On Multicast Capacity and Delay in CognitiveRadio Mobile Ad-hoc Networks

Yixuan Li, Xinbing WangShanghai Jiao Tong University, China

Email: lyx1990116, xwang8@sjtu.edu.cn

Chee Wei TanCity University of Hong Kong, Hong Kong

Email: cheewtan@cityu.edu.hk

Abstract—In this paper, we focus on the capacity and delaytradeoff for multicast traffic pattern in Cognitive Radio (CR)Mobile Ad-hoc Networks (MANET). In our system model,the primary network consisting of n primary nodes, overlapswith the secondary network consisting of m secondary nodesin a unit square. Assume that all nodes move according toan i.i.d. mobility model and each primary node serves as asource that multicasts its packets to kp primary destinationnodes whereas each secondary source node multicasts its packetsto ks secondary destination nodes. Under the cell partitionednetwork model, we study the capacity and delay for the primarynetworks under two communication schemes: Non-cooperativeScheme and Cooperative Scheme. The communication patternconsidered for the secondary network is Cooperative Scheme.Given that m = nβ (β > 1), we show that per-node capacityO(1/kp)

1 and O(1/ks) are achievable for the primary networkand the secondary network, respectively, with average delayΘ(n log kp) and Θ(m log ks). Moreover, to reduce the averagedelay in the secondary network, we introduce the RedundancyScheme and prove that a per-node capacity O(1/ks

√m log ks)

is achievable with average delay Θ(√m log ks). We find that the

fundamental delay-capacity tradeoff in the secondary networkis delay/capacity ≥ O(mks log ks) under both the cooperativescheme and redundancy scheme.

I. INTRODUCTION

Since the seminal work by Gupta and Kumar [1], thestudy of the capacity of wireless networks has received greatattention. Gupta and Kumar showed that the per-node capacityis Θ(1/

√n log n) for a unicast traffic pattern, where n is the

number of nodes in the network. Compared with unicast trafficpattern where packets are sent from one source node to onedestination node, multicast traffic pattern has several benefits.Using this pattern, packets from source nodes are delivered tomultiple destination nodes and certain links may be shared bydifferent destinations. Therefore, the multicast traffic pattern issupposed to reduce the frequency resources that are requiredto establish communication with all destination nodes.

Compared to the capacity scaling in the static networkmodel, mobility has been leveraged to improve the capacitybounds. In [2], M. Grossglauser et al. demonstrated that per-node capacity Θ(1) is achievable under the mobility model.However, they neglected another significant issue of wireless

1Given two functions f(n) > 0, g(n) > 0: f(n) =o(g(n)) means limn→∞ f(n)/g(n) = 0; f(n) = O(g(n)) mean-s limn→∞ sup f(n)/g(n) < ∞; f(n) = ω(g(n)) is equivalent tog(n) = o(f(n)); f(n) = Ω(g(n)) is equivalent to g(n) = O(f(n));f(n) = Θ(g(n)) means f(n) = O(g(n)) and g(n) = O(f(n)).

networks besides throughput (capacity): the communicationdelay. This motivated the later focus on the relationshipbetween delay and throughput. In [3], M. Neely et al. studiedthe capacity-delay tradeoff in cell partitioned MANET byintroducing the redundancy scheme which reduced delay bysacrificing capacity. They developed communication schemesto achieve per-node capacity Θ(1), Θ( 1√

n) and Θ( 1

n logn ) withaverage delay Θ(n), Θ(

√n) and Θ(log n), respectively, which

implies that the capacity-delay tradeoff is delaycapacity > O(n).

Later on, the capacity-delay tradeoff for unicast networks wasthoroughly studied using various mobility models such as therandom walk model [4], the random way point mobility model[5] and the constrained mobility model [6], [7].

For a multicast traffic pattern, X. Li et al. [8] studied thecapacity in static networks where each node sends messagesto k − 1 destinations. Based on their interference model,they demonstrated that the per-node capacity is Θ( 1√

kn logn)

if k = O( nlogn ) whereas the per-node capacity Θ( n

logn ) isachievable when k = Ω( n

logn ). This result is actually thegeneralization of the other two traffic patterns: unicast [1] andbroadcast [9]. Some relevant works concerning the capacity-delay tradeoff may also be found in [10], [11] and [12]. In[13], X. Wang studied the capacity and delay tradeoff in cellpartitioned MANET under a multicast traffic pattern. Theyproved that the per-node capacity and delay are O(1/k) andΘ(n log k) respectively if no redundant relay nodes are intro-duced, and O(1/k

√n log k) and Θ(

√n log k) respectively if

there are redundant relay nodes.

All the aforementioned results concentrate on the through-put scaling and delay analysis for a single network. Recently,the ever-growing demand for frequency resources has motivat-ed study of cognitive radio (CR) networks to efficiently utilizethe idle spectrum in the time and space domain. CR networkscan be regarded as a superposition of two independent net-works: primary network and secondary network. In cognitivenetworks, primary users have a higher priority when accessingthe spectrum, while secondary users could opportunisticallyaccess the licensed spectrum without harming the performanceof primary users. Specifically, the secondary network has“vacuum” space in which no transmissions can take place.In particular, secondary nodes within the interference range ofany primary nodes that are broadcasting or receiving messagescan not have spectrum opportunities. Unlike a single network,

2

the potential interaction between primary and secondary usersshould be considered when scaling the throughput and delayin CR networks. Therefore, the previous results of singlenetworks may not be effectively applied to CR networks.

The study of capacity and delay scaling laws for boththe primary and secondary network is a relatively new andchallenging field. In [14] and [15], the authors investigatedthe capacity scaling of a homogeneous cognitive networksfor unicast traffic by introducing a preservation region aroundprimary receivers. Wang et. al [16] extended the CR networkcapacity scaling to multicast traffic patterns under the Gaussianchannel model. Interestingly, all these works showed that boththe primary and secondary networks can achieve similar or thesame performance bounds as they are stand-alone networkswhen the secondary network has a higher density than theprimary one. Note that existing works mainly focus on staticCR networks while the performance of CR networks under amobility model has not been investigated to the best of ourknowledge.

Motivated by the fact that mobility can dramatically enhancethe throughput in stand-alone networks, we are interested infinding out:

• What throughput and delay scaling law can be achievedunder a multicast traffic pattern in Cognitive Radio Mo-bile Ad-hoc Networks (CR MANET)?

• What is the delay and capacity tradeoff in CR MANET?

In our approach, we first adopt a non-cooperative schemeto investigate the achievable per-node capacity and delay forthe primary network where one source is supposed to sendmessages to kp destinations. However, the per-node capacityreduces from O(1/kp) to O(1/k2p) as kp increases from o(n)to Θ(n), which motivates us to study the cooperative schemeamong primary destination nodes to improve the per-nodecapacity when kp = Θ(n). Next, we focus on the capacityand delay analysis for the secondary network under a cooper-ative scheme and a redundancy scheme, respectively. Finally,we introduce a destination oriented redundancy scheme toefficiently utilize the network resources for the secondarynetwork. The major contributions of this paper are as follows:

• We propose and investigate the mobility model for boththe primary network and the secondary network. Com-pared with the mobility model in [17] and [18], whichrequires the primary nodes to be static and only thesecondary nodes are allowed to move, our model ismore practical and general as both the primary andsecondary networks can be mobile. A novelty is to usethe transmission queues to analyze the capacity and delayfor CR MANET. Our results shows that mobility cansignificantly enhance the capacity performance for boththe primary and secondary networks.

• This paper is a nontrivial generalization of previous worksince we directly study the capacity and delay under amulticast traffic pattern. Our results can be specialized tothe unicast and broadcast traffic by setting the number ofdestinations to be 1 and n for the primary network (1 and

m for the secondary network), respectively.• We show that both the primary network and the secondary

network can achieve the same throughput as the optimalone established for a stand-alone MANET in [13], ifthe number of secondary users m is larger than that ofprimary users n scaled by m = nβ (β > 1). Specifically,under the cooperative scheme, per-node capacity O(1/kp)and O(1/ks) are achievable for the primary network andthe secondary network, respectively, with average delayΘ(n log kp) and Θ(m log ks).

• We introduce redundancy into the secondary network andprove that the tight bound of transmission delay canbe reduced to Θ(

√m log ks) with achievable per-node

capacity O(1/ks√m log ks) if there are redundant relay

nodes. To achieve the bound, a redundancy scheme isproposed. Moreover, we also proposed destination ori-ented redundancy scheme to help save wireless resourcesin practical operation. We find that the fundamentalcapacity-delay tradeoff in the secondary network is de-lay/capacity ≥ O(mks log ks) under both the cooperativescheme and the redundancy scheme.

The rest of the paper is organized as follows. In sectionII, we introduce our network model and give some necessarydefinitions. In section III, we present the communicationschedule for both the primary and the secondary network. Insections IV and V, we put forward our main results of capacityand delay in the primary network and secondary network,respectively. Finally, we conclude in section VI.

