qos-aware minimum energy multicast tree construction in wireless ad hoc networks

Ad Hoc Networks 2 (2004) 217–229

www.elsevier.com/locate/adhoc

QoS-aware minimum energy multicast tree constructionin wireless ad hoc networks

Song Guo *, Oliver Yang

School of Information Technology and Engineering, University of Ottawa, Ottawa, Ont., Canada K1N 6N5

Available online 28 April 2004

Abstract

Energy conservation is a critical issue in wireless ad hoc networks since batteries are the only limited-life energy

source to power the nodes. One major metric for energy conservation is to route a communication session along the

routes which require the lowest total energy consumption. Most recent algorithms for the MEM (Minimum Energy

Multicast) problem considered energy efficiency as the ultimate objective in order to increase longevity of such net-

works. However, the introduction of real-time applications has posed additional challenges. Transmission of video and

imaging data requires both energy and QoS-aware routing in order to ensure efficient usage of the networks. In this

paper, we only consider ‘‘bandwidth’’ as the QoS in TDMA-based wireless ad hoc networks that use omni-directional

antennas and have limited energy resources. We present a constraint formulation model for the QoS-MEM (QoS-aware

Minimum Energy Multicast) problem in terms of mixed integer linear programming (MILP), which can be used for an

optimal solution of the QoS-MEM problem. Experiment results show that in a typical static ad hoc network with 20

nodes, the optimal solutions can always be solved in a timely manner.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Wireless ad hoc networks; QoS routing; Minimum energy multicast; TDMA; Integer programming

1. Introduction

Ad hoc wireless networks are expected to bedeployed in a wide variety of civil and military

applications. The increasing use of collaborative

applications and wireless devices may further add to

the needs and usage of ad hoc networks. The com-

municating nodes might be distributed randomly

and are assumed to have the capacity of packet

forwarding to communicate with each other over a

shared radio channel. Building such networks poses

* Corresponding author.

E-mail address: [email protected] (S. Guo).

1570-8705/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.adhoc.2004.03.010

a significant technical challenge because of the

constraints imposed by the characteristics of the ad

hoc networks. Resources, including energy, band-width, processing capacity and memory, that are

relatively abundant in wired environments, are

strictly limited and must be preserved.

The emergence of real-time applications and the

widespread use of wireless devices have generated

the need to provide quality-of-service (QoS) support

in wireless ad hoc networking environments. QoS is

usually defined as a set of service requirements thatneed to be met by the network while transporting a

packet stream from a source to its destination(s).

The network needs are governed by the service

ed.

mail to: [email protected]

218 S. Guo, O. Yang / Ad Hoc Networks 2 (2004) 217–229

requirements specified by the end user applications.

The network is expected to guarantee a set of

measurable pre-specified service attributes to the

users in terms of end-to-end performance, such as

delay, bandwidth, probability of packet loss, delay

variance, etc. [1]. The QoSmetric bandwidth is moredifficult to guarantee in wireless ad hoc networks,

because the wireless bandwidth is the scarce re-

source and always shared among adjacent nodes.

This requires extensive collaboration between the

nodes, both to establish the route and to secure the

resources necessary to provide the QoS.

Since wireless nodes are generally dependent on

finite battery source, the routing protocol for QoSprovisioningmust also consider the residual battery

power and the rate of battery consumption in order

to increase longevity of such networks [2,3]. Thus

all the techniques for QoS provisioning should be

power-efficient. On the other hand, the ability to

provide QoS is heavily dependent on how well the

resources are managed at the MAC layer. A QoS

routing protocol developed for one type of MAClayer does not generalize to others easily. Among

the QoS routing protocols proposed so far, some

use generic QoS measures and are not tuned to a

particular MAC layer [4–6]. Some use CDMA to

eliminate the interference between different trans-

missions [7,8]. In [9], the authors develop a QoS

routing protocol for ad hoc networks using TDMA

in small networks. The protocol is based on AODV[10], and builds QoS routes only as needed.

Future networks must be adequately equipped

to handle multipoint communication in a fast and

economical manner. When the network is modeled

as a weighted, undirected graph, the problem is

that of finding a minimal Steiner tree for the

graph, given a set of destinations. The problem is

known to be NP-complete. Consequently, severalheuristics exist which provide approximate solu-

tions to the Steiner problem in networks [41]. In

[42], the authors present a random neural network

(RNN) model can be used to significantly improve

the quality of the Steiner trees delivered by the best

available heuristics that are the minimum spanning

tree heuristic and the average distance heuristic.

