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3rd Quarter FY07 Progress Report Task Highlights for the Quarter Q3’07 [Period: 4/1/07 - 6/30/07] Subtask 1 Scalable MANET Design and Analysis (Telcordia: Kant, Krishnan & Gopalakrishnan) (a couple of bullet items) Subtask 2: General Optimization Framework for Multi-User Flow Control (UMinn: Giannakis) (a couple of bullet items) Subtask 3: Co-operative Communications and Network Coding (UMd: Ephremedies) (a couple of bullet items) Subtask 4: Fundamental Properties and Cross-Layer Design of Wireless Sensor Networks: A Stochastic Optimization Approach (UCDavis: Zhao) (a couple of bullet items) Subtask 5: Non-Local Models for MANETs (UMd: La, Makowski, Martins, UMd) Subtask 5.1: Non-local models for MANETs We continued studying how the distribution of nodes affects the critical transmission range in wireless networks which can be modeled as geometric random graphs on intervals. In Q3FY2007 the emphasis was given to understanding the nature of associated phase transitions. We continued investigating the structural properties of the random connection model, a model recently introduced by Hekmat and van Mieghem in the context of wireless networks. This class of models offers the possibility of incorporating pathloss and fading which are more realistic than in the usual disk model. Subtask 5.2: Overhead in MANETs We continued our investigation of the minimum routing overhead in static MANETs. Using a family of strategies, we identified a trade-off between propagation of topology information in a local neighborhood (maintenance of routing information in a local neighborhood of destinations) and flooding of data packets outside the neighborhood with topology information. Subtask 5.3: Impact of MAC on network connectivity 1 3rd Quarter FY07 Progress Report We refined a model we developed in the previous quarter for studying the impact of medium access control on network connectivity in MANETs. Our model captures the inherent spreading of packet transmissions in space and in time. Task ID: SWMN-07-04 Task Name: Network Science Principal Investigators: Associate Investigators: Industry PI: Dr. Latha Kant Name: Dr. K.R.Krishnan, Telcordia Technologies Affiliation: Telcordia Technologies Citizenship: USA Address: RRC 1A 235, One Telcordia Drive, Task#: 1 Piscataway, NJ 08854 Email: lkant@telcordia.com Phone: 732.699.2428; Fax: 732.336.7026 Country of Citizenship: USA ARL PI: Dr. Ananthram Swami Name: Praveen Gopalakrishnan, Telcordia Affiliation: ARL Technologies Address: Citizenship: India Email: aswami@arl.army.mil Task#: 1 Phone: 301.394.2486 Country of Citizenship: USA Name: Dr. Georgios B. Giannakis, University of Minnesota Citizenship: USA Task #: 2 Name: Dr. Anthony Ephremides, University of Maryland Citizenship: USA 2 3rd Quarter FY07 Progress Report Task#: 3 Name: Dr. Qing Zhao Citizenship: P.R. China Task#: 4 Name: Dr. Richard J. La, University of Maryland Citizenship: USA Task#: 5 Name: Dr. Armand M. Makowski, University of Maryland Citizenship: Belgium Task#: 5 Name: Dr. Nuno Martins, University of Maryland Citizenship: Portugal Task#: 5 Research Issues Addressed Broadly speaking, we view “Network Science” as the “science of interconnections”. We then observe that when an “interconnected structure” or “network” exhibits properties that are not mere extrapolations of the properties of the individual elements of the structure, we are in the realm of network science. The objective of this task to identify and address issues that are fundamental to the realm of network science. Before we expand on the research issues identified, we observe that since the term “network” is very broad, for example, it can correspond to electronic communications networks, social networks, biological networks, etc., the focus of this technical area will be on electronic communications networks. In particular, since mobile ad hoc networks (MANETs) are becoming the basis of the future Army networks and the network centric warfare (NCW) paradigm due to their enormous flexibility and dynamism, the focus of this technical area will be on mobile ad hoc networking and MANET-related issues. The flexibility and dynamism of MANETs that render them very powerful also introduce fundamental research questions and render analyses very challenging. We identify below, basic questions and associated research issues that will be addressed by Tasks 1 through 5, under this technical area of Network Science. When a network of mobile ad hoc nodes need to communicate, what is the underlying network capacity? Note network capacity differs from an individual node’s capacity in wireless networks. Does an analogue of “Shannon’s Limit” exist for