ABSTRACT. Mobility is the most important factor in mobile ad-hoc networks (MANETs) and

Transcription

1 ABSTRACT KIM, SUNGWON. Super-Diffusive Behavior in Human Mobility and Finding Relevant Models in Mobile Opportunistic Networks. (Under the direction of Professor Do Young Eun.) Mobility is the most important factor in mobile ad-hoc networks (MANETs) and delay-tolerant networks (DTNs), and has posed a serious challenge to the analysis and design of protocols on such networks. The mobility pattern directly impacts time-varying contact/inter-contact dynamics among mobile nodes, which in turn affects the performance of any protocol built over these mobility patterns. Mobility models that fail to capture key characteristics in the movement patterns of mobile nodes will result in misleading guidelines on the design of new protocols and evaluations of their performance and thus prevent us from making a right decision on our choice. In this dissertation, we first study numerous GPS mobility traces of human mobile nodes and observe super-diffusive behavior in all GPS traces, which is characterized by a faster-than-linear growth rate of the mean square displacement (MSD) of a mobile node. Then, we investigate a large amount of access point (AP) based traces, and develop a theoretical framework built upon continuous time random walk (CTRW) formalism, in which one can identify the degree of diffusive behavior of mobile nodes even under possibly heavy-tailed pause time distribution, as in the case of reality. We then show that current synthetic mobility models are unable to produce various degrees of diffusive behavior, and propose to use a set of Lévy walk models as an alternative. In addition, we show that diffusive properties make a huge impact on contact-based metrics and the performance of routing protocols, and existing models such as random waypoint, random direction model or Brownian motion can predict the result overly optimistic or pessimistic

2 when diffusive properties are not properly captured. These results collectively suggest that the diffusive behavior of mobile nodes should be correctly captured and taken into account for the design and comparison study of network protocols. We then extend our study into the publish/subscribe case where each mobile node can be both a publisher and subscriber and opportunistic contacts are used for spreading out up-to-the-minute contents. This scenario has drawn a lot of attention recently due to the fast development of mobile devices and social networks with many viable applications, and the age (freshness) of contents would be the main interest in performance evaluation for this case. We first simplify the overall age dynamics in this scenario by two parameters, which are content update rate and contact rate among mobile users. Then, we show that, when content updates at each node and contacts among mobile nodes follow Poisson processes, the age dynamics can be obtained via a simple ordinary differential equation (ODE) in terms of the average content update rate and contact rate among mobile nodes. Finally, we provide a guideline where the ODE-based description can be conveniently used to predict the average age and discuss any deviation from the ODE solution in non-poisson contact mobility models, and discuss the degree of deviation from the ODE solution for non-poisson cases. Interestingly, our result shows a sharp contrast to many previous works that have mainly focused on the impact of mobility models, and provides a new perspective on proper mobility modeling in various network scenarios.

3 Super-Diffusive Behavior in Human Mobility and Finding Relevant Models in Mobile Opportunistic Networks by Sungwon Kim A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Computer Engineering Raleigh, North Carolina 211 APPROVED BY: Dr. Arne Nilsson Dr. Wenye Wang Dr. Alun Lloyd Dr. Do Young Eun Chair of Advisory Committee

4 DEDICATION To my parents To my brother and sister ii

5 BIOGRAPHY Sungwon Kim received the B.S. degree in Electrical Engineering from Ajou University, Suwon, South Korea, in 23, and received the M.S. degree in Computer Engineering from North Carolina State University, Raleigh, NC, in 26. Since 26, he has been a Ph.D. student in the Department of Electrical and Computer Engineering at North Carolina State University. His research interests include mobility modeling in mobile ad-hoc networks and delay tolerant networks. iii

6 ACKNOWLEDGEMENTS Firstly, I would like to thank my advisor Dr. Do Young Eun. It was truly my honor and privilege to receive his guidance on research during my M.S. and Ph.D study at North Carolina State University. Dr. Eun has consistently encouraged and inspired me by recommending the best research directions and showing me the big pictures on research topics. He has motivated me to make the best effort to pursue the high quality research, but he has been always forgiving. Thanks to his support and dedication, I could become a much better and confident researcher. I also thank Dr. Arne Nilsson, Dr. Wenye Wang and Dr. Alun Lloyd, for being my advisory committee members. They were always willing to spend time on giving me useful suggestions on my research. I have been very fortunate to have worked in Dr. Eun s group with Dr. Xinbing Wang, Dr. Yuh-Ming Chiu, Dr. Han Cai, Terrence van Valkenhoef, Chul-Ho Lee, Daehyun Ban, Boonyarith Saovapakhiran, Xin Xu and Jaewook Kwak. I would like to give my special thanks to Chul-Ho Lee who collaborated with me on several works and spent lots of time in heated discussions for research topics. Most of all, I cannot thank enough my family for their love and support. Thanks to the encouragement and prayers from my family, I could successfully get through difficulties during 7 year long journey in Raleigh with confidence and responsibility. iv