The recent proliferation of QoS-aware groupapplications over the wireless ad hoc networks has

accelerated the need for efficient multicast support.

In this paper, we only consider ‘‘bandwidth’’ as the

QoS and present a constraint formulation model

for the QoS-MEM (QoS-aware Minimum Energy

Multicast) problem in a TDMA-based ad hoc

network. In general, ‘‘bandwidth’’ in time-slotted

network system is measured in terms of theamount of ‘‘free’’ slots. Consequently, in order to

establish a bandwidth guaranteed QoS multicast

tree from a source to all destinations, we have the

following goals for this optimization problem:

1. The bandwidth allocated on each link of the

multicast tree should meet the bandwidth

requirement.2. A suitable scheduling of free slots for each link

of the multicast tree can be also obtained from

this model.

3. The total RF energy consumption on the band-

width-guaranteed multicast tree is minimized.

Clearly, such a joint power-minimization and

scheduling is a challenging optimization problem.In fact, either the scheduling problem with even a

single power level or the best-effort minimum en-

ergy multicast problem, is by itself known to be an

NP-hard problem [32,36]. Our simulation results

show that an optimal solution of the QoS-MEM

problem using our model can always be obtained in

a timely manner for networks with no more than 20

nodes. The remaining of this paper is organized asfollows. In Section 2, we overview related work

concerning QoS unicast/multicast routing and

minimum energy multicast routing in wireless ad

hoc networks. In Section 3, we give a network

model and the definition of BCMT (bandwidth-

constrained multicast tree). Section 4 derives the

linear constraint formulation for Problem QoS-

MEM systematically in a form of Mixed IntegerLinear Programming (MILP), and proves that it

produces the optimal solutions. Computational

results assessing the performance are given in Sec-

tion 5. Section 6 summarizes our finding and points

out several future research problems.

2. Related work

Over the recent few years, the design of energy-

efficiency routing algorithms has gained increasing

S. Guo, O. Yang / Ad Hoc Networks 2 (2004) 217–229 219

importance in wireless ad hoc networks. However,

the QoS awareness has never been considered so

far (to our best knowledge) in current research of

the minimum energy multicast problem. We are

inspired to study the joint optimization problem:

bandwidth guarantee and total RF energy mini-mization along the multicast routes. In the fol-

lowing, we give a brief literature review on each of

these two aspects.

2.1. QoS routing protocols

In traditional fixed wire networks, QoS routing

is usually performed through resource reservationin a connection-oriented communication in order

to meet the QoS requirements for each individual

connection. Many mechanisms have been pro-

posed for routing QoS constrained real-time mul-

timedia data [11–16,37]. Gelenbe et al. [15,37,39,40]

describe an experimental system which allows users

to take advantage of on-line measurements and

self-adaptation to seek network performancewhich approximates their QoS requirements in

quasi-real time. They also propose a self-aware

packet network design that uses smart and ACK

packets to collect and store data about network

state. Smart packets also search for routes using

QoS criteria suggested by users. Connections then

forward their payload using dumb packets along

routes that have been discovered by smart packets.Comparing with the abundant work on QoS

routing for fixed wire networks, QoS routing in ad

hoc networks has been studied only recently [5–

9,17–19]. A number of protocols have been pro-

posed for QoS routing in wireless ad hoc networks

taking the dynamic nature of the network into

account. Some promising work on QoS routing,

such as CEDAR [18], ticket-based probing [5], andQoS routing based on bandwidth calculation [9],

have been done and show good performance. Lin

[7,8] has proposed QoS routing protocols specifi-

cally designed for TDMA-based ad hoc networks.

It can build a QoS route from a source to desti-

nation with reserved bandwidth. The bandwidth

calculation is done hop-by-hop. CEDAR is an-

other QoS aware protocol, which uses the idea ofcore nodes (dominating set) of the network while

determining the paths [18]. Using routes found

through the network core, a QoS path can be

easily found. More recently, Chen et al. [19] de-

velop an on-demand link-state multipath QoS

routing protocol in a wireless mobile ad hoc net-

work. This protocol collects link bandwidth

information from source to destination in order toconstruct a network topology with the information

of link bandwidth at the destination. The band-

width calculation of the QoS route is determined

at the destination.