the MANET world and if so what is it? Recall, MANET capacity is a time varying quantity – a stochastic process whose quantification depends on a variety of factors. What specific information control techniques based on a systematic set of reasoning principles and algorithms vs. heuristic techniques need to be researched and developed in order to reliably deliver applications based on their Quality of Service (QoS) requirements over a collection of mobile ad hoc nodes? What is common to both Cooperative Communications and Network Coding and how can we exploit them to achieve more efficient communications amongst nodes that have random distribution (mobility)? Basic research in the area of Network coding for MANETs will be conducted to provide insights to the above problem. 3 3rd Quarter FY07 Progress Report How do the PHY and MAC layers interact when designing large scale sensor networks? The cross-layer nature of this topic combined with the uncertainty of the underlying network, fundamental research using stochastic optimization techniques will be employed to answer the above question. How is connectivity defined and determined amongst mobile ad hoc nodes? For example, edge connectivity should be determined by conditions involving several random variates such as node location and fading intensity, to name a few. Connectivity graph models should incorporate the important observation that a communication link between two nodes is determined by all active users, and not just by the pair of users, giving rise to non-local connectivity in these models. An additional question that arises is: “are there formal techniques to quantify MANET routing overheads and understand their scaling properties, and if so, what are they based on?” Subtasks 1 through 5 address each of the above-mentioned issues and more. In what follows below, we provide a status update on progress-to-date for each of the five tasks. Progress Summary SUBTASK 1: Scalable MANET Design and Analysis (Kant, Krishnan & Gopalakrishnan, Telcordia Technologies) (a few bullet items) SUBTASK 2: General Optimization Framework for Multi-User Flow Control (Giannakis, UMinn) (a few bullet items) SUBTASK 3: Cooperative communications and Network Coding (Ephremides, UMd) (a few bullet items) SUBTASK4: Fundamental Properties and Cross-Layer Design of Wireless Sensor Networks: A Stochastic Optimization Approach (Zhao, UC Davis) (a few bullet items) SUBTASK5: Non-Local Models for MANETs (La, Makowski, Martins, UMd) 4 3rd Quarter FY07 Progress Report Subtask 5.1: Non-local models for MANETs As a way to better understand the impact of mobility on connectivity, we have continued investigating how the distribution of nodes affects the critical transmission range for connectivity in geometric random graphs on intervals. In Q3FY2007 we further developed an asymptotic theory for the critical transmission range when the node placement density does not vanish. This was achieved by establishing a limit result for the maximal spacing associated with i.i.d. samples drawn from a distribution with compact support and non-vanishing density. This result complements the very strong zero-one law for connectivity (established in Q3FY2007) when the node placement distribution admits a non-vanishing density, and leads to exact asymptotics of the transition width for network connectivity. We have continued exploring the implications of these results for resource dimensioning. We have developed zero-one laws for the property of node isolation in the random connection model in two dimensions under uniform node placement. Subtask 5.2: Overhead in MANETs We continued our investigation of the minimum routing overhead in static MANETs: We considered a family of routing strategies (which can be viewed as a combination of on-demand and table-driven routing protocols) and identified a trade-off between propagation of topology information in a local neighborhood (maintenance of up-to-date routing information to destinations in local neighborhoods) and flooding of data packets outside the neighborhood around the intended destination with topology information. Subtask 5.3: Impact of MAC on network connectivity We refined the new model we developed in the second quarter for studying the impact of MAC protocol on network connectivity of a MANET with interference. The model captures (i) the time sharing aspect of MAC and (ii) the resulting interference as a function of the local density, i.e., the number of neighbors, in a neighborhood. Deliverables Due Milestone / Deliverable Status 1Q’07 Quarterly Report; conference papers (Telcordia, SubTask 1) Telcordia – Task 1: Complete (Quarterly Report). Conference papers in preparation. Report, conference, and