818 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 Connectivity, Power, and Energy in a Multihop Cellular-Packet System Sayandev Mukherjee, Senior Member, IEEE, Dan Avidor, Member, IEEE, and Katherine Hartman Abstract—In this paper, we study a large network of subscriber stations (SSs) with certain common wireless capabilities and base stations (BSs) having direct connections to the wired infrastructure in addition to common wireless capabilities. SSs can communicate with the “outside world” only through the BSs. Connections to SSs without a direct (i.e., a single hop) wireless connection to any BS are established, if possible, through other SSs serving as wireless repeaters. The locations of the SSs and the BSs follow independent homogeneous planar Poisson processes. The propagation channels exhibit signal attenuation with distance and log–normal shadowing. We evaluate exactly the probability of an SS to have a direct wireless connection to any of the BSs and a lower bound on the t-hop (t arbitrary) outage probability of an SS. We then define the minimal hop-count routing algorithm and calculate the mean number of hops for routes connecting SSs to BSs, when a maximum hop-count constraint is imposed. We compute next the probability distribution of the transmit power under the assumption of perfect power control. We conclude by calculating a bound for the total mean transmit energy required to transfer a data packet from an SS to a BS over a minimal hop-count route and show that this energy is significantly lower than the corresponding value in a single-hop network operating at the same outage probability. Index Terms—Multihop network, outage probability, routing, transmit energy. I. I NTRODUCTION E STUDY a large network of wireless transceivers that we call nodes. Two types of nodes are involved: subscriber stations (SSs) with certain common wireless capabilities and base stations (BSs) having direct wideband connection to the wired infrastructure in addition to the common wireless capabilities. SSs can communicate with the “outside world” only through the BSs. To augment connectivity, connections to SSs without a direct (i.e., a single hop) wireless connection to any BS are established, if possible, through other SSs serving as wireless repeaters, as long as the number of hops does not exceed a prescribed limit. Regular SSs are mobile or installed at customer premises, and their locations cannot be predicted ahead of time; therefore, we assume that their locations are random. We further assume that owing to practical constraints, W Manuscript received January 19, 2005; revised December 1, 2005 and March 11, 2006. Parts of this work were presented at the IEEE VTC 2005Fall, Dallas, TX, and IEEE SECON 2005, Santa Clara, CA. The review of this paper was coordinated by Dr. Q. Zhang. S. Mukherjee and D. Avidor are with Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974 USA (e-mail: sayan@lucent.com; avidor@ lucent.com). K. Hartman is with the Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail: khartman@mit.edu). Digital Object Identifier 10.1109/TVT.2007.891428 availability of high-speed wired connections, and economic considerations, BSs are sparse and often cannot be positioned based on coverage considerations only. To account for this reality, we assume that the BSs, like the regular SSs, are also placed randomly over the service area. This paper focuses on two related issues. The first is the probability that an SS in a fixed arbitrary location (or while passing such a location) has a “working” wireless connection to any of the BSs, evaluated as a function of the densities of the BSs and SSs, the statistical properties of the propagation channels, and the limit set on the maximum number of hops if any. Such a limit is typically set due to delay and possibly capacity considerations. We derive analytical results and lower bounds when exact results are not obtainable. To assess the tightness of the bounds, we compare them with simulation results. We focus next on the well-known minimal hop-count routing algorithm and calculate the mean number of hops for routes connecting SSs to BSs. We then turn to the second issue, which is the probability distribution of the transmit power under the assumption of perfect power control, i.e., transmitters transmit just enough power to be “properly” received. We conclude by calculating an expression for the total mean transmit energy required to transfer a data packet from an SS to a BS. We show that this energy is significantly lower than the corresponding value required in a single-hop network operating at the same outage probability. Multihopping can, therefore, save SS battery power. This paper is organized as follows: In Section II, we briefly discuss prior research work on connectivity in data networks. In Section III, we define the nomenclature to be used in this paper and the Poisson process that controls the spatial distribution of SSs and BSs in our system. In Section IV, we define the propagation model assumed in this paper. In Section V, we derive the distribution of the number of nodes (BSs or SSs, as the case may be) with a direct connection to an SS and an exact expression for the single-hop probability of an SS outage. We then proceed to derive a lower bound for the probability of t-hop outage. In Section VI, we describe an application where the derivations presented in the previous sections are utilized. The well-known minimal hop-count routing algorithm is presented in Section VII, where we calculate the mean number of hops conditioned on a preset limit on the maximum number of hops. Section VIII considers the transmit power on multihop links in a system with perfect power control. In Section VIII-A and -B, we derive the cumulative distribution function and the mean transmit power on a single hop from an SS to a BS or another SS, respectively. In Section VIII-C, we calculate the mean total transmit energy per packet on a minimum-hop 0018-9545/$25.00 © 2007 IEEE MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM route when at most tmax hops are allowed. In Section VIII-D, we compare, by way of example, the mean total energy per packet with that of a single-hop system operating at the same outage probability. Section IX presents the results and explains each figure and the way the data was derived. Section X presents our conclusions. II. R EVIEW OF R ELATED W ORK The issue of connectivity of wireless ad hoc networks has attracted many researchers. In this kind of network, the interesting question is the probability that the network of terminals forms a connected graph of nodes so that a route can be found from any node to any other node. A related issue is that of coverage: Can a wireless mobile moving randomly throughout the service area maintain continuous connectivity with a network of fixed wireless terminals? Several researchers focused on packet radio networks, in which transmitters with fixed range R are distributed over a d-dimensional space A = [0, l]d according to a homogeneous Poisson point process with intensity D. In [1], the authors prove that in a 2-D space and for
a given D, limA→∞ P{area is covered} = 0 when R = (1 − ǫ) ln A/(πD) for any ǫ > 0 and that limA→∞ P{area is covered} = 1 (i.e., all points of A are within distance R from a Poisson point) when R =
(1 + ǫ) ln A/(πD) for any ǫ > 0. They were also able to
prove that P{network is fully connected} = 0, if R = (1 − ǫ) ln A/(πD) for any ǫ > 0. Santi and Blough [2] study the topic of connectedness with high probability (WHP) for n independently and uniformly placed transceivers. In their formulation, n is fixed and not a random variable. They take the size A = [0, l]d of the deployment region as a parameter of the model and study the critical transmitting range required to achieve connectivity WHP when l goes to infinity. For a 1-D space of length l, they prove that, if Rn ≥ 2l ln l, the network is connected WHP, where R is the common radio range of the transceivers. The network is not connected WHP if Rn ≤ (1 − ǫ)l ln(l), for 0 < ǫ < 1. They generalize the sufficient condition for connectedness WHP for the 2-D and 3-D cases and give a necessary condition for connectedness WHP that is weaker than in the 1-D case. They show that, for 2-D and 3-D networks, reducing the communications range yields progressively “less connected” networks, making itself evident by the appearance of isolated nodes. Their results can be applied to both dense and sparse networks. Besides analytical results of static scenarios, they also present simulation results of mobility scenarios. They show that for a large range of parameters, mobile networks are effectively stationary as far as connectivity is concerned. For 2-D and 3-D spaces, they further investigate through simulations the minimal values of R and n that ensure either a connected graph or the formation of a single connected component that includes a large fraction (e.g., 90%) of the nodes. The above studies focused on pure ad hoc networks. Realizing that the capacity of ad hoc networks does not scale well with the number of nodes, some researchers have recently examined “hybrid networks” [3], [4], where a sparse network of BSs is placed on a regular grid within an ad hoc network. The 819 BSs are all connected via a high-data-rate wired infrastructure and serve as relays for data packets generated by the nodes of the ad hoc network. Dousse et al. [3] study