7 TABLE OF CONTENTS List of Tables vii List of Figures viii Chapter 1 Introduction Super-diffusive Behavior of Mobile Nodes and its Impact on Routing Protocols Age Invariant Regime for Multi-Source Content Update in Mobile Opportunistic Networks Organization Chapter 2 Preliminaries Super-Diffusive Property Mean Square Displacement (MSD) Super-Diffusion Content Update Scenario System description Content update Age metric Related Work Chapter 3 Super-Diffusive Behavior of Mobile Nodes and its Impact on Routing Protocol Performance Diffusive Behavior in GPS-based Traces and Synthetic Models GPS Traces Validation Diffusive Behavior in Synthetic Models Diffusive Behavior in AP-Based Traces Available AP-Based Traces MSD with Pause Time in AP-Based Traces Continuous Time Random Walk (CTRW) Numerical Results Metrics Routing Protocol Simulation Setup Impact of Different Diffusive Behavior Scenarios with Heterogenous Mobility Models Performance Evaluation by Existing Mobility Models Discussion v

8 3.5.1 Other Factors Toward the Super-diffusive Property Inter-meeting Time in Lévy Walk Chapter 4 Age Invariant Regime for Multi-Source Content Update in Mobile Opportunistic Networks Age Dynamics in Poisson Update and Contact Service provider content update case Publish/subscribe case Age Dynamics in Non-Poisson Case Simulation setup Age dynamics under mobility models with non-poisson contacts Deviation from ODE Equation Rule of thumb on validity of ODE-based description Chapter 5 Discussion Comparison Performance Evaluation Simulation Setup Results Interpretation Chapter 6 Conclusion References Appendix Appendix A vi

9 LIST OF TABLES Table 3.1 Summary of GPS traces Table 3.2 Summary of AP-Based traces Table 3.3 Notations and definitions used in Chapter Table 3.4 Estimated MSD exponents γ using GCTRW(µ, β, p) Table 3.5 Summary of model T parameters Table 5.1 Comparison between single message and publish/subscribe scenario. 79 vii

10 LIST OF FIGURES Figure 1.1 Figure 2.1 Various scenarios in mobile network. (a) Traditional one source and destination pair case; (b) Service provider content update case; (c) Multi-source content update case. Each mobile node A, B, C updates different type of contents (e.g., news, weather forecast and traffic info) MSD computation and sample trajectories of two nodes with different diffusive properties. Two nodes moving with the same constant speed (1.34 m/s) are simulated over the same duration (1 sec) Figure 3.1 MSD of human mobile nodes. In all cases, MSD increases faster than linear (super-diffusive behavior) Figure 3.2 Mean Square Displacement (MSD) of the existing synthetic mobility models (a) (c) and the Lévy walk models on a log-log scale Figure 3.3 Isotropic random walk with step-length L i and turning angle θ i. {L i } are i.i.d. with probability density f L (l), and {θ i } are i.i.d. with unif[, 2π] Figure 3.4 Illustration of properties/limitations of AP-based traces. Location of mobile nodes are mapped to AP-coordinates whenever they are associated with APs and unknown otherwise. A is the location where a mobile node pauses Figure 3.5 UCSD Traces: (a) CCDF of session time (P{T session > t}); (b) MSD (from 9AM) on a log-log scale. The inset is for the trace that started from 11AM. Power-law density of T session implies that T pause also follows a power-law with the same exponent, making the MSD grow very slowly; (c) MSD after randomization of location of mobile nodes inside APs. The MSD exponents remain untouched after randomization when compared with Figure 3.5(b) Figure 3.6 Dartmouth Traces: (a) CCDF of session time; (b) MSD on a loglog scale. Session time and diffusive behaviors are similar to those of UCSD traces in Figure Figure 3.7 MSD of GCTRW(µ, 1.38,.5) with µ = viii

11 Figure 3.8 Impact of diffusive properties on contact-based metrics: (a) total number of new contacts among all nodes after simulation time t = 4 seconds; (b) total number of new contacts during time interval t; (c) total number of contacts (including those among the same pair of nodes) after t = 4 seconds. When µ is smaller (more frequent long steps), nodes tend to spread out further from the starting points, thus creating larger number of new contacts with other nodes Figure 3.9 Impact of diffusive properties on message delivery ratio for different routing protocols. As the nodes tend to diffuse faster (smaller µ), the message delivery ratio becomes larger. This tendency holds for the performance of all six routing protocols Figure 3.1 Impact of diffusive properties on the performance of routing protocol with pause time considered. GCTRW(µ, β, p) model in Chapter 3.3, along with RWP and RD are used. While pause time induces longer message delivery, the ordering of performance is preserved as before Figure 3.11 Impact of diffusive properties on the performance of epidemic routing with different message TTL. The buffer size is set to 5 messages. The ordering of performance results in Figure 3.9 is preserved. 51 Figure 3.12 Impact of diffusive properties on the performance of epidemic routing with different Buffer size B. The message TTL is set to 4 seconds. The ordering of performance results in Figure 3.9 is preserved Figure 3.13 Impact of diffusive properties on message delivery ratio for real trace. Faster diffusive (µ =.8) nodes lead to higher message delivery ratio than slower diffusive (µ =.61). Note that pause time is included for the MSD slope in real trace Figure 3.14 Impact of diffusive properties on the number of new contacts and delivery ratio in epidemic routing under a heterogenous mix of mobility models. Mobile nodes following Lévy walk model with µ = 1.5 and that with µ = 2.5 coexist. We vary the fraction of nodes following Lévy walk model with µ = 1.5 from to 5, while keeping the total number of nodes the same (5 total). For example, legend µ(1.5), µ(2.5)(4,1) means that there are 4 nodes following Lévy walk with µ = 1.5 and 1 nodes with µ = Figure 3.15 MSD measurement under Model T [34]. The AP locations at UCSD and Dartmouth are used. This shows that trace based models can also lead to super-diffusive property of mobile nodes to some degree ix