journal papers on optimal congestion control in contention-based wireless tactical networks (UMinn – SubTask 2) Extension of the simple relay idea for MAC-level cooperation through Network Coding. (UMd – SubTask 3) 5 UMin – Task 2: Completed. Drafts + Matlab codes available upon request. See publication list below. UMd- Task 3: Our first milestone to convert throughputs to spectral efficiencies has been fully met. Next set of milestones with regards to examining cooperative relay techniques by means of Network Coding and 3rd Quarter FY07 Progress Report identifying similarities networks have begun. Quarterly report; SubTask 4) conference papers (UCD- to biological UCD – Task 4: Complete (Quarterly report); See publication list below. UMd – Task 5: One paper being prepared for submission to the IEEE Transactions on Information Theory. Development of a model for studying the minimum expected routing overhead in static MANETs (UMd - SubTask 5) One paper submitted to ISIT 2007. Complete 2Q’07 Telcordia – Quarterly Report Completed. Quarterly Report (Telcordia, Task 1) Preliminary Design Document on Capacity Analysis and Scalable (Telcordia, SubTask 1) MANET Routing Report , conference, and journal papers on joint congestion, routing and contention control algorithms (UMinn – SubTask 2) Use of Fountain Codes for wireless multicasting for both transmitter-based as well as receiver-based throughput measures. (UMd – SubTask 3) Quarterly report; conference and journal papers (UCD – SubTask 4) Telcordia - Completed and Delivered document on “Preliminary design document on MANET Capacity Analysis” along with this quarterly report. UMinn- Quarterly report completed. See references for conference and journal paper submission. UMd (Ephremedies) – Quarterly report complete. UCD – Task 4: Complete (Quarterly report); See also “Publications” for papers submitted. Study of SINR graphs with fading (onedimensional case); zero-one laws for node isolation. (UMd – SubTask 5) UMd – Task 5: Quarterly Report completed. routing overhead in static MANETs. (UMd – SubTask 5) One paper is being prepared for submission to the IEEE Transactions on Information Theory Development of a model for studying the effects of MAC protocols. (UMd – SubTask 5) Paper was accepted for inclusion in the program of ISIT 2007 (Nice, France, June 2007) Quarterly Report and (Telcordia, SubTask 1) . One paper is being prepared for submission to the IEEE Transactions on Information Initial investigation of the minimum expected Theory 3Q’07 Paper Submissions Report, conference, and journal papers on joint flow congestion control and multi-user scheduling in hybrid wireline-wireless tactical MANETs (UMinn – SubTask 2) 6 3rd Quarter FY07 Progress Report Conversion of throughput gains to Spectral Efficiencies for Cross-Layer Integration (UMd– SubTask3) Quarterly report; conference and journal papers (UCD – SubTask 4) Quarterly Report and Paper Submissions (UMd, SubTask 5) UMd – Task 5: Quarterly Report completed One paper was submitted to the IEEE Transactions on Information Theory One paper is being prepared for submission to the IEEE Transactions on Information Theory Study of SINR graphs with fading (twodimensional case); zero-one laws for node isolation (UMd – SubTask 5) Zero-one laws for one-dimensional geometric random graphs – non-uniform case (UMd SubTask 5) Exact asymptotics for width of phase transition have been derived Continuing investigation of the minimum expected routing overhead in static MANETs with a focus on identifying a lower bound on the routing overhead (as a function of the delay constraint T). (UMd – SubTask 5) A formula was obtained for quantifying a tradeoff between latency and transmission rate, due to routing overhead. Integration of carrier sensing in the model and initial study of the effects of carrier sensing on network connectivity. (UMd – SubTask 5) 4Q’07 Delayed Quarterly Report (Telcordia, SubTask 1) Final Design Document on MANET Capacity Analysis and Scalable Routing, and Paper Submissions (Telcordia, SubTask 1) Report, conference, and journal papers on distributed version of our joint congestion control and scheduling design (UMinn – SubTask 2) Exploration of Similarities to Biological Networks. (UMd – SubTask 3) Comparison of Cooperative vs Competitive Multiple Access through Network Coding and other diversity techniques and Connection to the Back-pressure algorithm ( UMd – SubTask 3) Quarterly report and Summary of GFY07 Progress; 7 A manuscript, for submission in the IEEE Transactions on Information Theory is under preparation. Delayed 3rd Quarter FY07 Progress Report conference and journal papers (UCD – SubTask 4) Study of SINR graphs with fading (onedimensional case); zero-one laws for connectivity. (UMd – SubTask 5) Initial investigation of routing overhead in mobile networks with a focus on identifying a lower bound on the overhead due to node mobility. (UMd – SubTask 5) Continuing study of the effects of carrier sensing on network connectivity and integration of the power control (with fixed modulation and coding schemes) in the model. (UMd – SubTask 5) Plans for Next Quarter SUBTASK 1: Scalable MANET Design and Analysis (Kant, Krishnan & Gopalakrishnan, Telcordia Technologies) (a couple of bullet items) SUBTASK 2: General Optimization Framework for Multi-User Flow Control (Giannakis, UMinn) (a few bullet items) SUBTASK 3: Cooperative communications and Network Coding (Ephremides, UMd) (a few bullet items) SUBTASK4: Fundamental Properties and Cross-Layer Design of Wireless Sensor Networks: A Stochastic Optimization Approach (Zhao, UC Davis) (a few bullet items) 8 3rd Quarter FY07 Progress Report SUBTASK5: Non-Local Models for MANETs (La, Makowski, Martins, UMd) Subtask 5.1: Non-Local Models for MANETs We shall continue to investigate the structural properties of the random connection model in two dimensions. In particular, building on progress made in Q3FY2007 we will develop zero-one laws for graph connectivity when the connection function has bounded support. We plan to begin a study of how the distribution of nodes affects the critical transmission range in wireless networks which can be modeled as geometric random graphs in two dimensions, say the original disk model of Gupta and Kumar. In Q4FY2007, as a preliminary step, emphasis will on understanding the existence of zero-one laws for the property that no isolated nodes exist. Subtask 5.2: Overhead in MANETs We will continue the investigation of the minimum routing overhead in static MANETs. In particular, we will focus on identifying a lower bound on the minimum expected routing overhead subject to a delay constraint with increasing network size. We will also start formulating the problem of studying routing overheads with mobile nodes. Subtask 5.3: Impact of MAC on network connectivity Using the model developed in 3QFY07, we will investigate the impact of the MAC protocol on network connectivity with a physical layer model (with interference). We will also extend the model to the case with power control by the nodes. Detailed Progress SUBTASK 1: Scalable MANET Design and Analysis (Kant, Krishnan & Gopalakrishnan, Telcordia Technologies) (detailed write-up goes in here) SUBTASK 2: General Optimization Framework for Multi-User Flow Control (Giannakis, UMinn) (detailed write-up goes in here) SUBTASK 3: Cooperative communications and Network Coding (Ephremides, UMd) (detailed write-up goes in here) 9 3rd Quarter FY07 Progress Report SUBTASK4: Fundamental Properties and Cross-Layer Design of Wireless Sensor Networks: A Stochastic Optimization Approach (Zhao, UC Davis) (detailed write-up goes in here) SUBTASK5: Non-Local Models for MANETs (La, Makowski, Martins, UMd) Subtask 5.1: Non-local models for MANETs We consider the random network where n points are placed independently on the unit interval [0,1] according to some probability distribution function F. Two nodes communicate with each other if their distance is less than some transmission range > 0. When F admits a continuous density f with f * inf( f ( x), x [0,1]) > 0, we have shown [HanMakowski2007a] by direct probabilistic methods that the property of graph connectivity for the underlying random graph admits a strong critical threshold given by n* 1 log( n) , n 2,3,... f* n (1) This amounts to the following statement: Let P(n;) denote the probability that the random graph with n nodes is connected when the communication range is . Then, we have lim n 0 if 0 c 1 P(n; c ) 1 if 1 c * n for any range function : N 0 R [HanMakowski2007a] for details. such that n ~ c n* for some c>0. See In Q2FY2007 we also showed that the property of graph connectivity admits a very strong threshold, and we identified the corresponding (very strong) threshold: Under mild conditions on the density function f, we showed that lim n 0 if lim n n P(n; ) 1 if lim n n * n when the range function : N 0 R is determined through 1 log( n) log(log( n)) n r n , n 1,2,... n with deviation function : N 0 R where r > 0 is some parameter indicating how “flat” this density is near its minimum; see [HanMakowski2007b] for additional details. 