MotionCast for Mobile Wireless Networksiwct.sjtu.edu.cn/personal/xwang8/paper/monograph.pdf ·...

Xinbing WangShanghai Jiaotong University, China

MotionCast for Mobile WirelessNetworks

– Monograph –

September 13, 2012

Springer

Contents

1 MotionCast: Delay and Capacity Tradeoff Analysis . . . . . . . . . . . . . . . . 11.1 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Capacity and Delay in the 2-hop Relay Algorithm Without

Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Upper bound of capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Lower bound of delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Capacity and Delay in the 2-hop Relay Algorithm With Redundancy 71.3.1 Upper bound of capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Lower bound of delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Capacity and Delay in the multi-hop Relay Algorithm WithRedundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.1 When m =Θ(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.2 When m = o(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 From i.i.d. mobility to random walk mobility . . . . . . . . . . . . . . . . . . . 131.5.1 Random Walk Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5.2 Capacity and Delay in the 2-hop Relay Algorithm without

Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5.3 Capacity and Delay in the 2-hop Relay Algorithm with

Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.6 Hybrid Random Walk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6.1 Capacity and Delay in the 2-hop Relay Algorithm WithoutRedundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6.2 Capacity and Delay in the 2-hop Relay Algorithm WithRedundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.7 The Impact of Node Density in the Network . . . . . . . . . . . . . . . . . . . . 241.7.1 Capacity and Delay in the 2-hop Relay Algorithm Without

Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.7.2 Capacity and Delay in the 2-hop Relay Algorithm With

Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

v

vi Contents

1.8 Random Way-point Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.8.1 Capacity and Delay in the 2-hop Relay Algorithm Without

Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.8.2 Capacity and Delay in the 2-hop Relay Algorithm With

Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.9 Applying Network Coding in 2-hop Relay Algorithm with

Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.9.1 Network coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.9.2 Upper bound of capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.9.3 Lower bound of delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.11 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2 MotionCast: General Connectivity in Clustered Wireless Networks . . 352.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1.1 Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1.2 Mobility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.1.3 Definition of (K, M)-Connectivity . . . . . . . . . . . . . . . . . . . . . . 382.1.4 Definition of Critical Transmission Range . . . . . . . . . . . . . . . 38

2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3 The Disconnected Probability

of a Cluster Member . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4 (K, M)-Connectivity under Random Walk Mobility Model . . . . . . . . 42

2.4.1 Disconnected Probability of a Cluster Member underRandom Walk Mobility Model . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.4.2 The Critical Transmission Range underRandom Walk Mobility Model with Simple V-Model . . . . . . 43

2.4.3 The Critical Transmission Range underRandom Walk Mobility Model with General V-Model . . . . . 45

2.4.4 The Critical Transmission Range underRandom Walk Mobility Model with HomogeneousVelocity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.5 (K, M)-Connectivity under i.i.d. Mobility Model . . . . . . . . . . . . . . . . 492.5.1 Necessary condition of Theorem 13 . . . . . . . . . . . . . . . . . . . . . 492.5.2 Sufficient condition of Theorem 13 . . . . . . . . . . . . . . . . . . . . . 50

2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.6.1 Explanation on the expression of the critical transmission

range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.6.2 Random walk mobility model with different velocity models 51

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3 MotionCast: A Survey on the Capacity Scaling of Wireless Networks 573.1 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1 Traffic Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.1.2 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Contents vii

3.1.3 Transmission Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.1.4 Capacity Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.1.5 Definitions of Related Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Capacity-Delay Tradeoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2.1 End-to-end Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2.2 Definitions of Related Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2.3 Capacity-delay Tradeoff in Static Wireless Networks . . . . . . 653.2.4 Capacity-delay Tradeoff in Mobile Wireless Networks . . . . . 663.2.5 Other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3 Random Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.3.1 Random Homogeneous Networks: Unicast, Multicast and

Broadcast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.3.2 Random Inhomogeneous Networks: Clusters . . . . . . . . . . . . . 703.3.3 Combination of Cellular System: Hybrid Networks . . . . . . . . 70

3.4 Arbitrary Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.5 Factors that influence capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.5.1 Network size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.5.2 Communication patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.5.3 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.5.4 power control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.6 Techniques to improve capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.6.1 Mobility increases capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.6.2 Using directional antenna improves capacity . . . . . . . . . . . . . 753.6.3 Multi-Input Multi-Output (MIMO) increases capacity . . . . . . 773.6.4 Network coding increases capacity . . . . . . . . . . . . . . . . . . . . . 813.6.5 MPT and MPR improve capacity . . . . . . . . . . . . . . . . . . . . . . . 833.6.6 Hybrid Network increases capacity . . . . . . . . . . . . . . . . . . . . . 85

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Acronyms

MIMO multiple input multiple outputMANET mobile ad hoc networkTDMA time division multiple accessFDMA frequency division multiple accessCSI channel status informationSINR signal to interference and noise ratioLSRM local-based speed-restricted modelGSRM global-based speed-restricted modelMMM multi-hop MIMO multicastDMM direct MIMO multicastCMMM converge based multi-hop MIMO multicastCDMM converge based direct MIMO multicastMT multicast treeSN sender numberRN request numberi.i.d. independently and identically distributedw.h.p. with high probabilityp.d.f. probability density function

ix

Preface

Wireless ad hoc networks are useful when there is a lack of infrastructure for com-munication. Such a situation may arise in a variety of civilian and military contextslike sensor network applications and communication in harsh environments. Sincethe seminal work by Gupta and Kumar [3], the study of wireless ad hoc network-s has focused on understanding its fundamental capacity limits. The capacity persource-destination (S-D) pair of random ad hoc network developed by Gupta andKumar is Θ(1/

√n logn) 1 , which is pessimistic because the capacity goes to 0 as

the number of nodes in a fixed area n → ∞. Since then there are three kinds of workthat focus on the study of capacity.

One of the topics is to extend the original work done by Gupta and Kumar. Thiskind of work includes completing the proofs on unicast [46], extending the numberof receivers to the case of multicast [9], [47], [148], broadcast [48] and convergecast[152], [153]. It also contains the extension of the unicast and the generalizationof different kinds of transmission possibilities in the real networks such as user-centric networks [145]. However, the results are also pessimistic because the per-node capacity tends to 0 as the number of nodes n → ∞.

Another topic is on the trade-off between capacity and other network variableslike delay and power consumption. In 2002 Grossglauser and Tse [2] found thatmobility can increase the capacity. The per-node capacity can be bounded by a con-stant according to the 2-hop scheme proposed in [2]. However, the end-to-end delayis very large when mobility is introduced. Following this work, there are a groupof people working on the capacity-delay tradeoff [13], [18], [147], [149], [150],

1 The following notations are used throughout our book.

1. f (n) = O(g(n))⇔ limsupn→∞f (n)g(n) < ∞,

2. f (n) = Ω(g(n))⇔ liminfn→∞ f (n)g(n) < ∞,3. f (n) =Θ(g(n))⇔ f (n) = O(g(n)) and g(n) = O( f (n)),4. f (n) = o(g(n))⇔ limn→∞ f (n)g(n) = 0,

5. f (n) = ω(g(n))⇔ limn→∞ g(n)f (n) = 0.

xi

xii Preface

[160]. There is also another issue that needs consideration in the network: energyconsumption [89]. Since we are quite interested in the capacity-delay tradeoff, wepresent this issue in section II. Due to limited time, we will not consider other typesof tradeoff in our report.

Thirdly, other works are related with changing the ad hoc network model. Theclassical model is so-called random homogeneous ad hoc networks. To have achange on this, some people studied arbitrary networks [50], some studied inho-mogeneous networks (clusters) [51], [52], [53], [159], some combined the cellularnetwork and ad hoc network and worked on hybrid networks [54], [55], [56], [157],some use directional antenna to enhance the capacity performance [155], some letnodes cooperate and build a hierarchical MIMO network [114][151], some stud-ied the scaling law for cognitive radio networks (CRNs) [144], [146], [154], [156],[158], still others used network coding [57], [58], [59], [60], [61] and MPR [62],[63] to improve the network capacity.

Connectivity is also a fundamental issue in wireless networks and has been exten-sively studied in recent years. Generally speaking, there are two types of definitionson connectivity. One is in the sense of percolation, i.e., the existence of a componentthat consists of infinite connected nodes. The other one is defined as the ability thateach node in the network can find at least one path to any other node, either directlyor with the help of several other nodes acting as relays. A network is said to havek-connectivity if there exist at least k mutually independent paths connecting eachpair of nodes. According to [23], this definition is equal to the statement that a net-work is k-connected if and only if removal of any k− 1 nodes does not disconnectthe graph.

Based on the definitions on connectivity, the research works mainly fall into twocategories. Some study connectivity from the percolation perspective. These worksconsider continuum percolation with the Poisson Boolean model. Let λ be the nodedensity. Then there exists a critical value λc for which percolation occurs. If λ > λc(supercritical case), there will be a cluster consisting of infinite connected nodesalmost surely. If λ < λc (subcritical case), the network has no infinite connectedcomponent and is separated into an infinite number of finite connected components.In literature, the accurate value of the percolation threshold has not been decid-ed yet, while [24] [25] demonstrate that the analytical upper and lower bounds forthe threshold are 10.526 and 2.195, respectively. [26] provides simulation results toshow that if connected nodes use cooperation techniques to further connect isolatednodes that can not be connected separately, the percolation threshold of the coop-erative network is less than that of the non-cooperative network for 2-D extendednetwork. [27] analytically obtains this result when the path loss exponent is less than4.