The multicast protocol is a primitive communi-

cation operation for sending the same message

from a source node to a group of destination nodes.

It is very significant for many wireless and mobileapplications. There are many existing multicast

protocols such as MAODV [20], CAMP [21], OD-

MRP [22], andDCMP [23] protocols for wireless ad

hoc networks. However, these multicast protocols

do not explicitly provide the QoS function. The

design difficulty of designing QoS multicast proto-

cols is much greater than for best-effort multicast

protocols in such networks due to the need to takebandwidth-reservation into consideration.

2.2. Minimum energy broadcast/multicast

In a wireless ad hoc network, each node has a

limited energy resource (battery), and operates in

an unattended manner. Consequently, energy

efficiency is an important design consideration forthese networks. Most recent work [24–30,38] has

been proposed for the problems of minimizing the

energy consumption for broadcasting and multi-

casting in wireless ad hoc networks, addressed as

the MEB (Minimum-Energy Broadcast) problem

and MEM (Minimum-Energy Multicast) problem,

respectively. Since both the MEB problem and the

MEM problem have recently been shown to beNP-hard [31,32], efficient heuristic algorithm de-

sign has received much more attention.

For the MEB problem, a straight greedy ap-

proach is the use of broadcast trees that consist of

the best unicast paths to each individual destina-

tion from the source node (broadcast session ini-

tiator). This heuristic first applies the Dijkstra’s

algorithm to obtain an SPT (Shortest Path Tree),and then to orient it as a tree rooted at the source

node. Similarly the MST (Minimum Spanning


Tree) heuristic first applies the Prim’s algorithm to

obtain an MST, and then to orient it as a tree

rooted at the source node. In [24,29], another

heuristic algorithm for the MEB problem called

BIP (Broadcast Incremental Power) was pre-

sented. The BIP algorithm is similar in principle tothe standard Prim algorithm for the formation of

minimum spanning trees. It maintains throughout

its execution a single tree rooted at the source

node. Initially, the rooted tree only includes the

source node. Subsequently the tree node that can

cover a new node outside the rooted tree with the

least incremental power expands its power range

to include this new node in the rooted tree. Thisoperation is repeated until all nodes are included

in the tree. BIP exploits the ‘‘wireless multicast

advantage’’ property 1 in the formation of the

broadcast trees, and thus provides better perfor-

mance than the greedy algorithms SPT and MST.

All the algorithms mentioned above are central-

ized. Recently, distributed algorithms RBOP (Re-

lated Neighbourhood Graph based BroadcastOriented Protocol) [33] and EWMA (Embedded

Wireless Multicast Advantage) [34] are shown to

have comparable performance to BIP. In most of

the literature, the MEM problem was studied in a

similar approach as the MEB problem except that

the final minimum energy multicast tree is ob-

tained by pruning from the minimum energy

broadcast tree all transmissions that are not nee-ded to reach the member of the multicast group.

... 1 2 K...

Control phase Data phase

i ...u

... 1 2 K... i ...v

Pvi

Pxi

3. The network model

Let us model the wireless ad hoc network by a

simple directed graph GðN ;AÞ, where N is a finite

node set, jN j ¼ n, and A is an arc set correspondingto the unidirectional wireless communication links.

Each node is equipped with a single omni-direc-

tional antenna. When considering uniform propa-

gation condition, we observe that all nodes within

the communication range of a transmitting node

1 The ‘‘wireless multicast advantage’’ property means that all

nodes within communication range of a transmitting node can

receive a multicast message with only one transmission if they

all use omni-directional antennas.

can receive its transmission, and the received signal

power varies as r�a, where r ðr > 1Þ is the distanceto the sender, and a is propagation loss exponent

that typically takes on a value between 2 and 4,

depending on the characteristics of the communi-

cation medium. We assume that any node u 2 Ncan choose its transmission power level continu-

ously up to some maximum value pmaxu . Therefore,

any directed arc ðv; uÞ 2 A if and only if pvu 6 pmaxv ,

where pvu presents the minimum power needed for

the link from node v to node u. For the convenienceof the reader, the notations introduced in this sec-

tion are summarized in Appendix A.

