuts for networks with mobiity due to the intractabiity of anaysis with existing mathematica methods. Due to the reativey recent emergence of sensor networks, many fundamenta questions remain unanswered. We address one of those questions, namey what are the conditions that must hod to ensure that a depoyed network is connected initiay and remains connected as nodes migrate? We address this question, and a number of reated ones, in probabiistic terms, i.e. we evauate the probabiities of various events reated to network connectedness. The specific conditions we evauate are how many nodes are required and what transmitting ranges must they have in order to estabish a wireess ad hoc network with a particuar property, e.g. connectedness. Determining an appropriate transmitting range for a given number of nodes is essentia to minimize energy consumption since transmitting power is proportiona to the square (or, depending on environmenta Proceedings of the Internationa Conference on Dependabe Systems and Networks (DSN 2) /2 $7. 22 IEEE

2 conditions, to a higher power) of the transmitting range. Our evauation of required transmitting range is aso usefu in directing various topoogy contro protocos, which try to dynamicay adjust transmitting ranges in order to minimize energy consumption at run time [6,9,]. The question of how many nodes are necessary for a given transmitting range is very important for panning and design of wireess ad hoc networks when devices empoy a fixed transceiver technoogy. Our primary anaytica resut in this paper shows that a -dimensiona network with nodes paced over a region of ength is connected if and ony if the product of the number of nodes and the transmitting range is on the order of at east og. This coses a gap between ower and upper bounds on this product that were estabished in an earier paper []. Note that the -dimensiona version of the probem does have important practica appications. The most notabe such appication is to cars on a freeway, which approximates a -dimensiona region. An oft-cited potentia use of mobie ad hoc networks is to have transmitters paced in cars that can transmit information about congestion or accidents to cars further back. By repeated reaying of such information, drivers far from the probem site can earn of the congestion and seect an aternate route without waiting for a centra notification system to earn of the event and post warning notices. We aso evauate 2-dimensiona networks with mobiity through extensive simuations. We compare two different mobiity modes, the random waypoint mode, which modes intentiona movement, and the drunkard mode, in which movement is random. In both mobiity modes, we have incuded a parameter that accounts for those situations in which some nodes are not abe to move. For exampe, this coud be the case when sensors are spread from a moving vehice, and some of them remain entanged, say, in a bush or tree. This can aso mode a situation where two types of nodes are used, one type that is stationary and another type that is mobie. The goa of our simuations is to study the reationship between the vaue of the transmitting range ensuring connected graphs in the stationary case and the vaues of the transmitting range ensuring connected graphs during some fraction of the operationa time. In this paper, we focus on the transmitting ranges needed to ensure connectedness during %, 9% and % of the simuation time. These vaues are chosen as indicative of three different dependabiity scenarios that the ad hoc network must satisfy. We aso consider the vaue of the transmitting range ensuring that the average size of the argest connected component is a given fraction of the tota number of nodes in the network. The rationae for this investigation is that the network designer coud be interested in maintaining ony a certain fraction of the nodes connected, if this woud resut in significant energy savings. Further, considering that in many scenarios (e.g. wireess sensor networks) the cost of a node is very ow, it coud aso be the case that dispersing twice as many nodes as needed and setting the transmitting ranges in such a way that haf of the nodes remain connected is a feasibe and cost-effective soution. The resuts of our simuations have shown the somewhat surprising fact that, from a stricty statistica view of connectedness and connected component size, there are no major differences between the two mobiity modes. We aso demonstrate that quite arge reductions in transmitting range can be achieved if brief periods of disconnection are aowed and/or the network is aowed to operate with ony a significant fraction of the nodes being connected. These resuts iustrate an energy vs. quaity of communication trade-off that can be achieved in ad hoc networks, whereby the extent of communication capabiity can be somewhat reduced without great impact on the appication and with the benefit of significanty reduced energy consumption. A fina interesting resut of