II. NETWORK MODEL

Cell Partitioned Network Model: Our network model isbased on the model utilized in [3]. Consider a unit squarewith a co-existing primary network and secondary network,where the two networks share the same space, time andfrequency domain but with different priorities in accessing thespectrum. The primary and secondary nodes are distributedaccording to a Homogeneous Poisson Process (HPP) of densityn and m, respectively. And we assume n and m satisfym = nβ (β > 1), which indicates that the secondary networkis more dense than the primary network. It can be provedthat the total numbers of primary nodes and secondary nodeswithin the unit area are of order Θ(n) and Θ(m), respectively(See Lemma 9). For simplicity, we will assume that thereare n mobile primary nodes and m mobile secondary nodesdistributed over the square throughout the paper, which doesnot influence our main results. We further divide the primarynetwork into w = Θ(n) non-overlapping cells with equal areaΘ( 1n ) and divide the secondary network into c = Θ(m) non-overlapping cells with equal area Θ( 1

m ) as illustrated in Figure1. The node density τp is n

w for the primary network and τsis m

c for the secondary network. Clearly, both τp and τs areof constant order Θ(1).

Traffic Pattern: In a multicast traffic pattern, we assume thatthere are a set Vp = v1p, v2p, . . . , vnp mobile primary nodesand another set Vs = v1s , v2s , . . . , vms mobile secondarynodes in the unit square. For each multicast group in the

3

1( )m

Θ

1( )n

Θ

Inactive secondary cells

Active secondary cells

Primary cells

Fig. 1. Illustration of cell partitioned network model: the blue primarycells represent the active cells when a pair of primary nodes (denoted bygreen spots) are transmitting within it. No spectrum opportunities will beavailable for secondary nodes located in the region of active primary cells.Thus we color the secondary cells blue to indicate that they are inactive dueto the presence of active primary cells. All the secondary nodes in the inactivesecondary cells are buffering their packets without transmitting.

primary network, there is a source node vi ∈ Vp and kpdestinations randomly and independently chosen nodes in Vp.Multicast groups in the secondary network can be similarlydefined.

Mobility Model: Assuming that time is slotted, we adoptthe ideal i.i.d. fast mobility model. The initial position of eachnode (primary or secondary) is equally likely to be in any ofthe (primary or secondary) cells independent of other nodes’positions. At the beginning of the next time slot, nodes canroam to a new cell randomly over all cells in the network.Since we adopt the fast mobility model, the time it takes for anode to move from one cell to another will not be accountedfor in the delay. Under the mobility model, packets are carriedby the nodes (source or relay nodes) until they reach theircorresponding destination nodes.

Interference: In CR MANET, there are three kinds ofinterference: inter-cell interference, intra-cell interference andinter-network interference. To avoid intra-cell interference,we assume that each cell (primary or secondary) allowsat most one transmission during a time slot. In order tomitigate inter-primary-cell interference, neighboring primarycells have to transmit over orthogonal frequency bands andfour bands are enough for the whole primary network to ensurethis condition[3]. The inter-secondary-cell can be avoided byadopting a 25-TDMA schedule for the secondary network.Since the primary nodes have higher priority in using the spec-trum, the secondary network has to operate without causinginterference to the primary network. To limit the inter-networkinterference, no spectrum opportunities will be available for asecondary node if it resides in the region of an active primarycell. 2 If a secondary node roams to an inactive secondary cell

2A primary cell becomes active when a transmission between two primarynodes takes place in it; otherwise, it is called an inactive primary cell.Accordingly, we define a secondary cell as active only when it is locatedin the region of inactive primary cells.

at a time slot, it will buffer its packets without transmitting.Capacity: We assume that the exogenous rate of packets to

each primary and secondary source node are both a PoissonProcess with rate λp and λs packets per slot, respectively.The network is stable when there exists a scheduling policy toensure that the packet queue for each node will not approachinfinity as time approaches infinity and each packet can bedelivered to its k (k = kp for the primary network and k = ksfor the secondary network) distinct destinations. We define λpand λs as the achievable per-node capacity for the primarynetwork and the secondary network, respectively.

Delay: In the multicast communication pattern, the delayin the primary network is defined as the time interval fromthe time that the packet departs from the primary (secondary)source node to the time that it arrives at all the kp (ks)destination nodes.

Redundancy: We say that there exists redundancy whenmore than one node is acting as a relay for transmitting asingle packet within one time slot.

Transmission Queues: We analyze the communication delaythrough queueing theory. For both the primary network and thesecondary network, packets are transmitted through source-to-destination queues, source-to-relay queues and relay-to-destination queues according to their specific communicationschemes.

Useful Notations: The key notations used in this paper arelisted in Table I.

Notations Definitionsλp Input rate of primary source node (per-node

capacity for the primary network).λs Input rate of secondary source node (per-node

capacity for the secondary network).λop Total service rate of primary source queue.

λos Total service rate of secondary source queue.

λisub Input rate of a primary source-to-destination

sub-queue.λosub Output rate of a primary source-to-destination

sub-queue.λos→r Output rate of a primary source-to-relay queue.

µos→r Output rate of a secondary source-to-relay queue.

Dps→d Delay of primary source-to-destination

communications.Dp

s→r Delay of primary source-to-relay communications.Dp

r→d Delay of primary relay-to-destinationcommunications.

Dss→r Delay of secondary source-to-relay communications.

Dsr→d Delay of secondary relay-to-destination

communications.Dp Total delay for the primary network.Ds Total delay for the secondary network.

TABLE IUSEFUL NOTATIONS

.

III. COMMUNICATION SCHEME

In this section, we will introduce the communicationschemes for both the primary network and the secondary

4

network. Due to the priority of the primary network, it operateswithout being aware of the existence of the secondary net-work. The secondary network adaptively chooses its protocolaccording to the given primary transmission scheme.

A. Communication Scheme for the Primary Network

In the primary network, we assume that the packets aredelivered using at most two hops. The source node eithersends packets directly to all the destinations or to one ofthe relays. Then the relay will forward the packet to all thecorresponding destinations. Each cell becomes active when atleast one successful transmission can happen. In the following,we consider two schemes: non-cooperative scheme 3 andcooperative scheme.

Non-cooperative Scheme:For an active cell containing at least two primary nodes, the

transmissions are conducted in the following way.1) If at least one primary source-destination pair can be

found in the cell, then randomly pick one pair to finishthe communication.

2) If no source-destination pairs can be found in onecell, perform the following two schedules with equalprobability:

• Source-to-Relay Transmission: If one primarysource node can be found in the cell and at leastone normal relay node 4 is available for the sourcenode in the same cell, pick the source node as thesender and a relay node as receiver to finish thetransmission.

• Relay-to-Destination Transmission: If one relay n-ode, carrying a packet destined for a primary desti-nation node, can be found in one cell and at least onecorresponding “pristine” primary destination node 5

stays within the same cell, pick the relay node asthe sender and one of these “pristine” destinationnodes as receiver to finish the transmission.

3) If neither of the two items mentioned above are satisfied,no transmissions will take place in this cell. A packetwill be discarded whenever all its kp primary destina-tions have received it.

Cooperative Scheme:According to the previous scheme, one primary destination

node can not serve as a relay for other destinations within thesame multicast group. That is to say, a certain destination nodeeither receives packets from corresponding source node oracts as a relay for other multicast groups. However, under thecooperative scheme, we will no longer discriminate between

3We use the term “non-cooperative” to describe the situation when adestination node can not serve as a relay for other destinations within thesame multicast group. Otherwise, the destination can act as a relay for anymulticast group, which is called “cooperative”.

4For a certain multicast group, all nodes in the area except for thosedestination nodes within this multicast group can be treated as normal relaynodes for this multicast group.

5A “pristine” primary destination node represents a destination node whichhas not received the desired packet, that the relay node is carrying. This isthe same with a “pristine” secondary destination node.

the destination nodes and normal relay nodes. The first nodethat the source encounters in its cell will be treated as a relaynode irrespective of whether it is a destination node or not.Once a destination node is chosen as a relay node, it can sendpackets to other destination nodes (possibly within the samemulticast group or not). Under the cooperative scheme for theprimary network, the following two transmission patterns areconducted with equal probability:

1) Source-to-Relay Transmission: If one primary sourcenode can be found in the cell and one relay node isavailable in the same cell, pick the source node as thesender and one relay node as the receiver to finish thetransmission. Note that the receiver is treated as a relaynode irrespective of whether it is a destination node ornot.

2) Relay-to-Destination Transmission: If one relay node,carrying a packet destined for primary destination n-odes, can be found in this cell and meanwhile at leastone corresponding “pristine” primary destination noderesides within the same cell, pick the relay node as thesender and one destination node as receiver to finish thetransmission.

If neither of the two items mentioned above are satisfied,no transmissions will take place in this cell.

B. Communication Scheme for the Secondary Network

Unlike the scheme in the primary network, a node inthe secondary network can only opportunistically transmitwhenever it is outside the region of active primary cells.Since a secondary cell is smaller than a primary cell, theregion of a primary cell can actually cover Θ(mn ) secondarycells. All these secondary nodes within a certain primary cellregion will transmit using the primary cell’s frequency. Hence,interference is likely to happen among the secondary cellswithin the same primary cell. To mitigate the interferencebetween secondary cells, we adopt a 25-TDMA scheme toschedule the secondary cells within the same primary cellregion. We assume the time slot of a secondary frame is equalto the time slot of a primary frame. Each secondary frame isfurther divided into 25 sub-frames. During each time slot, asecondary cell becomes active in one sub-frame.

