Node Density and Delay in Large-Scale Wireless Networks with Unreliable Links

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1 Noe Density an Delay in Large-Scale Wireless Networks with Unreliable Links Shizhen Zhao, Xinbing Wang Department of Electronic Engineering Shanghai Jiao Tong University, China Abstract We stuy the elay performance in large-scale wireless multi-hop networks with unreliable links from percolation perspective. Previous works [][3][] have showe that the en-to-en elay scales linearly with the source-to-estination istance, an thus the elay performance can be characterize by the elay-istance ratio γ. However, the range of γ, which may be the most important parameter for elay, remains unknown. We expect that γ may epen heavily on the noe ensity of a wireless multi-hop network. In this paper, we investigate the funamental relationship between γ an. Obtaining the exact value of γ() is extremely har, mainly because of the ynamically changing network topologies cause by the link unreliability. Instea, we provie both upper boun an lower boun to the elay-istance ratio γ(). Simulations are conucte to verify our theoretical analysis. Inex Terms Connectivity, Delay, Density I. INTRODUCTION Large-scale wireless multi-hop networks are fast becoming the most preferre way for communication in outoor environment []. Each user in a wireless multi-hop network coul initiate a communication in a multi-hop fashion without of the ai of any network infrastructure. Thus, the maintenance cost for communication networks can be significantly reuce. However, relays of a wireless multi-hop network may be also users, which are highly unreliable ue to the unexpecte user behavior an the time-varying wireless channel. Thus, it is highly possible that a connection may get lost uring ata transmission, in which case the ata has to wait at some noe an resume its transmission after the reestablishment of the network connection. The lost of connection coul cause significant elay to ata transmission. Therefore, it is very critical to provie some elay guarantee for wireless multi-hop networks. Note that the ata communication reliability can be improve by some back-up routes, an it is usually easier to fin more routes when the noe ensity is higher. Hence, we expect that the ata communication elay may epen heavily on the noe ensity. Our goal in this paper is to quantify the funamental relationship between noe ensity an elay. To make our stuy meaningful, we assume that the largescale wireless network is connecte. Full connectivity [] can ensure the successful communication between each noe pair in a wireless network. However, it is overly power consuming to achieve full connectivity in large-scale networks (i.e., the An earlier version of this paper appeare in ACM MobiCom [9] power require to maintain full connectivity increases with the size of the network). Hence, it is necessary to introuce a slightly weaker connectivity criterion, in which we only ensure that a large portion of the network noes are connecte via multiple hops with each other. Thanks to percolation theory [][5], it is possible to achieve this weaker connectivity in large-scale networks with power boune per noe. Percolation theory [], especially continuum percolation, is a powerful mathematical tool when analyzing the connectivity an the elay of wireless networks. An important an general moel of Continuum Percolation is Ranom Connection Moel (RCM). RCM escribes the behavior of connecte components in an infinitely large ranom geometric graph, in which noes are istribute accoring to poisson point process with noe ensity, an two noes share a link with probability h(r) (r is the istance between the two noes). A classical result of RCM points out a funamental phase transition effect at a critical noe ensity c (, ). For > c (supercritical), there exists a unique connecte component containing a large portion of noes in the infinitely large network (we also say the network is percolate). However, for < c (subcritical), all connecte component in the network are finite almost surely. Applying percolation theory to large-scale wireless multihop networks, we introuce two important concepts, i.e., the instantaneous connectivity an the long-term connectivity. The Instantaneous connectivity requires wireless networks percolate at all the time instances. While the long-term connectivity only requires wireless network percolate in the long run (we will elaborate it more clearly later in sectionii-b). The instantaneous critical ensity, enote by I, is the critical ensity for the instantaneous connectivity an the long-term critical ensity, enote by L, is the critical ensity for the long-term connectivity. Long-term connectivity is a weaker criterion for connectivity, thus L < I. To ensure connectivity, we assume that > L throughout this paper. In large-scale wireless networks, elay is mainly compose of the waiting elay an the propagation elay. The waiting elay is cause by the loss of connection at some time instances, which is ue to the lack of instantaneous connectivity. Packets must wait for some time until the connection is reestablishe. Usually, such a waiting time is in the orer of secons, minutes or more. The propagation elay is the time require for a packet to transverse a link whenever the link is active. Mostly, the propagation elay is in the orer of millisecons.

2 Therefore, it is negligibly small compare to the waiting elay. For ease of analysis, we first ignore the impact of propagation elay an will consier its effect in the last. We stuy the elay performance for flooing in large-scale wireless networks. Ignoring the propagation elay, previous works [][3][] have showe that if L < < I, the en-to-en elay scales linearly with the source-to-estination istance (γ() > ), an that if > I, the en-to-en elay scales sub-linearly with istance (γ() = ). However, these results are far from enough to provie reasonable elay guarantee. It remains extremely important to fin out the exact value or the lower an upper bouns of γ(). In this paper, we present a theoretical analysis about the flooing elay in wireless multi-hop networks with unreliable links. We first ignore the propagation elay an fin the upper an lower bouns of γ(). For the upper boun, we first fin a path between two noes using percolation theory. An then we calculate the number of hops along this path using another coupling technique an the elay at each hop. We obtain the result of upper boun by multiplying the above two items. For the lower boun, we first introuce a concept calle cluster to cluster transmission process an establish the relationship between elay an the cluster to cluster transmission process, which reveals the essence of the flooing elay in wireless multi-hop networks. Then, base on the elay of the cluster to cluster transmission, we obtain a lower boun of γ(). We then generalize the previous result to the case with nonzero propagation elay. Propagation elay will aggravate the elay performance in Large-Scale wireless Networks, making γ() > even when > I. Using similar methos, we present new upper an lower bouns to γ() for all > L. Finally, we conuct simulation to verify our theoretical results. The original contributions that we have mae in the paper are highlighte as follows: We present two properties to γ(), i.e., γ() = whenever propagation elay is an > I ; γ() is a monotone ecreasing function. Ignoring propagation elay, we provie the upper boun an the lower boun to reflect the range of variation on γ(). Taking propagation elay into consieration, we obtain further results. We conuct simulations to obtain experimental values of γ() in the above two cases. The new observation arises from our comparison between theoretical an simulation results is that the elay-istance ratio γ() can be approximate by the lower boun in relative ense networks while the experimental values of γ() get closer to the upper boun in relative sparse networks. This also justifies the sounness of our theoretical conclusion. The rest of the paper is organize as follows. In section II, we introuce our network moel, several useful mathematical tools an some important notations. In section III, we first give two properties of γ(), an then present our main results concerning the upper an lower boun of γ(). The analysis for obtaining the upper an lower bouns is given in section IV. Simulation results are presente in Section VI to justify our theoretical finings. We summarize the paper in Section VII. Some proofs of the theorems an lemmas are presente in section V. II. BACKGROUND AND NETWORK MODEL In this section, we introuce some backgroun knowlege an the network moel. A. Backgroun ) Poisson Point Process: In large-scale wireless multi-hop networks, it is extremely ifficult to preict the exact locations of ifferent users (noes) ue to the ranom behavior of ifferent users. To moel the location-ranomness, we use Poisson Point Process []. One way to visualize the Poisson Point Process is to assume that all users are istribute uniformly in a given area, e.g., n noes are evenly istribute in a n n area. If we let n, then the istribution of these ranom processes (inexe by n) will converge in istribution to Poisson Point Process with rate. Poisson Point Process has the following two properties: The superposition of two inepenent Poisson Point Processes is still a Poisson Point Process. The number of noes counte in isjoint areas are inepenent from each other (spatial inepenence). As reaers will see, the above two properties are extremely useful in our analysis. ) Ranom Connection Moel: Before introucing the network moel, we nee a brief introuction to Continuum Percolation. Connectivity is a significant issue in wireless network, which has been extensively explore by [][5][][7][][9]. In this paper, we present the efinition of connectivity in the percolation perspective. To make the results in this paper applicable to the most scenarios, we focus on a fairly general moel in Continuum Percolation Theory, i.e., the Ranom Connection Moel(RCM) [5]. In the RCM, noes are istribute accoring to Poisson point process in R. Here we only focus on the case of R with noe ensity >. Each noe x connects to another noe y accoring to some preefine connection function h(r), where r = x y is the istance between x an y. Specifically, noes x an y are connecte with probability h(r). We assume that h(r) satisfies < R h(r)xy <, which ensures the phase transition effect [5] in the Ranom Connection Moel. We enote a RCM by G(, r, h(r)), where is the noe ensity, r = sup{r h(r) > }, h(r) is the connection function. Then G(, r, h(r)) is a set of noes connecte by ranom links. For convenience, we assume that the origin G(, r, h(r)). Obviously, G(, r, h(r)) is compose of a set of isjointe connecte components. Let us enote W (A), A G(, r, h(r)), the set of noes attainable from noes in set A, i.e., W (A) = {x G(, r, h(r)) a A, a x}, In general, a RCM can be fully etermine by an h(r) only. We just use r to emphasize that we only focus on functions h(r) that are for r larger than some finite value r.