our simuations shows that if about ½ or fewer of the nodes are mobie, then the network appears equivaent, in terms of statistica connectedness, to one without mobiity. The properties we study in this paper are akin to a simpe form of avaiabiity for wireess ad hoc networks. Assuming that a network is up if a nodes are connected and down otherwise, then the percentage of time it is connected is an estimate of network avaiabiity. Since, in some appications, the network might be functiona if at east a given fraction of nodes are connected, we aso study the size of the argest connected component when the network is disconnected. For these appications, the percentage of time for which a sufficienty arge number of nodes are connected is an avaiabiity estimate. 2 Preiminaries A d-dimensiona mobie wireess ad hoc network is represented by a pair M d =(N,P), where N is the set of nodes, with N =n, and P: N T [,] d, for some >, is the pacement function. The pacement function assigns to every eement of N and to any time t T a set of coordinates in the d-dimensiona cube of side, representing the node s physica position at time t. The choice of imiting the admissibe physica pacement of nodes to a bounded region of R d of the form [,] d, for some >, is reaistic and wi ease the probabiistic anaysis of Section 3. If the physica node pacement does not vary with time, the network is said to be stationary, and function P can be redefined simpy as P: N [,] d. In this paper, we assume that a the nodes in the network have the same transmitting range r. With this assumption, the communication graph of M d induced at time t, denoted G M (t), is defined as G M (t)=(n,e(t)), where the edge (u,v) E(t) if and ony if v is at distance at most r from u at time t. If (u,v) E(t), node v is said to be a Proceedings of the Internationa Conference on Dependabe Systems and Networks (DSN 2) /2 $7. 22 IEEE

3 neighbor of u at time t. G M (t) corresponds to a point graph as defined in [2]. In the next section, we consider probabiistic soutions to the foowing probem for stationary ad hoc networks: MINIMUM TRANSMITTING RANGE (MTR): Suppose n nodes are paced in [,] d ; what is the minimum vaue of r such that the resuting communication graph is connected? Given the number of nodes, minimizing r whie maintaining a connected network is of primary importance if energy consumption is to be reduced. In fact, the energy consumed by a node for communication is directy dependent on its transmitting range. Further, a sma vaue of r reduces the interferences between node transmissions, thus increasing the network capacity [5]. Observe that we coud just as easiy have stated the probem as one of finding the minimum number of nodes to ensure connectedness given a fixed transmitting range. This formuation is of primary importance in many dimensioning probems arising in the design of wireess ad hoc networks. For exampe, soving this probem woud answer the foowing fundamenta question to the system designer: for a given transmitter technoogy, how many nodes must be distributed over a given region to ensure connectedness with high probabiity? In fact, our soutions typicay specify requirements on the product of n and r d that ensures connectedness. These soutions can, therefore, be used to sove either MTR, as specified above, or the aternate formuation where the number of nodes is the primary concern. It shoud be observed that the soution to MTR depends on the information we have about the physica node pacement. If the node pacement is known in advance, the minimum vaue of r ensuring connectedness can be easiy determined. Unfortunatey, in many reaistic scenarios of ad hoc networks the node pacement cannot be known in advance, for exampe because nodes are spread from a moving vehice (airpane, ship or spacecraft). If nodes positions are not known, the minimum vaue of r ensuring connectedness in a possibe cases is r d, which accounts for the fact that nodes coud be concentrated at opposite corners of the pacement region. However, this scenario appears to be very unikey in most reaistic situations. For this reason MTR has been studied in [,] under the assumption that nodes are distributed independenty and uniformy at random in the pacement region. Observe that connectivity probems with formuations simiar to MTR have aso been studied in [4,7]. However, in these papers the depoyment area is a fixed region (the unit disk in [4], or [,] 2 in [7]), and the number of nodes is increased to infinity. Thus, the asymptotic investigation is for networks with increasing node density, and is expected to be accurate in dense networks. On the contrary, the probem formuation used in this paper does not force the node density to asymptoticay increase to infinity. In the next section, we wi improve the resuts of [,] for the case d= by means of a more accurate anaysis of the conditions eading to disconnected communication graphs. The anaysis wi use some resuts of the occupancy theory [3], which