For the second type of definition on connectivity, extensive research investigatesfrom various aspects the critical conditions to ensure (k-)connectivity. One concernis to determine the critical transmission range, as presented in [28] [29] [30]. Inthis context, all nodes in the network possess uniform transmission power. In [28],by using the theory of continuum percolation, Gupta and Kumar provide the criti-cal transmission range for asymptotic connectivity 2-D dense network with n nodes

Preface xiii

independently and identically distributed in a disc of unit area. It is shown that ifeach single node has a transmission area πr2 = logn+c(n)n , the network is asymptot-ically connected with probability one if and only if c(n)→+∞. Then in [29], Wanand Yi offer a precise asymptotic distribution of the critical transmission radius fork-connectivity.

Another concern concentrates on the minimum number of neighbors [29] [31][32]. Each node is assumed to have the ability to adjust its transmission power so asto maintain direct connections with a certain number of neighbors. In [31], Xue andKumar point out that each node should be connected to Θ(logn) nearest neighborsto ensure the connectivity of the network with n uniformly and independently placednodes in a unit square. [32] regards the minimum node degree for connectivity of awireless multi-hop network.

The rest part of this book is summarized as follows.

1. In Chapter 1, we investigate the impact of base stations on the capacity of Mo-tionCast. Here MotionCast means multicast between mobile nodes. The mobilitypattern is assumed to be i.i.d. mobility. Three protocols are analyzed, i.e., 2-hoprelay algorithm without redundancy, 2-hop relay algorithm with redundancy andmultihop relay algorithm. This network model combines multicast, mobility andbase stations together and thus brings significant enhance to the capacity anddelay tradeoff.

2. In Chapter 2, we turn to the connectivity issues in clustered networks. A new kindof connectivity, (k,m)-connectivity, is defined. Its critical transmission range fori.i.d. and random walk mobility models are derived respectively. By the term of(k,m)-connectivity, we mean that in each time period consisting of m time slots,there exist at least k time slots, during any one of which every cluster membercan directly communicate with at least one cluster head. For random walk mo-bility, two heterogeneous models, velocity model with constant number of valuesand velocity model with constant number of intervals, are proposed and studied.For random walk mobility with either of the two heterogeneous velocity modelsand i.i.d. mobility model, under weak parameters condition, we provide bound-s on the probability that the network is (k,m)-connected and derive the criticaltransmission range for (k,m)-connectivity. For random walk mobility with veloc-ity model with constant number of values and i.i.d. mobility model, under strongparameters condition, we present a precise asymptotic probability distribution ofthe probability that the network is (k,m)-connected in terms of the transmissionradius.

3. In Chapter 3, we conduct a survey on existing scaling law results on wirelessnetworks. We will give you a global perspective about the researches on capacityin the past years. We set up a system model to analyze and compare the resultsfor wireless networks. First, we introduce the network models and some impor-tant definitions which have been widely used in the past researches. Then wediscuss the capacity-delay tradeoff problems in wireless networks. After that, thecapacity in random networks and arbitrary networks are illustrated respectively.Furthermore, based on the capacity discussion, we give some points on the fac-

xiv Preface

tors that can have a great impact on capacity. In the end, we come up with somepopular techniques and show how they contribute to capacity respectively.

Chapter 1MotionCast: Delay and Capacity TradeoffAnalysis

Abstract In this chapter, multicast capacity-delay tradeoff of the wireless networkcombining mobile wireless nodes with base stations is studied. m = nb base stationsare located in a square region and divide it into m super cells according to their cov-erage. Their transmission power is large enough to directly transmit to any nodesin the same super cell. We further assume that mobile nodes move in an indepen-dently and identically distributed (i.i.d.) pattern and each wants to send packets tok = nd distinctive destinations. Packets are delivered to destinations or base stationsvia node’s mobility. Under this model, we study capacity, delay and their tradeoff inthree transmission protocols: one uses two-hop relay algorithm without redundan-cy, another adopts the scheme of redundant packets transmissions to improve delay,and the third one cancels the constriction of 2-hop. The upper bound of per-nodethroughput of 2-hop relay algorithm without redundancy is O(n−min{1,1−b+d}) andits delay is Θ(nmin{1,1−b+d}). The lower bound is Θ(n1−b), with a smaller capacityO(nb−2). It obtains a better tradeoff than using ad-hoc only in [4] when k = Θ(n)or b > min{ 1+d2 ,

2−d2 }. For two-hop relay algorithm with redundancy, the biggest

capacity and its corresponding delay are O(n−min{1,1−b+d}) and Θ(nmin{1,1−b+d}

2 ).The smallest delay and the related capacity are Θ(n 1−b2 ) and O(n b−32 ). When d > 12or b > min{ 1+d3 ,1− d}, adding base stations will enhance the tradeoff mentionedin [4]. As to multi-hop relay algorithm with redundancy, if m = Θ(n), we preferpassing through base stations to ad-hoc only.

1.1 Network Model

The network model of this chapter is similar to that of [165] and is specified asfollows.

Cell Partitioned Network Model With Base Station: The system model extendsthe cell partitioned network model in [4] by adding base stations to it. Suppose thenetwork is a square with length a, and there are n mobile users, which are randomly

1

2 1 MotionCast: Delay and Capacity Tradeoff Analysis

distributed in it. Then we divide this square into C non-overlapping cells with equalsize as depicted in Figure 1.3. To reduce interference, the order of C is of n. Thus,the node density q = n/c scales as Θ(1). We assume that nodes can send packets ifboth sender and receiver lie in the same cell.

Then m = nb(0 ≤ b ≤ 1) BSs divide the original area into m super cells, as thewider lines shows in Figure 1.3. All these base stations are interconnected by wiredlines and hence they can communicate at no delay. Base station offers enough trans-mission power to cover the whole super cell which is enclosed by wider lines andmobile node can only transmit in a range of small cell enclosed by slenderer linesin Figure 1.3. Besides, uplink and downlink will use different bandwidth to avoidinterference. That means when base station is sending packets, all the other trans-mission between two nodes or one node and a base station can still work withoutany problems.

Mobility Model: We adopt i.i.d. mobility model. At the start point of every slot,all the nodes are reshuffled and therefore they are randomly distributed in the Ccells by equal possibility. And, all these nodes choose the cells to stay in the nextslot independently and identically.

1.2 Capacity and Delay in the 2-hop Relay Algorithm WithoutRedundancy

Due to the property of this network, transmissions can be carried out either in in-frastructure mode or in ad hoc mode. In [4], there is a result on capacity and delayin ad hoc mode under the condition 2-hop without redundancy. In the following, wewill discuss the scaling laws when packets are sent all by base stations.

1.2.1 Upper bound of capacity

Here, we will figure out the upper bound of the capacity scaling and its delay.Under the conditions above, we consider the following protocol:Two-hop Relay Without Redundancy - capacity scaling: During each lot, in each

cell, a node is selected as a source to base station uniformly and randomly. If thisnode has new packets to deliver, it sends them to the BS in the cell. Otherwise, itremains idle.

1.2.1.1 When k = ω(m)

First, we consider that the number of receivers is much larger than that of infras-tructures, i.e. 0 ≤ b < d ≤ 1. Thus, we have a lemma as follows.

1.2 Capacity and Delay in the 2-hop Relay Algorithm Without Redundancy 3

Lemma 1 In each big cell, there are 3km nodes at most w.h.p., where k = nd , the total

number of destinations, m = nb, total number of BSs, and b < d.

Proof. Let Xi be a random variable denoted as the number of destinations in super-cell i, and E[Xi] be the expectation of Xi. Obviously, we have E[Xi] = km .

We learn Chernoff bounds in [161]:For any δ > 0,

P(Xi > (1+δ )E[Xi])< e−E[Xi] f (δ ) (1.1)

where f (δ ) = (1+δ ) log(1+δ )−δ .Take δ = 2, which satisfies the assumption of Chernoff bounds, from (1.1), we

can obtainP(Xi > (1+2)

km)< e−

km f (2) (1.2)

where f (2) = (1+2) log(1+2)−2 > 1.Thus, the possibility that the number of destinations in one supercell is less than

or equal to 3km when k = ω(m) can be calculated as:

P(Xi ≤3km

;∀i)≥ 1−mP(Xi >3km)> 1−me−

f (2)km (1.3)

Since, m = nb and km = nd−b(d −b > 0), we have that P(Xi ≤ 3km ∀i)→ 1 as n → ∞.

We know that one base station can just send one packet in one slot. For anypacket, because of Lemma 1, there are 3km destinations which can be accessed by aBS at most. I.e., in one slot, one BS, w.h.p., can make at most 3km transmissions.

Each base station will use queue to buffer the packets. Assume input rate λi andoutput rate µi. The number of packets arriving at m BSs in the interval [0,T ] is λiT m.Packets have to be sent at least k times, which means the number of transmissionsis λiT mk. As above, for certain packets, one BS can make 3km transmissions in oneslot. We therefore have the following:

λiT mk ≤3km

T m (1.4)

i.e., λi ≤ 3m . As a result, the rate of input for each queue in BS is Θ(1m ). Hence,

delay for the 2-hop Relay Algorithm Without Redundancy D = 1/P =Θ(n).Since input rate of every queue in BSs is Θ( 1m ), during time interval [0,T ], the

number of packets sent to BSs is Θ( 1m )×T m. For a stable network, the throughputof the network cannot exceed the number of packets that BSs are able to handle intime interval [0,T ]. We have:

λT n ≤Θ( 1m)×T m (1.5)

i.e., λ ≤ Θ( 1n ). Therefore, an upper bound of the capacity scaling for 2-hop RelayAlgorithm Without Redundancy is thus Θ( 1n ).