From the capacity analysis for the primary network in sec-tion IV, we find that the cooperative scheme can achieve bet-ter performance than the non-cooperative scheme. Therefore,we only consider the cooperative scheme for the secondarynetwork without considering the non-cooperative scheme any-more. The cooperative scheme in the secondary network issimilar to that in the primary network.

Cooperative Scheme:Under the cooperative scheme for the secondary network,

the following two transmission patterns are conducted withequal probability for an active secondary cell:

1) Source-to-Relay Transmission: If one secondary sourcenode can be found in the cell and one relay node isavailable in the same cell, pick the source node as the

5

sender and one relay node as the receiver to finish thetransmission. Note that the receiver is treated as a relaynode irrespective of whether it is a destination node ornot.

2) Relay-to-Destination Transmission: If one relay node,carrying a packet destined for secondary destinationnodes, can be found in this cell and at the same timeat least one corresponding “pristine” primary destinationnode resides within the same cell, pick the relay nodeas the sender and one destination node as the receiverto finish the transmission.

If neither of the two items mentioned above are satisfied,no transmissions will take place in this cell. A packet will bediscarded when all its ks destinations have received it.

Note that in the cooperative scheme, only one relay node isused for sending a single packet. For the secondary network,we further propose a redundancy scheme in order to reducethe delay, which allows more than one relay to be used fordelivering a single packet. Then, we improve the redundancyscheme by avoiding introducing extra nodes other than thedestination nodes which have received packets to serve asrelays. This is referred to as destination oriented redundancyscheme. In this way, we are able to better utilize networkresources under this scheme. This communication schedulewill be discussed in more details later on in section V.

IV. CAPACITY AND DELAY ANALYSIS FOR THEPRIMARY NETWORK

In this section, we will study the capacity and delay tradeoffin the primary network. Denote the probability that there areat least two primary nodes in one cell by p1. Denote theprobability of finding a source-destination pair in the primarynetwork by p2. We first calculate p1 by

p1 = 1− (1− 1

w)n −

(n

1

)1

w(1− 1

w)n−1

= 1− (1− 1

w)n − n

w(1− 1

w)n−1

∼ 1− (1 + τp)e−τp .

To get p2, we need to model the primary nodes as mutuallyexclusive groups according to the following definition.

Definition 1: kp + 1-grouped Primary Network: Foreach source node vip ∈ Vp in the primary network, we putit and its kp destination nodes into a group, denoted by Gip.Then we will have n

kp+16 groups over the whole primary

network. We also assume that Gip∩

Gjp = ∅, for i = j, andi, j ∈ [1, n

kp+1 ].Note that in the kp + 1-grouped primary network, each

node within Gip can be a source node or destination node.Thus, any two nodes from the same group are a source-destination pair. However, for any randomly chosen two nodes

6Here we assume that n is divisible by kp + 1.

from different groups, they cannot be viewed as a source-destination pair. Then, we get

p2 = 1− [(1− 1

w)kp+1 +

(kp + 1

1

)1

w(1− 1

w)kp ]

nkp+1

= 1− (1− 1

w)

nkpkp+1 (1 +

kpn)

nkp+1

∼

1− e−τpeτ = 0, if kp = o(n);

1− (1 + τp)e−τp , if kp = Θ(n).

Next we move on to a capacity and delay analysis for thenon-cooperative scheme.

A. Capacity and Delay Analysis of the Non-cooperativeScheme

In this subsection, we will study the achievable capacity andcommunication delay in the primary network when destinationnodes can not relay packets to other destination nodes withinthe same multicast group. Our main results are presented inthe following.

Theorem 1: In a cell-partitioned network with overlappingn primary nodes and m secondary nodes, the achievableper-node capacity for the primary network under the non-cooperative scheme is λp = O(1/kp) with average delayE[Dp] = Θ(n log kp) when kp = o(n), and λp = O(1/k2p)with average delay E[Dp] = Θ(n log kp) when kp = Θ(n).

Proof: In each time slot, a new packet arrives at primarysource node vip in the group Gip with rate λp. Denote the ratethat the packet is transmitted to a primary destination node orhanded over to a relay node by R1 and R2, respectively. Thenwe have

λop = R1 +R2.

Since the transmission of source-to-relay and relay-to-destination are of equal probability, R2 is also equal tothe rate at which the relay nodes are sending packets toprimary destinations. Thus, in every time slot, the total rateof transmission opportunities over the primary network isn(R1+2R2). Meanwhile, a transmission occurs in any givencell with probability p1. Hence we obtain

wp1 = n(R1 + 2R2). (1)

Similarly, since p2 denotes the probability that a primarysource-destination pair can communicate with each other inone cell, we find

wp2 = nR1. (2)

Combining Equations (10) and (2), we have

R1 =p2τp

; R2 =(p1 − p2)

2τp.

Then we obtain that the total service rate for one primarysource queue is λop =

(p1+p2)2τp

.Now we will deal with the traffic delay for the primary

network using queueing theory techniques. There are twopossible routing strategies for a primary packet to reach itsdestination: the 1-hop source-to-destination path and the 2-hop source-to-relay-to-destinations path. For the first strategy,

6

it is clear that transmission delay consists of only the queueingdelay at the source since we neglect the propagation delay. Onthe other hand, in the second strategy, delay is composed ofthe queueing delay at both the source and the relay nodes.

If one packet is directly sent from the source node todestination nodes, it will wait at the source for a time periodDps→d before the source can find its corresponding destinations

to distribute this packet. Since a source node transmits packetsto kp destinations for multicast traffic, we assume that thereare kp identical copies of the packet in the buffer of thesource node. Thus, we can model a source queue as a setof kp independent sub-queues, in which each sub-queue isintended for one destination. Then the process of directlysending a packet to kp destinations can be viewed as the packetbeing transmitted through kp independent source-destinationroutings, as illustrated in Figure 2.

Source Packet

Destination 1

Destination 2

Destination 3

Destination

Destination

1kp−

kp

jP

2

jP

3

jP

1pk

jP−

pk

jP

pλ p pk λO

subλ

1

jP1

jP

2

jP

3

jP

1pk

jP−

pk

jP

Fig. 2. Illustration of the source-to-destination transmission process inthe primary network under the non-cooperative scheme. Since one packet isintended for kp destinations, we assume that there were kp identical copiesof the packet in the buffer of the source node. Then a source queue can beregarded as a set of kp independent sub-queues, in which each sub-queue isintended for one destination.

Denote the delay in each routing by Djs→d. Then we have

Dps→d = max

1≤j≤kpDj

s→d.

To calculate the delay in this scenario, we first obtain theinput and output rate for each source-to-destination sub-queuein the following lemma.

Lemma 1: Each source-to-destination sub-queue is aM/M/1 queue corresponding to one destination node withinput rate λisub =

λpkpR1

λop

and service rate λosub =R1

kp.

Proof: It is clear that the probability that the packet willbe sent directly from the source to the destination is R1

λop

. Asillustrated in Figure 2, for each sub-queue, we pick one nodefrom these kp nodes in the source pool as the sub-queue’ssource. Therefore, in each time slot, each sub-queue has aps→d = 1 − (1 − R1

λop)kp ∼ kp

R1

λop

probability of receiving a

packet. 7 Hence, the input rate of each sub-queue is

λisub = λpps→d =λpkpR1

λop.

The output rate of each sub queue is equal to the communi-cation rate of a source-to-destination pair λosub =

R1

kpbecause

the kp destinations are equally likely to get the packet directlyfrom the source.

Thus, the expected queueing time for each sub-queue is1

λosub−λ

isub

. According to Lemma 9, we can conclude that

Dps→d =

Θ

(1

R1kp

−λpkpR1λop

), kp = 1

Θ

(log kp

R1kp

−λpkpR1λop

), kp > 1

. (3)

On the other hand, if the packet is transmitted through arelay node, the delay is composed of both the queueing timeat source node and relay node. We denote the delay in the firstphase (from source to relay) by Dp

s→r and delay in the secondphase (from relay to destination) by Dp

r→d. Then we have thefollowing lemma.

Lemma 2: In the first phase of 2-hop routing strategy underthe non-cooperative scheme, the expected delay before a pri-mary source node can find a normal relay node to send a packetis equal to 1/(

R2pos→r

λop

−λp), where pos→r = 1−(1− 1w )

n−kp−1.Proof: The probability that a packet is handed over by a

relay in the primary network is R2

λop

. Thus, for a source-to-relayqueue with input rate λp and service rate λos→r, the averageresponse time (including both the waiting time and servicetime) is 1/(λos→r − λp).

Next we need to find λos→r. Consider a source node vip ∈ Gipthat is scheduled to send a packet to a relay node. According toour communication scheme, the source node can find a normalrelay node only when there exists at least one node vkp /∈ Gip inthe same cell as the source. The probability for this conditionis

pos→r , 1− (1− 1

w)n−kp−1.