3 3 where, a x means that noes a an x are in the same connecte component. Besies, we use W to represent the carinality of the set W. Further, we efine θ h () = P,h ( W ({}) = ) 3 an χ h () = E,h ( W ({}) ). Then, the critical ensity can be etermine in two ways, i.e., θ (h) = inf{ θ h () > }; () (a) Illustration of connection function g(r). (b) Illustration of connection function f(r). χ (h) = inf{ χ h () = }. () Fig.. Illustration of two connection functions. Accoring to Theorem. in [5], < θ (h) = χ (h) = c (h) <. Furthermore, there exists a unique infinite connecte component containing a large portion of the noes in G(, r, h(r)) with probability if > c (h) (supercritical). This infinite connecte component is also calle the giant component, enote by C(G(, r, h(r))). In this case, we call G(, r, h(r)) percolate. On the other han, if < c (h) (subcritical), all the connecte components are finite almost surely. Another useful moel in Continuum Percolation is Poisson Boolean Moel, enote by B(, r). In the Poisson Boolean Moel B(, r), noes are istribute accoring to Poisson Point Process with ensity, an two noes can communicate if an only if their istance is smaller than r. Poisson Boolean Moel can be viewe as a special case of Ranom Connection Moel. Thus, the conclusions for Ranom Connection Moel also hol for Poisson Boolean Moel. B. Moel ) Network Moel: We moel the large-scale wireless multi-hop network as a ranom graph with ynamically varying eges. We assume that the network noes are istribute accoring to Poisson Point Process with noe ensity in an infinite two-imensional space R. For each noe u, we use u to represent both this noe an its location without causing confusion. We say two noes share a link if an only if their istance is smaller than r. However, ue to the unreliability of the wireless channel an the unexpecte user behaviors, each link suffers the possibility to fail. We say a link open at a time slot if this link can successfully transmit a packet, an close otherwise. We moel the link failure as each link opening or closing intermittently. Specifically, we assume that time is slotte, an the uration of each time slot is. Consier a link with length r, at time slot t. We let it open with probability g(r) (Fig. ), inepenent of its former states. In reality, the farther two noes are apart, the more ifficult for a successful communication. Moveover, when r > r, there exists no link. Thus, it is reasonable to assume that g(r) is a monotone ecreasing function an g(r) = whenever r > r. Further, we place another constraint on g(r), i.e., > g() g(r) g(r ) >, r r. (3) As reaers will see, constraint (3) is use to ensure that the expecte traversing elay over each possible link is boune above. 3 P is the probability of a event. E(x) is the expectation of ranom variable x Base on the above iscussion, the network at each time slot t can be represente by a RCM G t (, r, g(r)). Here, we use subscript t to inicate that the network is ynamic. Note that if > c (g(r)), G t (, r, g(r)) is percolate for all t (we also say the network has instantaneous connectivity); while if < c (g(r)), G t (, r, g(r)) is not percolate for all t. Thus, the instantaneous critical ensity I = c (g(r)). Next, we introuce the concept of the long-term connectivity. We first construct a new graph. The location of all noes in this graph is the same as that in G t (, r, g(r)). Two noes x an y share a link in this graph if an only if there exist t, such that x an y share an open link in G t (, r, g(r)). Note that x an y has the potential to share a link in G t (, r, g(r)) for some t, whenever g(r) > (equivalently, r < r ). Thus this new geometric graph can be represente by a RCM G(, r, f(r)) (it can be also represente by Poisson Boolean Moel B(, r )). Here, f(r) = when r < r, an f(r) = when r > r (Fig. ). We say the wireless network has longterm connectivity if an only if G(, r, f(r)) is percolate, an the critical ensity L = c (f(r)) is efine as the longterm critical ensity. Note that the long-term connectivity graph G(, r, f(r)) is a super-graph of all the instantaneous connectivity graph G t (, r, g(r)). Thus, whenever G t (, r, g(r)) is percolate, G(, r, f(r)) is percolate. Base on this observation, it is easy to know that L I. Since the prerequisite for communication in large-scale wireless network is connectivity, it is enough to only focus on the case > L. ) Moeling Delay in Large-Scale Wireless Networks: The efinition of elay of large-scale network is base on the First Passage Percolation []. Specifically, given a Ranom Connection Moel G(, r, h(r)), we attach each link e of G(, r, h(r)) a ranom variable T c (e), representing the time neee to pass through the link e. Consier a path π, the passage time is efine as T p (π) = e π(t c (e)). We then efine the first-passage time T (x, y) for any two noes x an y (x, y are not necessarily ajacent) as the minimum elay among all possible routes, i.e., T (x, y) = inf {T p (π) : π is a path from x to y}. () We now give the etaile efinition of T c (e). Generally, the time neee to cross the link e is compose of two parts. The first part is calle waiting elay, which is cause by the unreliability of this link. The secon part is calle propagation