12 Figure 4.1 Average age of contents with Poisson content update rate µ and contact rate λ in service provider update case. Since service provider can directly distribute contents to every node, content update rate µ plays a major role, and opportunistic contacts hardly contribute to freshness of contents when µ is large Figure 4.2 Publish/subscribe case. Unlike the service provider update case, the age of content at the publisher is not zero when one of the subscribers meets this publisher Figure 4.3 Average age of contents with Poisson update rate µ and contact rate λ in publish/subscribe case. N is set to 5. For large λ/µ (µ.3 or λ 2 in Figures above), additional contacts (larger λ) hardly impact the average age Figure 4.4 A sample trajectory of Pearson walk (L=1m) and Random direction model during 1 sec. In Pearson walk, a mobile node moves straight with constant step length L m before it chooses the direction uniformly (and repeat), and in RD, a mobile node changes the direction at the boundary Figure 4.5 Average age of contents under Pearson walk with different L. We vary contact rate λ (.25.17) by adjusting the number of nodes N from 1 to Figure 4.6 Average age of contents under Lévy walk model with different µ. We can observe the same trend as in Figure Figure 4.7 Age distribution for N = 5 in Pearson walk. When λ/µ is larger, the age distribution is almost identical to RD model. Note that for µ = 1 ( 1 4 ), the age distribution is almost matched. The trend is almost identical to that of Figure Figure 4.8 Age distribution for N = 7 in Pearson walk. Comparing with Figure 4.7, the difference of age distribution among different model is a lot more decreased since λ/µ is larger in this case Figure 4.9 Age distribution in Lévy walks for N = Figure 4.1 Age distribution in Lévy walks for N = Figure 4.11 Average age comparison between numerical results and theory.. 76 Figure 4.12 Rule of thumb on validity of ODE-based description. For large λ/µ regime, mobility models do not play a role on freshness of contents. Note that diffusive property of mobility model decides the slope a. 77 Figure 5.1 End-to-end delay under Pearson walk. Unlike the publish/subscribe scenario in Figure 4.5, the impact of mobility models are more noticeable x

13 Figure 5.2 End-to-end delay under Lévy walks. Unlike the publish/subscribe scenario in Figure 4.6, the impact of mobility models are more noticeable xi

14 Chapter 1 Introduction Mobility is the most important factor in mobile ad-hoc networks (MANETs) and delaytolerant networks (DTNs), and has posed a serious challenge to the analysis and design of protocols on such networks. The mobility pattern directly impacts time-varying contact/inter-contact dynamics among mobile nodes, which in turn affect the performance of any protocol built over these mobility patterns [12]. Mobility models that fail to capture key characteristics in the movement patterns of mobile nodes will result in misleading guidelines on the design of new protocols and evaluations of their performance and thus prevent us from making a right decision on our choice. For better mobility models, numerous approaches have been put forth, ranging from various synthetic mobility modelings with certain desired properties, to the numerical study of MANET protocols using mobility traces obtained from real-world measurements. Synthetic mobility models [11, 21], such as random waypoint models, random direction models, random walk or Brownian motion on a square or a sphere, and their variations, have been developed mainly for the purpose of simplicity and the ease of analysis, but subsequently been criticized for their unrealistic behaviors [23, 76]. Another common 1