are presented next. The occupancy probem can be described as foows: assume we have C ces, and n bas to be thrown independenty in the ces. The aocation of bas into ces can be characterized by means of random variabes describing some property of the ces. The occupancy theory is aimed at determining the probabiity distribution of such variabes as n and C grow to infinity (i.e., the imit distribution). The most studied random variabe is the number of empty ces after a the bas have been thrown, which wi be denoted µ(n, in the foowing. Under the assumption that the probabiity for any particuar ba to fa into the i-th ce is /C for i=,...,c (uniform aocation), the foowing resuts have been proved : i - P( µ ( n, = ) = ( ) - E[ µ ( n, ] C i= C i C = C - Var[ ( n ] = C( C ) n n 2 2 µ, + C C, C C C where E[µ(n,] and Var[µ(n,] denote the expected vaue and the variance of µ(n,, respectivey. The asymptotic behaviors of P(µ(n,=k), E[µ(n,] and Var[µ(n,] depend on the reative magnitudes of n and C as they grow to infinity. The foowing theorem has been proved: Theorem. For every n and C, [ µ ( n ] Ce E,, where α=n/c. Furthermore, if n, in such a way that α=o(, then: - E[ µ ( n, ] = Ce α e 2 - Var[ µ ( n, ] = Ce ( + α ) i C n ( α ) α + e + O C n 2n ( ) ( ) e + O α + α e e + C Using the asymptotic formuas of Theorem, we can distinguish five different domains such that n,, for which the asymptotic distribution of the random variabe µ(n, is different. These domains are: - the centra domain (CD for short), when n=θ(; A the resuts presented in this section are taken from [3]. Proceedings of the Internationa Conference on Dependabe Systems and Networks (DSN 2) /2 $7. 22 IEEE

4 - the right-hand domain (RHD for short), when n= Θ(Cog; - the eft-hand domain (LHD for short), when n=θ( C ); - the right-hand intermediate domain (RHID for short), when n=ω( but C ogc >> n; 2 - the eft-hand intermediate domain (LHID for short), when n=o( but n>> C. The foowing theorem describes the imit distribution of µ(n, in the different domains. Theorem 2. The imit distribution of the random variabe µ(n, is: - the norma distribution of parameters (E[µ(n,], [ ( n ] Var µ, ) in the CD, RHID and LHID; - the Poisson distribution of parameter λ in the RHD, λ = im E µ n, C. where [ ( )] n, Furthermore, in the LHD the imit distribution of the random variabe η(n,=µ(n,-(c-n) is the Poisson ρ = im Var µ n, C. distribution of parameter ρ, where [ ( )] n, 3 Probabiistic anaysis of MTR for stationary networks Consider the probabiity space (Ω,F,P ), where Ω =[,], F is the famiy of a cosed subsets of Ω and P is a probabiity distribution on Ω. In this paper, we assume that P is the uniform distribution on Ω. Under this setting, nodes in N can be modeed as independent random variabes uniformy distributed in [,], which wi be denoted Z,,Z n. We say that an event V k, describing a property of a random structure depending on a parameter k, hods asymptoticay amost surey (a.a.s. for short), if P(V k ) as k. In the foowing we consider the asymptotic behavior of the event CONNECTED on the random structures (Ω,F,P ) as. Informay speaking, event CONNECTED corresponds to a the vaues of the random variabes Z,..,Z n for which the communication graph is connected. The foowing upper bound on the magnitude of rn ensuring a.a.s. connectedness has been derived in []. Theorem 3. Suppose n nodes are paced in [,] according to the uniform distribution. If rn Θ( og ) and r>>, then the communication graph is a.a.s. connected. 2 Notation f(x)<<g(x) (resp., f(x)>>g(x)) is used to denote the fact that f(x)/g(x) (resp., ) as x. Observe that the constraint r>> in the statement of the theorem is not restrictive, since we are interested in investigating the magnitudes of r such that <<r<<. In [], a ower bound on the magnitude of rn ensuring a high probabiity of connectedness is derived by anayzing the probabiity of existence of an isoated node. In fact, the existence of an isoated node impies that the resuting communication graph (which is a point graph) is disconnected. However, the cass of disconnected point graphs is much arger than the cass of point graphs containing at east one isoated node. For this reason, the bounds estabished in [] are not tight, and the gap between the ower and upper bounds on the magnitude of rn is in the order of og. In [], it is conjectured that the upper bound stated in Theorem 3 is actuay tight. This intuition has been experimentay confirmed by the resuts of extensive simuations [,]. In what foows, we prove that the conjecture stated in [] is true for -dimensiona networks. The resut derives from a more accurate approximation of the cass of disconnected point graphs, which is based on occupancy theory. This aows us to cose the gap, proving the tightness of the bound stated in Theorem 3. r Figure. Node pacement generating a disconnected communication graph. In order to derive the ower bound, we consider the foowing subdivision of the pacement region into ces. We