hybrid networks, where BSs positioned on a fixed square grid are added to a low-density ad hoc network with nodes placed according to an isotropic Poisson point process. They take the transmission range R to be fixed and allow the intensity of the Poisson process to change. Otherwise, their system model is basically the same as in [1], but they also consider 1-D space and the case of a strip of infinite length and fixed finite width. Like [1], [2], and [5], they allow any number of hops. As aforementioned, the objective is to achieve connectivity between any two arbitrarily chosen nodes, allowing the use of a pair of BSs as repeaters. For the hybrid 1-D model, they provide a bound on the maximum spacing between consecutive BS to keep the probability of finding a node with no connection below a prescribed value. For the 2-D and the “strip” models, they provide simulation results of the probability of having a disconnected node. Assuming nodes distributed in the plane according to a 2-D spatial Gaussian distribution around the origin, Miller [6] provides an analytical approximation, as well as a curvefitting function derived by regression to simulated data, for the probability of a double-hop connection between two randomly selected nodes. He also provides an upper bound to the probability of a t-hop connection. However, Miller’s model deals with a single cluster of nodes. Chandler [7] considers a network with node locations given by points of a homogeneous Poisson process and derives an expression for the probability that two nodes at a distance D apart cannot communicate in t hops or less. Although not stated explicitly in his paper, his expressions for t ≥ 3 are actually lower bounds on the exact probabilities (see [8] for details). The above studies do not consider fading. For the case of nodes distributed according to a planar homogeneous Poisson process with log–normal shadowing, Bettstetter and Hartmann [9] derive an expression for the node isolation probability (single-hop outage) but do not obtain a closed form for this expression. Orriss and Barton [10] derive an exact closed-form expression for the node-isolation probability with log–normal shadowing for arbitrary densities. They also allow two-slope propagation models and varying spatial densities (in this paper, we provide a different and independent derivation of an exact closed-form expression (18) for the single-hop outage probability, which matches the one in [10]). All of the above studies ignored the effects of cochannel interference. Zorzi and Pupolin [11] consider randomly located cochannel interferers and account for log–normal shadowing and Rayleigh fading. However, in this paper, the authors address a different issue, namely, the probability that a data packet is received correctly by the intended receiver under interference caused by concurrent transmissions of a random set of other nodes with no central control, no power control, and under the assumption that all the nodes have the same probability of being on (i.e., transmitting) or off (i.e., not transmitting). They apply the results to the computation of the throughput of a packet radio network. It is, therefore, conceptually different from the multihop connectivity that we study in our paper. 820 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 III. L OCATION M ODEL FOR BS S AND SS S We begin with the following model assumptions. 1) BSs are points of a homogeneous Poisson process on the plane with intensity λBS , i.e., i) The number of BSs in any finite region A, NBS (A), is a Poisson random variable with mean given by λBS × area (A), i.e., n [λBS × area(A)] , n! SS KSS SS PT,max δ rij SS 10zij /10 > Pmin ⇔ rij < rSS 10zij /(10δ) , with rSS ≡ P{NBS (A) = n} = e−λBS ×area(A) × 6) SS i has a direct connection to (i.e., is one hop away from) another SS j if, and only if, the received power exceeds SS , i.e., if, and only if some given threshold Pmin  SS SS PT,max KSS SS Pmin 1/δ (4) n = 0, 1, . . . . (1) ii) Conditioned on a given number of BSs in a chosen region, say NBS (A) = n, the locations (X1 , Y1 ), . . . , (Xn , Yn ) of these n BSs are independent and identically distributed (i.i.d.) uniformly over that region: (X1 , Y1 ), . . . , (Xn , Yn ) i.i.d. with common probability density function (pdf):  1/area(A), (x1 , y1 ) ∈ A fX1 ,Y1 (x1 , y1 ) = (2) 0, otherwise. iii) The numbers of BSs in any two disjoint finite regions A and B are independent, i.e., A ∩ B = ∅ ⇒ (∀m)(∀n)P{NBS (A) = m, NBS (B) = n} = P{NBS (A) = m} × P{NBS (B) = n}. where δ is the distance-loss exponent, 10zij /10 is the shadow fade between SSs i and j, rij is the distance SS between the SSs, and KSS is a constant taking into account the parameters of an SS-to-SS link like antenna gain, antenna height (again assumed equal for all SSs), etc. Observe that (4) reduces to rij < rSS if there is no shadow fading, i.e., if zij ≡ 0. 7) An SS i has a direct connection to a BS b if, and only if, the received power at the BS on the uplink exceeds some BS given threshold Pmin and the received power at the SS on SS the downlink exceeds Pmin , i.e., if, and only if BS KSS SS PT,max δ rib (3) 2) SSs are points of a homogeneous Poisson process on the plane with intensity λSS , i.e., (1)–(3) hold with NSS (·) and λSS in place of NBS (·) and λBS , respectively. 3) The BS process is independent of the SS process, i.e., for any two regions A and B (which may not be disjoint in general), NBS (A) and NSS (B) are independent, and given NBS (A) = n, the distribution (2) of the location of the BSs in A is the same, regardless of the number and location of the SSs in B. IV. R ADIO P ROPAGATION M ODEL We make the following assumptions. 1) There is attenuation with distance and log–normal shadow fading. 2) The shadow-fade attenuations between any BS and SS, and between any two SSs, are i.i.d. log–normal.1 3) The shadow-fade attenuation between any two nodes is the same, regardless of which node is the transmitter and which is the receiver. 4) All SSs are identical, and the maximum transmit power SS of any SS is PT,max . 5) All BSs are identical, and the maximum transmit power BS of any BS is PT,max . 1 The i.i.d assumption is widely used in the literature, even though field measurements seem to indicate that the shadow fades between two links with a common node are correlated [12]–[15]. However, the authors are not aware of any correlation model, which has been widely endorsed by the scientific community. BS KSS BS 10zib/10 > Pmin and BS PT,max δ rib SS 10zib/10 > Pmin ⇔ rib < rBS 10zib /(10δ) (5)  BS rBS ≡ KSS min  SS BS PT,max PT,max , BS SS Pmin Pmin 1/δ (6) where 10zib /10 is the shadow fade between the BS and the SS, rib is the distance between the BS and the SS, and BS SS KSS is defined similarly to KSS but takes into account the antenna gains and heights of both the BS and the SS. Observe that (5) reduces to rib < rBS if there is no shadow fading, i.e., if zib ≡ 0. V. O UTAGE P ROBABILITY In this section, we derive expressions for the probability that an arbitrarily located SS cannot communicate with any BS or with any SS. We then show how to obtain lower bounds on the probability that an arbitrary SS cannot communicate to any BS in t hops or less. It will be seen that the analysis of connectivity that follows applies equally to the connectivity with BSs and with other SSs. For brevity, in the derivation of these results, we use the term “node” for either “BS” or “SS.” Accordingly, wherever a subsequent result has a quantity with the subscript “n” (for “node”), the corresponding result is true if “n” is replaced with either “SS” or “BS” throughout the expression. MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM Now, observe that as r0 → ∞, we have A. Distribution of the Number of Nodes With a Direct Connection to an SS    lim r02 pn (r0 ) = lim EZ min r02 , rn2 exp(2hZ/δ) Define the disk with radius r centered at (x, y) by  ′ ′ ′ ′ 2 2 B(x, y; r) = (x , y ) : (x − x) + (y − y) ≤ r r0 →∞ 2 and the punctured disk obtained by deleting the center = rn2 exp(2h2 σ 2 /δ 2 ) B (x, y; r) = B(x, y; r)\ {(x, y)} . For brevity, we shall write B(r) and B ′ (r) in place of B(0, 0; r) and B ′ (0, 0; r), respectively. Recall our use of the subscript “n” in place of either “SS” or “BS” throughout an expression. Consider an SS at (0, 0), and let Nn′ (r0 ) ≡ Nn (B ′ (r0 )) be the number of nodes (BSs or SSs, as the case may be, with range rn and corresponding intensity λn ) whose distance from this SS is at most r0 (where r0 is assumed to be very large). From the theory of homogeneous Poisson point processes, we know that Nn′ (r0 ) ∼ Poiss λn πr02 . Let R denote the distance of an arbitrary node in B ′ (r0 ) from the SS at (0, 0). Since the node is in B ′ (r0 ), it follows from the theory of homogeneous Poisson processes that its location must be uniformly distributed in B ′ (r0 ) and independent of the location of any other node(s) in B ′ (r0 ). Thus, the pdf of R is  2r , 0 < r ≤ r0 0, r > r0 . r02 (7) Let 10Z/10 be the shadow-fade attenuation on the link between an arbitrary node in B ′ (r0 ) and the SS at (0, 0). Since the shadow fades are assumed i.i.d. across links and the locations of the nodes in B ′ (r0 ) are also i.i.d., it follows that the event that there is a connection between the arbitrary node and the SS at (0, 0) is also i.i.d. across nodes, with the common probability being given by pn (r0 ) = P{R < rn 10Z/(10δ) } = P{R < rn