4 elay 5, which is the time require for a packet to transverse the link e whenever the link is on. We assume that the uration of a time slot is long enough for a packet to be successfully transmitte over a link. Equivalently, the propagation elay over a link is smaller than time slot. For ease of analysis, we assume that all active links can transmit simultaneously with the same propagation elay, where < time slot. Base on the above iscussion, we can express the crossing time T c (e) as a geometrically istribute ranom variable. Specifically, assume that the length of the link e is < r < r, then we have P(T c (e) = k + ) = ( g(r)) k g(r). (5) Now we are reay to introuce the elay-istance ratio γ(). Consier two noes x, y C(G(, r, f(r))). Previous works [][3][] have prove that, the two its T (x,y) E(T (x,y) (x,y) an (x,y)) (x,y) (x,y) exist almost surely. Furthermore, T (x, y) (x,y) (x, y) = (x,y) E(T (x, y)), a.s. () (x, y) We enote the above it by γ(). γ() characterizes the first passage elay T (x, y) with respect to the istance (x, y). We nee to emphasize that γ() epens on. In fact, previous results also show that if = an > I, γ =. Otherwise, γ >. However, none of the previous works stuies the exact relationship between γ() an, especially, how γ() varies with respect to in the region ( L, I ]. In this paper, we will give a more etaile quantification of γ(). C. Useful Notations Some useful notations are liste as follows. (Section II-A) G(, r, h(r)) is a Ranom Connection Moel, an h(r) is the connection function; B(, r) is the Poisson Boolean Moel; we use C(G(, r, h(r))) an C(B(, r)) to represent the giant component of G(, r, h(r)) an B(, r) respectively. (Section II-B) G t (, r, g(r)) is the instantaneous geometric graph at time slot t an its critical ensity is I ; G(, r, f(r)) is the long-term geometric graph an its critical ensity is L. P( ) represents the probability of some event; E( ) represents the expectation of a ranom variable; z x (z y ) represents the x(y)-coorinate of z; (u, v) = u v is the Eucliean istance between noe u an v. (Section IV-B) H(z, a) is a circular region efine as H(z, a) = {z = (z x, z y ) R z z < a}. The ranom variable S g,t,u () is efine as S g,t,u () = sup{a noe v H c (u, a), v an u are connecte in G t (, r, g(r))}. In fact, all S g,t,u () s (inexe by t an u) have the same istribution. Thus, we write the expectation of S g,t,u () as E(S g ()) for short. 5 To be precise, this propagation elay inclues both the processing elay at the sening noe, an the propagation elay along the link Such a giant component exists with probability, because we focus on the case > L in this paper. (Section II-B) T c (e) is the passage time for a link e; T p (π) is the passage time for a path π; T (x, y) is the first passage time from noe x to y; (Section IV-B) T p (Π) is the passage time for a cluster to cluster transmission process Π; (Section IV-A) N ((u, v)) is the minimum number of hops from noe u to v. π represents a path; Π represents a cluster to cluster transmission process. III. MAIN RESULTS In this section, we first give two properties on the elayistance ratio γ(). After that, we present our main results concerning the relationship between noe ensity an γ(), in which an upper boun an a lower boun for γ(), are given. A. Properties of γ() γ() can be viewe as a function mapping from ( L, ) to R. The properties of γ() are liste below. Theorem. γ() has the following two properties: ) if =, then > I, γ() = ; ) γ() is a monotone ecreasing function. Proof: The first property has alreay been prove by previous literatures [][][3][7]. Thus, we only present the proof of the property () here. Given >, consier two Ranom Connection Moels G t (, r, g(r)) an G t (, r, g(r)). We use coupling technique to prove γ( ) γ( ). Noes in G t (, r, g(r)) an G t (, r, g(r)) are istribute accoring to Poisson Point Processes, enote by Γ an Γ, with noe ensities an, respectively. Note that Γ can be viewe as the superposition of Γ an another Poisson Point Process Γ with noe ensity. Consier noes x, y Γ, since Γ Γ, we obtain x, y Γ. For any path π connecting x an y in G t (, r, g(r)), this path also exists in G t (, r, g(r)). By the efinition of T (x, y), we must have T (x, y) T (x, y). Divie the above inequality by (x, y), an let (x, y), we obtain B. Main results on γ() γ( ) γ( ). We have obtaine several properties of γ(). Now we are reay to present our main results.

5 5 Theorem. Given a RCM G t (, r, g(r)) with L ( + ϵ) < I (ϵ > ) an =, the corresponing γ() satisfies E(S g () + r ) γ() κ ϵ inf [ L (+ϵ),] L where κ ϵ is a constant inepenent of. ) g (r,(7) L (+ϵ) Theorem ignores the propagation elay. If we take propagation elay into consieration, we have the following results. Theorem 3. Given a RCM G t (, r, g(r)) with L (+ϵ) (ϵ > ) an < <. Then the corresponing γ() satisfies E(min{S g (), r γ() }) + r inf [ L (+ϵ),] where κ ϵ is a constant inepenent of. κ ϵ / L ) () g (r L (+ϵ) We nee to emphasize that the two κ ϵ s in Theorem an Theorem 3 are the same. The rigorous efinition of κ ϵ is postpone to Section IV-A (see Lemma 3). Our results provie a way to estimate elay in Large-scale wireless multi-hop networks. Our result is also very general. By changing the connection function g(r), our results can be easily applie to ifferent scenarios in real networks. IV. UPPER AND LOWER BOUNDS OF γ() In this section, we first give an upper boun to the elayistance ratio, γ(). Then, we make further analysis on first passage elay an introuce a concept calle cluster to cluster transmission. Using this concept, we erive a lower boun. Finally, we take propagation elay into consieration, an formulate its impact on γ(). A. Upper Boun of γ() Recall the efinition of γ() (Eqn. ) that γ() = T (x,y) (x,y) (x,y), where x, y belongs to the giant component of G(, r, f(r)). To compute such a it, we o not have to calculate γ() for all x, y C(G(, r, f(r))). The correctness of this assertion is assure by the following lemma. Lemma. Given a convergent sequence {x k }, k =,,..., an k x k = x. {y k }, k =,,..., is a subsequence of {x k }, an k y k = y. Then x = y. It is easy to see that the number of noes in C(G(, r, f(r))) is countable. We enumerate for all noes. We ranomly select a noe an label it as x, an then label other noes accoring to the istance from x (larger subscript means larger istance from x ). Define sequence {m k, k =,,..., }, m k = T (x,x k ) (x,x k ), then k m k = γ(). Accoring to lemma, we only nee to fin a subset of noes of C(G(, r, f(r))) (the carinality of this subset must be infinity), an calculate γ() from this subset. This technique is use in eriving the upper boun. Now we are to obtain the upper boun of γ(). The elay T (x, y) is efine as the minimum elay along all paths connecting noes x an y. Thus, it must be smaller than or equal to the elay along one specific path. In this part, we first fin a subset of noes of C(G(, r, f(r))). Then, we fin a path for each noe pair in