assume that a ine of ength is subdivided into C=/r segments of equa ength r. With this subdivision, if there exists an empty ce c i separating two ces c i-,c i+ that each contains at east one node, then the nodes in c i- are unabe to communicate to those in c i+, and the resuting communication graph is disconnected (see Figure ). The foowing emma, whose immediate proof is omitted, estabishes a sufficient condition for the communication graph to be disconnected. Lemma. Assume that n nodes are paced in [,], and that the ine is divided into C=/r segments of equa ength r. Assign to every ce c i, for i=,..,c-, a bit b i, denoting the presence of at east one node in the ce. Without oss of generaity, assume b i = if c i is empty, and b i = otherwise. Let B={b...b C- } be the string obtained by concatenating the bits b i, for i=,..,c-. If B contains a substring of the form {*}, where * denotes that one or more s may occur, then the resuting communication graph is disconnected. Observe that the condition stated in Lemma is sufficient but not necessary to produce a disconnected graph. In fact, there exist node pacements such that B does not contain any substring of the form {*}, but the resuting communication graph is disconnected. Proceedings of the Internationa Conference on Dependabe Systems and Networks (DSN 2) /2 $7. 22 IEEE

5 Let us denote with CONNECTED, DISCONNECTED, * and E the events corresponding to a the vaues of the random variabes Z,..Z n such that the resuting communication graph is connected, disconnected, or a substring of the form {*} occurs in B, respectivey. The subscript indicates that we are considering these events in the case that the ength of the ine is. Since * CONNECTED =Ω -DISCONNECTED and E DIS- CONNECTED, it is immediate that a necessary condition * im P E =. for a.a.s. connectedness is that ( ) * In order to evauate im P( E ), we decompose the * event E by conditioning on the disjoint events {µ(n,=k}, for k=,..,c; i.e., P C * * ( E ) = P( E { ( n, = k} ) P( µ ( n, = k) k = * Observe that when goes to infinity ( ) µ (). P is defined as the sum of an infinite number of non-negative terms t,t 2,. Ceary, if there exists at east one term t k such * that im t = ε >, then im P ( E ) ε >. In what foows, k we prove that if <<rn<< og and k = E [ ( n, ] k µ, then im t = ε >, thus impying that the resuting communication graph is not a.a.s. connected. We start with a emma that characterizes the asymptotic * P E µ n, C = k as goes to infinity. behavior of { ( ) } ( ) * im P( E { n, C = k} ) = Lemma 2. If <k<<c, then ( ) Proof. See Appendix. E µ. We now state the main theorem of this section. Theorem 4. Assume that <<rn<< og. Then * im P E ε >. ( ) Proof. See Appendix. Combining the resut stated in Theorem 4 with the bound of Theorem 3, we concude this section with the foowing theorem. Theorem 5. Suppose n nodes are paced in [,] according to the uniform distribution, and assume <<r<<. The communication graph is a.a.s. connected if and ony if rn Ω( og ). The resut stated in Theorem 5, for random distribution of nodes, shoud be compared to the transmitting ranges necessary with worst-case and best-case pacements. To iustrate this, consider the case where the number of nodes is inear with the ength of the ine,. In the worst-case, nodes are custered at either end of the ine and the transmitting range must be Ω() for the network to be connected. In the best-case pacement, nodes are equay spaced at intervas of /n, which in this case is a constant. Hence, a constant transmitting range is sufficient in the best case. Theorem 5 s resut yieds a transmitting range of Ω(og ) with random pacement. Thus, there is a substantia reduction in transmitting range from the worstcase but aso a significant increase compared to the bestcase. 4 Evauation of MTR for mobie networks In this section, we consider the mobie version of MTR, which can be formuated as foows: MINIMUM TRANSMITTING RANGE MOBILE (MTRM): Suppose n nodes are paced in [,] d, and assume that nodes are aowed to move during a time interva [,T]. What is the minimum vaue of r such that the resuting communication graph is connected during some fraction, f, of the interva? A forma anaysis of MTRM is much more compicated than that of MTR and is beyond the scope of this paper. In this section, we study MTRM by means of extensive simuations. The goa is to study the reationship between the vaue of r ensuring connected graphs in the stationary case (denoted r stationary ) and the vaues of the transmitting range ensuring connected graphs during some fraction of the operationa time. In this paper, we focus on the transmitting ranges needed to ensure connectedness during %, 9% and % of the simuation time (denoted r, r 9 and r, respectivey). These vaues are chosen as indicative of three different dependabiity scenarios that the ad hoc network must satisfy. In the first case, the network is used for safety-critica or ife-critica appications (e.g., systems to detect physica