Then we have λos→r =R2p

os→r

λop

. Therefore, we obtain thedelay in the first phase as

Dps→r =

1

λos→r − λp=

1R2pos→r

λop

− λp. (4)

Thus we conclude this lemma.Next we will calculate the delay in the second phase Dp

r→d.Note that the probability that the packet is sent to a relay nodeis pis→r =

λos→r

n−kp−1 =R2p

os→r

λop(n−kp−1) , because all these n−kp−1

primary nodes outside the group Gip are equally likely to bechosen as a relay under the non-cooperative scheme. Then

7Actually, according to Taylor’s formula, we have 1 − (1 − R1λop)kp =

kpR1λop

+ o(kpR1λop). Since the smaller part o(kp

R1λop) makes no difference

to our results, we will omit it in paper and assume that ps→d = 1 − (1 −R1λop)kp = kp

R1λop

.

7

the input rate for a relay-to-destination queue in the primarynetwork λir is

λir = λppis→r =

λpR2pos→r

λop(n− kp − 1).

A relay node carrying packets from one source node willforward the packet to all the kp destinations. Hence, similarto the previous discussion, we can model this process as apacket transmitted through kp independent sub-queues andeach destination is associated with one certain sub-queue, asshown in Figure 3.

Relay Packet

Destination 1

Destination 2

Destination kp

ii

ii

rλi

p rk λ

jP

1

jP1

jP1

1pn k− −

2

3

2pn k− −

2

jP

3

jP

1pk

jP−

pk

jP

2

jP

pk

jP

Fig. 3. Illustration of the relay-to-destination transmission process in theprimary network under the non-cooperative scheme. Since one packet isintended for kp destinations, we assume the designated relay will duplicate thepacket into kp identical copies and then distribute them to the correspondingdestinations. Then a relay queue can be modeled as a set of n − kp − 1independent sub-queues, among which there are kp sub-queues intended fordestinations.

Thus, using the same technique as in the previous analysis,the input rate of one sub-queue is kpλir and the output rate is

R2

n−kp−1 because one primary node can relay packets for allthe other n−kp−1 nodes except the kp+1 nodes in its group.Therefore, the expected queueing time for each sub-queue inthe relay node is 1

R2n−kp−1−kpλi

r

. By Lemma 9, we can obtain

Dpr→d =

Θ

(1

R2n−kp−1−kpλi

r

), kp = 1

Θ

(log kp

R2n−kp−1−kpλi

r

), kp > 1

. (5)

Combining Equations (3), (4) and (5), we can obtain that thetotal expected delay for the primary network under multicasttraffic (kp > 1) is

E[Dp] =R1

λopDps→d +

R2

λop(Dp

s→r +Dpr→d)

=R1

λop·Θ

(log kp

R1

kp− λpkpR1

λop

)

+R2

λop

(1

R2pos→r

λop

− λp+Θ

(log kp

R2

n−kp−1 − kpλir

)).

(6)

In the mathematical expression of E[Dp] in Equation (6),the first item represents the delay of the 1-hop source-to-destination transmission and the second item represents the

delay of the 2-hop source-to-relay-to-destination transmission.From the previous results, we find that the value of kp (whetherkp equals o(n) or Θ(n)) makes a major difference on thetransmission pattern of the primary network since both R1

and R2 are determined by kp. Therefore, we need to studythe capacity and delay tradeoff based on the value of kp.

1) If kp = o(n), then p2 ∼ 0 and pos→r ∼ 1 − e−τp .Thus, R1 ∼ 0 and R2 = λop ∼ 1−(1+τp)e

−τp

2τp, which

implies that the dominant communication pattern forthe primary network is the 2-hop source-to-relay-to-destination transmission strategy. Therefore, E[Dp] isdetermined by the second item of Equation (6). Sincethe input rate of a stable queue must be smaller than itsoutput rate, we haveλp <

R2pos→r

λop

∼ (1−e−τp )R2

λop

;

kpλir ∼

(1−e−τp )λpkpR2

λop(n−kp−1) < R2

(n−kp−1) .

The constraint of the above two inequalities allowsthe per-node capacity λp <

λop

(1−e−τp )kp, which means

a per-node capacity O(1/kp) for the primary networkis achievable when kp = o(n). A delay E[Dp] =Θ(n log kp) is achievable in this case.

2) If kp = Θ(n), then we have p1 ∼ 1 − (1 + τp)e−τp .

Thus, R2 ∼ 0 and R1 = λop ∼ 1−(1+τp)e−τp

τp, which

implies that the 1-hop source-to-destination transmissionstrategy is the dominant communication pattern for theprimary network. Therefore, E[Dp] is determined bythe first item of Equation (6). Similarly, we have thefollowing inequality

λpkpR1

λop<

R1

kp.

Thus, the per-node capacity λp = O(1/k2p) and delayE[Dp] = Θ(kp log kp) = Θ(n log kp) are achievable forthe primary network when kp = Θ(n).

This concludes the proof of Theorem 1.Note that when kp = Θ(n), the per-node capacity decreases

to O(1/k2p) but the delay is still on the order of Θ(n log kp),which implies that the capacity performance of our com-munication scheme is relatively poor as kp increases to thesame order as n. This motivates us to improve the schemeby allowing packets to be transmitted from one primarydestination node to another, which will be discussed in thefollowing subsection.

B. Capacity and Delay Analysis of the Cooperative SchemeIn this subsection, we bring forward the cooperative trans-

mission scheme for the primary network. We will show that, byutilizing the cooperation among destinations within the samemulticast group, the capacity of the primary network can beimproved when kp = Θ(n).

It is clear that under the cooperative scheme, packets canbe transmitted only through the 2-hop source-to-relay-to-destination pattern. The difference from the previous subsec-tion is that the relay node for each packet is selected from n−1

8

nodes rather than from n − kp − 1 nodes. Our main resultsunder this scenario are presented in the following theorem.

Theorem 2: In a cell-partitioned network with overlappingn primary nodes and m secondary nodes, the achievable per-node capacity for the primary network is λp = O(1/kp) withaverage delay E[Dp] = Θ(n log kp) under the cooperativescheme.

Proof: The proof is similar as that of Theorem 1. Toavoid confusion, we will adopt the same mathematical symbolsas the previous subsection. Since only the 2-hop source-to-relay-to-destination pattern is available under the cooperativescheme, we have R2 = λop = p1

2τp. For the source-to-relay

delay Dps→r, we have the following lemma.

Lemma 3: In the first phase of 2-hop routing strategy underthe cooperative scheme, the expected delay before a primarysource node can find a relay node to send a packet is

Dps→r =

1

(R2pos→r

λop

− λp)=

1

pos→r − λp, (7)

where pos→r = 1− (1− 1w )

n−1.Proof: The proof is quite similar to that of Lemma 2 and

we omit it to avoid triviality.Now we will discuss the delay in the second phase Dp

r→d.Under the cooperative scheme, when destination nodes canserve as relay nodes for any multicast group, the input ratefor a relay-to-destination queue λir is

λir =λpR2p

os→r

λop(n− 1)=λpp

os→r

n− 1.

If a destination node is chosen as a relay node, then it onlyneeds to transmit packets to the remaining kp − 1 destinationnodes. Otherwise, a normal relay node must send packets toall the kp destination nodes. It is clear that whether kp− 1 orkp destination nodes that are supposed to receive the packetmakes no difference to the delay in an order sense. Therefore,we assume that a relay node has to relay packets for kpdestination nodes regardless of whether it is a destinationnode or not. Hence, using the same technique as the previoussubsection, we have that the input rate for a sub-queue is kpλirand the output rate is

λop

n−1 . Therefore, the expected queueingtime for each sub-queue in the relay node is 1

R2n−1−kpλi

r

.

According to Lemma 9, we can obtain

Dpr→d =

Θ

(1

R2n−1−kpλi

r

), kp = 1

Θ

(log kp

R2n−1−kpλi

r

), kp > 1

. (8)

Combining Equations (7) and (8), we have that the totalexpected delay under multicast traffic (kp > 1) is

E[Dp] = Dps→r +Dp

r→d

=1

R2pos→r

λop

− λp+Θ

(log kp

R2

n−1 − kpλir

). (9)

From Equation (9), it is clear that the following twoinequalities hold:λp <

pos→rR2

λop

∼ (1−e−τp )R2

λop

;

kpλir ∼

(1−e−τp )λpkpR2

λop(n−1) < R2

(n−1) .

By solving the above two inequalities, we have that the per-node capacity λp <

λop

(1−e−τp )kpwhich indicates that per-node

capacity O(1/kp) is achievable under the cooperative schemeregardless of the order of kp. Note that the expected delayE[Dp] is still in the order of Θ(n log kp). Therefore we haveshowed that the cooperative scheme is able to improve theper-node capacity for the primary network when kp = Θ(n).This concludes the proof for Theorem 2.

From the above results, we find that the per-node capacity ofthe primary network is always scaled by O(1/kp) irrespectiveof the value of kp under the cooperative scheme, which impliesthat the cooperative scheme performs better than the non-cooperative scheme.

V. CAPACITY AND DELAY ANALYSIS FOR THESECONDARY NETWORK

In this section, we will discuss the capacity and delaytradeoff for the secondary network. Unlike the scheme in theprimary network, a node in the secondary network can onlyopportunistically transmit whenever it is outside the region ofactive primary cells. Thus, when scaling the capacity and delayfor the secondary network, we should consider the impactof the primary network and the transmission interruption ofsecondary nodes. In the following, we first analyze the averagecapacity and delay under the cooperative scheme. We thenintroduce the redundancy scheme and destination orientedredundancy scheme to help reduce the delay and efficientlyutilize the network resources for the secondary network.