this subset. After that, we calculate the elay along this path. Finally, iviing the elay by istance, we obtain an upper boun of γ(). Before proceeing, we nee the following lemma. Lemma. Consier Poisson Boolean moels in R. Let c (r) enote the critical ensity in the case where the transmission range is r. Then it is the case that where r, r >. c (r )r = c (r )r, Proof: See Proposition. in [5]. The long-term critical ensity L is also the critical ensity of Poisson Boolean Moel with transmission range r. Consier the network with ensity > L, accoring to lemma, we immeiately know that when r > L r, i.e., r > L r, Poisson Boolean Moel B(, r) is percolate. Let r = r L (+ϵ), then B(, r) is percolate. Further, since L ( + ϵ), we must have r r. Note that, the Ranom Connection Moel G(, r, f(r)) is essentially a Poisson Boolean Moel B(, r ) with r r. Hence, B(, r) is a subgraph of G(, r, f(r)). (Here, B(, r) has the same noe locations as G(, r, f(r)).) We enote the giant component of B(, r) by C(B(, r)). Accoring to the uniqueness of giant component in supercritical case, there must be C(B(, r)) C(G(, r, f(r))). Accoring to lemma, when calculating γ(), we only nee to focus on the case that both noes belong to C(B(, r)). Assume that noes u, v C(B(, r)). Then there exists at least one path in B(, r) from u to v. We choose the path with minimum number of hops, an enote it by π m. Up to now, we have foun a path connecting u an v. Next, we are to calculate the elay along this path. To start with, we nee to compute the number of hops, enote by N ((u, v)), in π m. We can fin a relationship between N ((u, v)) an using the following scaling arguments. Scale the network B(, r) up by L, then the network B(, r) becomes another network B( L, r + ϵ). Further, the original istance (u, v) between u an v becomes (u, v) L. Then to compute N ((u, v)), it is equivalent to compute N L ((u, v) L ). Next, we present the lemma concerning N L (). Lemma 3. Given B( L, r + ϵ), an u, v C(B( L, r + ϵ)), the minimal number of hops among all paths from u to v is N L ((u, v)). Then there exist κ ϵ such that N L ((u, v)) = κ ϵ. (u,v) (u, v)

6 Lemma 3 gives the rigorous efinition of κ ϵ, which is a useful parameter in Theorem an Theorem 3. The proof of Lemma 3 is base on a conclusion on subaitivity an is given in section V-A. Accoring to lemma 3, we immeiately get N ((u, v)) (u,v) (u, v) N L ((u, v) = (u,v) (u, v) L ) = κ ϵ L. (9) Then we calculate the elay T p (π m ) along path π m. Accoring to the Strong Law of Large Numbers, with probability, we have (u,v) Therefore, T p (π m ) N ((u, v)) = (u,v) e π m T c (e) N ((u, v)) = E[T c(e)]. γ() = T (u, v) (u, v) T p (π m ) (u, v) = T p (π m ) (u,v) N ((u, v)) N ((u, v)) (u, v) = κ ϵ L E[T c (e)]. () From the efinition of the path π m, we know that the length of each hop is smaller than r = r L (+ϵ). Besies, the connection function g(r) is monotone ecreasing. Thus, for a link e whose length is r, there must be E[T c (e)] E[T c (e )] = kp(t c (e ) = k) = k= k= k( g( r)) k g( r) =. () g( r) Thus, ( ) γ() = κ ϵ E[T c (e)] κ ϵ L L g( r) = κ ϵ L g (r L (+ϵ) ). Furthermore, from property () of theorem, we know γ() is a monotone ecreasing function. Thus, γ() inf κ ( ) ). [ L (+ϵ),] L g (r L (+ϵ) B. Lower Boun of γ() ) Cluster to Cluster Transmission: In section IV-A, we have obtaine the upper boun of γ() by calculating the elay along one path. However, the metho use in section IV-A cannot be use to stuy the lower boun of γ(). It is har to obtain the number of paths connecting two noes. The elay along each path is ranom an may correlate with that of other paths. Therefore, it is usually intractable to obtain the elay from Eqn. (). To overcome this ifficulty, we introuce the concept of cluster to cluster transmission. Consier the packet transmission process from noe u to noe v. Assume that at time slot t, noe u (u = u) transmits a packet to other noes. Since we have ignore the propagation elay, all noes connecte to u in G t (, r, g(r)), enote by W, receive the information instantaneously. Then, the transmission process stops. This is because all outgoing connections from W are unavailable at this time instance. The transmission process will restart at time slot t > t 7, when at least one noe in W fin the opportunity to forwar the packet to a new noe, enote by u. At this time slot, u transmits the packet to a set of new noes (not in W ) which are connecte to u in G t (, r, g(r)), enote by W, instantaneously. This process goes on, until at time slot t M, noe u M an the estination noe v are in the same connecte cluster an the packet is transmitte to noe v instantaneously. We can see that the cluster to cluster transmission as a series of outbursts. During each outburst, a set of new noes receive the packet. W k, k =,,..., M is the set of noes which receive the information right at the kth outburst. u k W k is the starting noe at the kth outburst. It is possible that at time t k, two noes u k an u k fin transmission opportunity simultaneously. Choosing ifferent starting noes will lea to ifferent cluster to cluster transmission processes. A cluster to cluster transmission process can be fully escribe by Π = {(t, u ), (t, u ),..., (t M, u M )}. Packets can be transmitte from u to v through Π. Define the passage time for the cluster to cluster transmission process Π as T p (Π) = t M t. Then, we have the following lemma. Lemma. Given noes u, v C(G(, r, f(r))), the first passage time T (u, v) = inf{t p (Π) Π is a cluster to cluster transmission process from u to v}. () Proof: For convenience, we use L to enote the set of all the cluster to cluster transmission processes from u to v. Then Eqn.() can be rewritten as T (u, v) = inf{t p (Π) Π L }. It is easy to see that for each cluster to cluster transmission process Π from u to v, Thus, Next, we show that T p (Π) T (u, v). inf{t p (Π) Π L } T (u, v). (3) inf{t p (Π) Π L } T (u, v). 7 Here, we o not require t to be the smallest, i.e., there may exist t < t < t, such that at least one noe in w have the opportunity to forwar the information to a new noe at time slot t.