exp(hZ/δ)}   = EZ  min{r0 ,rnexp(hZ/δ)} 0 where h = ln 10 10 Nn′′ ∼ Poiss(µn exp(2α2 )) (12) µn = λn πrn2 (13) where α= hσ . δ (14) Note that µn is the mean number of connected “neighbors” in the absence of fading (σ = 0). From (12), it follows that the probability that an SS at (0, 0) has no direct connection to any of the “nodes” in the network (where the “nodes” can be either the BSs or the SSs) is given by   qn ≡ P {Nn′′ = 0} = exp −µn exp(2α2 ) . (15) Thus, taking the “nodes” to be the BSs, the probability that an SS has no direct connection to any BS is given by where ′′ qBS = P {NBS = 0}   = exp −µBS exp(2α2 ) (16) 2 µBS = λBS πrBS (17) and similarly, taking the “nodes” to be the SSs, the probability that an SS has no direct connection to any other SS is given by (8) 2r  dr r02 where (9) It follows from the theory of Poisson processes [16, Sec. 5.1, para. 3, p. 53] that Nn′′ (r0 ), which is the number of nodes in B ′ (r0 ) with a connection to the SS at (0, 0), has the following Poisson distribution: Nn′′ (r0 ) ∼ Poiss λn πr02 pn (r0 ) . (11) where we use the fact that Z ∼ N (0, σ 2 ) and the known moment generating function of the Gaussian distribution. From (11) and (10), we see that Nn′′ , which is the number of nodes in the entire plane with a connection to the SS at (0, 0), has the following distribution:     1 = 2 EZ min r02 , rn2 exp(2hZ/δ) . r0 r0 →∞   = EZ rn2 exp(2hZ/δ)  ′ fR (r) = 821 (10) ′′ qSS = P {NSS = 0}   = exp −µSS exp(2α2 ) (18) 2 µSS = λSS πrSS . (19) B. Application: Triangulation-Based Positioning The above analysis may be applied to the case where the location of a mobile SS in a region served by multiple BSs is to be determined through triangulation. For this, we require the SS to have a connection to at least three BSs. Taking the 822 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 SS’s location to be the origin, the probability that the SS’s location cannot be determined is just the probability that ′′ < 3} = qBS P {NBS  2  µBS exp(2α2 ) . 1+ µBS exp(2α )+ 2  2 therefore, the complementary cdf F̄T (t) ≡ 1 − FT (t). Clearly, F̄T (1) = qBS . For arbitrary t ≥ 2, we can write F̄T (t) = P{T > t} ′′ > t − 1} = P{T > 1, T1 > t − 1, . . . , TNSS ′′ > t − 1|T > 1} = qBS × P{T1 > t − 1, . . . , TNSS (21) C. Probability Distribution of the Distance Between an SS and a Node With a Direct Connection to the SS For future reference, we derive here the pdf of the distance between an SS and a node with a direct connection to the SS. Recall the definition of the indicator function 1A (·) of an arbitrary set A:  1, if x ∈ A . 1A (x) = 0, if x ∈ A For the same situation as analyzed in Section V-A, the pdf of the distance R between the node and the SS, conditioned on there being a direct connection between them, i.e., R < rn exp(hZ/δ), is given by fR|R<rn exp(hZ/δ) (r) P {R ≤ min[r0 , r], R < rn exp(hZ/δ)} d lim dr r0 →∞ pn (r0 )    ln(R/rn ) 1 E (R)Q R (0,min[r ,r]) 0 α d = lim dr r0 →∞ pn (r0 ) = 1 d lim 2 = dr r0 →∞ r0 pn (r0 ) 2rQ =  ln(r/rn ) α rn2 exp(2α2 )  r 0 2r′ Q  ln(r′ /rn ) α  dr′ = ∞ n=0 [from (11)] ≥ ∞ x = exp(−u2 /2) √ du. 2π D. t-Hop Outage Probability of an SS In hybrid multihop ad hoc networks, with a limit imposed on the maximum allowed number of hops, the probability that an SS is in outage is the probability that it cannot reach a BS in a given maximum number of hops. We call this the multihop outage probability, as opposed to the single-hop outage probabilities computed in the previous sections. Let T be the minimum number of hops required for an arbitrary SS (assumed located at the origin) to connect to a BS. Then, T is a random variable taking values in the set {1, 2, . . .}. Let the cumulative distribution function (cdf) of T be FT (t). The t-hop outage probability of the SS is, ′′ P {NSS = n} ′′ × P {T1 > t − 1, . . . , Tn > t − 1|T > 1, NSS = n} (20) where Q(x) = ′′ where NSS ∼ Poiss(µSS exp(2α2 )) is the number of SSs with ′′ which the SS at (0, 0) has a direct connection, and T1 , . . . , TNSS are, respectively, the minimum number of hops required for ′′ (which have a direct connection to the SS at SS1 , . . . , SSNSS the origin) to connect to a BS. ′′ For any NSS = n, it is clear that T1 , . . . , Tn are identically distributed. However, we know from (20) that these SSs are located in the geographical proximity of the SS at the origin and, thus, are in the vicinity of one another as well. Consequently, T1 , . . . , Tn are not independent, and the probability that all of them are larger than t − 1 is higher than it would have been if they were randomly located somewhere on the (infinite) plane. Nevertheless, in order to proceed further, we make the following crucial assumption. Assumption (Conditional Independence): Conditioned on the SS at the origin having no direct connection to any BS (and not counting paths crossing the SS at the origin), the event that any SS with a direct connection to the SS at the origin cannot connect to any BS in ≤ t hops is independent across all such SSs (and has the same probability for all such SSs, by symmetry) for all t. Note that this assumption leads, as discussed above, to a lower bound on the probability of t-hop outage. We have   ′′ > t − 1|T > 1 P T1 > t − 1, . . . , TNSS ∞ ′′ P {NSS = n} (P{T1 > t − 1|T > 1})n n=0 1−P{T >t−1|T >1} qSS 1 (22) ′′ where we use the fact that NSS ∼ Poiss(−ln qSS ). In the Appendix, we prove that P{T1 > 1|T > 1} = βqBS = βP{T1 > 1} (23) where, from (77) β= ∞ 0 2r 1 2 exp(2α2 ) Q α ln rSS × exp  λBS 2π ∞ −v 2/2 e  ∞ r rSS  2 e−w /2 −∞ −∞ αw × g(r; rBS e αv  , rBS e )dwdv dr (24) MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM able to connect to a BS is lower bounded by limt→∞ qt . This may be obtained numerically by solving for q from the following transcendental equation obtained by setting qt−1 = qt = q in (29): and g(r; a, b) is given by (72). Then, we may write P{T1 > t − 1|T > 1} = P{T1 > 1|T > 1}P{T1 > t − 1|T1 > 1, T > 1} = β P{T1 > 1} P{T1 > t − 1|T1 > 1, T > 1}. 823 1−βq q = qBS qSS . (25) Furthermore, we note that the SS at (0, 0) is a neighbor of SS 1. Suppose SS 1 has no direct connection to any BS. Then, knowing in addition that the SS at (0, 0) also has no direct connection to any BS, increases the probability that SS 1 has no connection to any BS in t − 1 hops or less over its corresponding value if we do not know that the SS at (0, 0) has no direct connection to any BS. In other words, we have P{T1 > t − 1|T1 > 1, T > 1} ≥ P{T1 > t − 1|T1 > 1}. This gives us P{T1 > 1} P{T1 > t − 1|T1 > 1, T > 1} ≥ P{T1 > 1} P{T1 > t − 1|T1 > 1} (31) Note that q = 0 is not a solution of the above equation. In other words, a randomly located SS has a nonzero probability of not being able to connect to any BS, even if it is allowed an arbitrary number of hops to do so. Remark: Observe that the sequence {qt }∞ t=1 in (29) is monotonically decreasing, as can be shown by mathematical induction. 1−βq1 1−βq1 < 1) From (78), we see that qSS < 1, so q2 = q1 qSS q1 . 2) Suppose qt < qt−1 < · · · < q1 ≡ qBS for some t. 1−βq 1−βqt 1−βqt 3) Then, qSS < qSS t−1 , so qt+1 = qBS qSS < 1−βqt−1 = qt , i.e., qt+1 < qt < qt−1 < · · · < q1 ≡ qBS qSS qBS . = P{T1 > t − 1} = F̄T (t − 1) (26) where, in the final step, we make use of the fact that the unconditional t-hop outage probability of SS 1 is the same as that of the SS at (0, 0). Substituting (26) into (25), we thus obtain P{T1 > t − 1|T > 1} ≥ β F̄T (t − 1). (27) Substituting (27) into (22), we may rewrite (21) as follows: 1−β F̄T (t−1) F̄T (t) ≥ qBS qSS . (28) Note that (28) is a sharper bound than the one previously derived in [17] and can itself be generalized to SSs located according to a “clumped Poisson” process, as shown in [18]. We have previously proven that the single-hop outage probability is given exactly by qBS . Define q1 = qBS , 1−βqt−1 qt = qBS qSS , t = 2, 3, . . . . (29) Then, it is easy to show by induction that we have the probability of t-hop outage in (28) to be lower bounded by qt , as defined by (29): F̄T (t) ≡ 1 − FT (t) ≥ qt , t = 2, 3, . . . . 1−β F̄T (t) Consider the same model as discussed in Sections III and IV, but the SSs, aside from their wireless capabilities, are sensors designed to monitor certain localized events and report to any of the BSs whenever an event is detected. Multihopping is acceptable, but there is a limit tmax on the number of hops allowed. tmax reflects the “real time” requirement of the application and the need to flush out old reports, which might otherwise clog the system. Nothing is known about possible locations of events, and we therefore assume that the location of each event is uniformly distributed over the area and is independent of the SS process. Assume further that each sensor can detect (with probability 1) an event occurring within a fixed (nonrandom) distance rsensor from it but cannot detect further events. The task is to estimate the probability that an event will not be reported to BS. Consider an event occurring at an arbitrary location and denote it as the origin. The probability that this event cannot be detected by any sensor is just the probability that there is no sensor within rsensor of the point (0, 0). As we have seen above, this probability is 2 . exp −λSS πrsensor (30) Proof: Since 1 − FT (1) = q1 , it follows from (28) that 1−βq1 1 − FT (2) ≥ qBS qSS = q2 , so (30) holds for t = 2. Now, suppose it holds for some arbitrary t, so that F̄T (t) ≥ qt . Then, from (28) 1 − FT (t + 1) ≥ qBS qSS