In this paper we shall develop a constraintformulation for the QoS-MEM problem in ad hoc

networks using TDMA, in which all the nodes are

synchronized. We assume that any node can only

receive a single transmission at a time and cannot

transmit and receive simultaneously. The band-

width is partitioned into a set of time slots S ¼f1; 2; . . . ;Kg which consist the data part of a frame

as shown in Fig. 1. The information concerningavailable bandwidth (in number of free time slots)

between two nodes is critical. It is used to select a

route that satisfies the QoS requirement. In addi-

tion, it is also used to determine whether a new

connection request is allowed into the network.

Let Pui ðu 2 NÞ be the power level in the slot iassigned to node u, where 06 Pui 6 pmax

u and

16 i6K. The transmission schedule TSu of nodeu 2 N is thus defined as the power assignment in

each time slot, i.e. TSu ¼ ðPu1; Pu2; . . . ; PukÞ. For anynew traffic request, based on its current transmis-

... 1 2 K... i ...x

Fig. 1. Illustration of frame structure and data transmission in

a TDMA-based wireless ad hoc network.


sion schedule TSu, a transmission from node u can

be only scheduled in a set of free time slots FSu,define as FSu ¼ fi jPui ¼ 0; i 2 Sg. Note that FSu is

only an alternate set for scheduling, and may not

guarantee conflict-free transmissions. We say a

transmission from node v is successfully received atnode u in the slot i (as shown in Fig. 1) if and only if

the signal-to-interference plus noise ratio (SINR)

at u is not less than the minimum required thres-

hold c, i.e.

Pvi=ravugþ

Pðx;uÞ2A;x6¼v ðPxi=raxuÞ

P c; ð1Þ

where rvu is the distance between nodes v and u,and g is the thermal noise at every receiver. Thismodel is commonly known as the Physical Inter-

ference Model [35].

We consider a source-initiated multicast in

wireless ad hoc networks. Any node is permitted to

initiate multicast sessions. Multicast requests and

session durations are generated randomly at the

network nodes. The set of nodes M that support a

multicast session includes the source node and alldestination nodes. Multicast employs a tree struc-

ture in the network to efficiently deliver the same

data stream to a group of receivers. We assume that

no power expenditure is involved in signal recep-

tion and processing activities. Thus the total power

is expended completely on transmission at each

node in the tree. Obviously, leaf nodes do not

contribute to this quantity because they do notrelay traffic to any other nodes. Hence, we evaluate

performance in terms of total RF power from all

transmitting nodes required to maintain the tree.

Any multicast tree is a rooted tree. We define a

rooted tree as a directed acyclic graph with a

source node s called root with no incoming arcs,

and all its other nodes with exactly one incoming

arc. A property of rooted tree is that for any nodeu in the tree, there exists a single directed path

from s to u in the tree. A node with no out-going

arcs is called a leaf node, and all other nodes are

internal nodes, or relay nodes. The minimum-en-

ergy multicast problem is to find a multicast tree

with the minimum power consumption. Doing so

involves the choice of transmission power level and

relay nodes. The relay nodes may be multicastmembers or may not.

Formally, we define TsðN 0;A0Þ to be a band-

width-constrained multicast tree (BCMT) of

GðN ;AÞ rooted at s with a multicast node set

N 0 � N , and an arc set A0 � A, if and only if the

following constraints are satisfied:

1. RTC (Rooted Tree Constraint). This constraint

requires Ts to be a rooted tree and span all the

multicast members from node s, i.e. M � N 0.

2. BWC (Bandwidth Constraint). This constraint

requires that the bandwidth allocated on each

link of the multicast tree should meet the band-

width requirement (B slots per frame), and the

scheduling should be conflict-free.

4. Constraints formulation

The definition of bandwidth-constrained mul-

ticast tree allows us to formulate the QoS-MEM

Problem as an MILP (Mixed Integer Linear Pro-

gramming) model. The main idea is to extract asub-graph T �

s from the original graph G, such that

T �s is a BCMT with minimum energy consumption.

In order to formulate the problem, we define the

following decision variables:

(i) Zvu is a binary variable which is equal to one if

the arc (v; u) is in the sub-graph T �s of G, and

zero otherwise.(ii) Fvu is a non-negative continuous variable that

only represents fictitious flow produced by the

multicast initiator s going through arc(v; u),and thus helps prevent loops.