15 approach is to rely on real mobility traces [1] and use them as inputs to a simulator for the study and comparison of protocols [44, 76]. This approach, however, suffers from lack of the amount of available traces on a fine time/space scale; most existing traces show only partial or filtered information about the real trajectories of mobile nodes such as access point (AP) association information or just contact duration with others, not the actual spatial-temporal information of the mobile users on a fine scale. While [44, 76] have tried to extract meaningful metrics and reconstruct detailed mobility patterns out of those filtered traces using some heuristic algorithms, such reconstructed traces are applicable only for the particular setting under consideration (e.g., the same campus) and are highly sensitive on the choice of the reconstruction algorithm. In this dissertation, we observe super-diffusive movement pattern in mobility traces as a key property in human mobility, and study its impact on network performance extensively. By investigating the location of mobile nodes, and how it changes over time, we find that there is a common and distinctive characteristic observed in all mobility traces, super-diffusive movement pattern, which is characterized by a faster-than-linear growth curve of the mean square displacement (MSD). The mean square displacement (MSD) average square distance traveled by a mobile node over time duration t can be used easily to quantify the diffusive property both in synthetic models and mobility traces, and does not require the accurate location coordinates of mobile nodes to find diffusive property. We use all the available GPS traces as well as AP-based traces to investigate the diffusive property in human mobility, and super-diffusive property is observed in all cases. Our findings are also in line with recent results in other scientific disciplines where almost all other living organisms invariably display super-diffusive property. We then study how can we easily generate different diffusive property. We use a set of Lévy walk models [68] since there is a one-to-one relationship between power-law step- 2

16 length exponent and the MSD slope in this model, and thus we can control the diffusive property easily by adjusting the step-length exponent. Then, we provide the results that show how varying degrees of diffusive properties affect the characteristics of several contact-based metrics and simulation results for network performance evaluation via various routing protocols by using a set of Lévy walk models. We consider various scenarios by adding pause time, limiting resources (e.g., buffer size and TTL), and observe the huge impact of diffusive property consistently in all different scenarios. Next, we extend the scope of our study into more scenarios. In the new network environment, mobile nodes generate their own messages (e.g., wireless ad hoc podcasting in [53]), spread them out to other nodes in the network by opportunistic contacts, and update them frequently to keep their contents as fresh as possible. Thus, unlike the traditional single message scenario where one source and one destination are pre-defined, the age of contents would be the main interest in performance evaluation for this scenario. Also, in this scenario, we have multiple destinations. This dynamic content update scenario can find various applications of sharing useful information such as news, weather forecast and traffic information, where the value of information depreciates over time. We first analyze how the age of time-evolving contents changes in the publish/subscribe case when Poisson update and contact are assumed, and show that there exists an ageinvariant regime where the average age does not depend on content update rate or contact rate. We then compare the dynamic content update scenario with the traditional single message scenario in view of the role of mobility models, and specifically show that the impact of the mobility model can be minimal in content update scenario under certain conditions in terms of message generation rate and contact rate among mobile nodes. This result is in sharp contrast to previous works that highlighted the impact of mobility 3

17 models under single message scenarios. 1.1 Super-diffusive Behavior of Mobile Nodes and its Impact on Routing Protocols To find out key characteristics in movement patterns of mobile nodes, we first investigate numerous GPS-based mobility traces as well as AP-based traces. Unlike previous approaches using mobility traces, we specifically focus on the location of mobile nodes and how it changes over time. We then find that there is a common and distinctive characteristic observed in all mobility traces, super-diffusive movement pattern, which is characterized by a faster-than-linear growth curve of the mean square displacement (MSD), i.e., E{ Z t Z 2 } O(t γ ) with γ > 1, where Z t R 2 is the position of the mobile node at time t. The mean square displacement (MSD) average square distance traveled by a mobile node over time duration t is non-parametric and does not require any a priori specific mobility model for test, and is robust against the noise/error in the coordinates of mobile devices and the granularity of measurement time. We then study existing synthetic models and trace based models to find out whether these models can produce varying degrees of super-diffusive behavior as observed from all available GPS-based mobility traces as well as AP-based traces, and show that each model can generate only a limited range of diffusive properties or cannot be conveniently used to produce different degrees of diffusive property in practice. As a viable alternative, we use a set of Lévy walk models [68] as simple, easy-to-generate, yet still versatile mobility models. The Lévy walk model is an isotropic two-dimensional random walks, whose super-diffusive behavior (super-linear growth in MSD) is easily controlled via a single parameter the exponent of its power-law step-length distribution. Since there is a one- 4

18 to-one relation between the exponent of the step-length distribution and the MSD slope, we can easily control the diffusive property through Lévy walk model in performance evaluation. In particular, for AP-based traces, we show that there is a way to extract the diffusive property of mobile nodes in AP-based traces where location information Z t of a mobile node is spatially quantized (to coordinates of APs) and sporadically time-sampled only when mobile nodes get inside the range of an AP. By capturing the tail behavior of the pause time of mobile nodes and with the help of the continuous time random walk (CTRW) formalism, we set out to extract key characteristics of the mobility patterns again from MSD measurements. Specifically, we analytically show that under the class of CTRW models, a class of Lévy walk models interspersed with power-law distributed pause time can easily capture diffusive behavior observed in various AP-based traces. Lastly, we provide numerical results that show how varying degrees of diffusive properties affect the characteristics of several contact-based metrics and simulation results for network performance evaluation via six different routing protocols. We observe that the number of new contacts is strongly related to the network performance, which implies that contact-based metrics can be used to predict network performance. We consider various scenarios by changing the resource constraints and node density, or by adding pause time. We also show the impact of the diffusive property on network performance by using different diffusive sets of mobility traces. In particular, we show that existing models such as random waypoint models, random direction models and Brownian motion models may lead to overly optimistic or pessimistic results when diffusive properties are not properly captured. Our results thus collectively imply that correct diffusive behavior of mobile nodes should be taken into account for the development of new protocols and comparison with existing ones. 5