7 7 Recall the efinition of T (u, v), i.e., T (u, v) = inf {T p (π) : π is a path from u to v}. Let π be the path with minimum elay from u to v, we prove that there exists a cluster to cluster transmission process Π, such that T p (Π ) = T p (π ). Assume that π = i i i...i K (i = u, i K = v). At time slot t, some noes in path π may be in the same connecte cluster as i. Let i η be the noe attainable from i with largest subinex, then the link between i η an i η must be off at this time slot. Let t > t be the first time slot that this link is on. At time slot t, let i η be the noe attainable from i η with largest subinex. Then the link between i η an i η must be off until time slot t 3 > t...at time slot t k, noe i ηk an estination noe v are in the same connecte cluster an the information transmit to v instantaneously. We enote by Π this cluster to cluster transmission process. An Π = {(t, i ), (t, i η ),..., (t k, i ηk )}. From the construction of Π, it is obvious that T p (Π ) = T p (π ). Thus, T (u, v) = T p (π ) = T p (Π ) inf{t p (Π) Π L }. () Combining Eqn.() an Eqn.(), we obtain T (u, v) = inf{t p (Π) Π L }. Base on the cluster to cluster transmission process, we are now reay to compute a lower boun of γ(). ) Compute the Lower Boun: In this section, we use the concept of cluster to cluster transmission to erive a lower boun of γ(). To start with, we nee to introuce a ranom variable S g,t,u () (g is the connection function, u is a noe) to represent the raius of the connecte cluster W ({u}) in the instantaneous geometric ranom graph G t (, r, g(r)). We establish a cartesian coorinate in R. We efine H(z, a) as H(z, a) = {z = (z x, z y ) R z z < a}. The ranom variable S g,t,u () is efine as S g,t,u () = sup{a noe v H(u, a), v u at time slot t}. Accoring to the spacial stationarity an time inepenence of our ynamic RCM G t (, r, g(r)), the istribution of S g,t,u () is inepenent of t an u. Thus, we can write S g,t,u () as S g () if causing no confusion. Now, given two noes u (source) an v (estination), consier a cluster to cluster transmission process(fig. ) Π = {(u (), t ), (u (), t ),..., (u (M), t M )}, where u () = u, an u (M), v are in the same connecte cluster at the time instance t M. Then the elay along this cluster to cluster transmission process is T p (Π) = t M t = (t k+ t k ) M. M k= Fig.. S S ()( ) gt,, u ()( ) gt,, u r r r Illustration of a cluster to cluster transmission process. S ( M) ( ) gt, M, u Note that, k =,,..., M, u (k+) is connecte to a noe in W k, enote by u. Then, u (k+) u (k) u (k+) u + u u (k) As for k = M, S g,tk,u (k)() + r. v x u (M) S g,tm,u (M)(). Combining the above two inequalities together, we obtain, (u, v) = v u < M k= M k= u (k+) u (k) + v u (M) (S g,tk,u (k)() + r ) + S g,tm,u (M) x () M (S g,tk,u (k)() + r ). (5) k= For each k, it is easy to check that S g,tk,u(k)() amits the same istribution as another ranom variable S g (). Further, we can prove the following lemma. Lemma 5. With probability, we have M k= S g,t k,u (k)() = E(S g ()). () M M We nee to emphasize that Lemma 5 cannot be obtaine irectly from the Strong Law of Large Number, because S g,tk,u(k)() s may not be inepenent for ifferent k s. However, we can show that the ranom process {S g,tk,u (k)()} k is in fact ergoic. Then, accoring to Birkhoff Ergoic Theorem, the time average of S g,tk,u(k)() (left han sie of Eqn. ()) will be equal to the spacial average of S g,tk,u(k)() (right han sie of Eqn. ()). See Section V-E for the etaile proof. Accoring to Lemma 5, ϵ >, M ϵ, such that M > M ϵ (this conition is satisfie for large enough (u, v)), we have M k= (S g,t k,u (k)() + r ) < E(S g () + r ) + ϵ. M Combine with Eqn. (5), we have (u, v) < M(E(S g () + r ) + ϵ ).

8 Then T p (Π) M > (u, v) E(S g () + r ) + ϵ. Note that the right han sie of the above inequality oes not epen on the selection of the cluster to cluster transmission processes. Thus, Therefore, T (u, v) (u, v) E(S g () + r ) + ϵ. γ() = T (u, v) (u,v) (u, v) E(S g () + r ) + ϵ (7) Let ϵ, we finally obtain γ() C. Impact of Propagation Delay E(S g () + r ). The elay in large-scale Wireless Networks is compose of two parts, i.e., the waiting elay an the propagation elay. In previous sections, we ignore the propagation elay. Now, we will consier its impact on γ(). The propagation elay will not have much influence on γ() when is relatively small, because the first passage elay is ominate by the waiting elay. However, when is relatively large, e.g., > I, the network connectivity is maintaine at each time slot, an thus the propagation elay will increase γ() substantially (γ() is now boune below away from ). The proof of Theorem 3 is in fact a generalization of that of Theorem. Proof: We first consier the upper boun. We have alreay obtaine Eqn. (), i.e., γ() κ ϵ L E[T c (e)], where the length of link e is smaller than r = r L (+ϵ). Using similar metho in eriving Eqn. (), we obtain E[T c (e)] E[T c (e )] = (k + )P(T c (e ) = k) < = k= (k + )( g( r)) k g( r) = g( r) k= g (r L (+ϵ) ) () where e is link whose length is r. The inequality above is slightly ifferent from Eqn. (). This is because we have taken propagation elay into consieration. Thus, γ() κ ϵ ). L g (r L (+ϵ) Note that γ() is a monotone ecreasing function, thus γ() inf κ ϵ ). (9) [ L (+ϵ),] L g (r L (+ϵ) Then we consier the lower boun. Similar to the previous part, we still focus on the cluster to cluster transmission. Consier a cluster to cluster transmission process Π = {(u (), t ), (u (), t ),..., (u (M), t M )}. Similarly, we have, k =,,..., M, As for k = M, u (k+) u (k) S g,tk,u (k)() + r. v u (M) S g,tm () < S,u (M) g,tm,u (M)() + r. x Besies, the istance transmitte is also ite by the finite hops in one time slot. Since each hop takes fraction of a time slot, then a packet can transmit at most hops in one time slot. As a result, the longest istance transmitte in one time slot is upper boune by r. Then k =,,..., M, u (k+) u (k) r + r. As for k = M, v u (M) r < r + r. Integrating the above four inequalities, we obtain, k =,,..., M, u (k+) u (k) min{s g,tk,u (k)(), r } + r. As for k = M, v u (M) min{s g,tm,u (M)(), r } + r. Again, using the metho in section IV-B, we immeiately obtain γ() E(min{S g (), r }) + r. A. Proof of Lemma 3 V. PROOF The metho use in this proof is similar to that use by Dousse et al. in [] an Kong et al. in []. We first construct a cartesian coorinate system. Without loss of generality, we assume that there is a noe at the origin. Let z n = arg min z C(B(L,r +ϵ)) {(z, (, n))}. We efine N L (m, n) = N L ((z n, z m )). We first prove that N L (, m) scales linearly with respect to m (Lemma ). As we will see later, Lemma 3 follows irectly from Lemma. Lemma. There exists κ ϵ, such that N L (, m) = κ ϵ. m m To prove Lemma, we use Liggett s subaitive ergoic theorem, which is formally restate below.