1.2.1.2 When k =Θ(m)

When k = Θ(m), i.e. b = d, we will demonstrate that the condition is very similarto k = ω(m) by the following Lemma.

Lemma 2 Suppose the total number of BSs and the total number of destinations areof the same order, nb. Destinations are uniformly located into m big cell, the numberof BSs which contain one destination at least, written as v = nt w.h.p., are of ordernb.

According to Lemma 2, we get λ =Θ( 1n ) and D =Θ(n).To sum up the first and second condition, we have the following theorem:

Theorem 1 Under two-hop without redundancy scheme, with infrastructure modeonly, if k = Ω(m), i.e. d ≥ b, the capacity and delay scaling of the network are 1n , n,respectively. Hence the tradeoff is λ/D = 1n2 .

1.2.1.3 When k = o(m)

Clearly, when k = o(m), the expected total number of big cells which contain atleast one destination, E(v) = k, w.h.p. because of Lemma 3.

Lemma 3 When k = o(m) and m → ∞, for any big cell Si containing at least onedestination, Si will contain exactly one destination w.h.p..

Proof. ∀Si, assume ki is the number of destinations in Si, then the probability for Sicontaining at least one destination is

P(ki ≥ 1) = 1− (1−1m)k (1.6)

In addition, the probability for Si containing at least two destinations is

P(ki ≥ 2) = 1− (1−1m)k − k(1− 1

m)k−1

1m

(1.7)

Then for any supercell Si which contains at least one destination, the probability forSi containing at least two destinations is

P(ki ≥ 2 | ki ≥ 1) = 1−(1− 1m )

k−1 km

1− (1− 1m )k(1.8)

In addition, we know when m → ∞ and km → 0 from k = o(m). Thus,

P(ki ≥ 2 | ki ≥ 1) = 1−km (1−

1m )e

− km

1− e− km(1.9)

Obviously, P(ki ≥ 2 | ki ≥ 1)→ 0, when m → ∞ and km → 0. This finishes the proof.

1.2 Capacity and Delay in the 2-hop Relay Algorithm Without Redundancy 5

Assume ε = b− d − δ > 0, where δ is a tiny positive constant. Then k× nε =nb−δ = o(nb) = o(m). Applying Lemma 3, these knε destinations, w.h.p., are dis-tributed in knε different supercells and each supercell only contains one destination.Hence, during one slot, the whole network can make nε successful passages at least.The output rate in every BS turns out to be n

ε

nb = n−d−δ . The probability that a packet

is sent from source to a BS turns out to be P =Θ(mC ×n−d−δ )× 1q =Θ(n

ε), whereq denotes the density of the network, which is of a constant order. And capacityλ =Θ(nε−1) and delay D =Θ(n1−ε). Because δ can be any tiny positive constant,ε can be quiet close to b−d, i.e., ε → b−d. Thus, we have the following theorem:

Theorem 2 Under 2-hop without redundancy protocol, exploiting infrastructuremode only, if k = o(m), i.e. d < b, the throughput capacity and delay of networkare Θ(nb−d−1), Θ(n1−b+d), respectively. And the tradeoff is λ/D =Θ(n2b−2d−2).

1.2.2 Lower bound of delay

Two-hop Relay Algorithm Without Redundancy - delay scaling : View all n queuesin n nodes as a big queue. Packets arrived in every queue are sent step by step,meaning that each slot has only one packet in the network. The order is decided bythe famous rule of first comes, first sends. If two packets reach in different nodes inthe same slot, we will choose the packet from the session i that maximizes (tp + i)mod N to be sent first, others will wait in the queue.

Lemma 4 Suppose the inputs to one single server queue are Poisson process withsub memoryless service times that are bounded in expectation by a certain value,say TN . Denote the arrival rate as λ , with λ < 1/TN , then the average delay willsatisfy:

D ≤ 12+

TN1−ρ

(1.10)

where ρ , λTN . The expression on the R.H.S. of the above inequality is a standardexpression for the delay in the M/M/1 queue with an i.i.d. service times, TN , whichare restricted to begin on time slot boundaries.

According to Lemma 7, we have:

Theorem 3 For Poisson inputs with rates λi for every source i, the network underthe scheme is stable iff Σiλi < 1/TN , and average end to end delay must satisfy:

D ≤ 12+

TN1−ρ

(1.11)

where ρ , ΣiλiTN . Note that TN = nm . Hence, when all sources have same input ratesλ , stability and logarithmic delay is achieved when λ = O( mn2 ).

Since m = nb, the capacity/delay tradeoff is λ/D = O(n2b−3).


Comment 1 1.2.3 Discussion

As a conclusion, the results of 2-Hop relay without redundancy using infrastructuremode are showed in the following table.

In [4], the author provided the result of 2-Hop relay without redundancy usingad-hoc mode. The capacity and delay of network are Θ( 1k ), Θ(n logk), respectively.And the tradeoff is λ/D =Θ( 1kn logk ).

Under largest capacity algorithm, if k = Θ(n), no matter what m is, using in-frastructure mode will lead to a better tradeoff than ad-hoc mode. If k = o(n), infirst two situation, tradeoff is Θ( 1n2 ), which is worse than Θ(

1kn logk ) =Θ(

1n1+d logn ),

where 0 ≤ d < 1. When k = o(m), tradeoff is Θ(n2b−2d−2). The ratio between trade-off in ad-hoc mode and infrastructure mode is Ta/Tf =Θ( 1n1+d logn )/Θ(n

2b−2d−2) =

Θ( n1+d−2blogn ). If b >

1+d2 , Ta/Tf → 0, when n → ∞, which means that infrastructure

mode has a better tradeoff. If b ≤ 1+d2 , Tf /Ta → 0, when n → ∞, which means thatad-hoc mode has a better tradeoff.

As to smallest delay algorithm, the capacity and delay tradeoff is O(n2b−3). Ob-viously, when b > 2−d2 , the tradeoff is larger than that of ad-hoc mode.

To sum up, the number of destinations k = nd , the number of base stations m= nb,the following table illustrates the final result:

Table 1.1 Capacity And Delay Tradeoff of 2-Hop Relay w.o. redundancy Algorithm using infras-tructure mode

mode k-m capacity delay tradeofflarge capacity k = ω(m) Θ( 1n ) Θ(n) Θ(

1n2 )

k =Θ(m) Θ( 1n ) Θ(n) Θ(1n2 )

k = o(m) Θ(nb−d−1) Θ(n1−b+d) Θ(n2b−2d−2)small delay O(nb−2) Θ(n1−b) O(n2b−3)

Table 1.2 Capacity And Delay Tradeoff of 2-Hop Relay w.o. redundancy Algorithm using infras-tructure mode

condition infrastructure ad-hocd = 1

√

d < 1 b > 1+d2 or b >2−d

2√

otherwise√

1.3 Capacity and Delay in the 2-hop Relay Algorithm With Redundancy 7

1.3 Capacity and Delay in the 2-hop Relay Algorithm WithRedundancy


In order to obtain biggest capacity, the relationship between the total number ofdestinations and the total number of BSs is taken into consideration and because thesituations when k = ω(n) and when k =Θ(n) are quite similar, we unite them intok = Ω(n) for simplification. Thus, we divide the problem into two parts and buildtwo schemes respectively.

1.3.1.1 When k = Ω(m)

Two-hop Relay Algorithm With Redundancy - capacity scaling I: Divide slots intotwo parts. One is odd, the other is even.

1. During each odd timeslot, do following:

• From Source to Relay Transmissions: The sender transmits packets SN, anddoes so in each opportunity until

√n repeats have been delivered to different

relays, or the k destinations have already obtained SN. Afterwards, the sendernumber is increased to SN + 1. Provided the sender does not own any newpackets to transmit, it stays idle.

• From Relay to Destination Transmissions: When a node will transmit relaypackets to desired destinations, the following steps will be done:– The receiver transmits its RN number for the packets.– The transmitter sends packets RN to the receivers. If the transmitter does

not own the requested packets RN, it will remain idle.– If all k destinations received RN, the transmitter will delete the packets

which have SN number identical to RN in the buffer.

2. During any even time slots, do following things: In every cell containing a BS, anode is selected as a source to BS uniformly and randomly. If this node has onepacket to transmit, it will randomly pick a packet and send this packet to the BSinside the cell. Otherwise, remain idle.

Next, we will demonstrate the good performance of this protocol by providingthe following theorem 4.

Theorem 4 When k = Ω(m), Two-hop Relay Algorithm with Redundancy schemewill achieve Θ(

√n) delay bound with the capacity in order of O( 1n ).

Proof. There are two main parts contributing to delay. One is duplicating a packet toΘ(

√n), denoted as T1, and the other is passing this packet to destinations, denoted

as T2. According to Lemma 1 in [13], T1 = O(√

n). We just need to calculate T2.


Same as the situation without redundancy, the output rate and input rate of thequeue in base stations is Θ( 1m ) when k = Ω(m). Assume every node contains Rdifferent packets on average as a relay. the probability that a given packet is sentfrom source to a BVS can be calculated as following:

P = [1− (1− mC)√

n]×Θ( 1m)× 1

q× 1

R(1.12)

→√

n× mC×Θ( 1

m)× 1

q× 1

R(1.13)

where q is the density of nodes in network, which is in a constant order. Recall thatm = nb and C =Θ(n), P = 1RΘ(

1√n ). Thus, delay T2 = 1/P = RΘ(

√n). In order to

achieve the delay bound Θ(√

n), R should be constant order, which results in delayD = T1 +T2 =Θ(

√n).