intrusions in a home or business), and network connectedness during the entire operationa time is a vita requirement. In this scenario, the potentiay high price (expressed in terms of increased energy consumption) to be paid to keep the network aways connected is a secondary issue. In the second case, temporary network disconnections can be toerated, especiay if this is counterbaanced by a significant decrease of the energy consumption with respect to the case of continuous connectedness. This scenario is pausibe in many appications of wireess ad hoc networks, e.g. when the network is used to connect a squad of workers in an oi patform. In the atter case, the network stays disconnected most of the time, but temporary connection periods can be used to exchange data among nodes. This coud be the case of wireess sensor networks [8] used for environmenta monitoring [3], where environmenta data (e.g., temperature, pressure, air poution eves) are gathered by sensors, which periodicay exchange these data with the other nodes in order to buid a goba view of the monitored Proceedings of the Internationa Conference on Dependabe Systems and Networks (DSN 2) /2 $7. 22 IEEE

6 region. In this setting, reducing energy consumption is the primary concern, and temporary connectedness is sufficient to ensure that the data sent by a sensor is eventuay received by the other nodes in the network. 4. Simuation modes To generate the resuts of this section, we extended the simuator used in [,] for the stationary case by impementing two mobiity modes. The simuator distributes n nodes in [,] d according to the uniform distribution, then generates the communication graph assuming that a nodes have the same transmitting range r. Parameters r, n, and d are given as input to the simuator, aong with the number of iterations to run and the number, #steps, of mobiity steps for each iteration. Setting #steps= corresponds to the stationary case. The simuator returns the percentage of connected graphs generated, the average size of the argest connected component (averaged over the runs that yied a disconnected graph) and the minimum size of the argest connected component. A of these parameters are reported with reference both to a singe iteration (in this case, the averages are over a the mobiity steps) and to a the iterations. In a simuations reported herein, we set d=2, as the two-dimensiona setting is an appropriate mode for many appications of wireess ad hoc networks. Two mobiity modes have been impemented. The first mode is the cassica random waypoint mode [2], and is used to mode intentiona movement: every node chooses uniformy at random a destination in [,] d, and moves toward it with a veocity chosen uniformy at random in the interva [v min,v max ]. When it reaches the destination, it remains stationary for a predefined pause time t pause, and then it starts moving again according to the same rue. In the simuator, t pause is expressed as the number of mobiity steps for which the node must remain stationary. We have aso incuded a further parameter in the mode, namey the probabiity p stationary that a node remains stationary during the entire simuation time. Hence, ony (-p stationary )n nodes (on the average) wi move. Introducing p stationary in the mode accounts for those situations in which some nodes are not abe to move. For exampe, this coud be the case when sensors are spread from a moving vehice, and some of them remain entanged, say, in a bush or tree. This can aso mode a situation where two types of nodes are used, one type that is stationary and another type that is mobie. The second mobiity mode resembes a drunkard-ike (i.e., non-intentiona) motion. Mobiity is modeed using parameters p stationary, p pause and m. Parameter p stationary is defined as above. Parameter p pause is the probabiity that a node remains stationary at a given step. This parameter accounts for heterogeneous mobiity patterns, in which nodes may move at different times. Intuitivey, the higher is the vaue of p pause, the more heterogeneous is the mobiity pattern. However, vaues of p pause cose to resut in an amost stationary network. If a node is moving at step i, its position in step i+ is chosen uniformy at random in the disk of radius m centered at the current node ocation. Parameter m modes, to a certain extent, the veocity of the nodes: the arger m is, the more ikey it is that a node moves far away from its position in the previous step. 4.2 Simuation resuts for increasing system size The first set of simuations was aimed at investigating the vaue of the ratio of r (respectivey, of r 9 and r ) to r stationary for vaues of ranging from 256 to We aso considered the argest vaue r of the transmitting range that yieds no connected graphs. In both mobiity modes, n was set to. The vaue of r stationary is obtained from the simuation resuts for the stationary case reported in [,], whie those for r, r 9, r and r are averaged over 5 simuations of steps of