VI. A PPLICATION : S ENSOR N ETWORKS FOR E VENT D ETECTION 1−βqt ≥ qBS qSS = qt+1 which completes the proof of (30) by induction. As t → ∞, i.e., as we allow an SS to connect to a BS through more and more hops, the probability that the SS will not be (32) Next, suppose there are NSS (rsensor ) = n sensors in B(rsensor ). In this case, the event is detected by all of these n sensors. However, no report will reach a BS if none of these sensors can reach a BS in tmax hops. Allowing n to take all possible values, the probability of this event is ∞ n=1 P {NSS (rsensor ) = n} × P{None of these n SSs can connect to a BS in ≤ tmax hops}. (33) 824 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 2 Since NSS (rsensor ) ∼ Poiss(λSS πrsensor ), we can follow the same sequence of steps as in the derivation of (28) to show that (33) can be lower bounded for every n by ∞ n=1 P {NSS (rsensor ) = n} (P{T > tmax })n   2 = exp −λSS πrsensor (1 − P{T > tmax }) . (34) Let pnr be the probability that an event is not reported to any BS. Then, pnr is the probability that either the event is not detected at all by any sensor or that it is detected by one or more sensors, none of which can report it to any BS within tmax hops. Using (32), (34), and (30), we obtain the following lower bound on pnr :   2 2 + 1 − exp −λSS πrsensor pnr ≥ exp −λSS πrsensor   2 (1 − qtmax ) (35) × exp −λSS πrsensor connection possible to any BS, the SS chooses a minimum power connection to one of the SSs to which it knows it can connect (from the route-exploration phase). Note that in the routing-path-exploration and setup phase, if each SS at each hop-distance from some BS broadcasts the minimum transmit energy per packet to reach a BS starting from itself, then each SS could have chosen the minimum total transmit-energy routing path instead of the minimum hop-count routing path to a BS. The minimum hop-count and minimum transmit-energy routing paths are two extreme choices in the spectrum of possibilities arising from trading off transmit energy (affecting the battery life of SSs and self-interference, which affects the throughput) versus latency (as measured by the hop count). It is possible to conceive of a whole family of routing algorithms that choose paths that optimize a weighted combination of transmit energy and hop count, within a certain level of complexity. However, in this paper, we focus only on the minimum hop-count routing algorithm. where qtmax is given by (29) with t = tmax . B. Properties of the Minimum Hop-Count Routing Path VII. N UMBER OF H OPS ON A P ATH F ROM AN SS TO A BS Recall that T is the hop count of the minimal hop-count route between an SS and a BS, and the maximum transmit power SS capability of the SS is PT,max . A. Routing Scheme In this section, we define the minimal hop-count routing algorithm, some properties of which we then analyze in subsequent sections. This same routing scheme was studied extensively before, for instance in [19]. For the reader’s convenience, we present here one simple implementation. We begin with each BS broadcasting (with maximum power) a beacon or pilot signal that, when received by any SS, indicates to that SS that it is just a single hop away from the particular BS. These SSs now broadcast signals (also with maximum power) giving their own identification and saying that they are one hop away from a BS. The SSs that receive these signals, and have not received any BS directly, now know that they are two hops away from a BS, and so on. This models the signaling in the IEEE 802.16-2004 mesh-networking mode, for example, in [20]. A given SS may receive such signals from multiple SSs that advertise themselves as being at some number of hops away from a BS. The given SS focuses on those SSs that advertise the minimum number of hops and establishes a routing path to the particular SS among those that it can reach with minimum transmit power. This process guarantees that if a path exists at all between the given SS and BS, one that contains the minimum number of hops will be found. If the given SS has no path connecting it to any BS, then it is in outage and will be assumed not to transmit at all. While the route exploration phase requires beacon broadcasts with maximum power, the actual route followed by a packet from any given SS is as follows: The SS first checks to see if it can connect directly to a BS (even if this requires using the maximum power available). If it can, it connects to the BS that requires least transmit power to do so. If there is no Recall that a given SS is in t-hop outage if there is no path at all connecting this SS to any BS in t or fewer hops and that the probability of this event is lower bounded by qt , as computed earlier. In other words, for a given t ∈ {1, 2, . . .}, the probability of t-hop outage is the probability that the number of hops required for the SS at (0, 0) to connect to a BS is at least t + 1, or P{T > t}. Thus, the values of the t-hop outage probability for t = 1, 2, . . . give us the complementary cdf of the random variable T . However, we do not know the exact distribution of T , since (29) is only a lower bound on the t-hop outage probability. Note that we may write the expected value of T as follows: E[T ] = ∞ i P{T = i} i=1 = ∞ i P{T = i} i=1 t=1 = ∞ ∞ P{T = i} t=1 i=t = ∞ t=1 =1 + P{T ≥ t} ∞ t=2 =1 + ∞ P{T > t − 1} P{T > t} (36) t=1 but this sum does not converge, since P{T > t} ≥ qt and, as shown above, limt→∞ qt > 0. In other words, since an SS has a nonzero probability of never being able to connect to a BS, regardless of the number of hops allowed, the expected minimum number of hops required to connect to a BS is infinite. This situation happens if, and only if, the SS belongs to a MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM connected cluster of SSs that are isolated from all the BSs and is modeled by setting T = ∞. Suppose now that we permit a maximum of tmax hops for an arbitrary SS to connect to some BS. The limit tmax is expected to be set by delay considerations. An SS that cannot reach a BS in at most tmax hops is not served. The quantity of interest is, therefore, E[T |T ≤ tmax ]. Clearly, if tmax = 1, then E[T |T ≤ tmax ] = 1. For tmax > 1, E[T |T ≤ tmax ] may be computed as follows: i=1 tmax i P{T = i|T ≤ tmax } i = i=1 t=1 tmax tmax t=1 i=t tmax = t=1 =1 + =1 + tmax −1 t=1 =1 + tmax −1 t=1 = rSS (p) ≡ P{T ≥ t|T ≤ tmax } t=1 P{t < T ≤ tmax } 1 − P{T > tmax } rBS (p) ≡ P{T > t} − P{T > tmax } 1 − P{T > tmax } (37) 1 1 − tmax P{T > tmax } 1 − P{T > tmax } tmax −1 tmax −1 P{T > t} (42) t=1 qt . (43) t=1  SS pKSS SS Pmin 1/δ = rSS  p SS PT,max 1/δ . Furthermore, assume that if an SS can reach a BS on the uplink, then the BS can always reach the SS on the downlink. Then, from (6), the range of the SS to the BS is given by P{T > t|T ≤ tmax } + 1 1 − qtmax ≥ P{T = i|T ≤ tmax } tmax −1 tmax −1 1 1 − P{T > tmax } E[T |T ≤ tmax ] ≥ From (4), we see that if an SS transmits with maximum power p, then in the absence of fading, its range to another SS is given by P{T = i|T ≤ tmax } = then (40) reduces to VIII. M EAN T OTAL T RANSMIT E NERGY PER P ACKET ON A P OWER -C ONTROLLED M INIMUM -H OP P ATH F ROM AN SS TO A BS tmax E[T |T ≤ tmax ] = 825 P{T > t} .  BS pKSS BS Pmin 1/δ = rBS  p SS PT,max 1/δ . (44) A remark on notation: Observe that rBS is identical to SS rBS (PT,max ) as defined in (44) and, similarly, for rSS . Simi′′ larly, we shall shortly define NBS (p), µBS (p), T (p), qBS (p), qSS (p), and µSS (p) in such a way that the quantities rBS , rSS , ′′ NBS , µBS , T , qBS , qSS , and µSS defined in the previous sections may now be interpreted as the corresponding functions of p SS . evaluated at the argument p = PT,max t=1 (38) If we estimate P{T > t}, t = 1, . . . , tmax for some tmax from simulations, then (38) can be used to estimate E[T |T ≤ tmax ]. Alternatively, we could look at the effect of tmax on E[T |T ≤ tmax ] as follows: If tmax is such that the probability of tmax -hop outage is small, i.e., if P{T > tmax } < ǫ (39) for some small ǫ, then from (38), we have E[T |T ≤ tmax ] ≥ 1 1 − P{T > tmax }  t × 1 − tmax ǫ + Consider an SS that can receive from one or more BSs on the downlink. What is the minimum transmit power on the uplink from this SS to any of these BSs? If the SS transmits with power p, the number of BSs to which it can connect is given by ′′ 2 NBS (p) ∼ Poiss µBS (p) exp(2α2 ) , µBS (p) ≡ λBS πrBS (p). Let P0 be the minimum required transmit power to connect the SS directly to at least one BS. Then, the probability that P0 > p is F̄P0 (p) = P{P0 > p} max −1 t=1  P{T > t} . (40) In particular, if ǫ ≤ 1/tmax , i.e., if P{T > tmax } < A. Minimum Transmit Power From an SS to a BS 1 tmax (41) ′′ = P {NBS (p) = 0}   = qBS (p) ≡ exp −µBS (p) exp(2α2 ) (45)    2/δ p 2   = exp −µBS exp(2α ) SS PT,max 2/δ p̃ = qBS (46) 826 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 where where p̃ ≡ p SS PT,max . γ(a, z) = z e−v v a−1 dv (48) 0 Suppose that the SS always transmits just enough power to establish a connection to a BS, if at all possible, and does not SS transmit at all if P0 > PT,max , i.e., if T > 1. Thus, the actual SS and zero transmit power is P0 if, and only if, P0 ≤ PT,max otherwise (i.e., if T > 1, which occurs with probability qBS ). Thus, the cdf of the power transmitted by the SS is FP0 |P0 ≤PTSS,max (p) = FP0 |T =1 (p) 1 − F̄P0 (p) 1 − qBS  p̃2/δ qBS 1 SS − 1−qBS , 0 ≤ p ≤ PT,max = 1−qBS SS 1, p > PT,max = fP0 |P0 ≤PTSS,max (p) = ζa (x) = γ (a + 1, ln(1/x)) 1 , (1 − x) [ln(1/x)]a we may rewrite (47) as   SS E P0 |P0 ≤ PT,max SS PT,max d SS F (p) dp P0 |P0 ≤PT ,max    1 2/δ p̃2/δ 2 1 1 SS ln qBS qBS , 0 ≤ p ≤ PT,max 1−qBS δ p p̃ lim ζa (x) = 0. lim ζa (x) = lim x→1 y→0 y→0 ay a−1 = lim y→0 = = PTSS ,max pfP0 |P0 ≤PTSS,max (p)dp  1    2 1 p̃2/δ dp̃ p̃2/δ qBS ln δ qBS 1 = 1 − qBS 1  × ln = = 0 1 = 1 − qBS uδ/2 exp [−u ln(1/qBS )] 0 qBS  1 du [u = p̃2/δ ] ln(1/q  BS ) (1 − qBS ) [ln(1/qBS )]δ/2 0   1 γ 2δ + 1, ln qBS 1 =  δ/2 (1 − qBS ) 1 ln qBS y a exp(−y) [1 − exp(−y)] + y a exp(−y) 1 1+ a exp(y)−1 y 1 1 + a limy→0 exp(y)−1 y 1 1 + a limy→0 1 1 < a+1 2 exp(y) 1 for all [l’Hôpital’s rule again] a>1 which means that for any δ > 2, even if qBS is very close to 1, the average transmit power when the SS does transmit and reach a BS is less than half the maximum transmit power available to it. Remark: It follows from the discussion in Appendix I that when the BS and SS location processes are independent, if the BSs are not located according to a homogeneous process, then the probability of SS isolation increases. Since ζa (x) increases monotonically with x (as proven in Appendix II), this also implies that if the BSs location process is not homogeneous, i.e., if the BSs are not spread uniformly, but are, for instance, clustered, the mean minimal transmit power of the SS is larger than if the BS location process is homogeneous. 0 1 [y = ln(1/x)] [l’Hôpital’s rule] SS PT,max SS PT,max γ(a + 1, y) y a [1 − exp(−y)] = lim   SS E P0 |P0 ≤ PT,max = (50) x→0 Thus, the mean power transmitted by the SS (normalized by the SS ) is given by maximum transmit power PT,max 1 = ζδ/2 (qBS ). (49) From (49), we see that SS p > PT,max . 0,  0<x<1 From (50), it follows that for any x and a, ζa (x), being the ratio of the average of P0 to its maximum, must be ≤ 1. In fact, we have with the corresponding pdf given by = is the incomplete Gamma function.2 Defining v δ/2 e−v dv (47) 2 Some authors use this name for the function Γ(a, z) = [21, eq. (8.350), p. 890]. !∞ z e−v v a−1 dv MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM B. Minimum Transmit Power From an SS to Another on the Minimum-Hop Route Consider an SS, say SS∗ , that is not within t − 1 hops from any BS. Given this fact, its location is not independent from those of the BSs; SS∗ tends to be somewhat removed from BSs, especially if t is large. We are interested in the mean minimum transmit power required for SS∗ to connect, if at all possible, with an SS that, in turn, is exactly t − 1 hops away from a BS. Note that in computing the mean of the minimum transmit power, we also need to average over all arbitrary locations (X, Y ) of SS∗ . Let Pt∗ denote the minimum power for SS∗ to transmit to one of the SSs that are exactly t − 1 hops from a BS (assuming no constraint on the maximum transmit power). Then, the probability that Pt∗ > p is the probability that transmitting at p, SS∗ cannot reach any SS that is exactly t − 1 hops away from a BS. Although these SS locations do not form a homogeneous process, we know from the remark at the end of Appendix I that the outage probability of SS∗ from these SSs is lower bounded by that obtained by assuming that these SSs are points of a homogeneous Poisson process (with intensity λSS P{T = t − 1}), which is qSS (p)P{T =t−1} . Therefore, we have  P{T =t−1} P {Pt∗ > p} ≥ qSS (p)P{T =t−1} = qSS p̃2/δ . Recall again that SS∗ actually transmits if, and only if, there is at least one such SS to which it can connect, the probability P{T =t−1} of which is 1 − qSS . Following the same steps as in the calculation of (50) shows that the mean SS transmit power SS (normalized by its maximum value PT,max ) has the following lower bound: E [PT∗ |can reach SS for which T = t − 1] SS PT,max   P{T =t−1} . (51) ≥ ζδ/2 qSS C. Computation of Mean Total Transmit Energy per Packet on a Minimum-Hop Path With at Most tmax Hops From an SS to a BS Consider a given SS and another SS, which is t ≤ tmax hops away from the “closest” BS in the routing scheme described above. Denote it by SSt . Let SSi , i = 1, . . . , t − 1 be i hops away from the BS on the path connecting SSt to the BS. For tmax = 2, and when the distance between the given SS and the BS is known, the authors have computed an exact expression for cdf of the total transmit energy per packet [22]. Here, we deal with the general case where tmax is arbitrary and the distance between the SS and the BS is not known. From (50), SS1 transmits to some BS with average-to-peak transmit power ζδ/2 (qBS ). In order to calculate the mean transmit power from SSi+1 to SSi , i = 1, . . . , t − 1, recall from the definition of the minimum-hop routing algorithm that SSi+1 cannot connect to a BS directly and also cannot directly connect 827 to any SS that is ≤ i − 1 hops away from a BS, so it then looks among the SSs that are i hops away from a BS and chooses the one (SSi ) that it can reach with minimum transmit power. From (51), we see that SSi+1 transmits with average-to-peak power P{T =i} ). lower bounded by ζδ/2 (qSS Assume that the transmission of a single packet on each hop occupies a single time slot of duration Ts . Then, conditioned on T = t ≤ tmax , the expected value of the total transmit energy Etot for the packet in the path has the following lower bound: E[Etot |T SS = t] ≥ Ts PT,max  t−1 ζδ/2 ζδ/2 (qBS ) + i=1  P{T =i} qSS   so the expected total transmit energy per packet conditioned on the path being at most tmax hops is lower bounded as follows: E[Etot |T ≤ tmax ] = ET {E[Etot |T ]|T ≤ tmax } = 1 1 − P{T > tmax } tmax t=1 SS ζδ/2 (qBS ) + ≥ Ts PT,max tmax t−1 × t=1 i=1 tmax −1 i=1 tmax −1 i=1 1 1 − P{T > tmax }   P{T =i} ζδ/2 qSS SS = Ts PT,max ζδ/2 (qBS ) + × 1 1 − P{T > tmax }   P{T =i} P{T = t} ζδ/2 qSS SS = Ts PT,max ζδ/2 (qBS ) + × E[Etot |T = t]P{T = t} tmax P{T = t} t=i+1 1 1 − P{T > tmax } (P{T > i} − P{T > tmax })   P{T =i} × ζδ/2 qSS . (52) We can estimate P{T > i}, i = 1, . . . , tmax from simulation and use these values to evaluate the right-hand side of (52). D. Comparison of Mean Total Transmit Energy per Packet for Single-Hop and for Minimum-Hop Count-Routed MultiHop We will show now, by way of example, that allowing multihop makes it possible to obtain the same connectivity as in a single-hop system but with a significant reduction in transmit energy per data packet. Suppose we have a system with a given ratio of intensities λSS /λBS . We assume that the SSs have maximum transSS mit power capability PT,max = PT . We now compare the following. 1) We allow a multihop with a maximum of tmax hops. Then, the tmax -hop outage probability is P{T > tmax }, 828 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 where we emphasize that qBS is as defined in (54) for transmit power PT . The mean energy per packet over this single hop is then and the mean energy per packet under the minimum hopcount routing scheme is lower bounded by (52) SS Ēmultihop ≥ Ts PT,max ζδ/2 (qBS ) + × 1 1 − P{T > tmax } tmax −1 i=1  P{T =i} (P{T > i} − P{T > tmax }) ζδ/2 qSS  (53) Ēsingle-hop = Ts PT′ ζδ/2 (P{T > tmax })  δ/2 ln P{T > tmax } = Ts PT ζδ/2 (P{T >tmax }) . ln qBS where (58)   qBS = exp −λBS π exp(2α2 )rBS (PT )2   qSS = exp −λSS π exp(2α2 )rSS (PT )2 (λ = qBSSS (54) /λBS )[rSS (PT )/rBS (PT )]2 (55) and rBS (p) = Thus, the ratio of the mean energy per packet for the single-hop versus the multihop path with the same outage probability is upper bounded by Ψ   BS 1/δ KSS BS Pmin p1/δ , rSS (p) =   SS 1/δ KSS SS Pmin p1/δ . (56) SS PT,max Note that because = PT , rBS (PT ) = rBS and, similarly, for rSS . In practice, SS-to-SS links often do not have the same range capability as SS-to-BS links. The reasons are likely to be better placements, higher towers, better antennas (for instance, diversity reception at the BSs), and other enhancements that are common at BSs but might be impractical at the SS side of the link due to cost or space. As a result, rSS might be significantly smaller than rBS [see (4) and (6)]. Note that the ratio ρ≡ rSS (p) = rBS (p)  SS KSS SS Pmin  BS Pmin BS KSS  1/δ is not a function of SS transmit power. We may then rewrite (55) as (λ qSS = qBSSS /λBS )ρ2 . This value of qSS can then be substituted into (53) to obtain a lower bound on Ēmultihop . 2) Now, suppose we aim to achieve the same outage probability P{T > tmax } on a single hop to a BS by increasing the maximum SS transmit power capability from PT to PT′ . From (45), we see that in order to have the single-hop outage probability match the tmax -hop outage probability P{T > tmax }, the maximum SS transmit power capability must be increased to PT′ such that   2 exp −λBS π exp(2α2 )rBS (PT′ ) = P{T > tmax }. (57) Comparing (54) and (57) and using (56), we see that PT′ = PT  ln P{T > tmax } ln qBS δ/2  λSS , qBS , tmax , δ λBS ≡  Ēsingle-hop Ēmultihop ζ δ (P{T > tmax }) 2 ≤ tmax −1 ζ δ (qBS ) + 2 " i=1  ln P{T >tmax } ln qBS (P{T >i}−P{T >tmax })ζ δ 2   δ2 ρ2 λSS P{T =i} λBS qBS . 