Consider a time interval T =Θ(√

n), which is equal to delay, λnT new packetscome to the network and each of them at most is duplicated

√n times. Thus, there

are at most λnT√

n packets in the network. As mentioned above, every node holdsR different packets on average. We get the following inequation:

λnT√

n ≤ Rn (1.14)

Replacing T and R by Θ(√

n) and Θ(1) respectively, we know the capacity of thenetwork is at most Θ( 1n ). i.e. λ = O(

1n ). This finishes the proof.

1.3.1.2 When k = o(m)

The scheme when k = o(m) is quite analogous to that when k = Ω(m). There is justa little change on the number of duplications.

Two-hop Relay Algorithm With Redundancy - capacity scaling II: Divide the timeslots into two parts. One is odd, and other is even.

1. During each odd timeslot, do following:

• From Source to Relay Transmissions: The sender transmits packets SN, anddoes so in each opportunity until n

1−b+d2 repeats have been delivered to dif-

ferent relays, or the k destinations have already obtained SN. Afterwards, thesender number is increased to SN +1. Provided the sender does not have anynew packets to transmit, it stays idle.

• From Relay to Destination Transmissions: When a node will transmit relaypackets to desired destinations, the following steps will be done:– The receiver transmits its RN number for the packets.– The transmitter sends packets RN to the receivers. If the transmitter does

not own the requested packets RN, it will remain idle.– If all k destinations received RN, the transmitter will delete the packets

which have SN number identical to RN in the buffer.

1.3 Capacity and Delay in the 2-hop Relay Algorithm With Redundancy 9

2. During any even time slots, do following things: In every cell containing a BS, anode is selected as a source to BS uniformly and randomly. If this node has onepacket to transmit, it will randomly pick a packet and send this packet to the BSinside the cell. Otherwise, it remains idle.

Theorem 5 When k = o(m), two-hop Relay Algorithm With Redundancy protocolcan achieve Θ(n 1−b+d2 ) delay bound with the capacity in order of O(nb−d−1), wherek = nd and m = nd .

The proof of theorem 5 is the same as that of theorem 4. We just need to replace√n in theorem 4 by n

1−b+d2 .


To get the minimum delay, we apply the following algorithm:2-hop Relay Algorithm With Redundancy - delay : View all n queues in n nodes

as one large queue. Packets that arrive in every queue are sent one by one. Thismeans in each time slot there is just one packet transmitting. The transmission orderis determined by the rule of first comes, first goes. If two packets arrive in differentnodes in exactly the same slot, select the packet from the session i which will max-imize (tp + i) mod N to be sent first, while others keep waiting in the queue. Theselected one is sent by the following two steps:

• First, if the node holding the original packet finds another node has not got itspacket in the same cell, it sends the packet to the node without the packet. Else re-main idle. This process will not end until there are Θ(n 1−b2 ) nodes in the networkthat contain the packet.

• Second, if one of these Θ(n 1−b2 ) nodes with the packet to be sent and a basestation are within the same cell. The node will send the packet to the BS and theBS transmits it to all the destinations. Else, BS remains idle.

In order to calculate the expectation of delay, we calculate the delay for singlepacket, TN . TN is combined by two parts: the time spending on duplicating Θ(n

1−b2 )

packets, denoted as T1 and the expectation of time that spends on passing one ofthese packets to a base station, denoted as T2.

First, we calculate T1. In each time slot, at least Θ(n− n1−b

2 ) nodes does notcontain the packet. the possibility that one of these nodes and the source are in thesame cell is at least p = 1− (1−1/C)n−

√n. Thus, the expectation of T1 is:

E(T1)≤n

1−b2

1− (1−1/C)n−n1−b

2→ n

1−b2

1− e−d(1.15)


To compute T2, note that the probability that one of the n1−b

2 meets the base

station is 1− (1−m/C)n1−b

2 =Θ(n b−12 ). Thus, we have:

E(T2) =1

Θ(n b−12 )=Θ(n

1−b2 ) (1.16)

As we regard the network as a single queue with n input streams of rates λ1, λ2,λ3..., λn which share a single server with service times TN . From lemma 7, we have:

Theorem 6 For Poisson inputs that have rates λi for every source i, the networkunder the scheme is stable iff Σiλi < 1/TN , and the average end-to-end delay willsatisfy:

D ≤ 12+

TN1−ρ

(1.17)

where ρ , ΣiλiTN . Note that TN = Θ(n1−b

2 ). Hence, when all sources have sameinput rates λ , stability and logarithmic delay can be achieved when λ = O(n b−32 ).

The tradeoff between capacity and delay is λ/D = O(nb−2).

Comment 2 1.3.3 Discussion

To sum up, the results of 2-Hop relay with redundancy using infrastructure modeare showed in the following table.

In [4], the author provided the result of 2-Hop relay with redundancy using ad-hoc mode. The capacity and delay of network are Ω( 1k√n logk ), Θ(

√n logk), respec-

tively. And the tradeoff is λ/D = Ω( 1kn logk ).First, we consider the algorithm for large capacity. Recall that k = nd and m =

nd . When d > 12 , no matter what m is, using infrastructure mode will lead to a bettertradeoff than ad-hoc mode. If d ≤ 12 , in first case, k = Ω(n), tradeoff of capacityand delay is Ω(n− 32 ), which is worse than Ω( 1kn logk ). But in second case, k = o(n),

tradeoff of capacity and delay is Ω(n3(b−d−1)

2 ). The ratio between tradeoff in ad-hoc

mode and infrastructure mode is Ta/Tf =Θ( 1n1+d logn )/Ω(n3(b−d−1)

2 ) = O( n1+d−3b

2logn ).

If b > 1+d3 , Ta/Tf → 0, when n → ∞, which means that infrastructure mode has abetter tradeoff. If b ≤ 1+d3 , Tf /Ta → 0, when n → ∞, which means that ad-hoc modehas a better tradeoff.

Second, we notice that in the algorithm for smallest delay, the tradeoff isO(nb−2). We could easily conclude that when b > 1 − d, transmitting by infras-tructure is a better choice.

As a conclusion, the number of destinations k = nd , the number of base stationsm = nb, the following tables demonstrate the final result in 2-hop relay algorithmwith redundancy:

1.4 Capacity and Delay in the multi-hop Relay Algorithm With Redundancy 11

Table 1.3 Capacity And Delay Tradeoff of 2-Hop Relay with redundancy Algorithm using infras-tructure mode

mode k-m capacity delay tradeofflarge capacity k = ω(m) Ω( 1n ) Θ(

√n) Ω(n− 32 )

k =Θ(m) Ω( 1n ) Θ(√

n) Ω(n− 32 )k = o(m) Ω(nb−d−1) Θ(n 1−b+d2 ) Ω(n

3(b−d−1)2 )

small delay O(nb−3

2 ) Θ(n 1−b2 ) O(nb−2)

Table 1.4 Infrastructure V.S. Ad-hoccondition infrastructure ad-hoc

d > 12√

d ≤ 12 b >1+d

3 or b > 1−d√

otherwise√

1.4 Capacity and Delay in the multi-hop Relay Algorithm WithRedundancy

The main difference between multi-hop and 2-hop is that since transmissions areallowed between two relays, packets can be well flooded in multi-hop model.

multi-hop Relay Algorithm With Redundancy : View all the n queues in n nodesas a big queue. Packets arrived in every queue will be sent one by one. This meansin each slot there is just one packet sending. The order is then determined by thefamous rule of first comes, first goes. If two packets arrive in different nodes inexactly the same time slot, select the packet in the session i which will maximize(tp + i) mod N to be sent, others keep waiting in the queue. The selected one is sentby the following: each time slot and in every cell, provided there is one node holdingthe packet and a BS, the node must transmit the packet to the BS and the BS willbroadcast it to all the destinations. Afterwards next transmission starts. If there isone node holding one packet and one node does not have this desired packet, floodsit to all nodes in that cell. Otherwise, keep idle.

1.4.1 When m =Θ(n)

We still consider the delay for a single packet. The probability that a node with thepacket meets a base station is p = m/C. Given m = Θ(n), C = Θ(n), p = Θ(1).It means the packet will be transmitted to base station in a constant time slot, i.e.TN =Θ(1). From lemma 7, we conclude:

Theorem 7 For Poisson inputs with rates λi for every source i, the network underthe scheme is stable iff Σiλi < 1/TN , and the average end-to-end delay will satisfy:


D ≤ 12+

TN1−ρ

(1.18)

where ρ , ΣiλiTN . Note that TN =Θ(1). Thus, when all sources have identical inputrates λ , stability and logarithmic delay is achieved when λ = O( 1n ).

The tradeoff between capacity and delay is λ/D = O( 1n ).

1.4.2 When m = o(n)

Similar to the situation m = o(n), we have the following lemma:

Lemma 5 Under multi-hop Relay Algorithm With Redundancy II, the expectationof delay for a single packet, TN , cannot be a constant order if m = o(n).

Proof. If TN = Θ(1), in a constant time, the expectation of duplications of the o-riginal packet can only be a constant order, denoted as s. The probability that anode with the packet meets a base station is p = 1− (1−m/C)s. Since m = o(n),C = Θ(n), s = Θ(1), p = o(1). Thus, the expectation of delay for a single pack-et, E(TN) = 1/p = ω(1). It is a contradiction with the assumption that TN =Θ(1).Therefore, TN cannot be a constant order.