19 1.2 Age Invariant Regime for Multi-Source Content Update in Mobile Opportunistic Networks Source (a) Destination Weather Breaking News Pair wise contact rate λ P Service Provider A update rate µ A B µ B Traffic C µ C (c) (b) Figure 1.1: Various scenarios in mobile network. (a) Traditional one source and destination pair case; (b) Service provider content update case; (c) Multi-source content update case. Each mobile node A, B, C updates different type of contents (e.g., news, weather forecast and traffic info). In many mobile ad-hoc network (MANET) and delay tolerant network (DTN) scenarios, a source and destination pair is chosen first, and then, the source delivers a single message to the destination. This single message scenario illustrated in Figure 1.1(a) has been widely used in the study of forwarding/routing protocols over such networks [36, 51, 63, 74, 77]. In this case, message delivery ratio and end-to-end delay have been used as the main metrics. Recently, the publish/subscribe message paradigm has emerged as a viable application in MANET and DTN [25, 83]. Unlike the single message case, the publisher (information provider) updates the same type of contents frequently, and these are delivered to subscribers by the publisher directly or by opportunistic contacts with 6

20 other subscribers. In this publish/subscribe paradigm, it is more reasonable to assume that subscribers would receive messages frequently from the publisher, so we would be interested in how fresh the contents are, or the age of the contents. Also, in this scenario, all the mobile nodes in the network can be subscribers of contents. There are several recent literatures that have focused on the age of contents, but in a different context. In [22], it is assumed that mobile nodes belong to one class depending on the spatial location, and move between classes in content update scenario. However, in this work, the specific mobility models that are used in many literatures have not been studied. In our work, by using several key mobility models, we investigate the age dynamics and the degree of impact depending on mobility models. [32] has studied the optimal allocation and scalability in mobile network where the freshness of contents matters. In this work, a single service provider, which can communicate directly with all the mobile nodes in the network, updates contents, and opportunistic contacts among mobile nodes are used to share more recent contents as briefly illustrated in Figure 1.1(b). In contrast, in our publish/subscribe scenario as shown in Figure 1.1(c), each and every mobile node updates its own content. Thus, a mobile node i acts like an individual service provider that generates a sequence of contents of type i with different time stamps and wants to get updated by all other types of contents via opportunistic contacts only. This scenario would be more realistic and useful in the future due to the fast growing users with portable communication devices and the development of social community that shares various information. In this work, we investigate the dynamics of content age in the publish/subscribe case where every node generates contents over time and is interested in receiving all other types of contents afresh as often as possible. We first show that, when content updates at each node and contacts among mobile nodes are Poisson, the age dynamics can be 7

21 obtained via a simple ordinary differential equation (ODE) in terms of the average content update rate µ and contact rate λ among mobile nodes, by modification of the results in [22]. We compare the age dynamics in our publish/subscribe case with that under the single service provider update case, and observe that there exists an age-invariant regime in which more frequent content updates or contacts among mobile nodes hardly contribute to the average age of the contents in both cases and specify this regime in terms of the ratio of content update rate µ to contact rate λ. Interestingly, we observe a sharp contrast of age-invariant regimes in two cases. We then extend our investigation into far more general settings by considering mobility models with non-poisson contacts, for which the aforementioned ODE-based description is no longer applicable. In the literature, these realistic mobility models generating non-poisson contacts are known to lead to very different performance results (e.g., average end-to-end delay) when compared with the case of Poisson contacts [19, 2, 46] under the single message generation from a given source. In our publish/subscribe scenario, however, we find that there still exists an age-invariant regime specified by the contact rate among mobile nodes and the content update rate, in which the choice of mobility models (thus non-poisson contacts) are largely irrelevant. Also, for the deviation from the ODE solution for non-poisson mobility models, we discuss the degree of deviation for each model. Interestingly, in this age-invariant regime, every contact including the repeated ones with the same node is equally likely to contribute to the average age dynamics as the new contact, implying that the total number of contacts is a more meaningful metric than the number of new contacts (excluding the repeated ones), which is opposite to the case of single message generation scenario [46] in which only the number of new contacts matters to properly differentiate performance under various mobility models with different diffusive properties. We also provide a guideline where 8