mobiity each. First, we considered the random waypoint mode, with parameters set as foows: p stationary =, v min =., v max =., and t pause =2. This setting modes a moderate mobiity scenario, in which a the nodes are moving, but their veocity is rather ow. The effect of different choices of the mobiity parameters on the vaues of r, r 9 and r is studied in the next sub-section. The vaues of the ratios are reported in Figure 2. Figure 3 reports the same graphic obtained for the drunkard mode, with p stationary =., p pause =.3 and m=.. This is aso a moderate mobiity scenario, but more heterogeneous than the other: a sma percentage of the nodes remain stationary, and mobie nodes are stationary for 3% of the simuation time (on average). The graphics show the same quaitative behavior: as increases, the ratio of the different transmitting ranges for mobiity to r stationary tends to increase, and this increasing behavior is more pronounced for the case of r. However, even when the system is arge, a modest increase to r stationary (about 2% in the random waypoint and about 25% in the drunkard mode) is sufficient to ensure connectedness during the entire simuation time. Comparing the resuts for the two mobiity modes, we can see somewhat higher vaues of the ratios for the drunkard mode, especiay for the case of r. This seems to indicate that more homogeneous mobiity patterns hep in maintaining connectedness. However, it is surprising that the resuts for the two mobiity modes are so simiar. This indicates that it is more the existence of mobiity rather than the precise detais of how nodes move that is significant, at east as far as network connectedness is concerned. It shoud aso be observed that r 9 is far smaer than r (about 35-4% smaer) in both mobiity modes, Proceedings of the Internationa Conference on Dependabe Systems and Networks (DSN 2) /2 $7. 22 IEEE

7 ,4,4,2,2 r r9 r r9,6 r r,6 r r,4,4,2,2 256 K 4K 6K 256 K 4K 6K Figure 2. Vaues of r x/r stationary for increasing vaues of in the random waypoint mode. Figure 3. Vaues of r x/r stationary for increasing vaues of in the drunkard mode.,6 r9,6 r9 r r,4 r,4 r,2,2 256 K 4K 6K 256 K 4K 6K Figure 4. Average size of the argest connected component (expressed as a fraction of n) for increasing vaues of in the random waypoint mode. independenty of the system size. Hence, substantia energy savings can be achieved under both modes if temporary disconnections can be toerated. When the requirement for connectedness is ony % of the operationa time, the decrease in the transmitting range is about 55-6%, enabing further energy savings. However, if r is reduced to about 25% to 4% of r stationary, the network becomes disconnected during the entire simuation time. The average size of the argest connected component when the transmitting range is set to r 9, r and r was aso investigated. Simuation resuts are dispayed in Figures 4 and 5. Once again, the graphics show very simiar behaviors: the ratio of the average size of the argest connected component to n increases as increases. When the transmitting range is set to r 9 and is sufficienty arge, this ratio is very cose to (about.98 in both mobiity modes). This means that during the short time in which the network is disconnected, a vast majority of its nodes forms a arge connected component. Hence, on the average disconnection is caused by ony a few isoated nodes. This fact is confirmed by the pots for r : even when the network is disconnected most of the time, a arge Figure 5. Average size of the argest connected component (expressed as a fraction of n) for increasing vaues of in the drunkard mode. connected component (of average size about.9n for arge vaues of ) sti exists. However, if the transmitting range is further decreased to r, the size of the argest connected component drops to about.5n. We aso considered the vaue of the transmitting range ensuring that the average size of the argest connected component is.9n,.75n and.5n, respectivey. The corresponding vaues of the transmitting range are denoted r 9, r 75 and r 5. The mobiity parameters and n were set as above. The rationae for this investigation is that the network designer coud be interested in maintaining ony a certain fraction of the nodes connected, if this woud resut in significant energy savings. Further, considering that in many scenarios (e.g. wireess sensor networks) the cost of a node is very ow, it coud aso be the case that dispersing twice as many nodes as needed and setting the transmitting ranges in such a way that haf of the nodes remain connected is a feasibe and cost-effective soution. The vaue of the ratio of r 9, r 75 and r 5 to r stationary for increasing vaues of in the random waypoint mode is shown in Figure 6. Simuation resuts have shown that whie r 9 /r stationary tends to decrease with increasing vaues of, converging to about.52, the ratios r 75 /r stationary and Proceedings of the Internationa Conference on Dependabe Systems and Networks (DSN 2) /2 $7. 22 IEEE