Again applying lemma 7, we have:

Theorem 8 For Poisson inputs with rates λi for every source i, the network underthe scheme is stable iff Σiλi < 1/TN , and average end-to-end delay will satisfy:

D ≤ 12+

TN1−ρ

(1.19)

where ρ , ΣiλiTN . Note that TN = Ω(1). Hence, when all sources have same inputrates λ , stability and logarithmic delay is achieved when λ = o( 1n ).

The tradeoff between capacity and delay is λ/D = o( 1n ).

Comment 3 However, in [13], Neely gives a tradeoff attaining Θ( 1n(logn)2 ). Thatmeans, only when the number of base stations and the number of mobile users arein the same order, motioncast with base stations would be able to provide a bettercapacity and delay tradeoff. The following table will show the choice.

Table 1.5 Infrastructure V.S. Ad-hoccondition infrastructure ad-hoc

b = 1√

b < 1√

1.5 From i.i.d. mobility to random walk mobility 13

1.5 From i.i.d. mobility to random walk mobility

In this section, we consider another mobility model, random walk mobility, whichis more realistic than i.i.d. mobility. First, we will describe this mobility model andexplain the underlying difference between the two mobility models. Then, we willcalculate the bound of capacity and delay in several different scenarios as before.

1.5.1 Random Walk Mobility

As in the i.i.d. mobility model, the unit square is divided into n squares of area 1/neach, resulting in a discrete torus of size

√n×

√n. Time is divided into slots of

equal duration. Initially, each node is equally likely to be in any of the n subcells,independent of the other nodes. At the beginning of a slot, a node jumps from itscurrent cell to one of its adjacent cells, which is chosen in an uniformly randomfashion. By adjacent cell we mean the following: Let (i, j) : i, j = 0,1, ...,

√n−1, be

a numbering of the cells of the 2-D torus, as shown in Figure 1.5. The cells adjacentto cell (i, j) are the cells (i+1, j),(i−1, j),(i, j+1),and (i, j−1), where the additionand subtraction operations are performed modulo n.

Recall the definition of first hitting time in [18]. Definition of First Hitting Time:The first hitting time for the set of states A ⊂ SX is given by τAH = inf{t ≥ 0 : X(t) ∈A} with X(0) being distributed according to ΠX .

Next, we recall the result concerning the first hitting time for a single state incase of a 2-D torus of size

√n×

√n.

Lemma 6 Let H denote the first hitting time for a single state on a 2-D torus of size√n×

√n, then E{H}=Θ(n logn).

We note that the expectation of the first hitting time under the i.i.d. mobility is n,which is lower than that under random walk mobility by a factor of logn.

1.5.2 Capacity and Delay in the 2-hop Relay Algorithm withoutRedundancy

Before we proceed to analyze the capacity and delay, we need to simplify our net-work model so as to make our analysis convenient. We could look on a supercellas a 2-D torus of size

√ nm ×

√ nm and a mobile node is in this single supercell. If a

mobile node traverses the verge of a supercell and enter another supercell, since twodifferent base stations are the same for the mobile node, we could place the originalsupercell on a sphere whose base station is located on the north pole and the travers-ing is equal to reaching the sphere’s south pole and moving on. Hence the motionof nodes on a 2-D torus of size n× n with m base stations regularly distributed isequivalent of motion on a 2-D torus of size

√ nm ×

√ nm with a single base station.


1.5.2.1 Upper bound of capacity

When k = Ω(m)

We analyze the delay under the 2-hop relaying protocol. We denote the time by D.In order to transmit a packet to a base station, the source node first needs to enter thesame cell as a base station. Then two conditions should be satisfied for the successof transmission: the source node is scheduled to transmit the packet ; the base stationis scheduled to receive the packet. The probability that the source node is scheduledto transmit the packet to the base station is 1mq , where q is the density of nodes inthe network.Observe that

D = τ11

mq+ . . .+(τ1 + . . .+ τi)i−1

1mq

+ . . . (1.20)

where τ1 is the time required by the source node to meet a base station, henceforthdenoted by first meeting time; and τi for i ≥ 2 are the successive inter-meeting time.It is easy to see that the mean first meeting time is of the order of mean first hittingtime of a single state, in case of a random walk on a 2-D torus of size

√ nm ×

√ nm .

Using lemma 6, it follows that E{τ1}=Θ( nm logn). Further, the mean inter-meetingtimes are of the order of mean first return time(see, for example, [[162], Chap 2,p.2]) of a random walk on a 2-D torus of size

√ nm ×

√ nm , which is well know to

be nm . We therefore have E{τi} = Θ(nm ) for i ≥ 2. Taking the expectations on both

sides of 1.20, and performing some simple algebraic manipulations, we obtain

E{T} = E{τ1}+E{τ2}mq (1.21)

= Θ(nm

logn)+Θ(nm)mq (1.22)

→ Θ(n) (1.23)

Since input rate of each queue in base stations is not changed by the mobilitypattern of mobile nodes, which is still Θ( 1m ), during time interval [0,T ], the totalnumber of packets sent to base stations is Θ( 1m )×T m. To guarantee a stable net-work, the throughput of whole network cannot exceed the packets that base stationsare able to serve in time interval [0,T ]. We have:

λT n ≤Θ( 1m)×T m (1.24)

i.e., λ ≤ Θ( 1n ). Therefore, the upper bound of capacity in 2-hop Relay AlgorithmWithout Redundancy is Θ( 1n ).


When k = o(m)

The probability that a base station is scheduled to serve a packet is 1k . The probabilitythat in the cell, a source node is scheduled to transmit a packet to the base stationis 1q . Hence the probability that the source node is scheduled to transmit the packetto the base station is 1kq . Similar to the previous analysis, we have the followingequation:

D = τ11kq

+ . . .+(τ1 + . . .+ τi)i−11kq

+ . . . (1.25)

Taking the expectations on both sides of 1.25, we obtain

E{T} = E{τ1}+E{τ2}kq (1.26)

= Θ(nm

logn)+Θ(nm)kq (1.27)

→ Θ(n1+d−b) (1.28)

Since input rate of each queue in base stations is Θ( 1k ), during time interval[0,T ], the total number of packets sent to base stations is Θ( 1m )×T m. To guaranteea stable network, the throughput of whole network cannot exceed the packets thatbase stations are able to serve in time interval [0,T ]. We have:

λT n ≤Θ(1k)×T m (1.29)

i.e., λ ≤ Θ(nb−d−1). Therefore, the upper bound of capacity in 2-hop Relay Algo-rithm Without Redundancy is Θ(nb−d−1).

For this part, we could conclude that random walk mobility does not affect theresults. By looking at equation(1.23) and (1.48), we could find that the second partis greater than the first part in order and is thus dominant in deciding the order ofE{T}. The second part consists of the first return time and the inverse of probabilityof successful transmission for a source node and a base station. We note that the firstreturn time of random walk is the same with the first return time under i.i.d. mobilitymodel, which explains why the result doesn’t change while we employ random walkmobility.

Comment 4 For the upper bound of capacity, we note that in the previous analysisof this part, it is assumed that mobile nodes spend little time to reach a base station.In other words, mobile nodes are actually static and they always stay in the samecell of one of the base stations. When k = Ω(m), we have the following scheduling:in n time slots, each of the n nodes will be scheduled once to deliver a packet tothe base station located in the same cell. Then the base station can flood the packetto all of other base stations and then all the base stations can put the packet onthe downlink to mobile nodes, which is the end of a successful transmission. Hence,the capacity and delay of network are 1n , n, respectively. When k = o(m), we havesimilar analysis, which does not change the results under the i.i.d. mobility model. Itis easily understood that in the assumption, we have excluded the mobility of nodes,


so whether a node moves according to i.i.d. or random walk does not affect theresults.

1.5.2.2 Lower bound of delay

For the lower bound of capacity, first we calculate the delay TN when there is onlyone packet to be sent. TN is equal to expectation of time for a mobile node to meetany base station(a meeting means that the mobile node and base station are in thesame cell for a time slot).

According to Lemma 6, the expectation of first hitting time for a single state isnm log

nm . Since m = n

b, TN =Θ( nm logn).Then we exploit the following lemma to figure out the delay and capacity.

Lemma 7 Suppose the inputs to one single server queue are a Poisson process withsub-memoryless service times which are bounded in expectation by a certain valueTN . Denote the arrival rate as λ , , λ < 1/TN , then average delay will satisfy:

D ≤ 12+

TN1−ρ

(1.30)

where ρ , λTN . The expression on the R.H.S. of the above inequality is a standardexpression for the delay in an M/M/1 queue which has i.i.d. service times ,TN , thatare restricted to begin on time slot boundaries.

According to Lemma 4, we have:

Theorem 9 For Poisson inputs with rates λi for every source i, the network underthe scheme is stable iff Σiλi < 1/TN , and average end-to-end delay satisfy:

D ≤ 12+

TN1−ρ

(1.31)

where ρ , ΣiλiTN . Note that TN = nm logn. Thus, when all sources have identicalinput rates λ , stability and logarithmic delay is achieved when λ = O( mn2 logn ).

1.5.3 Capacity and Delay in the 2-hop Relay Algorithm withRedundancy

In this subsection, we will discuss the situation in the 2-hop relay algorithm withredundancy under the random walk mobility model.

First, we need to understand how redundancy decreases the delay in randomwalk mobility model. To begin with, we consider the situation of random walk in1-D torus: The set of states is S = {0,1,2, ..., p}. At the beginning of every timeslot, the mobile node jumps to one of its adjacent states, which is chosen uniformly.