A. Capacity and Delay Analysis of the Cooperative Scheme

The following lemma indicates that, with the communica-tion schemes defined previously, the secondary nodes (whethersource nodes or relay nodes) have finite opportunities todeliver their packets. And later we can prove that the wholesecondary network appears to have the same capacity anddelay performance as a stand-alone MANET in order sense.

Lemma 4: With the proposed communication scheme, thesetwo results hold:

1) In each time slot, a constant fraction of the secondarycells is outside the region of active primary cells, whichcan be scheduled successfully for transmission.

2) Each individual secondary cell has a finite spectrumopportunity to transmit.

Proof: Let cq(n) be the fraction of primary cells with q n-odes ( q ≥ 0 ). According to Lemma 1 in [19], cq(n) = e−1/q!w.h.p.. Then 1 − 2e−1 ≈ 0.26 fraction of the primary cellscontains at least two primary nodes w.h.p.. This implies thatthe rest fraction of primary cells may not be active and allow

9

the secondary nodes access to the spectrum for transmitting.Thus we conclude the first part of the lemma.

The above part implies a finite transmission opportunity forthe overall secondary cells. We have to further consider thetransmission opportunity of each individual secondary cell.Recall that the secondary network adopts a 25-TDMA schemewith adjacent-neighbor communication. In each primary timeslot, we have a complete secondary TDMA frame in ourcommunication scheme. Each active secondary cell will beassigned with at least one active secondary TDMA slot withineach secondary frame. This completes the proof for the lemma.

Lemma 5: The total service rate for a secondary sourcequeue is λos =

p32τs

, where p3 ∼ 1− (1 + τs)e−τs .

Proof: Denote the probability that there are at least twosecondary nodes in one cell by p3, which can be calculatedby

p3 = 1− (1− 1

c)m −

(m

1

)1

c(1− 1

c)m−1

= 1− (1− 1

c)m − m

c(1− 1

c)m−1

∼ 1− (1 + τs)e−τs .

Similar to the cooperative scheme in the primary network,packets are transmitted only through the 2-hop source-to-relay-to-destination pattern under the cooperative scheme in thesecondary network. Since the transmission of source-to-relayand relay-to-destination are of equal probability, λos is alsoequal to the rate at which the relay nodes are sending packetsto secondary destinations. Thus, in every time slot, the totalrate of transmission opportunities over the secondary networkis 2mλos. Meanwhile, a transmission occurs in any given cellwith probability p3. Hence we obtain

cp3 = 2mλos. (10)

Then we obtain that the total service rate for a secondarysource queue is λop =

p32τs

. Thus we conclude this lemma.Our results on achievable per-node capacity and transmis-

sion delay are given in the following.Theorem 3: In a cell-partitioned network with overlapping

n primary nodes and m secondary nodes, the achievable per-node capacity for the secondary network under the cooperativescheme is λs = O(1/ks) with average delay E[Ds] =Θ(m log ks).

Proof: For the 2-hop source-to-relay-to-destination com-munication strategy, both the queueing time at the secondarysource node and relay node are accounted for in the delay.We denote the delay in first phase (from a secondary sourcenode to a relay node) by Ds

s→r and delay in second phase(from a relay node to all the ks secondary destination nodes)by Ds

r→d. Then, similar to Lemma 2 in previous section, wehave the following lemma.

Lemma 6: In the first phase of the cooperative scheme inthe secondary network, the expected delay for a secondarysource to send one packet to a relay node equals 25/c1(1 −e−τs − λs), where c1(0 < c1 < 1) is a constant.

Proof: Since a secondary frame is divided into 25 sub-frames, the transmission rate for a secondary cell decreasesby a factor of 25. Moreover, according to Lemma 4, thereare a constant fraction of secondary cells in the unit areathat can access the spectrum. Here we use constant c1 torepresent the overall likelihood that the secondary cells will besuccessfully scheduled during one second. Therefore, the inputrate and output rate for a secondary source-to-relay queue arec125λs and c1

25µos→r, with average delay 1

c125 (µ

os→r−λs)

. Then weneed to calculate the service rate µos→r. The probability thata secondary source node is able to send its packets to a relaynode is 1−(1− 1

c )m−1 ∼ 1−e−τs since any of the secondary

nodes in the same cell with the secondary source node canserve as relays. Therefore µos→r = 1− e−τs and then we canconclude that

Dss→r =

25

c1(1− e−τs − λs). (11)

Now we need to get the delay in the second phase Dsr→d.

Note that a secondary node relays a packet from a secondarysource node with probability pss→r = 1−e−τs

m−1 because all thesecondary nodes are equally likely to be chosen as a relay.Then the input rate for a relay-to-destination queue is

µir =c125λsp

ss→r =

c1(1− e−τs)λs25(m− 1)

.

Similar to the previous section, we model the processthat a relay node distributes one packet from a secondarysource node to all the ks secondary destinations as the packetbeing transmitted through ks independent sub-queues and eachdestination node is associated with one certain sub-queue.

Therefore, we can conclude that the input rate of onesub-queue is ksµ

ir and the output rate is c1λ

os

25(m−1) becauseone secondary node can relay packets destined for all theother m − 1 nodes with equal probability. Therefore, theexpected queueing time for each sub-queue in the relay nodeis 25

c1(λos

m−1−ksµir)

. By Lemma 9, we can obtain

Dsr→d =

Θ

(25

λos

c1(m−1)−ksµi

r)

), ks = 1

Θ

(25 log ks

λos

c1(m−1)−ksµi

r)

), ks > 1

. (12)

Combining the Equations (11) and (12), we get the totalexpected delay for the secondary network under multicasttraffic (ks > 1) is

E[Ds] = Dss→r +Ds

r→d

=25

c1(1− e−τs − λs)+ Θ

(25 log ks

c1(λos/(m− 1)− ksµir)

).

(13)

To make sure that the number of packets waiting to betransmitted in each queue does not increase to infinity withtime, the following two constraints must be satisfied.

10

λs < 1− e−τs ;

ksµir =

c1(1−e−τs )λsks25(m−1) <

c1λos

25(m−1) .

From the above two inequalities, we can obtain λs <λos

(1−e−τs )ks. Thus per-node capacity λs = O(1/ks) and av-

erage delay E[Ds] = Θ(m log ks) are achievable for thesecondary network. Thus we conclude the Theorem 3.

Note that in our system model and communication scheme,the primary network and secondary network achieve per-nodecapacity O(1/kp) and O(1/ks), respectively. Therefore, theirco-existence and mutual interference make no difference fortheir throughput scaling and the two overlapping networks canachieve similar capacity and delay scaling as if they were twoindependently stand-alone networks.

B. Capacity and Delay Analysis of the Redundancy Scheme

In this part, we will discuss the capacity and delay tradeoffwhen more than one secondary node can serve as a relay, i.e.,the redundancy scheme, for the secondary network.

Intuitively, the time needed for a packet to reach its desti-nations can be reduced by repeatedly sending this packet tomany other secondary nodes, i.e., using more than one nodeas a relay. In this way, the chances that some node carrying anoriginal or duplicate version of the packet finds a destinationwill increase. Thus, it is supposed that adopting the redundan-cy scheme can reduce communication delay although this maynot help to improve the network throughput. Previous works[3] and [13] have also introduced the redundancy scheme fora stand-alone ad hoc network in which the source node sendspackets to more than one relay node. In CR networks, theredundancy scheme is also applicable for both the primarynetwork and the secondary network. However, considering thatsecondary nodes suffer from a larger average-delay than theprimary nodes, we namely introduce this redundancy schemeto the secondary network to help effectively reduce its end-to-end delay.

Here, we first show the lower bound of average delay, whichis given in the following.

Theorem 4: In a cell-partitioned network with overlappingn primary nodes and m secondary nodes, no communicationscheme with redundancy can achieve an average delay lowerthan O(

√m log ks) for the secondary network.

Proof: We prove this result by considering an idealsituation where the secondary network is empty except thatone secondary source node sends a packet to ks destinations.The optimal scheme for the source is to send duplicate versionsof the packet whenever it meets new nodes that have neverreceived the packet before. These duplicate-carrying nodes willthen act as relays to help transmit the packet to the destinationsas soon as they enter the same cell as one of the destinations.

During the time slots 1, 2, . . . , i, let ϑj be the totalnumber of intermediate relay nodes at the beginning of timeslot j(1 ≤ j ≤ i). Clearly, ϑ1 ≤ ϑ2 ≤ · · · ≤ ϑi since thenumber of relays is non-decreasing across time. Note that arelay can only be generated by the secondary source node,

hence at most one node can be a new relay in every time slot.Thus ϑi ≤ i for all i ≥ 1. Furthermore, denote the probabilitythat a destination node can meet at least one relay node duringtime slots 1, 2, . . . , i by p(i), we have

p(i) = 1−i∏

j=1

(1− 1

c)ϑj

= 1− (1− 1

c)∑i

j=1 ϑj ≤ 1− (1− 1

c)iϑi

≤ 1− (1− 1

c)i

2

.