22 ODE-based description can be conveniently used to predict the average age and discuss any deviation from ODE solution in some mobility models. We also compare the publish/subscribe scenario with single message scenario in depth. We first enumerate the differences of these two scenarios and characterize each scenario in various aspects (e.g., metric, relevant scenario, viable applications). Then, by using the same simulation setup, we show that the impact of mobility models can be minimal in publish/subscribe scenario, which is not the case in the single message scenario. This finding is a sharp contrast to many previous works that have mainly focused on the impact of mobility models. 1.3 Organization The rest of the dissertation is organized as follows. In Chapter 2, we provide preliminary background and related work. For background, we explain Mean Square Displacement (MSD) and Super-diffusion in detail, which are the key concepts in this work. Then, we provides the definitions and assumptions in content update scenario. In Chapter 3, we study super diffusive behavior of mobile nodes and its impact on routing protocol performance. In Chapter 4, we study the age dynamics of contents under the publish/subscribe scenario. In Chapter 5, we compare the publish/subscribe scenario with single message scenario and discuss the difference. We then conclude in Chapter 6. 9

23 Chapter 2 Preliminaries In this chapter, we first present background on the mean square displacement a metric to capture the rate at which mobile nodes spread out, and the super-diffusion. We then give notations, assumptions and the metric of interest in content update scenario. Finally, we summarize the related works. 2.1 Super-Diffusive Property Mean Square Displacement (MSD) One way to characterize the movement of a mobile node is to measure how far it is away from its current position after time t. This diffusive behavior or the rate at which the mobile node spreads out can be described and quantified by so-called the mean square displacement (MSD) [15, 68, 69]. Specifically, if we define Z t R 2 to be the position of the mobile node at time t, then the MSD becomes M(t) E{ Z t Z 2 }, (i.e., the second moment of the displacement Z t Z between the current position at time t and the position at time ) and M(t) gives typical amount of displacement of the 1

24 Y 1 Normal diffusive (γ=1.) Super diffusive (γ=1.5) t 2 5 t 1 Z 1 Z 2 Y(m) 5 (,) Z 3 t 3 (a) MSD computation X X(m) (b) Super-diffusive node Figure 2.1: MSD computation and sample trajectories of two nodes with different diffusive properties. Two nodes moving with the same constant speed (1.34 m/s) are simulated over the same duration (1 sec). mobile node after time t. For example, for a class of isotropic random walks with finite step-length 1 variance, the MSD will grow linearly with t, i.e., M(t) t, provided that the speed of the mobile node is O(1) (or constant). 2 In general, we have M(t) O(t γ ) for some γ >. The slope of M(t) in a log-log scale (γ) characterizes how fast a node spreads out in a simple way. Figure 2.1(a) shows how MSD can be measured. In this figure, as the mobile node starting from the origin follows the trajectory shown in dashed line, we can collect the displacement at each time instant t i and investigate how MSD grows with time t to uncover the diffusive property of mobile nodes Super-Diffusion When the step-length L has infinite variance (σl 2 = ), the mobile node tends to quickly spread out since longer step-lengths are generated more often. This behavior is called 1 Step-length is defined as the distance that a walker moves before changing its direction. 2 This is similar to the case of 2-D Brownian motion with its variance growing linearly with t. 11

25 super-diffusion [48, 69, 7, 81, 82], while for σl 2 < it is called normal diffusion. The varying degrees of diffusive properties of mobile nodes can be conveniently captured by the slope (γ) of M(t) in a log-log scale (i.e., M(t) t γ ). For example, we have γ = 1 for a normal diffusive case, while γ > 1 for super-diffusive case (faster-thanlinear growth of the MSD). Figure 2.1(b) shows typical sample trajectories of two mobile nodes with different diffusive properties (different γ). While both nodes have the same constant speed (1.34 m/s) and run over the same duration (1 sec), the super-diffusive node (γ = 1.5) spreads out from the origin much farther than the normal-diffusive node (γ = 1.). As Figure 2.1(b) illustrates, the occasional long jumps are key characteristics of super-diffusive movement patterns. 2.2 Content Update Scenario System description Our system consists of N mobile users, denoted by the set S = {1, 2,..., N}. User i updates its content (of type i) frequently, and whenever it does so, it inserts the timestamp information into its content to indicate when it is updated from the source (user i) originally. Figure 1.1(c) illustrates an example of our scenario. When there are N mobile nodes (publishers) that update their contents (of type i =1,..., N), each mobile user i will act as a publisher for type i content and as a subscriber to all other contents of type j (j i). Each user keeps the most recent (latest time-stamped) copies of contents whenever possible, so each user will eventually have all N contents in its buffer. We assume that buffer size at each user s device is large enough to hold up to N contents at the same time. 12

26 2.2.2 Content update User i (publisher) updates its content (type i) with rate µ. We denote by ts ij (t) the time-stamp of type i content (originally generated from user i) that is currently stored in user j s buffer at time t. In our publish/subscribe scenario, opportunistic contacts among mobile nodes are the only way to share contents with other nodes and keep them updated. Contact rate between user i and j is set to λ p (here subscript indicates pairwise contact), while we use λ to denote the aggregate contact rate per user, i.e., λ = (N 1)λ p when there are N mobile nodes in the network. When there is a contact between users i, j at time t, both of which having type k content (generated by user k) with time stamp ts ki (t) and ts kj (t), respectively, they both update this content by sharing more fresh one with time stamp given by max{ts ki (t), ts kj (t)} (and delete the old one from the buffer) Age metric To measure the freshness of contents, we need a proper metric. In this dissertation, we adapt the freshness metric defined in [32, 33]. For the type i content that is currently stored at node j with time stamp ts ij (t), we define the age of the content as A ij (t) = t ts ij (t) for i, j S. Thus, the age of each content increases linearly with time and jumps down if the user contacts other users having fresher content (larger time stamp) of the same type. We then define the average age over all nodes and all content types as A(N, t) = 1 N N N A ki (t). (2.1) i=1 k=1 Throughout this work, we will use A(N, t) as our main metric, unless otherwise specified. 13