By adjacent states, we mean that for state i, its adjacent states are i−1, i+1, wherethe addition and subtraction operations are performed modulo p+ 1. We have thefollowing transition graph for the 1-D random walk.

If there is only one mobile node in this 1-D torus, we have the following result.

Lemma 8 Assume that ki is the expectation of first hitting time of state 0 for thenode starting from state i, then

ki = (p− i+1)i (1.32)

Proof. According to Theorem 1.3.5 in [163], we have the following system of linearequations:

ki =

0 i = 0

1+ 12 ki−1 +12 ki+1 1 < i < p

1+ 12 k2 i = 1

1+ 12 kp−1 i = p

(1.33)

Solving the formula, the result follows.

According to the result, we could know that the largest expectation of first hittingtime of state 0 is (p+1)

2

4 , when the starting place isp+1

2 if p is odd. If p is even, thelargest expectation of first hitting time is (p+2)p4 when the starting place is either

p2

or p2 +1.If there are q(q ≥ 2) mobile nodes in this 1-D torus and they moves according to

the random walk model independently. We give the definition of first hitting time ofa state in this situation. Definition of First Hitting Time 2: The first hitting time forthe set of states A ⊂ SX and the set of mobile nodes B = {1,2,3, ...,q} is given byτAH = min{τ

A,iH : i ∈ B} where τ

A,iH = inf{t ≥ 0 : Xi(t) ∈ A} and Xi(t) is the Markov

Chain for i ∈ B with Xi(0) being distributed according to ΠXi .Since it is complicated to get the exact result of the first hitting time of state 0 for

multiple nodes moving simultaneously, we derive a lower bound of the expectationof the first hitting time. We note that the q mobile nodes “hit” state 0 when theminimum of the distances from the mobile nodes to state 0 becomes 0. So insteadof looking at q independent Markov Chains, we are considering a single MarkovChain for the minimum of these distances. We denote this variable by Z. The set ofstates for Z is A = {0,1,2, ..., p2}. Z transits from its current state i to state i− 1 ifthere are at least one node in the q mobile nodes that transit from state i (or p− i) tostate i−1 (or p− i+1). We denote the probability of transition to a smaller distanceby α . Then it is direct to get: α ≤ 1− ( 12 )

q.Thus we have the following transitiongraph for Z.

Then we get the following lemma, indicating the largest first hitting time of state0.

Lemma 9 Assume that kgi is the expectation of first hitting time of state 0 for gnodes starting from state i, the first hitting time of state 0 for Z starting from p2 is


kgp2=

p2

2α −1+

1−α(2α −1)2

(1−α

α)

p2 +

2α2 −2α(2α −1)2

(1.34)

Comment 5 The first hitting time of state 0 for Z starting from state 1 is

kg1 =[

1− (1−αα

)n]

12α −1

(1.35)

where α ≤ 1− ( 12 )g.

Proof. According to Theorem 1.3.5 in [163], we have the following system of linearequations:

kqi =

0 i = 0

1+αkqi−1 +(1−α)kqi+1 1 < i <

p2

1+(1−α)kq2 i = 1

1+ kqp2 −1

i = p2

(1.36)

Solving the formula, the result follows.

From the result, we could infer that when q is large enough, kqp2= p2 . This result

means that when there is a large number of nodes on the 1-D torus moving accordingto random walk mobility, it is with high probability that every time there is at leastone node moving towards state 0.

Next, we consider random walk on 2-D torus. We have the following approximateresult about the first hitting time of random walk on 2-D torus.

Lemma 10 The time for the q nodes that carry the packet to meet a base station is2kqp

2.

Proof. Let us call the walk Xn and write X+n and X−n for the orthogonal projections

of Xn on the diagonal lines y =±x (Figure 1.6).Then X+n and X

−n are independent simple random walks and Xn = 0 if and only

if X+n = 0 = X−n . We divide the process of hitting state (0,0) into 2 stages. The first

stage is that X−n becomes 0. The second stage is that X+n becomes 0. Figure 1.2

(this is when there are two nodes moving on the 2-D torus) illustrates the transitionprobabilities for X+n and X

−n , respectively.

Observe that X−n is the same with the Markov Chain of Z we have describedbefore. Although in the figure X+n is the same with random walk of one node on 1-Dtorus, we should note that when X−n hits 0, X

−n ’s transition matrix will be the same

with Z as well. The first hitting time of state 0 for X−n is kqp2

while that for X+n is also

kqp2. The result follows.

Comment 6 (Although these two stages happen simultaneously, here we are con-sidering the upper bound. While in the 2nd stage, it is natural to ask whether X+ncan become larger than 0. We should note that when q becomes large enough, X+n


Xn

Xn

Xn-

+

Fig. 1.1 Projection of random walk

3/8

1/8

1/8

3/8

3/4

1/4

1/2

1/2

XnXn

-+

Fig. 1.2 Transition probability of two orthogonal random walk


would stay close to 0 when X−n is doing random walk. As long as X−n is starting at

a distance of an even integer from state 0 in the 2nd stage, the nodes will surely hitstate(0, 0) when X−n hits 0).

In the 2-Relay algorithm with redundancy, the source node first needs to meeta number of nodes, to which the source node transmits the packet. Then the relaynodes help the source node to transmit the packet to one of the base stations. Next,we calculate a upper bound on the time the source node need to meet q distinctnodes on 2-D torus.

Lemma 11 The time for source node to meet g distinct nodes on 2-D torus is atmost Θ(g2).

Proof. If the source node moves away from its initial position by a distance of q,the source node will meet q distinct nodes, w.h.p., either vertically or horizontally.We denote the time for the source node to exit the area, of which the bound is at adistance of a from the node’s origin, by τaE . Let (x0,y0) be the cell containing theorigin. Also, let (xt ,yt) be the cell in which node i lies at time t. Further, we definethe following two variables:

τa−x∆= inf{t ≥ 0 : (xt − x0)≤−a}

τa+x∆= inf{t ≥ 0 : (xt − x0)≥ a}

and τa+y , τa−y be similarly defined with yt , y0 in place of xt and x0, respectively.Observe that

P(τaE ≤ m)≤ P(τ+x ≤ m or τ−x ≤ m or τ+y ≤ m or τ−y ≤ m) (1.37)

for m ≥ 0. Using the union bound and appealing to the symmetry of node motion,we obtain

P(τaE ≤ m)≤ 4P(τa+x ≤ m) (1.38)

Now, observe that before time τaE , xt has the following form:

xt = x0 +t

∑i=1

si

where si are i.i.d. random variables taking values in {−1,0,1} with probabilities{1/4,1/2,1/4}, respectively. Then we have:

P(τa+x ≤ k) = 2P(x⌊k⌋− x0 > a)+P(x⌊k⌋− x0 = a)≤ 2P(x⌊k⌋− x0 ≥ a) (1.39)

for k ≥ 0, where ⌊•⌋ denotes the greatest integer function. Since each si has mean 0and variance 1/2, a straightforward application of Lemma 8 in [18] gives

P(xt − x0 ≥ a)≤ e−a22t f or t ≤ a (1.40)

1.6 Hybrid Random Walk Models 21

for t ≥ a. When t = a28 , P(τa+x ≤ a2)≤ e−4. Then P(τaE ≤ a2)≤ 8e−4.

So P(τaE ≥ a2) ≥ 1 − 8e−4. Thus E{τaE} ≥ a2. Substituting a by q, the resultfollows.

In sum, the time of the whole procedure, i.e., the source node meeting q distinctrelay nodes and relay nodes helping source node to meet base stations, is 2kgp

2+

Θ(g2) + η , where η is reciprocal of probability of successful transmission for asource node and a base station. Here we assume that the first return time is 1, whichmeans that there is always more than one node, among the g nodes, in the same cellwith a base station and η is the time for the source node to be scheduled to transmita packet to the base station. When k = Ω(m), η = mq; when k = o(m), η = kq,where q is the density of nodes in the network.

Our next question is how many relay nodes we should choose to make this delayas small as possible. Increasing g by 1, we get two parts of difference to the abovedelay: ∆1 =−Θ(( 12 )

g p), ∆2 =Θ(g) and ∆ = ∆1 +∆2, which is the total difference.When ∆ = 0, we get g ≈ log p.

In our model, p= n1−b

2 . With similar analysis in the previous section, we have thefollowing conclusion: considering the upper bound of capacity, when k = Ω(m), thedelay is D =Θ(nmax{ 1−b2 ,b}) and the network capacity is O( 1D logn ); when k = o(m),

the delay is D = Θ(nmax{ 1−b2 ,d}) and the network capacity is O( 1D logn ). Consider-

ing the lower bound of capacity, the delay is Θ(n 1−b2 ) and the network capacity isO(n

b−32 ).

1.6 Hybrid Random Walk Models

These models are parameterized by a single parameter β , which takes value between0 and 1/2. The unit square is divided into n2β squares of area 1/n2β (referred to asβ -Cell), resulting in a discrete torus of size nβ × nβ . Each β -Cell is then furtherdivided into n1−2β square cells of area 1/n each. As before, m = nb base stations areregularly distributed on the unit square and thus the unit square is also divided intom supercells. At each slot a node is assumed to be in one of the cells inside a β -Cell.Initially, each node is equally likely to be in any of the n cells, independent of theother nodes. At the beginning of a slot, a node jumps from its current cell to oneof the cells in an adjacent β -Cell, which is chosen in an uniformly random fashion.Note that for β = 0, the above mobility model is essentially the i.i.d. mobility model;and for β = 1/2, it is the same as the random walk model.