However, a destination meeting a relay does not necessarilyensure the transmission can be scheduled since the secondarycell in which they meet may not be active during that slot.Similar to previous discussion in Lemma 6, a constant factorc125 should be multiplied. Hence, the probability that a des-tination node can successfully receive a packet from a relayduring time slots 1, 2, . . . , i is c1p(i)/25. Then we have

Pr(Ds ≥ i) ≥ 1− (c125p(i))ks

≥ 1− (c125

)ks(1− (1− 1

c)i

2

)ks

∼ 1− (c125

)ks(1− e−τsi2

m )ks .

(14)

We choose i =√

m log ksτs

and let ks → ∞, we obtain that

Pr(Ds ≥ i) ≥ 1− (c125

)ks(1− e− log ks)ks

= 1− (c125

)ks(1− 1

ks)ks ≥ 1− c1

25e−1.

Therefore, the average delay in the secondary network withredundancy satisfies

E[Ds] ≥ EDs|Ds ≥ iPr(Ds ≥ i)

≥ (1− c125e−1)

√m log ksτs

(15)

as m and ks approach infinity, which concludes the theorem.

To achieve the above proved lower bound of delay, wepropose the following redundancy scheme for the secondarynetwork.

Redundancy Scheme: For an active secondary cell contain-ing at least two nodes, the following two transmission patternsare performed with equal probability:

1) Source-to-Relay Transmission: Randomly choose a nodethat the secondary source node meets in the same sec-ondary cell as a relay. Pick the source node as the senderand the relay node as receiver to finish the transmission.If the source node has forwarded the duplicate of thepacket to at least

√m log ks distinct relays (possibly be

some of the destinations), the packet is deleted from thebuffer of source node.

2) Relay-to-Destination Transmission: If one relay node,carrying a packet destined for secondary destination

11

nodes, can be found in this cell and meanwhile at leastone corresponding “pristine” secondary destination noderesides within the same cell, pick the relay node as thesender and the destination node as receiver to finish thetransmission. A packet will be discarded when all its ksdestinations have received it.

Then we have the following theorem.Theorem 5: In a cell-partitioned network with overlapping

n primary nodes and m secondary nodes, the achievable per-node capacity for the secondary network is O(1/ks

√m log ks)

with average delay E[Ds] = O(√m log ks) under the redun-

dancy scheme.Proof: Under the redundancy scheme, the expected time

required for a certain packet to reach its corresponding ksdestinations E[Ds] is less than E[D1

s ] + E[D2s ], where E[D1

s ]represents the expected time required to distribute the dupli-cates to

√m log ks different nodes, and E[D2

s ] is the expectedtime required to reach all the ks destinations given that√m log ks relays are holding the packet. We then upper bound

E[D1s ] and E[D2

s ], respectively.E[D1

s ] bound: For the duration of D1s , there are at least

m−√m log ks secondary nodes that do not have the packet.

Hence, in every time slot, the probability that at least one ofthese nodes is located in the same cell as the source is at least1 − (1 − 1

c )m−

√m log ks . Every time slot in the duration D1

s ,a source node can successfully deliver a duplicate packet to anew node with probability of at least p∗ given by

p∗ ≥ c125α1α2(1− (1− 1

c)m−

√m log ks),

where α1 is the probability that the source is selected fromall the other nodes in the same cell to be the transmittingnode, and α2 is the probability that this source is chosen tooperate the “source-to-relay” transmission which is equal to1/2. According to Lemma 6 in [3], α1 ≥ 1/(2 + τs). Thus,

p∗ → c125

1− e−τs

4 + 2τs

as m approaches infinity.The average time for a duplicate packet to reach a new

node is upper bounded by 1/p∗. Since there are√m log ks

duplicates to be transmitted, in the worst case,√m log ks of

these times are required. Therefore, E[D1s ] is upper bounded by√

m log ks/p∗. Hence, we have E[D1

s ] ≤ 25c1

4+2τs1−e−τs

√m log ks.

E[D2s ] bound: Every time slot in the duration of D2

s , thereare at least

√m log ks nodes holding the duplicates of the

packet. The probability that at least one other node is in thesame cell as the destination is 1− (1− 1

c )m−1. Each time slot,

a destination node can successfully receive a duplicate packetfrom one of these relay nodes with probability of at least p∗∗

given by

p∗∗ =c125β1β2β3(1− (1− 1

c)m−1),

where β1 is the probability that the destination is selectedfrom all the other nodes in the same cell to be the receiver,β2 is the probability that the sender is chosen to operate the

“relay-to-destination” transmission which is equal to 1/2, andβ3 represents the possibility that the sender is one of these√m log ks nodes possessing a duplicate packet intended for

the destination. Similarly, we have β1 ≥ 1/(2+ τs) and β3 =√m log ks/(m− 1) ≥

√log ks/m. Thus,

p∗∗ → c125

1− e−τs

4 + 2τs

√log ks/m

as m approaches infinity. The average time that a singledestination node receives a duplicate packet destined for itis upper bounded by 1/p∗∗. The time needed for all the ksdestination nodes to receive the packet is the maximum ofthem. Again according to Lemma 9, E[D2

s ] is upper boundedby log ks/p

∗∗. Hence, we have E[D2s ] ≤ 25

c14+2τs1−e−τs

√m log ks.

Finally, according to Lemma 2 in [3], the total delay canbe upper bounded by E[Ds] = O(

√m log ks) and we obtain

an achievable per-node capacity O(1/ks√m log ks). Thus we

conclude the theorem.

Theorem 6: In a cell-partitioned network with overlappingn primary nodes and m secondary nodes, the achievable per-node capacity for the secondary network is O(1/ks

√m log ks)

with average delay E[Ds] = Θ(√m log ks) under the redun-

dancy scheme.Proof: This can be directly obtained by combining The-

orem 4 and Theorem 5.

C. Capacity and Delay Analysis of Destination Oriented Re-dundancy Scheme

The above mentioned redundancy scheme can effectivelyreduce the end-to-end delay, however, the shortcoming is thatrepeatedly sending one packet to more than one relay node willconsume more network resources and increase interference. Toavoid this, we improve the original redundancy scheme andpropose a novel redundancy scheme, named the DestinationOriented Redundancy Scheme, which not only reduces delaybut also better utilizes network resources.

Destination Oriented Redundancy Scheme:For an active secondary cell containing at least two sec-

ondary nodes, the following two transmission patterns areperformed with equal probability.

1) Source-to-Relay Transmission: Choose the first node thatthe secondary source node meets as a relay irrespectiveof whether it is a destination node or not. Pick the sourcenode as the sender and the relay node as receiver to finishthe transmission.

2) Relay-to-Destination Transmission: If one relay node,carrying a packet destined for secondary destinationnodes, can be found in this cell and meanwhile at leastone corresponding “pristine” secondary destination noderesides within the same cell, pick the relay node as thesender and the destination node as receiver to finishthe transmission. For the destination nodes which havereceived the packet, they can also serve as relay nodesto conduct the relay-to-destination transmission.

12

In the above mentioned scheme, to reduce the delay bybringing in redundancy, we allow the destination nodes whohave received the message to perform as relay nodes. Notethat at most one node besides the source and ks destinationsis chosen as a relay, hence we do not introduce extra relaynodes to relay the packet. Thus we save the network resourceswith this improved scheme.

With this new redundancy scheme, we show that the lowerbound of communication delay for secondary network isO(

√m log ks) if we allow only one transmission in one time

slot. However, if we assume that all the available transmissionsamong different active cells can be conducted in one timeslot, we prove that the lower bound of communication delayis O(m log ks

ks). Formally, we define these two communication

patterns as Destination Oriented Solo-redundancy Scheme (use“Solo-redundancy Scheme” for short) and Destination Orient-ed Ensemble-redundancy Scheme (use “Ensemble-redundancyScheme” for short), respectively.

Definition 2: Solo-redundancy Scheme refers to thescheme when at most one destination node within the samemulticast group is allowed to receive packets in one timeslot, even though there may be more than one active cellin which a packet from a certain source secondary node canbe sent to a “pristine” secondary destination node. Whereasthe Ensemble-redundancy Scheme allows all the availabletransmissions within the same multicast group among differentactive cells to be conducted in one time slot.

We are now ready to present our main results in thefollowing.

Theorem 7: The lower bound of communication delay inthe secondary network under the destination oriented redun-dancy scheme is

1) B1 , Ω(√m log ks) if we adopt the solo-redundancy

scheme.2) B2 , Ω(m log ks

ks) if we adopt the ensemble-redundancy

scheme.

Proof: We start with the proof of the first item in Theorem7. Suppose during the time slots 1, 2, . . . , i, there are ψi(ψi ≤ ks) destination nodes that have received the packet.Denote the number of destination nodes who have receivedthe message in time slot j (1 ≤ j ≤ i) by ψj . It is clear thatψ1 ≤ ψ2 ≤ · · · ≤ ψi. Furthermore, denote the probability thatone destination node has not received the packet during thetime slots 1, 2, . . . , i by p§. Then p§ satisfies

p§ =i∏

j=1

(1− 1

c)ψj

= (1− 1

c)∑i

j=1 ψj ≥ (1− 1

c)iψi .