1−P{T >tmax } (59) In the absence of the exact expression for the distribution of T , (59) cannot be computed. However, we can estimate P{T > i} from simulation and substitute in (59) in order to evaluate Ψ. Alternatively, Ψ could be estimated directly from simulations but with an obvious cost in terms of running time. IX. A NALYTIC AND S IMULATION R ESULTS —D ESCRIPTION AND D ISCUSSION In this section, we plot analytic results and bounds, describe the simulation setup, and present some simulation results. BS BS SS SS ) = (PTSS KSS /Pmin )= /Pmin We set δ = 4 and (PTSS KSS BS BS SS 12 (PT KSS /Pmin ) = 10 , such that from (5) and (4), respectively, we have rBS = rSS = 1000 m. In other words, when σ = 0 dB, the limiting reception range is 1000 m on both SS-to-SS and SS-to-BS links (this choice is kept true for all subsequent figures, except Fig. 4, where we specify otherwise). We present in Fig. 1 the probability of an SS not having a direct connection to any BS versus λBS , where the linkfading model is defined by σ = 0, 4, and 8 dB. These results were derived from (16). Note that σ = 0 dB corresponds to the geometric model, where the connectivity is deterministically ensured for all nodes within the range rBS . We notice that as σ increases, the probability of an SS not having a direct connection to any BS, i.e., the single-hop outage probability, decreases for a given choice of λBS . The shadow-fading blocks connections to some close-by BSs, while permitting direct connections to some BSs that are farther away than rBS , with positive-net effect on connectivity. MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM 829 Fig. 1. Plot of the single-hop outage probability given by (16) versus λBS for σ = 0, 4, and 8 dB. Fig. 3. Plots of the lower bound on t-hop outage probability given by qt from (29) for two different choices of the parameters (qBS , qSS ) and β = 1. Fig. 2. Simulation results for the t-hop outage probability for t = 2, 3, and 4 compared with the corresponding analytic lower bounds given by (30). Fig. 4. Plots of the analytical lower bound (30) on double-hop outage probability for a fixed SS-to-SS link budget and three cases of SS-to-BS link budget, equal to that of the SS-to-SS link, 10-dB better, and 20-dB better. To find the t-hop outage probability for t = 2, 3, and 4 hops versus λSS and compare the results with the corresponding analytic lower bounds, we ran simulations as described below. For each choice of λSS , the simulation makes repeated placements of 100 λSS a2 SSs over a large square of size 10a for some chosen a. This corresponds to a “partial” [23] realization of the Poisson SS process, where we choose a certain number of SSs to be placed in the region of interest, and these SSs are now placed independently and uniformly over the given region. Similarly, for each choice of λBS , we place 100 λBS a2 BSs over the large square. There are i.i.d. shadow fades between all pairs of nodes. For each placement of SSs and BSs, we count the fraction of SSs that are more than t hops away from all BSs and average this fraction over all placements to yield our estimate of the t-hop outage probability. To eliminate the dependence on edge effects, we collected statistics only on the nodes in a small square (with side of length a) at the center of the large square. Fig. 2, derived from simulations, plots the t-hop outage probability for t = 2, 3, and 4 hops versus λSS and compares them to the bound computed in (29) for σ = 8 dB. Note that β does not depend on λSS and only needs to be computed numerically once from (77). We see that the benefit of allowing additional hops diminishes quickly. This is supported by Fig. 3, derived analytically from the lower bound on t-hop outage probability given by (29). For both examples, three hops are sufficient to achieve most of the possible benefit. Note that once qBS and qSS are given, link parameters like σ, λBS , and λSS are immaterial. Fig. 4 compares the lower bound on double-hop outage probability given by (30) for a fixed SS-to-SS link budget given by rSS = 1000 m and three different values of rBS for the SS-to-BS links. The upper curve corresponds BS BS BS SS to the case (PTSS KSS /Pmin ) = (PTBS KSS /Pmin ) = 1012 or 830 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 Fig. 5. Plot of E[T |T ≤ tmax ] for tmax = 5 given by (38) versus the ratio λSS /λBS , as estimated in two ways. (a) By obtaining P{T > t} via simulation and (b) by approximating P{T > t} by the lower bound qt , as given by (29). We have also shown the ratio of the latter (approximate) value to the former (exact) value. The BS intensity was fixed at λBS = 10−7 m−2 , and the other parameters of the simulation were σ = 8 dB, δ = 4, and rBS = rSS = 1 km. rBS = 1000 m, the middle curve corresponds to rBS = 1013/4 m and the lower curve to 1014/4 m. In other words, the upper curve corresponds to the case where the SS-to-BS link budget is equal to that of the SS-to-SS link, the middle curve is 10 dB better, and the lower curve is 20 dB better. This figure allows us to compare the power of different techniques with respect to their ability to enhance connectivity. For instance, it allows us to compare the multihop scheme discussed in this paper to using higher towers or better antennas at the BSs in a single-hop system. In Fig. 5, we fix λBS = 10−7 m−2 and plot E[T |T ≤ tmax ] versus λSS /λBS for tmax = 5 hops, as calculated from (38) in two ways: 1) using values for P{T > t} obtained from simulation and 2) approximating P{T > t} by qt as given by (29). The other parameters are σ = 8 dB, δ = 4, and rBS = rSS = 1 km. Note that the use of qt in place of P{T > t}, t = 2, . . . , tmax , provide estimates that are mostly within 10% of the correct values. In Fig. 6, we plot ζδ/2 (q) given by (49) as a function of q, for several choices of the distance-loss exponent δ. The ratio of average to peak SS transmit power required to connect to a BS is given by ζδ/2 (qBS ) if the peak SS transmit power is chosen such that the probability of the SS having no direct connection to any BS is qBS . Note that (50) provides the exact value of the desired ratio and is not a bound. In Fig. 7, we compare the plots of E[Etot |T ≤ tmax ] and the right-hand side of (52) versus λSS for σ = 8 dB, tmax = 5 hops, λBS = 10−7 m−2 , δ = 4, and rBS = rSS = 1 km, so that qBS = qSS = 0.6187. The right-hand side of (52) was calculated employing the values of P{T > t} obtained from simulation, while E[Etot |T ≤ tmax ] was obtained directly through simulation. The reason for the behavior of E[Etot |T ≤ tmax ] Fig. 6. Plot of the function ζδ/2 (q) given by (49) as a function of q for several choices of the distance-loss exponent δ. The ratio of average-to-peak SS transmit power required to connect to a BS is given by ζδ/2 (qBS ) if the peak SS transmit power is chosen such that the probability of the SS having no direct connection to any BS is qBS . versus λSS is as follows: We are only considering SSs that have a connection to a BS within at most tmax hops. When λSS is very low, SSs are few and far between, so connections to BSs, when present, are principally just one hop in length. As λSS grows, more SSs gain a connection to a BS, but they require a larger number of hops in order to do so, which leads to a higher total transmit-energy requirement per data packet. As λSS keeps growing further, better routes to BSs can be found, thereby reducing the total transmit-energy requirement MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM 831 Fig. 7. Comparison of E[Etot |T ≤ tmax ]/(Ts PT , maxSS ) (derived directly from simulations) and the right-hand side of (52) (calculated using the values of P{T > t} obtained from simulation) when both are plotted against λSS for σ = 8 dB, tmax = 5 hops, λBS = 10−7 m−2 , δ = 4, and rBS = rSS = 1 km, so that qBS = qSS = 0.6187. Fig. 8. Plot, derived from the simulation, of the tmax -hop outage probability P{T > tmax } for tmax = 5 hops and the ratio Ψ of total transmit energy per packet for single-hop versus minimum hop-count-routed paths with the same outage probability for different choices of the ratio of intensities λSS /λBS and δ = 4. In each hop, the probability of not being able to connect to any BS is qBS = 0.6187. from before, while the number of SSs with a connection to a BS within tmax hops saturates (as P{T > tmax } becomes very small). We note that the bound in (52), while not tight, tracks the exact average transmit energy per packet reasonably well over the range of SS intensities simulated. In Fig. 8, we plot P{T > tmax }, which is obtained from the simulation versus λSS /λBS for tmax = 5, ρ = 1, and qBS = 0.6187, the latter being obtained from (16) with λBS = 10−7 m−2 , rBS = rSS = 1 km, σ = 8 dB, and δ = 4. We also plot the ratio Ψ of the mean single-hop to multihop transmit energies as derived from the simulations. It is seen that at least for this choice of parameters, the minimum hop-count multihop path requires much less total transmit energy per packet than the single-hop path for the same outage probability. Fig. 8 demonstrates that by replacing a single-hop system with a multihop system, where both operate at the same outage probability, it is possible to reduce the total mean transmit energy required per packet by a large factor. This happens even though some packets undergo repeated transmissions and receptions. Saving on transmit energy is very important in systems where the SSs are battery operated. Good examples are the mobiles of the cellular system and certain sensor systems. Another aspect of the same feature is the reduction in the mean interference emission generated per packet. This could be exploited in many ways, but we do not discuss this subject further here. X. C ONCLUSION In this paper, we investigate the possible benefits of multihopping in cellular packet-data networks, where the SSs’ and BSs’ locations follow independent