1.6.1 Capacity and Delay in the 2-hop Relay Algorithm WithoutRedundancy


Since the mobility pattern does not affect the upper bound of capacity, next we onlycalculate the delay.

When k = Ω(m)

When b ≥ 2β , the number of base stations in a β -Cell is N1 = nb−2β . The numberof cells that a β -Cell contains is N2 = n1−2β . So when a node chooses a cell in aβ -Cell randomly, the probability that the node enters a cell with base station is N1N2 =nb−1. As we have analyzed before, in each slot the probability that a base station ischosen to serve the packet is 1m . Hence the probability of successful transmission isP = nb−1 · 1m =

1n . Thus, the delay is D = 1/P =Θ(n).

When b < 2β , on average every n2β−b β -Cells contain a single base station. The2-D torus of size n×n with m base stations could be simplified into a 2-D torus ofsize nβ−

b2 ×nβ− b2 with one base station. Again, we have the following equation:

D = τ11

mq+ . . .+(τ1 + . . .+ τi)i−1

1mq

+ . . . (1.41)

Taking the expectations on both sides of equation (1.41), we obtain

E{D} = E{τ1}+E{τ2}mq (1.42)= Θ(n2β−b logn)+Θ(n2β−b)mq (1.43)→ Θ(n2β ) (1.44)

When k = o(m)

When b ≥ 2β , the number of base stations in a β -Cell is N1 = nb−2β . The numberof cells that a β -Cell contains is N2 = n1−2β . So when a node chooses a cell in aβ -Cell randomly, the probability that the node enters a cell with base station is N1N2 =nb−1. As we have analyzed before, in each slot the probability that a base station ischosen to serve the packet is 1m . Hence the probability of successful transmission isP = nb−1 · 1k = n

b−d−1. Thus, the delay is D = 1/P =Θ(n1−b+d).When b < 2β , we have the following equation:

D = τ11kq

+ . . .+(τ1 + . . .+ τi)i−11kq

+ . . . (1.45)

1.6 Hybrid Random Walk Models 23

Taking the expectations on both sides of equation (1.45), we obtain

E{D} = E{τ1}+E{τ2}kq (1.46)= Θ(n2β−b logn)+Θ(n2β−b)kq (1.47)→ Θ(n2β−b+d) (1.48)


When b ≥ 2β , the result under the hybrid random walk model is the same with thatunder i.i.d. model.

When b < 2β , we calculate the time required for a packet to be sent from anode to a base station. Applying Lemma 6, we obtain that TN = n2β−b logn. Further,according to Theorem 9, the lower bound of delay is n2β−b logn and the capacity is

1n2β−b+1 logn

.

1.6.2 Capacity and Delay in the 2-hop Relay Algorithm WithRedundancy

Since we have found that when b ≥ 2β , the hybrid random walk model is the samewith i.i.d. model, we only consider the scenario when b < 2β .

As we have already discussed, the procedure of a successful transmission usingthe 2-hop algorithm with redundancy is divided into two steps: the first step is for thesource node to meet g nodes which have not received the packet before; the secondstep is that all the g+1 nodes that have the packet, including the source node, moveto meet a base station in the network. The time for the first step in the hybrid randomwalk model is Θ(g) if β < 1/2. If β = 1/2, the model is transformed into randomwalk model, in which the time is Θ(g2). The time for the second step is 2kgp

2+η ,

where η is reciprocal of probability of successful transmission for a source nodeand a base station and p is the side length of the 2-D torus. Here we assume that thefirst return time is 1, which means that there is always more than one node, amongthe g nodes, in the same cell with a base station and η is the time for the source nodeto be scheduled to transmit a packet to the base station. When k = Ω(m), η = mq;when k = o(m), η = kq, where q is the density of nodes in the network. When weconsider the lower bound of delay, η = 0. Thus, the time for the whole procedure is2kgp

2+Θ(g)+η .

Increasing g by 1, we get two parts of difference to the above delay: ∆1 =−Θ(( 12 )

g p), ∆2 =Θ(1) and ∆ =∆1+∆2, which is the total difference. When ∆ = 0,we get g = log p. The total delay is: D =Θ(p).

In the hybrid random walk model, p = nβ−b2 . Our result is as follows: Consid-

ering the upper bound of capacity, when k =Ω(m), the delay is D=Θ(nmax{β− b2 ,b})


and the network capacity is O( 1D logn ); when k= o(m), the delay is D=Θ(nmax{β− b2 ,d})

and the network capacity is O( 1D logn ). Considering lower bound of capacity scenari-

o, the delay is D =Θ(nβ− b2 ) and the network capacity is O(n b2−β−1).

1.7 The Impact of Node Density in the Network

Some related works have assumed a similar network model of 2-D torus, but witha different node density, i.e., the number of nodes in a cell. If a node has a largertransmission power, the node can transmit a packet to more neighbor nodes and thusthere is a higher density of nodes. In other words, the node density has a strong rela-tionship with wireless transmission power. In this section, we consider the impact ofnode density on our previous results under purely i.i.d. mobility model. The analysisin this section could be easily extended to include the impact of transmission power.

We assume that the node density is q, which is not necessarily Θ(1). The numberof cells in the network is c = nq . The quantities of other parameters are used withoutchange.



We consider the scenario when k = Ω(m). The probability that a node enters a basestation’s cell in each slot is mc . The probability that a specific base station, among allbase stations, is chosen to serve a packet is 1m . Note that there are q nodes in a basestation’s cell, so the probability that a node is chosen to transmit a packet to the basestation is 1q . Therefore, the probability of successful transmission is P=

mc ×

1m ×

1q =

1n . The delay for the 2-hop algorithm without redundancy is D =

1P = Θ(n). The

upper bound of capacity is λ =Θ( 1n ). With the same method of analysis, we obtainthat when k = o(m), the upper bound of capacity is λ =Θ(nb−d−1) and the delay isD =Θ(n1+d−b). We could conclude that the choice of node density does not affectthe upper bound of capacity and the corresponding delay.


It is easy to get that the time required for a packet to be sent from a node to a basestation is TN = cm =

nmq . According to Theorem 9, the lower bound of delay is

n1−bq

and the capacity is qn2−b , which both contains the parameter q.

1.7 The Impact of Node Density in the Network 25

When q = Θ(n1−b), the delay is Θ(1) and the capacity is Θ( 1n ). In this case, acell is equivalent to a supercell. Each slot, any node is able to communicate with abase station so the smallest delay is in the order of a constant.



First we consider the scenario when k = Ω(m). We assume that a source node needsto transmit a packet to g relays. The time required for this process is T1 =Θ(g). Thetime for all the relays to meet a base station is denoted by T2 and the delay is givenas follows:

D = T1 +T2

= g+(mc· 1

m· 1

q)−1 · 1

g

= g+ng

In order to make D as low as possible, we choose g to be√

n. Thus the delay isD =Θ(

√n). We have the following inequation:

λnDg ≤ Rn (1.49)

Replacing g, T and R by√

n, Θ(√

n) and Θ(1) respectively, we know the capacity ofthe network is at most Θ( 1n ). With the same method of analysis, we obtain that whenk = o(m), the upper bound of capacity is Θ(nb−d−1) and the delay is Θ(n 1−b+d2 ).Thus we could also conclude that the density node has no effects on the results.

However, since above analysis is based on the assumption that in each slot andeach cell there is only one sender and one receiver, we are interested in the scenariowhere a sender can transmit a packet to all nodes in a cell, which is natural in wire-less communication. Thus when looking for relays, the source node could transmitits packet to at most q nodes at one time slot. In this case, a slight change should bemade to the delay:

D = T1 +T2

=gq+(

mc· 1

m· 1

q)−1 · 1

g

=gq+

ng



nq. Thus the delay is

D =Θ(√

nq ). We have the following inequation:

λnDg ≤ Rn (1.50)


nq, Θ(√

nq ) and Θ(1) respectively, we know the ca-

pacity of the network is at most Θ( 1n ). With the same method of analysis, we ob-tain that when k = o(m), the upper bound of capacity is Θ(nb−d−1) and the delay

is Θ( n1−b+d

2√q ). Thus we could conclude that for the single-sender-multiple-receiver

case, the node density will decide the quantity of delay but still not affect the upperbound of capacity.


We consider the single-sender-single-receiver and single-sender-multiple-receivercases respectively.

For the single-sender-single-receiver case, the delay is:

D = T1 +T2

= g+(mc)−1 · 1

g

= g+n

mqg


nmq . Thus the delay is

D = Θ(√

nmq ). According to Theorem 9, the lower bound of delay is

n1−b

2√q and the

capacity is√

q

n3−b

2.

For the single-sender-multiple-receiver case, the delay is:

D = T1 +T2

=gq+(

mc)−1 · 1

g

=gq+

nmqg

In order to make D as low as possible, we choose g to be√ n

m . Thus the delay is

D =Θ(√ n

m1q ). According to Theorem 9, the lower bound of delay is

n1−b

2q and the

capacity is qn

3−b2

. Hence, even though there is only one receiver or multiple receivers,

the results in this part have relationship with node density.

1.8 Random Way-point Mobility 27

1.8 Random Way-point Mobility

We review the random way-point mobility model in [164]. In this model, at eachtime-step the mobile node chooses a random destination on the sphere and movestowards it at a random speed. The speed is chosen uniformly from the interval[vmin,vmax], where vmin(n) and vmax(n) are strictly positive. The movement is a-long the great circle that passes through the initial position and final destination.On reaching the destination, the node pauses for a random amount of time and theprocess repeats itself. In this work, we consider the RWMM with no pause times.The pause times can easily be accounted for with only a minor set of changes in theanalysis.