≥ (1− 1

c)i

2

( since ψi ≤ i under solo redundancy scheme ).

Then we have

Pr(Ds ≥ i) ≥ 1− [c125

(1− p§)](ks−ψi)

≥ 1− c125

(ks−ψi)[1− (1− 1

c)i

2

](ks−ψi)

∼ 1− c125

(ks−ψi)(1− e−τs

i2

m )(ks−ψi).

(16)

We choose i =√

m log ksτs

. Substituting the value of i intoEquation (16), we obtain that

Pr(Ds ≥ i) ≥ 1− c125

(ks−ψi)(1− e− log ks)(ks−ψi)

= 1− c125

(ks−ψi)(1− 1

ks)(ks−ψi)

≥ 1− e−1.

Therefore, the expected communication delay in the sec-ondary network under solo-redundancy scheme satisfies

E[Ds] = EDs|Ds < iPr(Ds < i)

+ EDs|Ds ≥ iPr(Ds ≥ i)

≥ EDs|Ds ≥ iPr(Ds ≥ i)

(since we omit the first item)

≥ (1− e−1)

√m log ksτs

∼ Ω(√m log ks).

Thus, we prove the first claim in Theorem 7. Next we provethe second claim. In this scenario, we have that p§ satisfies

p§ =

i∏j=1

(1− 1

c)ψj

= (1− 1

c)∑i

j=1 ψj ≥ (1− 1

c)iψi

≥ (1− 1

c)iks .

( since ψi ≤ ks is always satisfied )

(17)

Then we have

Pr(Ds ≥ i) = 1− c125

(ks−ψi)(1− p§)(ks−ψi)

≥ 1− c125

(ks−ψi)[1− (1− 1

c)iks ](ks−ψi)

∼ 1− c125

(ks−ψi)(1− e−τs

iksm )(ks−ψi).

(18)

Here, we choose i = m log ksτsks

. Substituting that value intoEquation (18), we obtain

Pr(Ds ≥ i) ≥ 1− c125

(ks−ψi)(1− e− log ks)(ks−ψi)

≥ 1− e−1.

Hence, the expected communication delay in the secondarynetwork under ensemble-redundancy scheme satisfies

13

E[Ds] ≥ EDs|Ds ≥ iPr(Ds ≥ i)

≥ (1− e−1)m log ksτsks

∼ Ω(m log ksks

).

Note that under ensemble-redundancy scheme, more trans-missions are allowed in one time slot compared to solo-redundancy scheme, thus the delay in ensemble-redundancyscheme should be smaller than that in the solo-redundancyscheme correspondingly. Therefore, the two delay lowerbounds B1 and B2 should satisfy B1 > B2. It is clear that

1) If ks = Θ(m), then B2 = Ω(m log ksks

) ∼ Ω(log ks).Thus we conclude B2 < B1. This is acceptable.

2) If ks = o(m), then B2 > B1, which implies that B2 isnot a very tight lower bound in this scenario.

For this case of B2 when ks = o(m), we can find anintuitive explanation according to our network model. To makethe explanation more straightforward, here we assume that thefirst relay node met by the source node is one destination node.Note that the “pristine” destination nodes can get packets onlywhen they meet other destination nodes carrying messagesin one active cell. Denote the probability that one “pristine”destination node receives packets in one time slot by p4. Thenwe have

p4 ≤ 1−(ks1

)1

c(1− 1

c)ks−1 − (1− 1

c)ks

= 1− (1 +ks − 1

c)(1− 1

c)ks−1

= 1− (1 +ks − 1

c)e−τs·

ks−1m .

Hence, if ks = o(m), p4 is close to zero, which impliesthat relay-destination pairs are less likely to be found in asingle active cell. Therefore, even though we allow moretransmissions to be finished in different active cells duringone time slot under ensemble-redundancy scheme, the actualnumber of transmissions conducted in one time slot may bevery small since it is difficult to satisfy the condition forsuccessful communication. Thus ψi is smaller than i with highprobability even under ensemble-redundancy scheme. Recallthe process of finding B2, we used the inequality ψi ≤ ks, avery loose constraint, while deducing Equation (17). So if weadopt a tighter constraint ψi ≤ i, we find B2 = Ω(

√m log ks)

which is much better than Ω(m log ksks

). Therefore, we maymodify our results on the lower bound of communication delayunder ensemble-redundancy scheme as follows:

1) B2 = Ω(√m log ks) if ks = o(m);

2) B2 = Ω(log ks) if ks = Θ(m).

Thus we conclude Theorem 7.

VI. CONCLUSION AND DISCUSSION

A. Results ReviewIn this paper, we have studied the capacity and delay

tradeoff in CR MANET. For the primary network, we firststudied a non-cooperative scheme. Our results showed thatthis scheme achieves a per-node capacity O(1/kp) and averagedelay Θ(n log kp) when kp = o(n). However, if kp = Θ(n),the per-node capacity reduces to O(1/k2p). To improve this,we introduced the cooperative scheme and proved that per-node capacity O(1/kp) is also achievable even if kp = Θ(n),which implies that cooperation among destinations within thesame multicast group improves the throughput scaling.

For the secondary network, we proved that per-node ca-pacity O(1/ks) and average delay Θ(m log ks) are achievableunder the cooperative scheme. Note that our results indicatedthat both the primary network and the secondary network canachieve nearly the same capacity and delay scaling in CR net-works as if they were two independent stand-alone networks.Furthermore, in order to improve the delay performance inthe secondary network, a redundancy scheme was consideredand we showed that the per-node capacity O(1/ks

√m log ks)

and average delay Θ(√m log ks) are achievable. Finally, we

improved the original redundancy scheme and proposed thedestination oriented redundancy scheme to efficiently utilizewireless resources and reduce the delay for the secondarynetwork users.

Formally, we present our results in the following threeTables II, III and IV.

Communication scheme Capacity DelayNon-cooperative scheme kp = o(n) O(1/kp) Θ(n log kp)Non-cooperative scheme kp = Θ(n) O(1/k2p) Θ(n log kp)

Cooperative scheme O(1/kp) Θ(n log kp)

TABLE IICAPACITY AND DELAY FOR THE PRIMARY NETWORK.

Communication scheme Capacity DelayCooperative scheme O(1/ks) Θ(m log ks)

Redundancy scheme O(1/ks√m log ks) Θ

√(m log ks)

TABLE IIICAPACITY AND DELAY FOR THE SECONDARY NETWORK.

Communication scheme Lower BoundSolo-redundancy scheme Ω(

√m log ks)

Ensemble-redundancy scheme ks = o(m) Ω(√m log ks)

Ensemble-redundancy scheme ks = Θ(m) Ω(log ks)

TABLE IVLOWER BOUND OF DELAY FOR THE SECONDARY NETWORK WITH

DESTINATION ORIENTED REDUNDANCY SCHEME

.

B. Comparison with Previous WorkCompared with the multicast capacity of static CR networks

developed in [16], we find that the capacity performance is

14

better when nodes are mobile. Also, compared with the partialmobility model developed in [17] and [18], which requires theprimary nodes to be static and only the secondary nodes areallowed to move, our model is more practical and generalas both the primary and secondary networks can be mobile.Moreover, compared with the results of CR network underunicast traffic in [18], we find that the capacity diminishesby a factor of 1/kp and 1/ks for the primary network and thesecondary network respectively under the cooperative scheme.This is because we need to forward a packet to kp primarydestinations (or ks secondary destinations). Particularly, ifkp = Θ(1) and ks = Θ(1), our results can be specializedto the unicast traffic; if kp = Θ(n) and ks = Θ(m), ourresults can be specialized to the broadcast traffic.

Furthermore, we find that although redundancy can reducethe transmission delay in the secondary network, it will leadto a decrease in the capacity. This suggests that transmissionswith redundant relays can reduce delay at the cost of thecapacity. The tradeoff between the delay and capacity alwayssatisfies delay/capacity ≥ O(mks log ks) under both the coop-erative scheme and the redundancy scheme in the secondarynetwork, as shown in Figure 4. However, if we schedule thetransmission in the secondary network by multiple unicastfrom source to ks destinations, the capacity will diminishby a factor of ks and the delay will increase by a factor ofks, which indicates that the delay-capacity tradeoff becomesdelay/capacity ≥ O(mk2s) in CR MANET. This demonstratesthat our tradeoff is better than that of naively extending thetradeoff for unicast to multicast.

BC

2

1

sk

1

sk

A

logs

m k

logs

m k

1

logs sk m k

( )sD m

( )smλ

A: Redundancy Scheme

B: Cooperative Scheme

C: Non-cooperative Scheme

Fig. 4. Illustration of capacity-delay tradeoff in the secondary network.

C. Rationality of System Model

In our system model, we consider an ideal i.i.d fast mobilitymodel, which allows nodes to choose new locations everytimeslot from overall cells in the network. Indeed, actualmobility is better characterized by Markovian Dynamics orRandom Walk mobility model, where nodes can only roam

to adjacent cells in every timeslot. However, analysis underthe ideal i.i.d mobility model provides a significant bound oncapacity and delay performance in the limit of infinite mobility[3]. Under the i.i.d mobility model, all the nodes’ locations cannot be predicted from time to time, hence the communicationschemes are required to be more robust and adaptable. Ourschemes are accordingly designed to better fit such featuresof the mobility model. Compared with other communicationschemes, our schemes need less current and future informationabout users’ locations. In addition, it has been proved in [20]that the network capacity using an i.i.d mobility model isequivalent to the capacity region of the networks using non-i.i.d mobility model. Hence the results obtained in our paperare not only theoretically rational but also practically usefulwhen applying to large-scale networks.