homogeneous planar Poisson processes and the propagation channels exhibit signal attenuation with distance and log–normal shadowing. We calculate the probability that an arbitrary SS will not have access to any BS and show how it depends on the density of BSs and SSs and how it diminishes as more hops are allowed. We develop exact analytical expressions and concise 832 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 lower bounds when this is not possible. In the later case, we produce simulation results for comparison with the analytic bounds. Our bounds are concise and lend themselves to easy evaluation. We demonstrate that the use of SSs as repeaters is particularly beneficial in fading environments. We calculate the mean number of hops achieved by a minimum hop-count router when only routes containing at most tmax hops are allowed. Then, we calculate the mean transmit energy per packet and show, by way of example, that with multihop, it is possible to achieve the same outage probability as a singlehop system with reduced total transmit energy per data packet, which is an important consideration for battery operated SSs. Our derivations allow experimentation with different channel models, for instance, macrocell models for the SS-to-BS links and picocell models for SS-to-SS links. We also provide several example applications, showing how the developed theory can be put to use. The common probability that each of these nodes in B(x, y; r0 ) has a direct connection to (x, y) is given by A PPENDIX I P ROBABILITY OF I SOLATION OF AN A RBITRARY SS F ROM A LL N ODES W HEN N ODE L OCATION P ROCESS I S I NHOMOGENEOUS P OISSON µ (B(x, y; r0 )) pn (x, y; r0 ) Suppose the node location process is Poisson but not homogeneous, i.e., the number of nodes Nn (A) in a given region A is Poisson with mean  λn (x′ , y ′ )dx′ dy ′ (60) µ(A) ≡ (x′ ,y ′ )∈A for some intensity function λn (·, ·) instead of λn × area(A) for a constant intensity λn , as in the homogeneous Poisson case. For any region A, given Nn (A) = k, these k nodes are i.i.d. over the region A with the pdf of the location (X ′ , Y ′ ) of any one of them being ′ ′ fX ′ ,Y ′ (x , y ) =  λn (x′ ,y ′ ) µ(A) 0 (x′ , y ′ ) ∈ A . otherwise (61) Compare (60) and (61) with (1) and (2), respectively, for the homogeneous Poisson process. As before, we assume that the numbers of nodes in two disjoint regions are independent Poisson random variables and that the node location process is independent of the SS location process. Consider an SS at location (x, y). The number of nodes in the disk B(x, y; r0 ) is Poisson with mean µ (B(x, y; r0 )) ≡ 2πr0 λn (x + r cos θ, y + r sin θ)rd rdθ. 0 0 pn (x, y; r0 ) = (62) EZ pn (x, y; r0 ) = P {R < rn exp(hZ/δ)} where R is the distance of the node from (x, y), and Z is the shadow fade on the link between this node and the SS at (x, y). Note that if (r, θ) are the polar coordinates representing the location of this node relative to (x, y), then from (61), we have  rλ (x+r cos θ,y+r sin θ) n , r ∈ [0, r0 ], θ ∈ [0, 2π) µ(B(x,y;r0 )) fR,Θ (r, θ) = 0, otherwise. (64) Substituting (64) into (63), we obtain the expression shown at the bottom of the page. Applying the same argument that led to (10) now yields Nn′′ (B(x, y; r0 )), which is the number of nodes in B(x, y; r0 ) with a direct connection to the SS at (x, y), to be Poisson with mean 2π = EZ dθ 0 0 min{r0 ,r n exp(hZ/δ)}  dr rλn(x+r cos θ, y+r sin θ) . 0 Letting r0 → ∞, we finally obtain Nn′′ (x, y), which is the number of nodes in the plane with a direct connection to the SS at (x, y) to be Poisson with mean given by EZ [µ(B(x, y; rn exp(hZ/δ)))]. Recognizing rn exp(hZ/δ) as the range of the SS at (x, y), we now define the locally averaged intensity of the node location process around the SS location (x, y) as follows: λ̄n (x, y) ! !  2π r exp(hZ/δ) EZ 0 0 n rλn (x+ r cos θ, y+ r sin θ)dr dθ ! !  ≡ 2π r exp(hZ/δ) EZ 0 0 n r drdθ  ! ! 2π r exp(hZ/δ) rλn (x + r cos θ, y + r sin θ)drdθ EZ 0 0 n . = πrn2 exp(2α2 ) (65) Thus, the probability of isolation of an SS at (x, y) from the nodes of the inhomogeneous Poisson process is given by   P {Nn′′ (x, y) = 0} = exp −πrn2 exp(2α2 )λ̄n (x, y) . (66) From (66), we see that the isolation probability of an SS in the presence of nodes located according to such an inhomogeneous Poisson process is not independent of the location of this SS. Note that the SS could be located at any arbitrary point (X, Y ) in the plane. Equivalently, the locally averaged intensity ! ! 2π min{r0 ,rn exp(hZ/δ)} 0 (63) rλn (x + r cos θ, y + r sin θ)dr dθ µ (B(x, y; r0 ))  MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM of the node location process around the random SS location may be seen as the random variable V , which is the following function of the random SS location coordinates (X, Y ): V ≡ λ̄n (X, Y ). (67) From (66), the probability that an SS at a random location (X, Y ) is isolated may be written as exp[−πrn2 exp(2α2 )V ], where V is as defined in (67). Thus, qn the probability that an arbitrarily located SS is isolated from all such nodes is the average of exp[−πrn2 exp(2α2 )V ] over all possible values of V ; the locally averaged intensity of the node-location process around the random SS locations:    qn ≡ EV exp −πrn2 exp(2α2 )V . (68) Noting that the negative exponential is strictly convex in its argument, we see from Jensen’s inequality applied to (68) that   qn ≥ exp −πrn2 exp(2α2 )E[V ] . (69) Now, suppose that this SS is a point of a homogeneous Poisson point process and that we know that it is located in a given (possibly large) region, say the disk B(0, 0; r1 ). Then, the pdf of its location (X, Y ), fX,Y (·, ·) is given and constant over B(r1 ). From (69), we know that of all possible intensities λn (·, ·) of the node-location process, the SS outage probability from the nodes of this process qn , is minimized if 1) λn (·, ·) maximizes E[V ] for the given fX,Y , and 2) we have an equality in (69). To find λ̄n (·, ·) achieving 1), note that we can write   E[V ] = EX,Y λ̄n (X, Y )  = λ̄n (x, y)fX,Y (x, y) dx dy B(r1 )   ≤  B(r1 )   ×  B(r1 ) (69). From 1) and 2), it follows that qn is minimized if λ̄n (·, ·) is a constant. Remark: From (65), λ̄n (·, ·) is constant if, and only if, λn (·, ·) is constant, i.e., if, and only if, the node-location process is homogeneous Poisson. Equivalently, if the nodelocation process is inhomogeneous, then qn is larger than the value obtained by assuming that the node-location process is homogeneous Poisson. A PPENDIX II P ROOF T HAT ζa (x) I S M ONOTONICALLY I NCREASING IN x FOR A NY a > 0 We use the well-known result that the expected value of a random variable is the area under the curve of the complementary cdf of that random variable in order to obtain   SS E P0 |P0 ≤ PT,max ζa (qBS ) = SS PT,max PTSS ,max = = 1/2  2 fX,Y (x, y) dx dy  1  SS PT,max 1 SS PT,max 0 1 0   1 − FP0 |P0 ≤PTSS,max (p) dp 1/a p̃ qBS − qBS dp̃. 1 − qBS Therefore, to prove that ζa (x) is monotonically increasing in x for a given a, it is sufficient to prove that for any w ∈ [0, 1], the function hw (x) ≡ xw − x 1−x is monotonically increasing for all x ∈ (0, 1). This result follows from the observation that 1/2  [λ̄n ]2 (x, y) dx dy  833 h′w (x) = fw (x) wxw−1 − 1 − wxw + xw = (1 − x)2 (1 − x)2 where (70) where the final inequality is obtained by applying the Schwarz Inequality to the functions λ̄n (·, ·) and fX,Y (·, ·), which are viewed as elements of the Hilbert space L2 (R2 ) of functions on R2 with the Euclidean norm. The Schwarz Inequality also says that the inequality in (70) is replaced by an equality, i.e., E[V ] is largest, if, and only if, λ̄n (·, ·) and fX,Y (·, ·) are identical except for a multiplicative constant, i.e., if, and only if, the locally averaged intensity of the node-location process about each SS location exactly matches (to within a multiplicative constant) the pdf of the SS location itself. Since the latter is constant, we require λ̄n (·, ·) to be constant. Then, 2) follows immediately because if λ̄n (·, ·) is a constant, then so is V , which, from Jensen’s Inequality, yields equality in fw (x) = (1 − w)xw + w x1−w −1 is monotonically strictly decreasing for all x ∈ (0, 1) because   w(1 − w) w(1 − w) 1 w(1 − w) ′ fw (x) = 1− − = <0 x1−w x2−w x1−w x for all x ∈ (0, 1). Thus, for all x ∈ (0, 1), we have fw (x) > fw (1) = 1 − w + w − 1 = 0 which in turn implies that h′w (x) > 0 for all x ∈ (0, 1) which proves that hw (x) is monotonically increasing for all x ∈ (0, 1). 