9 9 Lemma 7. (Liggett [])Let {S l,m } be a collection of ranom variables inexe by integers l m. Suppose {S l,m } has the following properties: ) S,m S,l + S l,m, l m; ) {S (m )k,mk, m } is a stationary process for each k; 3) {S l,l+k, k } = {S l+,l+k+, k } in istribution for each l; ) E[ S,m ] < for each m. E[S Then α,m ] E[S m m = inf,m ] m m ; S S,m m m exists with probability an E[S] = α. Furthermore, if 5. the stationary process {S (m )k,mk, m } is ergoic; then S = α with probability. It is easy to see that N L (, m) N L (, l) + N L (l, m)( l m). Then the first conition of lemma 7 is satisfie. Since Poisson Boolean Moel B( L, r + ϵ)) is homogeneous, the secon an the thir conitions of lemma 7 are also satisfie. Now, we only nee to prove that conitions an 5 are also satisfie. Lemma. E(N L (, m)) <. Note that E(N L (, m)) = i= P(N L (, m) i). If we can show that P(N L (, m) i) ecays exponentially with respect to i, then Lemma follows irectly. The basic iea is to fin a sequence of toruses circling noes z an z m using noes in the giant component of B( L, r + ϵ)). Then, we can show that the least number of hops N L (, m) from z to z m cannot be arbitrarily large (see Fig. 5). Due to the supercriticality of B( L, r + ϵ)), we can show that such toruses exist with probability close to. Therefore, the probability that N L (, m) is greater than a large threshol is close to. The etaile proof is provie in Section V-B. Then conition of lemma 7 is satisfie. Next, we prove that N L (m, n) satisfies conition 5. To emonstrate that N L (mk, (m + )k) is ergoic, we show that it is strong mixing, which is a stronger property. Lemma 9. N L (mk, (m + )k) is strong mixing. The strong mixing property essentially states that the epenence between N L (mk, (m+)k) an N L ((m+n)k, (m+ n + )k) becomes negligibly small as n becomes large. The strategy to prove this statement resembles some parts of the proof of Lemma. We first fin two toruses circling the shortest path from z mk to z (m+)k an the shortest path from z (m+n)k to z (m+n+)k, respectively. By the stationarity of Poisson Point Process, we know that the size of the two toruses o not epen on n. Hence, as n increases, the two toruses will become isjoint eventually. By the spatial inepenence of Poisson Point Process, we can see that N L (mk, (m + )k) an N L ((m + n)k, (m + n + )k) becomes approximately inepenent when n is large enough. The etaile proof is provie in Section V-C. Now, we have prove that N L (m, n) satisfy conitions 5 of Lemma 7. Thus, there exists κ ϵ, such that N L (, m) = κ ϵ. m m Before proving Lemma 3, we also nee the following lemma. Lemma. (z n, (, n)) < with probability. Intuitively, if (z n, (, n)) =, there oes not exist any giant component in B( L, r + ϵ)) (otherwise, (z n, (, n)) < ). This contraict to the supercriticality of B( L, r + ϵ)). The etaile proof is presente in section V-D. Now, we are reay to prove Lemma 3. Proof: Consier N L ((u, v)). Without loss of generality, we suppose that u is at the origin, v is at the +x axis. Assume that integer n satisfy n (u, v) < n +, then an N L ((u, v)) N L ((, z n )) + N L ((z n, v)), () N L ((u, v)) N L ((, z n )) N L ((z n, v)). () Note that (z n, v) (z n, (, n )) + ((, n ), v) (z n, (, n )) + <. Using similar metho in the proof of Lemma, we can prove E(N L ((z n, v))) < (to avoi verbosity, we o not elaborate it here). Therefore, N L ((z n, v)) < with probability. Further, by the stationarity of Poisson Point Process, N L ((z n, v)) oes not epen on (u, v). Then, ivie Eqn. () an Eqn. () by (u, v), an let (u, v). We immeiately obtain B. Proof of Lemma N L ((u, v)) = κ ϵ. (u,v) (u, v) Proof: Consier N L (, m). Let z = z +z m. We raw a series of squares centering at z (Fig. 3), an the sie lengths are,,,..., k,... z Fig. 3. A series of squares centering at z. We use R() to enote the rectangle with sie lengths an. We say R() is goo if an only if there exists a crossing connecting the two short sies in R() (Fig..(a)). We enote

10 the event that R() is goo by A R (). Note that the Poisson Boolean Moel B( L, r + ϵ) is percolate. By Corollary. in [5], we have P(A R()) =. () [z y k, z y+ k ], it must intersect with the circuit in C( k )(Fig 5). We can replace the part of the path ACB with ADB, then the resulting path is shorter. Thus, the shortest path from z to z m must be containe in [z x k, z x + k ] [z y k, z y + k ]. (a) Illustration of a goo rectangle R(). (b) Illustration of a goo square torus C(). Here, z = (z x, z y). Fig.. Illustration of a goo rectangle an a goo square torus. We use C() to enote the square torus ([z x, z x + ] [z y, z y + ]) \ ([z x, z x + ] [z y, z y + ]). We say C() is goo if an only if there exists a circuit in C() (Fig..(b)). We enote the event that C() is goo by A C (). From Fig..(b), we can see that C() is compose of four rectangles, i.e., R () = [z x, z x ] [z y, z y + ], R () = [z x+, x+] [z y, z y+], R z 3 () = [z x, z x+ ] [z y +, z y +], R () = [z x, z x +] [z y, z y ]. An we use A Ri ()(i =,, 3, ) to represent the event that R i () is goo. Obviously, if i =,, 3,, R i () is goo, C() must be goo. Thus, P(A C ()) P(A R () A R () A R3 () A R ()). Note that i =,, 3,, A Ri () is an increasing event. Accoring to the FKG Inequality (Theorem. in []), we have P(A C ()) P(A R () A R () A R3 () A R ()) P(A R ())P(A R ())P(A R3 ())P(A R ()) = P(A R ()). (3) Combine Eqn. () with Eqn. (3), an we obtain P(A C()) =. Thus, < ρ <, there exists ρ, such that,, P(A C ()) ρ. Let k = min{k k z m z, k ρ }, then for all k k, P(A C ( k )) ρ. Assume that C( k )(k k ) is goo, if the shortest path from z to z m is not containe in the square [z x k, z x+ k ] A ranom variable N is on the measurable pair (Ω, F ) is calle increasing if N(ω) N(ω ) whenever ω ω. Fig. 5. The contraiction if the shortest path from z to z m is not containe in [z x k, z x + k ] [z y k, z y + k ]. Suppose u, v, w are three consecutive noes along this shortest path. Then u w > ( + ϵ)r, or we can einate noe v, an get a shorter path. This also inicates that if we raw isks with raius (+ϵ)r centering at u an w respectively, the two isks are isjoint. Assume the number of hops of the shortest path is N, then we can raw L isjoint isks in total. An these isks are all locate in a square with sie length k+ + ( + ϵ)r. Thus, Then, L π(( + ϵ)r ) ( k+ + ( + ϵ)r ). L (k+ + ( + ϵ)r ) π(( + ϵ)r ). Note that N L (, m) is the minimum number of hops from x to x m, thus N L (, m) L (k+ + ( + ϵ)r ) π(( + ϵ)r ). Now, if N L (, m) > (k+ +(+ϵ)r ) π((+ϵ)r ), then none of C( k+ ), C( k+ ),..., C( k ) is goo. Thus, P(N L (, m) > (k+ + ( + ϵ)r ) π(( + ϵ)r ) ) k P(A c C( i )) ( ρ) k k. i=k + Let l k = (k+ +(+ϵ)r ) π((+ϵ)r ), then,