(iii) qui is a non-negative continuous variable

which represents the transmission power of

the node u at slot i.(iv) tvui is a binary variable which is equal to one if

node v is scheduled to transmit to node u at

slot i, and zero otherwise.

Let TSu ¼ ðPu1; Pu2; . . . ; PukÞ be the current con-

flict-free transmission schedule (before the multi-

cast request). We note that if there are certain time

slots already reserved in the network, for example

the slot i reserved for transmission from node v tou with transmission power Pvi, the values of the

decision variables qvi and tvui should be preset as


qvi ¼ Pvi and tvui ¼ 1. For those unscheduled slots,

the values of variables qvi and tvui would be ob-

tained after the optimization problem is solved.

We shall prove that if ðxÞ� is the optimal solution

of variable x obtained from this MILP model, then

the graph T �s ðN 0;A0Þ is the optimal tree associated

with this solution, i.e. T �s ðN 0;A0Þ is a BCMT of G

with minimum energy consumption. In the fol-

lowing we formulate all the constraints for the

Problem QoS-MEM.

4.1. Linear constraints for RTC

We want to provide a set of constraints thatwould guarantee that T �

s ðN 0;A0Þ obtained from the

formulation satisfies the rooted tree property. In

this graph, N 0 ¼ fu j9ðv; uÞ 2 A0 or ðu; vÞ 2 A0g is

its arc set, and, A0 ¼ fðv; uÞ jZ�vu ¼ 1g is its arc set.

It can be characterized that T �s ðN 0;A0Þ is a rooted

tree spanning all the multicast members, i.e.,

M � N 0, by the following constraints:

RTC (a). Every node u, u 2 N 0 � fsg, has exactlyone incoming arc, and node s has no

incoming arcs.

RTC (b). T �s ðN 0;A0Þ does not contain cycles.

The construction and interpretation of the lin-

ear constraints for these two properties are elab-

orated in the following theorems.

Theorem 1. T �s ðN 0;A0Þ is a directed graph in which

node s has no incoming arcs, and each other nodehas exactly one incoming arc, provided ProblemQoS-MEM satisfies the following constraints:

n1

n2

n3

nk

(a) (b) (c)

Fig. 2. Illustration of constraints: (a) any non-multicast mem-

ber in T �s must have exactly one incoming arc; (b) a connected

component of T �s may be a simple cycle and (c) a cycle with sub-

tree leaving out of it. (Solid nodes indicate multicast members,

and hollow nodes indicate non-multicast members.)

Xv:ðv;uÞ2A

Zvs ¼ 0; ð2Þ

Xv:ðv;uÞ2A

Zvu ¼ 1 8u 2 M � fsg; ð3Þ

Xv:ðv;uÞ2A

Zvu 6 1 8u 2 N �M ; ð4Þ

Xv:ðu;vÞ2A

Zuv 6 ðn� 1ÞX

v:ðv;uÞ2AZvu 8u 2 N �M : ð5Þ

Proof. Note thatP

v:ðv;uÞ2A Z�vu and

Pv:ðu;vÞ2A Z

�uv are

the in-degree and out-degree of node u in T �s ,

respectively. Therefore, the root node s and the

other multicast members satisfy this statement di-

rectly from the constraints (2) and (3), respectively.It remains to prove that any non-multicast mem-

ber in T �s supporting the multicast communications

must have exactly one incoming arc.

Assume u 2 N 0 is a non-multicast member in T �s ,

indicated by a hollow node in Fig. 2, its incoming

degree must be 1 or 0 from constraint (4). IfPv:ðv;uÞ2A Z

�vu ¼ 0, from constraint (5), it follows

thatP

v:ðu;vÞ2A Z�uv ¼ 0. That means u must be an

isolated node as shown in Fig. 2a, thus u 62 N 0.