D. Future Work

Note that in our network model, we achieve a “harmonious”co-existence of the primary network and the secondary net-work by assuming primary nodes confine their interference toone cell. For a more general and practical network, we canintroduce a “guard zone” to limit the interference received byprimary and secondary nodes. The analysis of capacity anddelay under this network model will be considered in futurework. Additionally, the per-node capacity under the destinationoriented redundancy scheme in CR MANET is still an openquestion. Finally, if the number of primary and secondarynodes are of the same order, it would be interesting to studythe impact of the interference between them.

REFERENCES

[1] P. Gupta, P. R. Kumar, “The Capacity of Wireless Networks,”in IEEE Transactions on Information Theory, vol. 46, no. 2, pp.388-404, Mar. 2000.

[2] M. Grossglauser, D. Tse, “Mobility increases the capacity of adhoc wireless networks,” in IEEE/ACM Transactions on Network-ing, vol. 10, no. 4, pp. 477-486, Aug. 2002.

[3] M. J. Neely, E. Modiano, “Capacity and Delay Tradeoffs forAd-Hoc Mobile Networks,” in IEEE Transactions on InformationTheory, vol. 51, no. 6, June 2005.

[4] X. Lin, G. Sharma, R. Mazumdar and N. Shroff, “Degeneratedelaycapacity trade-offs in ad hoc networks with brownian mo-bility,” in IEEE/ACM Transactions on Networking, vol. 52, no.6, pp. 2777-2784, June 2006.

[5] G. Sharma, R. Mazumdar. “Scaling Laws for Capacity and Delayin Wireless Ad Hoc Networks with Random Mobility,” in IEEEInternational Conference on Communications, Paris, June 2004.

[6] J. Mammen and D. Shah, ”Throughput and delay in randomwireless networks with restricted mobility,” in IEEE Transactionson Information Theory, vol. 53, no. 3, pp. 1108-1116, Mar. 2007.

[7] M. Garetto, E. Leonardi, “Restricted Mobility Improves Delay-Throughput Trade-offs in Mobile Ad-Hoc Networks,” in IEEETransactions on Information Theory, Vol.56, No.10, pp.5016-5029, October 2010.

[8] X.-Y. Li, S.-J. Tang, and O. Frieder. “Multicast capacity for largescale wireless ad hoc networks,” in Proc. of ACM Mobicom’08,San Francisco, CA, USA, Sept. 2008.

[9] A. Keshavarz-Haddad, V. Ribeiro, and R. Riedi, “Broadcastcapacity in multihop wireless networks,” in Proceedings of ACMMobiCom, Sept. 2006.

15

[10] L. Ying, S. Yang, R. Srikant, “Optimal Delay-ThroughputTrade-Offs in Mobile Ad Hoc Networks,” in IEEE Transactionson Information Theory, Vol. 54, No. 9, Sept. 2008.

[11] G. Sharma, R. R. Mazumdar, N. B. Shroff, “Delay and capacitytrade-offs in mobile ad hoc networks: A global perspective,” inProc. of IEEE INFOCOM’06, Bar., Spain, Apr. 2006.

[12] S. Zhou, Lei Ying, “On Delay Constrained Multicast Capacityof Large-Scale Mobile Ad-Hoc Networks,” in IEEE INFOCOM2010 mini-conference, San Diego, CA, 2010.

[13] X. Wang, W. Huang, S. Wang, J. Zhang, C. Hu, “Delayand Capacity Tradeoff Analysis for MotionCast,” in IEEE/ACMTransactions on Networking, Vol. 19, no. 5, pp. 1354-1367, Oct.2011.

[14] S. Jeon, N. Devroye, M. Vu, S. Chung, V. Tarokh, “CognitiveNetworks Achieve Throughput Scaling of a Homogeneous Net-work,” IEEE Transactions on Information Theory, vol.57, no.8,pp.5103-5115, Aug. 2011.

[15] C. Yin, L. Gao, S. Cui, “Scaling Laws for Overlaid WirelessNetworks: A Cognitive Radio Network vs. a Primary Network,”in IEEE/ACM Transactions on Networking, vol. 18, no. 4, pp.1317-1329, August 2010.

[16] C. Wang, X.-Y. Li, S. Tang, C. Jiang, “Multicast Capacity Scal-ing for Cognitive Networks: General Extended Primary Network,”in Proc. IEEE MASS 2010.

[17] Y. Li, X. Wang, X. Tian, and X. Liu, “Scaling Laws forCognitive Radio Network with Heterogeneous Mobile SecondaryUsers,” in Proc. IEEE INFOCOM’12, Orlando, USA, March,2012.

[18] L. Gao, R. Zhang, C. Yin, and S. Cui, “Throughput and DelayScaling in Supportive Two-Tier Networks,” in IEEE Journal onSelected Areas in Communications, vol. 30, no. 2, February, 2012.

[19] A. E. Gamal, J. Mammen, B. Prabhakar, and D. Shah,“Throughput-delay trade-offi n wireless networks,” in Proc. IEEEINFOCOM’04, Hong Kong, Mar. 2004.

[20] M. J. Neely, E. Modiano, and C.E. Rohrs, “Dynamic powerallocation and routing for time varying wireless networks.” inProc. IEEE INFOCOM’03, April 2003.

[21] R. Motwani, P. Raghavan, Randomized algorithms, CambridgeUniversity Press, 1995.

[22] H. Kobayashi, B. L. Mark, and W. Turin, “Probability, RandomProcesses and Statistical Analysis,” Cambridge University Press,2012.

APPENDIX AUSEFUL LEMMAS

Lemma 7: Let ntotal and mtotal denote the total numbersof primary nodes and secondary nodes within the unit area,then we have n

e ≤ ntotal ≤ en and me ≤ mtotal ≤ em w.h.p..

Proof: As for primary nodes, since ntotal is a Poissonrandom variable with mean n, by applying the Chernoffbound[21], we obtain

Pr(ntotal ≤n

e) ≤ e−n(en)

ne

(ne )ne

= e(2e−1)n,

which tends to 0 as n approaches infinity. In addition,

Pr(ntotal ≥ en) ≤ e−n(en)en

(en)en

= e−n,

which also tends to 0 as n approaches infinity. Hence

Pr(n

e≤ ntotal ≤ en) → 1

as n→ ∞, which completes the proof for primary nodes. Thenumber of secondary nodes can also be bounded in the similarway.

The following two lemmas are needed when scaling thethroughput under multicast traffic. The first lemma can befound in [13], and we include it here for completeness.

Lemma 8:k∑i=1

(−1)i−1

i

(ki

)= ln(k+1)+r, where k ≥ 1

and r is the Euler constant.Proof: Denote the left-hand side of the equation by S(k),

then we have S(k − 1) =k∑i=1

(−1)i−1

i

(k − 1i

). Notice that(

ki

)=

(k − 1i

)+

(k − 1i− 1

), and it follows

S(k)− S(k − 1) =k∑i=1

(−1)i−1

i

(k − 1i− 1

)

=1

k

k∑i=1

(−1)i−1

(ki

).

(19)

Recall that (1 − 1)k =k∑i=0

(−1)i(ki

)= 0, hence

we obtaink∑i=1

(−1)i−1

(ki

)= −

k∑i=1

(−1)i(ki

)=

−[k∑i=0

(−1)i(ki

)− 1

]= 1. Combining with Equation 19,

we get S(k)− S(k − 1) = 1k , then

S(k) = S(1) +k∑i=2

[S(k)− S(k − 1)]

= 1 +k∑i=2

1

k=

k∑i=1

1

k,

(20)

where the right-hand-side is the harmonic series. Hence thislemma holds.

Lemma 9: Suppose X1, X2, . . . , Xk are continuous i.i.dexponential random variables with expectation of 1/a. DenoteXmax = maxX1, X2, . . . , Xk, then

EXmax =

Θ(1/a), k = 1Θ(log k/a), k > 1

.

Proof: Consider the cdf of Xmax, which can be calculatedby [22]

FXmax(t) = PXmax ≤ t = (1− e−at)k.

Thus, the pdf of Xmax can be accordingly obtained by

fXmax(t) =dFXmax

dt= k(1− e−at)k−1 · ae−at.

16

Then, the expectation of Xmax is

EXmax =

∫ ∞

0

k(1− e−at)k−1 · ae−at · tdt

= ka

∫ ∞

0

k−1∑i=0

(k − 1i

)(−1)ie−a(i+1)t · tdt

=

k−1∑i=0

ka

(k − 1i

)(−1)i

1

[a(i+ 1)2]

=k∑i=1

ka

(k − 1i− 1

)(−1)i

1

a2i2

=k

a

k∑i=1

(−1)i−1

i2

(k − 1i− 1

)

=1

a

k∑i=1

(−1)i−1

i

(ki

).

(21)

Then according to Lemma 8, we conclude this lemma.