10 3rd Quarter FY07 Progress Report Very strong thresholds are forerunners of sharp phase transitions: For each n=2,3,…, P(n;) is monotone increasing in with a transition from P(n; ) 0 to P(n; ) 1 as varies across some critical range, and the sensitivity to “small” deviations implied by the last equation suggests that the width of this transition interval decreases rapidly with large n. It is therefore natural to seek estimates of how quickly this transition takes place. This issue is by now well understood under uniform node placement, e.g., see [HanMakowski2006]. In Q3FY2007 we have made some progress on this issue in the non-uniform case: Under the assumptions leading to the very strong zero-one law, we have derived a limit result on the maximal spacing associated with n i.i.d samples drawn from the distribution with density f. This result generalizes a well-known result of P. Levy for uniform samples, and leads very naturally to the conclusion that the transition width is of the form n a 1 C a 1 o f* n n log a . Here n a n 1 a n a where for log 1 a for all 0 a 1 / 2 with C a log each b in 0,1 , n b is the unique solution to the equation Pn, b, 0 1 . See [HanMakowski2007c] for details. From a network engineering viewpoint, a small transition width would imply that if the transmission range of the nodes is set slightly larger than the critical threshold in a large network, then the probability that the network remains connected will be high (i.e., close to one). On the other hand, if the transition width is large, then the transmission range may need to be set considerably larger than the critical threshold in order to ensure network connectivity for most of the time. It is noteworthy that the density function enters the critical thresholds through the inverse of its minimum, a quantity which is notoriously hard to estimate. Therefore, the lack of accurate estimates or simply ignorance of the true density function, may force rather large assignments, in fact much larger than the ones associated with the (very) strong zero-one laws. The work on the random connection model under uniform placement is still in a preliminary phase and will be developed further in Q4FY2007. However, some progress was made concerning the derivation of very strong zero-one laws for the property that there exists no isolated nodes under the assumption that the connection function has bounded support. Subtask 5.2: Overhead in MANETs The model we use to study the minimum routing overhead in static MANETs is as follows: Consider a network of n nodes that are just deployed over some domain D in R2 or R3. These nodes are assumed to be unaware of the locations of other nodes. Further, we assume that the nodes are placed independently of each other according to a common distribution and once deployed, the locations of the nodes are fixed, i.e., a static network. There is a fixed set of source-destination pairs that need to communicate with each other. The source and destination nodes are randomly selected and unaware of the location of each other. Each source needs to send a packet to the corresponding destination without any prior routing information. We impose a delay constraint that the packet must reach the destination within T seconds for all source- 11 3rd Quarter FY07 Progress Report destination pairs. One can view this delay as the initial delay incurred before the network can begin to deliver packets between the source and the destination. The distance between a pair of source and destination nodes will depend on both the locations of the nodes. Hence, the minimum overhead required to route the packet will be random. Instead of attempting to compute the minimum required overhead for every realization, we focus on the expected minimum overhead. The questions we are interested in investigating are: (i) For a fixed T, how does the expected minimum overhead in bits meter/second scale with increasing network size, especially in relation to network throughput? (ii) How does the expected minimum overhead behave as a function of the deadline T? Note that not every node needs to be involved in delivering packets, and a subset of nodes may suffice. Related to these questions is the following issue: How long will it take on the average before a network becomes operational in the sense that it can deliver packets between sources and destinations, as the network size increases? Our preliminary results for static networks show that, for any fixed time horizon T after the initial deployment, the fraction of the nodes that can, at least, communicate their label (or ID) to one neighbor converges to zero as the number of nodes grows. This fact clearly proves the importance of studying such an initial transitory behavior. At a larger timescale, there is also a trade-off between allocating the network resources for exchanging/propagating routing information or for transmitting information. First, consider the following two extreme cases: In the first case