This contradicts the original assumption. There-

fore node u has exactly one incoming arc. h

Note that ifP

v:ðv;uÞ2A Z�vu ¼ 1 for any non-mul-

ticast member u in T �s , constraint (5) becomes

redundant since the out-degree of node u is at most

n� 1. From constraints (2)–(4), we obtain the

following conclusion:Xv:ðv;uÞ2A

Zvu 2 f0; 1g 8u 2 N : ð6Þ

Example 1. A generic example of a 4-node net-

work G4 that we consider is shown in Fig. 3. It is

an asymmetric directed graph. For example, the

bi-directed arc ð1; 2Þ indicates that node 1 and

node 2 can reach each other, while the uni-directedarc ð1; 4Þ indicates that only node 1 can reach node

4 since node 4 may not have enough power to

1

1

2

3

321

2

3

4

Fig. 3. Example 4-node network G4: multicast group is {1, 2, 3}

and node 1 is the source.


reach node 1. We can now list the first set of

constraints corresponding to (2)–(5) for RTC (a)

as follows:

Z21 ¼ 0;

Z12 þ Z32 ¼ 1;

Z13 þ Z23 ¼ 1;

Z14 6 1;

3Z14 P 0:

We shall see in Theorem 2 that the introduction

of variable Fvu is to help to prevent loops in Ts, andthis variable only represents fictitious flow producedby the multicast initiator s going through arc (v; u).

Theorem 2. T �S ðN 0;A0Þ does not contain cycles, if

Problem QoS-MEM satisfies constraint (2)–(4) andthe following constraints:Xv:ðv;uÞ2A

Fvu �X

v:ðu;vÞ2AFuv ¼

Xv:ðv;uÞ2A

Zvu 8u 2 N � fsg;

ð7Þ

Zvu 6 Fvu 6 ðn� 1ÞZvu 8u 2 N � fsg; ðv; uÞ 2 A:

ð8Þ

Proof. From the constraints in (2)–(4), it follows

that the only connected components in T �s that

might contain cycles could be composed of either a

simple cycle shown in Fig. 2b, or a simple cycle

with sub-tree leaving out of it as shown in Fig. 2c.

We will show in the following that such topologiesare not feasible for Problem QoS-MEM.

Assume that the nodes (n1; n2; . . . ; nk; nkþ1 ¼ n1),k > 1, form a simple cycle in T �

s . Then from con-

straint (2), node s will never be included in such a

cycle. Constraint in (8) implies that F �vu could be

positive if and only if ðv; uÞ 2 A0. Letting F �n1n2

¼ f ,then from the constraints in (7) it follows thatF �nrnrþ1

¼ F �n1n2

�Pr�1

i¼1 Z�niniþ1

for r ¼ 1; . . . ; k. Each

node nr (r ¼ 1; . . . ; k) is in A0 as stated in the

assumption, i.e., Z�nrnrþ1

¼ 1. Therefore F �nrnrþ1

¼F �n1n2

�Pr�1

i¼1 Z�niniþ1

¼ f � ðr � 1Þ for r ¼ 1; . . . ; k.After substituting F �

nkn1¼ f � ðk � 1Þ into con-

straint (7), for u ¼ n1, we obtainP

v2N F �vn1�P

v2N F �n1v

¼ f � ðk � 1Þ � f ¼ 1� k < 0. On the

other hand,P

v2N F �vn1

�P

v2N F �n1v

¼P

v2N Z�vn1

P 0from constraints (6). Thus the constraints in (7)

are violated, and therefore simple cycles are not

possible in T �s . Similar reasoning shows that the

topology in Fig. 2c also violates the constraints in

(7), and therefore T �s cannot contain cycles. h

Example 2. Still referring to G4 in Fig. 3, we can

list the next set of constraints corresponding to (7)and (8) for RTC (b), which is expressed as follows:

F12 þ F32 � F21 � F23 ¼ Z12 þ Z32;

F13 þ F23 � F32 ¼ Z13 þ Z23;

F14 ¼ Z14;

Z12 6 F12 6 3Z12;

Z13 6 F13 6 3Z13;

Z14 6 F14 6 3Z14;

Z32 6 F32 6 3Z32;

Z23 6 F23 6 3Z23:

4.2. Linear constraints for BWC

The bandwidth constraints reflect the condi-

tions that bandwidth allocated on each link of the

multicast tree should be conflict-free and meet thebandwidth requirement, which can be character-

ized as follows.

BWC (a). Along each wireless link (v; u) in the

optimal multicast tree T �s (Z�

vu ¼ 1),

the transmissions for this multicast

session are scheduled in the set of free

time slots fi j t�vui ¼ 1; i 2 FSvg with


cardinality of B (the bandwidth

requirement), i.e.P

i2FSv t�vui ¼ B. For

non-multicast link (Z�vu ¼ 0), no addi-

tional bandwidth should be reserved,

i.e.P

i2FSv t�vui ¼ 0.