834 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 A PPENDIX III A REA OF I NTERSECTION OF T WO D ISKS W ITH A RBITRARY R ADII Define, for any x, y, r, a, b > 0 θ ∈ [0, 2π) A(x, y; r, θ; a, b) ≡ B(x + r cos θ, y + r sin θ; a) ∩ B ′ (x, y; b). (71) Then from [24, eq. (11)], we see that the area of this region does not depend on x, y, and θ and is given by (72), shown at the bottom of the page. A PPENDIX IV D ERIVATION OF P{T1 > 1|T > 1} Suppose we have an SS at (0, 0) and another SS, labeled SS 1, at a distance of R̃ = r̃ from (0, 0). Without loss of generality, we may take the location of SS 1 to be (r̃, 0). SS 1 is given to have a connection to the SS at (0, 0), i.e., the cdf of R̃ is given by (20) 2r Q fR̃|R̃<rSS exp(hZ̃/δ) (r̃) = 2 rSS exp(2α2 )  1 r̃ ln α rSS  where Z̃ is the shadow fade (in decibels) on the link between the SS at the origin and SS 1. Consider the BSs in B(r0 ), where r0 is very large. The number of BSs in B(r0 ) is NBS (r0 ) ∼ Poiss(λBS πr02 ). Consider an arbitrary BS in B(r0 ). Let the distance of this BS from (0, 0) be R and its distance from SS 1 be R′ . Denote the shadow fade (in decibels) between this BS and the SS at (0, 0) by Z and the shadow fade (in decibels) between this BS and SS 1 by Z ′ . The BS has a connection to SS 1 if, and only if, R′ < rBS exp(hZ ′ /δ), i.e., if, and only if, it lies in B(r̃, 0; rBS exp(hZ ′ /δ)), and it has a connection to the SS at (0, 0) if, and only if, it lies in B(0, 0; rBS exp(hZ/δ)). Since the shadow fades on all links are i.i.d., the event that an arbitrary BS in B(r0 ) has a connection to SS 1 but not to the SS at (0, 0) is independent across all BSs in B(r0 ). Furthermore, p′BS (r0 ) which is the probability that an arbitrary BS in B(r0 ) has a connection to SS 1 (at distance r̃ from the origin) given that this BS does not have a connection to the SS at (0, 0), is, therefore, the probability that this BS lies in B(r̃, 0; rBS exp(hZ ′ /δ)) \ B(0, 0; rBS exp(hZ/δ)). Since the location of the BS is uniformly distributed over the disk B(r0 ), this probability is just the fractional area of the above region    ′ area B r̃, 0; rBS ehZ /δ \B rBS ehZ/δ ′ pBS (r0 ) = EZ,Z ′ area [B(r0 )]    ′ EZ,Z ′ area B r̃, 0; rBS ehZ /δ \B rBS ehZ/δ = πr02 (73) where we assume that r0 is large enough to avoid edge effects. ′ (r0 ), which It follows from [16, Sec. 5.1, para. 3, p. 53] that NBS is the number of BSs in B(r0 ) that have a connection to SS 1 but not to the SS at (0, 0), is Poiss(λBS πr02 p′BS (r0 )). Furthermore, since lim πr02 p′BS (r0 )       ′ = EZ,Z ′ area B r̃, 0; rBS ehZ /δ \ B rBS ehZ/δ  2 = EZ,Z ′ πrBS exp(2hZ ′ /δ) r0 →∞ #     ′ − area B r̃, 0; rBS ehZ /δ ∩ B rBS ehZ/δ  2  exp(2hZ ′ /δ) = EZ ′ πrBS    ′ − EZ,Z ′ g r̃; rBS ehZ /δ , rBS ehZ/δ    ′ 2 = πrBS exp(2α2 ) − EZ,Z ′ g r̃; rBS ehZ /δ , rBS ehZ/δ (74) where g(r̃; a, b) is as defined in (72), we see that (conditioned ′ on SS 1 being at r̃ from the origin) NBS , which is the number of BSs in the plane that have a connection to SS 1 but which does not have a connection to the SS at the origin, has the following distribution:  ′ ∼ Poiss µBS exp(2α2 ) NBS  #   hZ ′ /δ hZ/δ , rBS e − λBS EZ,Z ′ g r̃; rBS e . Thus, P{T1 > 1|R̃ = r̃, T > 1}, which is the probability that there is no BS in the plane with a connection to SS 1 but g(r; a, b) ≡ area [A(x, y; r, θ; a, b)] = area [A(0, 0; r, 0; a, b)]  π (min{a, b})2 , if r < |a − b|         2 −1 r 2 +a2 −b2 2 −1 r 2 +b2 −a2 + b cos 2ra 2rb = a cos
 1  − (−r + a + b)(r + a − b)(r − a + b)(r + a + b), if |a − b| ≤ r < a + b   2 0, if r ≥ a + b (72) MUKHERJEE et al.: CONNECTIVITY, POWER, AND ENERGY IN A MULTIHOP CELLULAR-PACKET SYSTEM which does not have a connection to the SS at the origin, is ′ the probability that NBS =0 P{T1 > 1|R̃ = r̃, T > 1} ′ = P {NBS = 0}     ′ . = qBS exp λBS EZ,Z ′ g r̃; rBS ehZ /δ , rBS ehZ/δ (75) Averaging over the values of R̃, we have P{T1 > 1|T > 1} = βqBS where β = ER̃|R̃<rSS exp(hZ̃/δ)      ′ × exp λBS EZ,Z ′ g R̃; rBS ehZ /δ , rBS ehZ/δ (76) = ∞ 0 dr̃ 1 2r̃ 2 exp(2α2 ) Q α ln rSS × exp  λBS 2π ∞ −v 2 /2 dv e  ∞ r̃ rSS  dw e−w 2 /2 −∞ −∞ αw × g (r̃; rBS e # , rBS e ) . αv (77) Note that from (77) 1 < β = ER̃|R̃<rSS exp(hZ̃/δ)     ′ × exp λBS EZ,Z ′ g R̃; rBS ehZ /δ , rBS ehZ/δ    1 2 e2hZ/δ = ≤ exp λBS EZ πrBS qBS i.e., qBS < qBS β ≤ 1. (78) 835 [7] S. A. G. Chandler, “Calculation of number of relay hops required in randomly located radio network,” Electron. Lett., vol. 25, no. 24, pp. 1669–1671, Nov. 23, 1989. [8] S. Mukherjee and D. Avidor, “On the probability distribution of the minimal number of hops between any pair of nodes in a bounded wireless ad-hoc network subject to fading,” in Proc. Int. Workshop on Wireless Ad Hoc Networks 2005 (IWWAN 2005), London, U.K., May 2005. [9] C. Bettstetter and C. Hartmann, “Connectivity of wireless multihop networks in a shadow fading environment,” in Proc. 6th ACM Int. Workshop Model. Anal. and Simul. Wireless and Mobile Syst., 2003, pp. 28–32. [10] J. Orriss and S. K. Barton, “Probability distributions for the number of radio receivers which can communicate with one another,” IEEE Trans. Commun., vol. 51, no. 4, pp. 676–681, Apr. 2003. [11] M. Zorzi and S. Pupolin, “Outage probability in multiple access packet radio network in the presence of fading,” IEEE Trans. Veh. Technol., vol. 43, no. 3, pp. 604–610, Aug. 1994. [12] J. van Rees, “Cochannel measurements for interference limited small-cell planning,” Int. J. Electron. Commun. (Archiv fur Elektronik und Ubertragungstechnik), vol. 41, no. 5, pp. 318–320, 1987. [13] V. Graziano, “Propagation correlations at 900 MHz,” IEEE Trans. Veh. Technol., vol. VT-27, no. 4, pp. 182–189, Nov. 1978. [14] H. W. Arnold, D. C. Cox, and R. R. Murray, “Macroscopic diversity performance measured in the 800-MHz portable radio communications environment,” IEEE Trans. Antenans Propag., vol. 36, no. 2, pp. 277– 281, Feb. 1988. [15] K. Zayana and B. Guisnet, “Measurements and modelisation of shadowing cross- correlations between two base stations,” in Proc. Int. Conf. Universal Pers. Commun., Florence, Italy, 1998, pp. 101–105. [16] J. F. C. Kingman, Poisson Processes. Oxford, U.K.: Oxford Univ. Press, 1993. [17] S. Mukherjee and D. Avidor, “Connectivity, power and energy in a multihop cellular packet system,” in Proc. 62nd IEEE VTC—Fall, Dallas, TX, Sep. 2005, pp. 1702–1707. [18] ——, “Outage probabilities in Poisson and clumped Poisson-distributed hybrid ad-hoc networks,” in Proc. 2nd Annu. IEEE SECON, Santa Clara, CA, Sep. 2005, pp. 563–574. [19] Q. Ma and P. Steenkiste, “On path selection for traffic with bandwidth guarantees,” in Proc. Int. Conf. Netw. Protocols, 1997, pp. 191–202. [20] IEEE Standard for Local and Metropolitan-Area Networks, Part 16: Air Interface for Fixed Broadband Wireless Access Systems, 2004. Download from http://standards.ieee.org. Available: IEEE Std. 802.16. [21] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic, 2000. [22] D. Avidor and S. Mukherjee, “Connectivity and minimum total transmit energy per packet between any pair of nodes in a bounded wireless ad hoc network subject to fading,” in Proc. IEEE WCNC, Las Vegas, NV, Apr. 2006, pp. 372–378. [23] P. J. Diggle, Statistical Analysis of Spatial Point Patterns. London, U.K.: Academic, 1983. [24] E. W. Weisstein, From Mathworld–A Wolfram Math Resource. [Online]. Available: http://mathworld.wolfram.com/Circle-CircleIntersection.html ACKNOWLEDGMENT The authors would like to thank P. Polakos and C. Papadias for their continued support of this research. R EFERENCES [1] T. K. Philips, S. S. Panwar, and A. N. Tantawi, “Connectivity properties of a packet radio network model,” IEEE Trans. Inf. Theory, vol. 35, no. 5, pp. 1044–1047, Sep. 1989. [2] P. Santi and D. M. Blough, “The critical transmitting range for connectivity in sparse wireless ad hoc networks,” IEEE Trans. Mobile Comput., vol. 2, no. 1, pp. 25–39, Jan.–Mar. 2003. [3] O. Dousse, P. Thiran, and M. Hasler, “Connectivity in ad-hoc and hybrid networks,” in Proc. IEEE INFOCOM, New York, 2002, pp. 1079–1088. [4] B. Xu, S. Hischke, and B. Walke, “The role of ad hoc networking in future wireless communications,” in Proc. ICCT, 2003, pp. 1353–1358. [5] P. Piret, “On the connectivity of radio networks,” IEEE Trans. Inf. Theory, vol. 37, no. 5, pp. 1490–1492, Sep. 1991. [6] L. E. Miller, “Probability of a two-hop connection in a random mobile network,” in Proc. 35th Conf. Inf. Sci. and Syst. (CISS 2001), Baltimore, MD, Mar. 2001. Sayandev Mukherjee (S’92–M’97–SM’05) was born in Bangalore, India, in 1970. He received the Bachelor of Technology degree from the Indian Institute of Technology, Kanpur, India, in 1991, and the M.S. and Ph.D. degrees from Cornell University, Ithaca, NY, in 1994 and 1997, respectively, all in electrical engineering. Since 1996, he has been a member of the Technical Staff with the Wireless Research Laboratory, Bell Laboratories, Lucent Technologies, Murray Hill, NJ. His research interests include stochastic models, wireless-system simulations, and connectivity issues in ad hoc wireless networks. 836 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007 Dan Avidor (M’06) was born in Tel Aviv, Israel, on October 18, 1936. He received the B.Sc. degree in electrical engineering from the Technion–Israel Institute of Technology, Haifa, Israel, in 1958, and the M.Sc. and Ph.D. degrees in engineering from the University of California, Los Angeles, in 1970 and 1981, respectively. After completing his studies for the Ph.D. degree, he returned to Israel, where he served as a Research and Development Department Head in the Israeli defense forces. He is currently a distinguished member of the Technical Staff with the Wireless Research Laboratory, Bell Laboratories, Lucent Technologies, Murray Hill, NJ. His current research interests include intelligent antennas, scheduling in high-speed downlink packet access (HSDPA) systems, ad hoc and sensor networks, and digital processing. Katherine Hartman was born in Albany, NY, in 1986. She is currently a sophomore working toward the B.Sc. degree in materials science and engineering at the Massachusetts Institute of Technology (MIT), Cambridge. She was an Intel Science Talent Search Finalist in 2004, with a research project in astronomy. She is currently an undergraduate member of an MIT team researching the electrodeposition of iron–tungsten. Her current interests include solar-cell research and carbon nanotubes.