the network topology information is broadcast throughout the network to all nodes. This allows the nodes to construct a network map and to use an efficient routing algorithm, e.g., Dijkstra's algorithm. In the second case, on the other hand, most of the network resource is allocated for the communication between nodes, and only little resource is assigned to propagating network topology information. The first scheme may demand that most of the network resource be dedicated to propagation of network topology information, leaving very little or no resource for the transmission of information between nodes. In the latter scheme, the lack of information about the network topology will preclude the use of efficient routing algorithms, but instead force the use of flood strategies, leading to low throughput. Our initial studies show that, in general, this problem persists even if we allow strategies inbetween these two extreme cases. In particular, we can show that there exists a universal lower bound on the expected minimum delay after deployment, before which no communication can take place. In addition, for any given positive throughput, there is a minimum delay before which such a throughput is not feasible. This finding can be summarized by the following equation: n min A latency Tn min Rn log 2 n Dn min 2 Rnmin C n Rn where Tnlatency is the required latency, Rnmin is the minimum radius around the destinations where the routing information to the destinations is available, Cn is the network transport capacity, A is the area of the domain, and , , are positive constants. When Gupta and Kumar’s results are used, we obtain latency Tn 12 n log 2 n 3rd Quarter FY07 Progress Report Subtask 5.3: Impact of MAC on network connectivity When distributed packet scheduling is employed in MANET, the interference at a receiver is unlikely to be known in advance. This is because the set of simultaneous transmitters is in general unknown to the receivers, and is hard to control without a centralized controller. This difficulty is further complicated by two additional factors: (i) Oftentimes there are constraints on the set of transmitters T when Medium Access Control (MAC) schemes are employed, and such constraints are difficult to model accurately, and (ii) the set of transmitters changes with time (i.e., T(n), n = 0, 1, …, in discrete time model), making it virtually impossible to predict the interference at a receiver accurately. First, many of MAC protocols (e.g., IEEE 802.11) employ carrier sensing; a node first listens to the channel to make sure that it does not hear any other transmissions taking place over the same channel (in its neighborhood). Carrier sensing tends to limit the number of transmissions in a local neighborhood and spread them out both in space and in time. Therefore, it imposes a constraint on the set of transmitters that access the channel simultaneously. Second, as the set of transmitters changes with time and is not known in general, it may be too restrictive, if not impossible, to insist that the realized SINR exceeds a certain threshold for all transmissions in order for two nodes to be neighbors (i.e., can communicate with each other). In practice, an automatic repeat request (ARQ) protocol will retry unsuccessful transmissions. Thus, it is not necessary for every transmission to be successful. Thus, we assume that there exists a link between two nodes (i.e., the two nodes are neighbors) if a transmission between them is successful with a high probability. This requirement can be written as Pi Gij P SINR N j Pk Gkj kT \{ i , j } where 0 < < 1, N j is the noise power at receiver j, Pk and G kj are the transmit power of node k and channel gain from node k to node j, respectively, SINR is the threshold on signal to interference plus noise ratio (SINR) for successful transmission, and T is a random set of transmitters. Note that this condition can be rewritten as Pi Gij P SINR N j I k Pk Gkj k i , j where I k 1 if k T and I k 0 otherwise. In general these indicator functions will not be mutually independent due to carrier sensing. Let T * 2 N be the set of all subsets of the set of nodes N. Now the question is what suitable probability distribution on T * one may assume. From the viewpoint of studying network connectivity, the problem is more interesting when we assume the worst scenario where the interference is as strong as it is allowed to be. One can 13 3rd Quarter FY07 Progress Report argue that the interference experienced at the receivers will be the largest when the network is congested and every node has packets to transmit and attempts to access the channel at all times. This is implicitly assumed throughout the rest. 1. Static