8 ,2,,6 r9 r75 r,4 r5,2,9 256 K 4K 6K,,2,3,4,5,6,7,9 Figure 6. Vaues of the ratio r 9, r 75 and r 5 to r stationary for ranging from 256 to 6384 in the random waypoint mode. Figure 7. Vaue of r /r stationary for different vaues of p stationary in the random waypoint mode.,2,2,, r r,9, ,,2,3,4,5 Figure 8. Vaue of r /r stationary for vaues of t pause ranging from to in random waypoint mode. r 5 /r stationary are amost independent of. In particuar, r 75 /r stationary is about.46 and r 5 /r stationary is about.4. Further, the reative differences between the three ratios decrease for increasing vaue of. This indicates that, whie for sma networks (few nodes distributed in a reativey sma region) the energy needed to maintain 9% of the nodes connected is significanty higher than that required to connect 5% of the nodes (r 5 is ess than haf of r 9 for =256), for arge networks the savings are not as great if the requirement for connectivity is ony 5% of the nodes (r 5 is 2% smaer than r 9 for =6384). 4.3 Simuation resuts for different mobiity patterns A second set of simuations was done to investigate the effect of different choices of the mobiity parameters on the vaue of r. We considered the random waypoint mode with =496 and n= =64. The defaut vaues of the mobiity parameters were set as above, i.e. p stationary =, v min =., v max =., and t pause =2. Then, we varied the vaue of one parameter, eaving the others unchanged. Figure 9. Vaue of r /r stationary for vaues of v max ranging from. to.5 in random waypoint mode. Figure 7 reports the vaue of r for vaues of p stationary ranging from (no stationary nodes) to (corresponding to the stationary case) in steps of.2. Simuation resuts show a sharp drop of r in the interva.4-.6: for p stationary =.4, r is about % arger than r stationary, whie for p stationary =.6 and for arger vaues of p stationary we have r r stationary. To investigate this drop more cosey, we performed further simuations by exporing the interva.4-.6 in steps of.2. As shown in Figure 7, there is a distinct threshod phenomenon: when the number of stationary nodes is about n/2 or higher, the network can be regarded as practicay stationary from a connectedness point of view. This resut is very interesting, since it seems to indicate that a certain number (abeit a rather arge fraction) of stationary nodes woud significanty increase network connectedness. With more than n/2 mobie nodes, the network quicky becomes equivaent to one in which a nodes are mobie. The effect of t pause and of the veocity on r is shown in Figures 8 and 9. Increasing vaues of t pause tend to decrease the vaue of r, athough the trend is not as pronounced as in the case of p stationary. A threshod phenomenon seems to exist in the interva 4-6 in this case aso. However, Proceedings of the Internationa Conference on Dependabe Systems and Networks (DSN 2) /2 $7. 22 IEEE

9 further simuations in this interva have shown that, athough the trend can be observed, no sharp threshod actuay exists. We beieve that the rationae for this is the foowing: whie the vaue of p stationary has a direct impact on the "quantity of mobiity" (which can be informay understood as the percentage of stationary nodes with respect to the tota number of nodes), the effect of the pause time is not so direct. In fact, in the random waypoint mode the "quantity of mobiity" depends heaviy on the node destinations, which are chosen uniformy at random: even if the pause time is ong and the veocity is moderate, a node coud be "mobie" for a ong time if its destination is very far from its initia ocation. So, an increased pause time tends to render the system more stationary, but in a much ess direct way than p stationary. As shown in Figure 9, the vaue of r is amost independent of the vaue of v max : except for ow veocities (v max beow.), r is sighty above r stationary. This surprising resut coud be due to the apparenty counterintuitive fact that the quantity of mobiity is ony marginay infuenced by the vaue of v max, and a arger vaue of v max tends to decrease the quantity of mobiity. In fact, the arger is v max, the more ikey it is that nodes arrive quicky at destination and remain stationary for t pause =2 steps. 5. Concusions In this paper, we considered a connectivity probem in the case of both stationary and mobie wireess ad hoc networks. For the stationary case, we have derived tight bounds on the magnitude of r, n and ensuring connectedness with high probabiity for -dimensiona networks. Our bounds improve on existing resuts, and prove a conjecture stated in a previous paper. We have aso investigated the mobie version of the probem for 2- dimensiona networks through extensive simuation. We impemented two motion patterns to mode both intentiona and non-intentiona movements, and we simuated 2-dimensiona networks of different sizes and using different mobiity parameters. Simuation resuts have shown that consistent energy savings can be achieved if temporary