First we give some definitions:

• S = {the locations of base stations on the sphere}• d(x,y) = The distance between the two points x and y on S2.• R=

∪s∈S {x ∈ S2 : d(x,s)≤ r(n)/2}

We start with the following lemma which shows that the probability that a lineconnecting two random points on S2 intersects with R is Θ(

√mr(n)).

Lemma 12 Let L be a line connecting two uniformly and independently chosenpoints on S2. Then, there exist strictly positive constants c1 and c2 such thatc1√

mr(n)≥ P(L intersects R)≥ c2√

mr(n), for large enough n.

Along the line of analysis in [164], we have the following result:

Theorem 10 Let I(n) denote the inter-meeting time of a mobile node and base s-tations under the RWMM. Then E{I(n)}=Θ( 1√mr(n)v(n) ). Here v(n) is the averagevelocity of the nodes.

Replacing r(n) and v(n) by 1√n and1√n , we obtain that E{I(n)}=Θ(

n√m ). Since

we have already been familiar with the method of analysis, we give the followingresults without providing the process of calculation.



When k = Ω(m), the delay is D = Θ(n1+ b2 ). When k = o(m), the delay is D =Θ(n1+d− b2 ).



The lower bound of delay is D =Θ(n1− b2 ) and the capacity is λ =Θ(n b2−2).



When k =Ω(m), the delay is D=Θ(n 12+ b4 ) and the capacity is λ =Θ( 1n

b2 +1

). When

k = o(m), the delay is D =Θ(n 1+d2 − b4 ) and the capacity is λ =Θ(n b2−d−1).


The delay is TN =Θ(n12−

b4 ) and the capacity is λ = O(n b4− 32 ).

1.9 Applying Network Coding in 2-hop Relay Algorithm withRedundancy

1.9.1 Network coding

Traditionally, when a source node needs to transmit B packets to its destinations,it should transmit the B packets one by one. Network coding enables the sourcenode to transmit linear combinations of the B packets to its destinations. As long asthe destination has received more than B combinations of the original packets, it isable to obtain the B original packets. For decoding purposes, the transmitting nodesalso send the random coding vectors as overhead within each packet. Each node vcollects the coding vectors for the packets it receives in a decoding matrix Gv. Areceived packet is said to be innovative if its coding vector increases the rank of thematrix Gv.

We have found that network coding does not affect the results of 2-hop relayalgorithm without redundancy, so next we analyze the 2-hop relay algorithm withredundancy. Under this scheme, the source node first need to meet a number of relaynodes. Every time the source node meets a relay node, it encodes the B packets andtransmits the packet. After all relay nodes have each received a packet from thesource node, which is a linear combination of original B packets, they move tomeet a base station. When B of all the relay nodes have transmitted their packets tobase stations, the base stations could cooperate to decode the packets and obtain the

1.9 Applying Network Coding in 2-hop Relay Algorithm with Redundancy 29

original B packets and then transmit to the destinations. We should note that the timefor the base stations to transmit B packets to destinations is at most B time slots butthe time for each packet is still in the order of constant. Thus we ignore this amountof delay in our analysis.


First we consider the scenario when k = Ω(m). We assume that a source node needsto transmit a packet to g relays. The time required for this process is T1 =Θ(g). Thetime for all the relays to meet base stations and transmit an innovative packet to basestations for B times is denoted by T2 and the delay is given as follows:

T = T1 +T2

= g+(mc· 1

m· 1

q)−1 · 1

g·B

= g+Bng

In order to make T as low as possible, we choose g to be√

Bn. Thus the delay isT =Θ(

√Bn). We have the following inequation:

λB

nT g ≤ Rn (1.51)


Bn, Θ(√

Bn) and Θ(1) respectively, we know the ca-pacity of the network is at most Θ( 1n ). The per-packet delay is D =

TB =

√ nB . With

the same method of analysis, we obtain that when k = o(m), the upper bound of

capacity is Θ(nb−d−1) and the per-packet delay is Θ( n1−b+d

2√B

).


We have the following equation for T :

T = T1 +T2

= g+(mc)−1 · 1

g·B

= g+Bnmg


In order to make T as low as possible, we choose g to be√

Bnm . Thus the delay is

T = Θ(√

Bnm ). The per-packet delay is D =

TB = Θ(

n1−b

2√B). According to Theorem

9, the capacity is Θ(√

Bnb−3

2 ).We could conclude that the employment of network coding in 2-hop relay al-

gorithm with redundancy brings benefit in the per-packet delay and thus leads to abetter capacity-delay trade-off.

1.10 Discussion

In Section 4, Section 5 and Section 6, we calculate the upper bound of capacity andlower bound of delay in 2-hop without redundancy model, 2-hop with redundancymodel and multi-hop with redundancy model respectively. Moreover, we presentalgorithms, which are able to achieve these bounds, to fulfill the task of MotionCast.In this section, we draw a comparison of the capacity and delay tradeoffs with theformer results.

The capacity and delay tradeoffs that we obtain in this chapter can be summarizedin Table 1.6.

From the table, we see that delay of the 2-hop algorithm with redundancy isbetter than that of the 2-hop algorithm without redundancy when they both achievetheir same upper bound of capacity. It means redundancy decreases delay and due toBSs, without an cost of the capacity. The tradeoff for the capacity and delay tradeoff

for these two schemes are O(n−2min{1,1−b+d}) and Θ(n−3min{1,1−b+d}

2 ), respectively.Redundancy obviously offers a better tradeoff. Furthermore, if we expect to achievethe corresponding lower bound of delay, we show that redundancy enhances bothper-node throughput and delay. By applying redundancy, the tradeoff is improved tonb−2 from n2b−3.

The profile of tradeoff-destination line in Figure 1.8 is analogous to Figure 1.7,but the tradeoff is bigger with a same number of destinations. Compared with theconclusions in [4], transmission through BSs is supposed to be one better selectionwhen d > 12 or b > min{

1+d3 ,1−d} than exploiting ad-hoc in [4].

In [13], Neely gives a tradeoff attaining Θ( 1n(logn)2 ) in multi-hop relay algorithmwith redundancy. That means, only when the number of base stations and the num-ber of mobile users are in the same order, motioncast with base stations would beable to provide a better capacity and delay tradeoff.

To sum up, it turns out if we have enough base stations, our tradeoff is better thanthat in [13] and [4].

1.11 Conclusion and Future Work 31

1.11 Conclusion and Future Work

In this chapter, we study capacity and delay tradeoffs for MotionCast using infras-tructure. We present the performance of the 2-hop relay algorithm without redun-dancy and then utilize redundancy to improve the tradeoff largely. After that, weallow multi-hop transmission instead of 2-hop. By comparing with the results inad-hoc mode, denoting the number of destinations k = nd and the number of base s-tations m = nb, we found that in 2-hop relay algorithm without redundancy scheme,when b > min{ 1+d2 ,

2−d2 } or d = 1, using infrastructure mode is better than ad-

hoc mode. And in 2-hop relay algorithm with redundancy scheme, when d > 12 orb > min{ 1+d3 ,1−d}, using infrastructure mode is better than ad-hoc mode. Howev-er, when it comes to multi-hop model, infrastructure fails to make an amazing per-formance. Only when m = Θ(n), using infrastructure mode will improve tradeoffby a factor log(n)2. We have not taken into account the effect of different mobilitypatterns yet, such as random walk, random way-point mobility and so forth. More-over, in our model, the base stations can modify their transmission range to coverthe whole area. What is the situation if transmission range of base stations is limitedso that they could only reach a fraction of the area sometimes? These may be ourfuture work.

Fig. 1.3 Network Model

Comment 7


S

R-N

R-N

R-N

R-B

R-B

R-B

D

D

D

D

D

D

D

D

D

D

D

Fig. 1.4 Traffic Pattern

Table 1.6 Capacity and Delay Tradeoffs

scheme condition capacity delay

2-hop relay w.o. redund upper λ O(n−min{1,1−b+d}) Θ(nmin{1,1−b+d})2-hop relay w.o. redund lower D O(nb−2) Θ(n1−b)

2-hop relay w. redund upper λ O(n−min{1,1−b+d}) Θ(nmin{1,1−b+d}

2 )

2-hop relay w. redund lower D O(nb−3

2 ) O(n1−b

2 )

m-hop relay w. redund m =Θ(n) O( 1n ) Θ(1)m-hop relay w. redund m = o(n) o( 1n ) ω(1)

1.11 Conclusion and Future Work 33

Fig. 1.5 Random Walk Model


Fig. 1.6 Queue in Base Station

Fig. 1.7 Tradeoff against destinations in 2-hop w.o. redundancy

Fig. 1.8 Tradeoff against destinations in 2-hop w. redundancy

Chapter 2MotionCast: General Connectivity in ClusteredWireless Networks

Abstract We propose a novel concept of (k,m)-connectivity in mobile clusteredwireless networks, in which there are n mobile cluster members and nd static clus-ter heads. (k,m)-connectivity signifies that in each time period consisting of m timeslots, there exist at least k time slots, during any one of which every cluster mem-ber can directly communicate with at least one cluster head. We investigate thecritical transmission range of asymptotic (k,m)-connectivity when cluster membersmove according to random walk or i.i.d. mobility model. Under random walk mod-el, we propose two general heterogeneous velocity models which characterize aninherent property of many applied wireless networks that cluster members movewith different veloc

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