27 2.3 Related Work Due to the importance of mobility modeling for proper network performance evaluation, numerous approaches have been put forth. In the literature, numerous synthetic mobility models [38, 62] have been proposed to help network designers evaluate and compare protocols on MANETs and DTNs. Examples include random waypoint mobility model, random direction mobility model, random walk mobility model and its many derivatives such as Brownian motion model on a sphere, among others. (See [11, 21] for a survey.) While these models are easy to generate and provide a quick platform to compare the performance of network protocols, they are mainly for the sake of simplicity and ease of comparison. Recent empirical results also indicate that current synthetic mobility models are not able to capture the characteristics of the real mobility patterns [23, 27, 4, 44, 76]. On the other hand, a large set of traces measured in various environments [1, 7, 56, 76] have been used to extract the key characteristics of mobile nodes and construct realistic mobility models [34, 44, 52, 76]. This collection of traces provides useful information when many participants are involved for a long observation period. In [44, 76], authors extract key features in mobile traces of their own campus, and propose realistic mobility models. [35] studied the time and space domain characteristics based on device registrations of mobile users at APs. In their extended work, Model T [34] included the space registration patterns of mobile users, and Model T++ [52] incorporated both the time and space registration patterns in their model. [29] proposes a mobility model that features time variance and periodic reappearance in mobile traces. In [57], social network theory is employed to construct their community based mobility model and tested with real traces. In our work, we find super-diffusive movement patterns in mobile traces as the key characteristic, and study other models in the view of producing diffusive properties. 14

28 The mobility model has also been a central topic in other scientific disciplines and various attempts to explain the movement of living organisms in nature have been made [24, 47, 65]. In particular, the so-called Lévy walk model, whose defining characteristic is its super-diffusive property, has been adopted for modeling the movement patterns of living organisms such as Microzooplankton [14], seabirds (albatross) [78], reindeer [55], jackals [1], and monkeys [16, 58], as well as capturing their foraging patterns [67, 78, 79]. Anthropologists also have started to pay attention to Lévy walk pattern to analyze the mobility phenomenon in their field [17]. Quite recently, the super-diffusive behavior of mobile nodes has also received attention in representing human mobility pattern via a limited number of GPS-based traces [45, 61]. In contrast, in this dissertation, we investigate the vast amount of AP-based traces [1, 7, 56] (for which GPS-coordinate information is unavailable) through the CTRW formalism with heavy-tailed nodes pause time [76], and find that the super-diffusive behavior of nodes movement is a universal property in all the mobility traces under our study. Contents update scenario in mobile opportunistic network has drawn a lot of attention recently due to the fast development of mobile devices, the need of up-to-the-minute information and the fast growing social networks. The scenario that podcast is disseminated is introduced in [53]. In [9, 32], the age dynamics is studied when mobile nodes update the contents from the base station and opportunistic contacts among mobile nodes. In [22], for the large number of mobile nodes, it is shown that the age of contents approaches a deterministic mean-field regime. In this work, authors assume that each mobile node belongs to one spatial class and it moves among different classes. Optimizing caching policies that can minimize the delay in mobile opportunistic networks was mentioned in [6]. Authors find item replication ratio that can maximize the social welfare. In [33], a novel mechanism for determining the caching policy that maximizes the system s social 15

29 welfare in content distribution scenario for heterogeneous environment is proposed. In our work, we consider more general contents update scenario called publish/subscribe case, where each mobile node can be both a publisher and subscriber and opportunistic contacts are used for spreading out up-to-the-minute contents. Also, we specially focus on the impact of mobility models in content update scenario by using well-known models, and compare the results with previous literatures that has mainly focused on the impact of mobility models under single message scenario. 16

30 Chapter 3 Super-Diffusive Behavior of Mobile Nodes and its Impact on Routing Protocol Performance There have been numerous approaches to find key characteristics in human mobility using mobility traces. Statistical information such as speed, pause time and inter-contact time of mobile nodes has been investigated in previous works [23, 27, 4, 44, 76] based on mobility traces and its impact on network performance is studied. In this chapter, we find super-diffusive property as key characteristic in human mobility. The mean square displacement (MSD) average square distance traveled by a mobile node over time duration t is used to measure diffusive property, and this metric is easy to use and robust against the noise/error in the coordinates of mobile devices and the granularity of measurement time. In Chapter 3.1, we investigate GPS mobility traces, and characterize the superdiffusive behavior of mobile nodes in terms of MSD. Then, we analyze existing mobility 17