disconnections can be toerated or if connectedness must be ensured ony for a arge fraction of the nodes. Regarding the infuence of mobiity patterns, simuation resuts have shown that connectedness is ony marginay infuenced by whether motion is intentiona or not, but it is rather reated to the quantity of mobiity, which can be informay defined as the percentage of stationary nodes with respect to the tota number of nodes. For exampe, when about n/2 nodes are static, the network can be regarded as stationary from a connectivity point of view. Further investigation in this direction is needed, and is a matter of ongoing research. References [] D.M. Bough, P. Santi, "The Random Point Graph Mode for Ad Hoc Networks and its Appication to the Range Assignment Probem", Tech. Rep. IMC-B4--5, Istituto di Matematica Computazionae de CNR, Pisa - Itay, Dec. 2. [2] D. B. Johnson, and D. A. Matz, Dynamic Source Routing in Ad Hoc Wireess Networks, in Mobie Computing, Kuwer Academic Pubishers, 996, pp [3] V.F. Kochin, B.A. Sevast yanov, and V. P. Chistyakov, Random Aocations, V.H. Winston and Sons, Washington D.C., 978. [4] P. Gupta, and P. R. Kumar, Critica Power for Asymptotic Connectivity in Wireess Networks, Stochastic Anaysis, Contro, Optimization and Appications, (W.M. McEneany, G. Yin, and Q. Zhang, eds.), Birkhauser, Boston, 998, pp [5] P. Gupta, P.R. Kumar, The Capacity of Wireess Networks, IEEE Trans. Inf. Theory, vo. 46, n. 2, pp , March 2. [6] L. Li, J. H. Hapern, P. Bah, Y. Wang, R. Wattenhofer, Anaysis of a Cone-Based Distributed Topoogy Contro Agorithm for Wireess Muti-hop Networks, Proc. ACM Symp. on Principes of Distributed Computing (POD, pp , August 2. [7] T.K. Phiips, S.S. Panwar, A.N. Tantawi, Connectivity Properties of a Packet Radio Network Mode, IEEE Trans. Inf. Theory, vo. 35, n. 5, pp , Sept [8] G. J. Pottie, W. J. Kaiser, Wireess Integrated Network Sensors, Communications of the ACM, vo. 43, no. 5, pp. 5-58, May 2. [9] R. Ramanathan, R. Rosaes-Hain, Topoogy Contro of Mutihop Wireess Networks using Transmit Power Adjustment, Proc. IEEE Infocom 2, pp , 2. [] V. Rodopu, T. H. Meng, Minimum Energy Mobie Wireess Networks, IEEE J. Seected Areas in Comm., vo. 7, n. 8, pp , Aug.999. [] P. Santi, D. M. Bough, and F. Vainstein, A Probabiistic Anaysis for the Range Assignment Probem in Ad Hoc Networks, Proc. ACM Symposium on Mobie Ad Hoc Networking and Computing (MobiHoc), Long Beach, CA, October 2, pp [2] A. Sen, and M. L. Huson, A New Mode for Scheduing Packet Radio Networks, Proc. IEEE Infocom 96, pp. 6 24, 996. [3] D. C. Steere, A. Baptista, D. McNamee, C. Pu, J. Wapoe, Research Chaenges in Environmenta Observation and Forecasting Systems, Proc. ACM MOBICOM 2, pp , 2. Appendix Proof of Lemma 2. * Consider the compementary event of E, i.e. * E = Ω E. It can be easiy seen that E corresponds to a the vaues of the random variabes Z,..Z n such that the -bits in B are consecutives. Given the hypothesis of Proceedings of the Internationa Conference on Dependabe Systems and Networks (DSN 2) /2 $7. 22 IEEE

10 independence of the random variabes Z,..Z n, when exacty k ces out of C are empty (i.e., k bits in B are ), P( E { µ ( n, = k} ) corresponds to the ratio of a configurations of (n-k) consecutive -bits over a possibe configurations of k -bits in C positions, i.e.: k + P( E { ( ) }) µ n, C = k =. C k Since C=/r and r<<, we have: * k + im P( E { µ ( n, = k} ) = im P( E { µ ( n, = k} ) = im C k We can rewrite the ast imit as: k + ( k + )! im = im C C( C )...( C k + ) k Since k<<c, we have: ( k + )! ( k + )! im = im k C( C )...( C k + ) C Taking the ogarithm, we obtain: ( k + )! im n = im n( k + )! k n C k C = im k n k k n C = im k n k n C ( ) Since <k<<c, we concude that k( n k n = im, hence: k + im =, C k and the Lemma is proved. Consider k = E [ ( n, ] α α( α + ) µ. From Theorem, we have that + e k Ce e O, (2) 2 C where α=n/c=rn/. Since <<rn<< og, we have <<α=f()<<og, and (2) can be rewritten as k, with < << k << = C, f () re r r hence the first condition is satisfied. Observe that the condition <<rn<< og impies that C<<n<<C og C, i.e. we are in the RHID. By Theorem 2, it foows that the imit distribution of µ(n, as n,c go to infinity is the norma distribution of parameters (E[µ(n,], Var [ µ ( n, ]). By Theorem, Var[µ(n,] can be rewritten as Var[ µ ( n, ] = Ce ( + α ) e + Oα + α e e + + f ( ) Hence, we have: P ( µ ( n, = k ) ( ) ( ) 2πVar [ µ ( n, ] 2π re f ( ) C Let us choose r = δ, for some <δ 2π. Observe that f () e this choice of r is consistent with the hypothesis <<r<<, since we have << <<. With this choice of r, we can f ( ) e write the imit as foows: f ( ) re δ im P( µ ( n, = k ) = im = = ε >, 2π 2π and the theorem is proved. re f ( ) e f ( ) Proof of Theorem 4. By Equation () and Lemma 2, it is sufficient to show that there exists a vaue k such that: - < k <<C, and im P µ n, C = k ε >. - ( ) ( ) Proceedings of the Internationa Conference on Dependabe Systems and Networks (DSN 2) /2 $7. 22 IEEE