model: First order approximation without dynamics: When every node always has packets to transmit, the fraction of time a node gets to access the channel will be (roughly speaking) inversely proportional to the number of its neighbors whose transmissions it can sense. One may be able to approximate the number of such neighbors within a given carrier sensing range using the neighbors that lie within a fixed radius from the node (similar to the disk model). The variable denotes the sensing range of a node. Let N i denote the number of neighbors within radius of node i. One can now assume that the fraction of time i 1 node i is active is proportional to N , i.e., the fraction of time node i transmits is / N i _for some 0<< 1. We now replace the previous condition with Pi Gij P SINR Pk Gkj N j k k i , j N This is equivalent to replacing the indicator functions in the previous condition with their expected values. This model attempts to capture the effects of carrier sensing of MAC schemes: A node in a dense neighborhood may experience stronger interference from its neighbors that are active. However, in such a dense neighborhood, the number of simultaneously active nodes will be limited by carrier sensing. Refinement: The previous model we focused on in the previous quarter attempts to model the first-order effect of scheduling in that it limits the fraction of time a node accesses the channel as a function of the number of neighboring nodes whose transmission it can sense. However, it suffers from an obvious shortcoming that it tends to overestimate the interference from the neighbors within the sensing range of a receiver. This is because the unconditional distribution of interference at the receiver j is likely to be considerably different from that conditional on the event that it is receiving a packet. In other words, when the receiver broadcasts a control packet, e.g., CTS, to reserve the channel, most (if not all) of its neighbors that can detect the control packet will remain silent. We model this by introducing a function f(d) in the previous condition as follows: Pi Gij P SINR f (d kj ) Pk Gkj N j k N k i , j where d kj is the distance between nodes k and j, and f(d) is an increasing function with f(0) = 0 and f(d) = 1 for all d > . The function f(d) models the probability that another node within the sensing range of the receiver will access the channel at the same time as 14 3rd Quarter FY07 Progress Report a function of their distance. If carrier sensing is perfect, we should have f(d) = 0 if 0<d < and f(d) = 1 otherwise. 2. Probabilistic model: As mentioned earlier, a source of difficulty is that the set of simultaneous transmitters that causes interference at a receiver is random. There has been some recent work on trying to understand the distribution over the sets of simultaneous transmitters T * (e..g, work by Durvy, Dousse and Thiran, IEEE Infocom 2007). In this quarter, we started developing a probabilistic model that first identifies the distribution over T * and allows us to use the first criterion for determining the one-hop connectivity. Publications & Presentations Journal Papers: [HanMakowski2007a] G. Han and A.M. Makowski, “A strong zero-one law for connectivity in onedimensional geometric random graphs with non-vanishing densities," submitted to IEEE Transactions on Information Theory (2007). [HanMakowski2007b] G. Han and A.M. Makowski, “A very strong zero-one law for connectivity in onedimensional geometric random graphs with non-vanishing densities," submitted to IEEE Transactions on Information Theory (2007). [HanMakowski2007c] G. Han and A.M. Makowski, “On the width of phase transition for connectivity in one-dimensional geometric random graphs with non-vanishing densities," in preparation and to be submitted to IEEE Transactions on Information Theory (2007). Conference Papers: [HanMakowski2006] G. Han and A.M. Makowski, “Very sharp transitions in one-dimensional MANETs," in Proceedings of the International Conference on Communications (ICC 2006), Istanbul (Turkey), June 2006. Presentations: A.M. Makowski, “On the critical communication range in one-dimensional geometric random graphs: Sensitivity to node placement distribution," Workshop on Mathematical Modeling and Analysis of Computer Networks, Ecole Normale Superieure, Paris (France), June 2007. “On the critical communication range in one-dimensional random networks under non-uniform node placement," Seminaire Reseaux, INRIA Sophia-Antipolis, Valbonne (France), June 2007. 15 3rd Quarter FY07 Progress Report Summer Research Institute, School of Computer and Communication Sciences, Ecole Polytechnique Fed\'erale de Lausanne, Lausanne (Switzerland), July 2007. 16