31 models in the context of their diffusive properties, and introduce Lévy walk models as good candidates for producing various degrees of diffusive behavior. In Chapter 3.2, we investigate AP-based traces, and characterize the diffusive properties when pause time is included. In Chapter 3.3, we introduce CTRW and generalized MSD for a class of isotropic random walks with heavy-tailed pause time. In Chapter 3.4, we provide simulation and numerical results to show the impact of diffusive properties on contact-based metrics and network performance. We consider various scenarios with and without pause time, and change the resource constraints by adjusting the message TTL and buffer size. We also consider the different sets of diffusive mobility traces. In Chapter 3.5, we discuss the issue of other factors toward the super-diffusive property and how to incorporate the observed property in traces into the set of Lévy walk models. In addition, we show that even trace-based models can generate super-diffusive property. 3.1 Diffusive Behavior in GPS-based Traces and Synthetic Models In this chapter, we examine various GPS traces with different movement patterns and show all of them display the super-diffusive behavior. We then study whether the existing synthetic models can effectively capture the observed diffusive properties GPS Traces Validation In this part, we investigate various GPS-based mobility traces for human beings and divide them into several categories based on the movement patterns walking, running, inline skating and bicycling, in view of future applications such as MetroSense [4]. 18

32 Table 3.1: Summary of GPS traces source duration sample # of nodes Web-Walking [3] 19 hours 3 6sec 11 Web-Running [3] 27 hours 3 6sec 22 Web-Inline Skating [3] 42 hours 3 6sec 17 Web-Bicycling [3] 36 hours 3 6sec 13 UW-Walking [1] 2 hours 1 sec 1 NCSU-Walking [5] 3 weeks 1 sec 1 Available GPS Traces Table 3.1 is the summary of GPS traces of human beings under our consideration for their diffusive properties. Below is the detail about the GPS traces sources. [3] is a GPS device users community website where users share GPS traces. This website has an extensive amount of data from about 5 countries for various activities. We use this as our main source of GPS traces. University of Washington: The GPS trace of University of Washington (UW) was collected by one of the authors in [39] for about two hours. This trace provides the x, y co-ordinates of the mobile node every second even inside of buildings by using Place Lab [6]. NCSU: NCSU GPS traces [5] were collected from the present authors school campus, where one student carried a GPS device (Garmin etrex [2]) to collect GPS traces. Extracting MSD from GPS traces We investigate diffusive behavior of mobile nodes from all the available GPS traces in Table 3.1. In order to decouple any effect of pause time from nodes diffusive behavior, 19

33 we first removed all the pause time in the GPS traces, and then computed MSD from the resulting traces. M(t) (m 2 ) Walking Running Inline Skating Bicycling γ = 2 γ = time (sec) γ > 1.48 Figure 3.1: MSD of human mobile nodes. In all cases, MSD increases faster than linear (super-diffusive behavior). Figure 3.1 shows the MSD of mobile nodes with different movement patterns from GPS traces on a log-log scale. In all cases, MSD increases faster than linear with γ > 1, which implies the super-diffusive behavior of mobile nodes. Different values of γ conveniently captures the degree of diffusive behavior. For instance, mobile nodes with inline skate (γ = 1.88) tend to spread out quickly and show almost ballistic movements during certain time range, while walking mobile nodes (γ = 1.48) tend to spread out a little slower than mobile nodes with inline skate, but still faster than the normal diffusion with γ = 1 (the case of Brownian motion). Regardless of movement patterns, MSD grows faster than linear (γ > 1) in all cases. 2

34 3.1.2 Diffusive Behavior in Synthetic Models In this part, we first study diffusive behavior in current synthetic mobility models via their MSD and show that most of them are not suitable for capturing varying degrees of super-diffusive pattern. We then propose to use a set of Lévy walk models as a simple alternative to current synthetic models. Correlated Random Walks Among existing synthetic mobility models, we first consider several correlated random walk models to see if they can capture super-diffusive behavior. Since a mobile node in a correlated random walk tends to move along the same (or similar) direction for a while before changing its direction, these models might be good candidates for capturing super-diffusive behavior. Here, we consider the following set of correlated random walk models currently used in MANET simulations. Correlated Random Walk on Grid (CRWG): A two-dimensional correlated random walk model on grid is proposed in [13] as a generalization of Manhattan mobility model [12]. In this model, a mobile node takes a step to the same direction as the previous one with probability p and opposite direction (moving backward) with probability q, while the probability of turning right or left is r satisfying p+q+2r =1. By assigning different values of p, q, we can control the degree of tendency for a mobile node to follow the same direction. Gauss-Markov Mobility Model: A Gauss-Markov mobility model [21] first assigns an initial speed and direction of a mobile node. After moving for a fixed amount of time t, the mobile node updates its speed and direction based on the previous speed and direction (much like an auto-regressive recursion), where the speed and turning angle 21