11 E(N L (, m)) = P(N L (, m) i) i= l k = P(N L (, m) i) + i= l j+ j=k i=l j + P(N L (, m) i) l k + P(N L (, m) > l j ) (l j+ l j ) j=k < l k + ( ρ) j k (j+ + ( + ϵ)r ) π(( + ϵ)r ) j=k <. C. Proof of Lemma 9 Proof: In previous analysis, we have prove that P(A C ( k )) ρ whenever k k. Summing over k yiels P(A C ( k )) = ρ =. () k=k k=k Since A C ( k ), k = k, k +,... are inepenent events, accoring to the Borel-Cantelli Theorem, with probability, there exist k < 9, such that A C ( k ) occurs. We now construct squares B an B centere at x mk +x (m+)k an x (m+n)k+x (m+n+)k with sie length k + k + an respectively, such that the path with minimum number of hops from x mk to x (m+)k, an the path with minimum number of hops from x (m+n)k to x (m+n+)k are containe in B an B. Due to the stationarity, k oes not rely on n. Besies, k an k are all finite(we enote this event by A f ) with probability. Thus, when n is large enough, B an B are isjointe. Hence, N L (mk, (m+)k) an N L ((m+n)k, (m+n+)k) become inepenent. Therefore, P({N n L (mk, (m + )k) < i} {N L ((m +n)k, (m + n + )k) < i}) = n P({N L (mk, (m + )k) < i} {N L ((m +n)k, (m + n + )k) < i} A f )P(A f ) + n P({N L (mk, (m + )k) < i} {N L ((m +n)k, (m + n + )k) < i} A c f )P(A c f ) = P({N L (mk, (m + )k) < i} A f ) P({N L ((m +n)k, (m + n + )k) < i} A f ) = P({N L (mk, (m + )k) < i}) P({N L ((m+ n)k, (m + n + )k) < i}). Then, N L (mk, (m + )k) is strong mixing. 9 Actually, the number of such k is infinite. D. Proof of Lemma Proof: Similar to that in the proof of lemma, we construct a series of squares centere at (, n) with sie length,,,..., k,...(fig. 3). C() is efine similarly. We say C() is perfect if an only if C() is goo an the circuit in C() belongs to the giant component. Again, we apply Borel-Cantelli Theorem to Eqn. (). With probability, there exists {k }, such that event A C ( k ) occurs. Then, (z n, (, n)) < k < with probability. E. Proof of Lemma 5 Proof: Accoring to Birkhoff Ergoic Theorem, we only nee to show that {S g,tk,u (k)()} k is ergoic. In fact, we will show that {S g,tk,u (k)()} k is strong mixing, an thus ergoic. Consier S g,tk,u (k)() an S g,t k+n,u(k+n)(). We only nee to show that for any constants s, s >, P({S g,t n k,u (k)() < s } {S g,tk+n,u (k+n)() < s }) = P(S g,tk,u (k)() < s )P(S g,tk+n,u (k+n)() < s ). (5) Note that all S g,tk,u(k)() s are equal in istribution to a ranom variable S g (), an E(S g ()) < ue to the subcriticality of RCM G t (, r, g(r)). Therefore, for any fixe ϵ >, there exists M ϵ, such that P(S g () > M ϵ ) < ϵ. We efine Ŝg,t k,u (k)() = min{s g,t k,u (k)(), M ϵ} for k =,,... Then, Ŝ g,tk,u(k)() only epens on the realization of noes an links in the circler region H(u (k), M ϵ ). Consier the istance (u (k), u (k+n) ) between noe u (k) an u (k+n). By the construction of u (k) s, we know that as n increases to infinity, (u (k), u (k+n) ) also increases to infinity. We choose N >, such that for all n > N, (u (k), u (k+n) ) > M ϵ. Then H(u (k), M ϵ ) an H(u (k+n), M ϵ ) will become isjoint. By the spacial inepenence of Poisson Point Process, we then obtain that for any n > N, P({Ŝg,t k,u (k)() < s } {Ŝg,t k+n,u (k+n)() < s }) = P(Ŝg,t k,u (k)() < s )P(Ŝg,t k+n,u (k+n)() < s ).() From the construction of Ŝg,t k,u(k)(), we know that for s M ϵ, P(S g,tk,u (k)() < s) = P(Ŝg,t k,u(k)() < s); while for s > M ϵ, P(M ϵ S g,tk,u(k)() < s) ϵ. Then, using Eqn. (), it is easy to show that P({S g,t n k,u (k)() < s } {S g,tk+n,u (k+n)() < s }) P(S g,tk,u (k)() < s )P(S g,tk+n,u (k+n)() < s ) < ϵ. Let ϵ, we then obtain Eqn. (5).

12 VI. DISCUSSION In this section, we make simulations to uphol our theoretical results. First, we give a further iscussion on some parameters in our expressions. Then, enormous simulations are one to justify several assertions in this paper. Our theoretical results are base on a relatively general moel, Ranom Connection Moel. Many Network Moels can be converte to a Ranom Connection Moel, making our results applicable to many ifferent cases. The ifference is that the connection functions are ifferent in ifferent cases. In the following iscussion, we simply let r =, an the connection function g be efine as { g(r) = ( r) : r : r > Moreover, if we take propagation elay into consieration, we let =.. A. Discussion on Several Parameters In our expression of theoretical bouns Eqn. (7), two terms κ ϵ an E(S g ()) are applie. Besies, In Eqn. (), E(min{S g (), r }) is applie. κ ϵ is a constant, an we simply obtain its value through simulation; while E(S g ()) an E(min{S g (), r }) are functions epening on, an we fin two analytical expressions to approximate them. We first focus on κ ϵ (efine in lemma 3). Here, we set ϵ =.. In our simulation, we simulate 3 points in a region. The noe ensity is = L., an the transmission range is r + ϵ.. A packet is originate from a noe locate at the center of the region, we recor own the minimum number of hops an the istance from the source for each noe, an plot it in Fig..(a). N () (a) Illustration of the relationship between N () an (b) The probability istribution graph of N () Fig.. Simulation results on κ. The first Figure reveals the linear relationship between N () an, an the secon inicates κ ϵ.753. From Fig..(a), we can see that N () grows linearly with. To fin κ, we calculate N () for each noe, an present its probability istribution graph in Fig..(b). It can be seen from Fig..(b) that the probability N () =.753 is the largest. Thus, κ ϵ.753. Next, we turn to E(S g ()). The physical meaning of E(S g ()) is the average size of the connecte component intersecte with the origin. It is obvious that when =, E(S g ()) =. An when = I, E(S g ()) = since the network is percolate in this case. Thus, we give a conjecture about the analytical expression of E(S g ()), i.e., E(S g ()) = c I. (7) We make enormous numerical computations to fin the experimental values of E(S g ()) with respect to ifferent, ranging from.5 ( L ( + ϵ) <.5) to.. We then rewrite Eqn. (7) as E(S g ()) = I c. c Using the least square fitting, we can easily obtain c., I.. Then we make a comparison between the fitting value an the experimental value of E(S g ()). From Fig. 7, It can be seen that there is a goo agreement between fitting an experimental results. Fig. 7. E(S g ()) Experimental Result Fitting Result Comparison between experimental an fitting value of E(S g ()). Now, we come to E(min{S g (), r }). Obviously, E(min{S g (), r }) = whenever =. Besies, E(min{S g (), r }) r,, an E(min{S g (), r }) is monotone increasing with. Thus, we conjecture that the analytical expression of E(min{S g (), r }) has the format E(min{S g (), r }) = c c 3 +. () Similarly, we conuct numerical computations to fin the experimental values of E(min{S g (), r }) with respect to ifferent, ranging from. to. We rewrite Eqn. () as E(min{S g (), r }) = c 3 c +. c Using the least square fitting, we can easily obtain c.5, c 3.3. Then we make a comparison between the fitting value an the experimental value of E(min{S g (), r }). From Fig., It can be seen that there is a goo agreement between fitting an experimental results. B. Comparison between Two Bouns This paper is originate from the iea that the elay-istance ratio γ() may epen on the noe ensity. We calculate γ() uner ifferent noe ensities. From Fig. 9, we can see T that γ(.) = ()., γ(.9).35, γ(.)

