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Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links Shizhen Zhao, Luoyi Fu, Xinbing Wang Department of Electronic Engineering Shanghai Jiao Tong University, China Qian Zhang Department of Computer Scien Engineering Hong Kong, China

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Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links 2 Outline Background Background Large-scale Networks Large-scale Networks Random Connection Model Random Connection Model First Passage Percolation Model First Passage Percolation Model Network Model & Objective Network Model & Objective Main Results and Intuitions Main Results and Intuitions Simulation Results Simulation Results Concluding Remarks Concluding Remarks

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3 Large-scale Networks Network size is growing. Network size is growing. Number of users is growing Number of users is growing Need more base stations Need more base stations Unreliable links. Unreliable links. Communication between adjacent nodes is not always available Communication between adjacent nodes is not always available Cause of the unreliability. Cause of the unreliability. Increased interference Increased interference Severe environment Severe environment Sleep-wake scheduling Sleep-wake scheduling Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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4 Random Connection Model (RCM) Ramdom Connection Model Ramdom Connection Model Stationary point process (e.x. Poisson point process) Stationary point process (e.x. Poisson point process) Connection function Connection function 1. A non-increasing function h(·) defined on positive reals 2. An edge exists between nodes x 1 and x 2 with probability h(|x 1 - x 2 |) Phase transition. Phase transition. Condition: Condition: There exists a critical node density, such that There exists a critical node density, such that If, an infinite large cluster exists. If, an infinite large cluster exists. 1. if, all clusters are finite almost surely. Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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5 Random Connection Model (RCM) Poisson Boolean Model Poisson Boolean Model Special case of RCM with Special case of RCM with Two nodes are connected if and only if their distance is smaller or equal to Two nodes are connected if and only if their distance is smaller or equal to Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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6 First Passage Percolation Model First Passage Percolation Model. First Passage Percolation Model. Assign a random variable to each link. Assign a random variable to each link. Define the passage time for a path : Define the passage time for a path : Define the first passage time between nodes and : Define the first passage time between nodes and : Remark. Remark. Use first passage time to model delay Use first passage time to model delay Use the random variable to model the unreliability of links Use the random variable to model the unreliability of links Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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7 Outline BackgroundBackground Network Model & ObjectiveNetwork Model & Objective Network Model Network Model Objective Objective Main Results and IntuitionsMain Results and Intuitions Simulation ResultsSimulation Results Concluding RemarksConcluding Remarks Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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8 Network Model-Connectivity Delay is a basic concern in designing and implementing large scale wireless networks. However, the premise of communication is connectivity. Delay is a basic concern in designing and implementing large scale wireless networks. However, the premise of communication is connectivity. In large scale wireless network with unreliable links, it is too costly to maintain full connectivity. Therefore, we consider a slightly weaker connectivity-connectivity in percolation sense. In large scale wireless network with unreliable links, it is too costly to maintain full connectivity. Therefore, we consider a slightly weaker connectivity-connectivity in percolation sense. Random Connection Model (RCM) Random Connection Model (RCM) Instantaneous and Long-term Connectivity Instantaneous and Long-term Connectivity Instantaneous Critical Density ( ) and Long-term Critical Density ( ) Instantaneous Critical Density ( ) and Long-term Critical Density ( ) Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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9 Network Model-Connectivity Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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10 Network Model-Delay Usually, the time needed for links to change state is much larger than the time scale used in scheduling, routing, etc. Therefore, we assume that it won't take much time for transmission between connected node pairs. In this paper, we mainly focus on the delay caused by the lack of instantaneous connectivity which is closely related to the node density in such a netwok. Usually, the time needed for links to change state is much larger than the time scale used in scheduling, routing, etc. Therefore, we assume that it won't take much time for transmission between connected node pairs. In this paper, we mainly focus on the delay caused by the lack of instantaneous connectivity which is closely related to the node density in such a netwok. Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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11 Network Model-Delay We assign the delay to each link in the following way, We assign the delay to each link in the following way, where is the connection function, and is the length of link. where is the connection function, and is the length of link. Applying first passage percolation model here, we could define the first passage time for node and : Applying first passage percolation model here, we could define the first passage time for node and : Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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12 Objective Previous result: Previous result: We study: We study: Detailed relationship between and node density. Detailed relationship between and node density. Two steps: Two steps: Ignore the propagation delay ( ). Ignore the propagation delay ( ). Consider the propagation delay ( ). Consider the propagation delay ( ). Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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13 Outline BackgroundBackground Network Model & ObjectiveNetwork Model & Objective Main Results and IntuitionsMain Results and Intuitions Main results Main results Intuitions Intuitions Impact of Propagation Delay Impact of Propagation Delay Simulation ResultsSimulation Results Concluding RemarksConcluding Remarks Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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14 Main Results Properties of, Properties of, There exists, such that There exists, such that For any, For any, is a monotone decreasing function is a monotone decreasing function Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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15 Main Results Upper bound and lower bound of Upper bound and lower bound of is a constant is a constant g(·) is the connection function g(·) is the connection function is the size of a cluster is the size of a cluster E(·) is the expectation E(·) is the expectation Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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16 Intuitions-Upper Bound For two nodes and, the first passage time is bounded by the the passage time along any path connecting and. For two nodes and, the first passage time is bounded by the the passage time along any path connecting and. least number of hops average delay of one hop distantce of one hop Basic idea: and are connected by a sequence of hops with distance smaller or equal to Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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17 Intuitions-Upper Bound Existance of such paths Existance of such paths The long-term critical density can be viewed as the critical density of the Poisson Boolean Model with transmission range The long-term critical density can be viewed as the critical density of the Poisson Boolean Model with transmission range (Proposition 2 in [14]) Let be the critical density for the Poisson Boolean Model when the transmission range is, then (Proposition 2 in [14]) Let be the critical density for the Poisson Boolean Model when the transmission range is, then The critical density for is, and. Therefore, such a path exists. The critical density for is, and. Therefore, such a path exists. Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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18 Intuitions-Upper Bound Least Number of hops of such paths Least Number of hops of such paths Scale the network up by, then each hop of such paths becomes, and Scale the network up by, then each hop of such paths becomes, and We can show that We can show that Then, we can see that Then, we can see that Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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Intuitions-Lower Bound Waiting occurs at the boundry of a cluster Waiting occurs at the boundry of a cluster 19 M M M M M M M M M Cluster to Cluster transmission Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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20 Intuitions-Lower Bound Delay between and Delay between and Motivation: The minimum number of clusters between and gives a lower bound of delay Motivation: The minimum number of clusters between and gives a lower bound of delay Let the size of a cluster be, Let the size of a cluster be, then the minimum number of then the minimum number of clusters is lower bounded by clusters is lower bounded by number of clusters-1 Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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21 Impact of Propagation Delay Upper Bound Upper Bound The average delay of one hop changes The average delay of one hop changes Lower Bound Lower Bound The minimum number of The minimum number of clusters is lower bounded clusters is lower boundedby average delay of one hop changes Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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22 Outline BackgroundBackground Network Models & ObjectiveNetwork Models & Objective Main Results and IntuitionsMain Results and Intuitions Simulation ResultsSimulation Results without Propagation Delay without Propagation Delay with propagation delay with propagation delay Concluding RemarksConcluding Remarks Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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23 Without Propagation Delay Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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24 With Propagation Delay Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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25 Outline IntroductionIntroduction K-hop Clustered Network ModelsK-hop Clustered Network Models Main Results and IntuitionsMain Results and Intuitions The Impact of MobilityThe Impact of Mobility Concluding RemarksConcluding Remarks Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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26 Concluding Remarks We have studied the first passage delay of large scale network with unreliable links. We have studied the first passage delay of large scale network with unreliable links. In two senarios (with propagation delay and without propagation delay), we In two senarios (with propagation delay and without propagation delay), we sketch by excavating its properties; sketch by excavating its properties; provide upper bound and lower bound to. provide upper bound and lower bound to. Fundamental Relationship between Node Density and Delay in Wireless Ad Hoc Networks with Unreliable Links

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