13 3 3. Experimental Result Fitting Result Expeimental Results Upper Boun Lower Boun. The experimental results get closer to the upper boun when is small γ() E(min{Sg(),r/}) T()/ T()/ 5 () =.5 Simulation results on ifferent.., γ(.5). This justifies the fact that γ() is monotone ecreasing with respect to. Now, we are reay to compare our theoretical bouns an the experimental values of γ(). We first ignore the propagation elay, Fig. shows the comparison between the experimental value an our theoretical value of both the upper boun an the lower boun. In our simulations, We work out the experimental values of γ() where is set to be evenly istribute in the interval of [L ( + ϵ), I ]. From Fig., we fin that the experimental values are right boune by both the upper an the lower bouns. Then, we take propagation elay into consieration. In the following part, we take =.. We first examine the effect of introucing propagation elay to the elay-istance ratio γ(). We consier two networks, whose noe ensities are.5 an. (.5 is near L,. is near I ) respectively. For each network, we make simulations to fin γ() in the cases = an =.. The result is show in Fig.. From Fig., we can also see that when is small, the influence of to γ() is small; an the influence becomes more significant as grows larger. This also inicate that when the noe ensity is small, elay is mainly cause by (c) =. (a) =.5, =, γ() (b) =.5, =., γ() Fig. 9. (b) =.9 T()/ T()/ (a) =.. Fig.. Comparison between upper boun an lower boun(propagation elay is ignore). T()/ T()/ T()/ I Fig.. Comparison between experimental an fitting value of E(min{Sg (), r }). 3 The experimantal results get closer to the lower boun when get closer to T()/. (c) =., =, γ() () =., =., γ().. Fig.. γ(). The influence of propagation elay to the elay-istance ratio the waiting elay (waiting elay is mainly cause by the loss of wireless connections); while when the noe ensity is large, elay is mainly cause by the propagation elay. Next, we compare our theoretical bouns an experimental results of γ() in the case =.. In our simulation, the noe ensities are chosen from [.5, ]. Since the change of γ() is larger when is small, we choose more simulation points for smaller noe ensity. The comparison between our theoretical bouns an experimental results is shown in Fig.. The experimental values are right boune by both the upper an the lower bouns. From Fig. an Fig., it can be seen that when is large, γ() is much closer to the lower boun. An explanation to this phenomena is that the larger the noe ensity is, the larger the size of clusters in the cluster to cluster transmission process. Larger cluster size provies more opportunity for forwaring packets. Thus, when the noe ensity is large enough, it is probably that the message can transmit again right at the next time slot. This makes our lower boun more accurate. However, our upper boun is a little bit loose. But

14 3 Experimental Results Upper Boun Lower Boun The Experimantal Results get closer to the upper boun when is small γ() The Experimantal Results get closer to the lower boun when is large Fig.. Comparison between upper boun an lower boun(propagation elay is consiere). we can see that γ() gets closer to the upper boun when the noe ensity ecreases. This is because our upper boun is obtaine from the elay of just one path. An the smaller is, the smaller the number of paths connecting two noes. This makes γ() get closer to the upper boun. VII. C ONCLUSION In this paper, we stuy the funamental relationship between γ() an using percolation theory. We point out that unreliability of wireless connections brings about waiting elay an analytically characterize the upper an lower boun for γ(). Then we take propagation elay into consieration, an obtain further results. Finally, through simulations base on the exact value of γ(), we further obtain a new observation that the lower boun serves as a goo approximation to the value of γ() in ense networks. An γ() gets closer to the upper boun when ecreases. Simulation results conform our theoretical finings. VIII. ACKNOWLEDGMENT This paper is supporte by National Funamental Research Grant (No. CB37); NSF China (No. 79); Shanghai Basic Research Key Project (No. JC5); China Ministry of Eucation Fok Ying Tung Fun (No. ); China Ministry of Eucation New Century Excellent Talent (No. NCET--5). R EFERENCES [] Next-Generation Wireless Mesh Networks: Combining a multi-raio architecture with high-performance routing to optimize vieo surveillance an other multimeia-grae applications, in Aruba Networks, Mar.. [online]. Available: gy/whitepapers/wp wirelessmesh.pf [] J.M.Hammersley an D.J.A.Welsh, First Passage Percolatin, Subaitive Processes, Stacastic Networks, an Generalize Renewal Theory, in Proc. Internat. Res. 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Lin, Mobility Increases the Connectivity of K-hop Clustere Wireless Networks, ACM Mobicom 9, Beijing, Sept. 9. [] D. Goeckel, B. Liu, D. Towsley, L. Wang an C. Westphal, Asymptotic connectivity properties of cooperative wireless a hoc network, ACM Mobihoc, May.. [9] O. Dousse, F. Baccelli an P. Thiran, Impact of Interferences on Connectivity in A Hoc Networks, in Proc. of IEEE Infocom, pp. 7733, Apr. 5. [] G. R. Grimmett, Percolaiton, Springer, 999. [] O. Dousse, P. Mannersalo an P. Thiran, Latency of Wireless Sensor Networks with Uncoorinate Power Saving Mechnisms, in Proc. ACM MobiHoc, pp. 9-, May.. [] Z. Kong, an E. M. Yeh. Connectivity an Information Dissemination in Large-Scale Wireless Networks with Dynamic Links, submitte to IEEE Trans. Inform. Theory,. [3] D. J. Daley an D. Vere-Jones, An Introuction to the Theory of Point Process, New York: Springer, 9. [] J.F.C. Kingman, Poisson Processes, Clarenon Press, Oxfor, 993. [5] R. Meester an R. Roy, Continuum Percolation, NewYork: Cambrige University Press, 99. [] T. Liggett, An Improve Subaitive Ergoic Theorem, Annals of Prob., vol. 3, pp. 79-5, 95. [7] W.Ren, Q.Zhao, an A.Swami, On the Connectivity an Multihop Delay of A Hoc Cognitive Raio Networks to appear in Proc. of IEEE International Conference on Communications(ICC), May,. [] A. Baeley, Spatial point processes an their applications, Lecture Notes in Mathematics, 9:-75, 7. [9] S. Zhao, L. Fu, X. Wang, Q. Zhang, Funamental Relationship between Noe Density an Delay in Wireless A Hoc Networks with Unreliable Links, ACM MobiCom, Las Vegas, Sept. Shizhen Zhao receive the BE egree in electronic engineering from Shanghai Jiao Tong University, China in, an is pursuing a Ph.D. egree at the school of ECE, Purue University, West Lafayette, IN, USA. His research interests focus on the scheuling problem in wireless networks an smart gri. Xinbing Wang receive the B.S. egree (with hons.) from the Department of Automation, Shanghai Jiaotong University, Shanghai, China, in 99, an the M.S. egree from the Department of Computer Science an Technology, Tsinghua University, Beijing, China, in. He receive the Ph.D. egree, major in the Department of electrical an Computer Engineering, minor in the Department of Mathematics, North Carolina State University, Raleigh, in. Currently, he is a professor in the Department of Electronic Engineering, Shanghai Jiaotong University, Shanghai, China. Dr. Wang has been an associate eitor for IEEE/ACM Transactions on Networking an IEEE Transactions on Mobile Computing, an the member of the Technical Program Committees of several conferences incluing ACM MobiCom, ACM MobiHoc, 3, IEEE INFOCOM 9-.