BWC (b). If the current transmission schedule

(before the multicast request) TSu ¼ðPu1; Pu2; . . . ; PuKÞ is conflict-free, then

the transmission schedule TS 0u ¼ ðq�u1;

q�u2; . . . ; q�uKÞ is also conflict-free after

the multicast with required bandwidth

is allowed into the network.

The following theorem explains how the BWC

(a) can be achieved.

Theorem 3. T �s ðN 0;A0Þ satisfies the bandwidth

requirement, if the formulation of Problem QoS-MEM includes the constraints (9) and (10):

tvui 6 Zvu 8ðv; uÞ 2 A 8i 2 FSv; ð9ÞXv:ðv;uÞ2A

Xi2FSv

tvui ¼ B �X

ðv;uÞ2AZvu 8u 2 N : ð10Þ

Proof. The BWC (a) requires that any free time

slot could be reserved for the transmission along

the arc (v; u) under the bandwidth requirement, i.e.

t�vui ¼ 1 ði 2 FSvÞ, only if (v; u) is included in the

multicast tree T �s , i.e. Z

�vu ¼ 1. This is equivalent to

constraint (9).

Recall thatP

v:ðv;uÞ2A Z�vu is the in-degree of node

u in T �s and could only be 1 or 0. For any arc (v; u)

in T �s , i.e. Z

�vu ¼ 1, we have

Py:ðy;uÞ2A Z

�yu ¼ Z�

vu ¼ 1

and Z�xu ¼ 0 for any ðx; uÞ 2 A and x 6¼ v, resulting

in t�xui ¼ 0 ði 2 FSxÞ from constraint (9). After

substituting the values of Z�vu;Z

�xu, and t�xui into

constraint (10), we obtain

Xi2FSv

t�vui ¼X

x:ðx;uÞ2Ax6¼v

Xi2FSx

t�xui þXi2FSv

t�vui ¼X

x:ðx;uÞ2A

Xi2FSx

t�xui

¼ B �X

ðx;uÞ2AZ�xu ¼ B � Z�

vu ¼ B:

Similarly, we can verify that if Z�vu ¼ 0, thenP

i2FSv t�vui ¼ 0. h

The last set of constraints we need to build

up is the conflict-free condition, which requires

that at any time slot and any receiving node the

S1NR requirement (Eq. (1)) should be satisfied.

That is, the new reservation for the multicast

session would not result in any conflicts eitherwith reservations established earlier or within

the multicast traffic itself allowed into the net-

work.

Theorem 4. The new transmission schedule (q�u1;q�u2; . . . ; q

�uk) is conflict-free, if the formulation of

Problem QoS-MEM includes constraints (11)–(13),where b is a relatively large number.

Xv:ðu;vÞ2A

tuvi 6 ðn� 1Þ 1

�

Xv:ðv;uÞ2A

tvui

!

8i 2 S 8u 2 N ; ð11Þ

qviravu

� c gþX

x:ðx;uÞ2Ax6¼v

qxiraxu

0BBB@

1CCCAP bðtvui � 1Þ

8i 2 S 8ðv; uÞ 2 A; ð12Þ

06 qui 6 pmaxu 8i 2 S 8u 2 N : ð13Þ

Proof. Note that ftxyijðx; yÞ 2 A and y ¼ ug andftxyijðx; yÞ 2 A and x ¼ ug present all possible

transmissions to node u and from node u at slot i,respectively. Since each node is equipped with a

single antenna, a node u can only receive a single

transmission at a time, i.e.P

v:ðv;uÞ2A t�vui 2 f0; 1g,

and cannot transmit and receive simultaneously,

i.e. ifP

v:ðv;uÞ2A t�vui ¼ 1 then

Pv:ðu;vÞ2A t

�uvi ¼ 0. This is

equivalent to constraint (11). We also note that ifPv:ðv;uÞ2A t

�vui ¼ 0 for any node u, constraints (11)

becomes redundant since its simultaneous receiv-

ers at slot i is at most n� 1.

In order to guarantee a transmission from node

v is successfully received at node u in the slot i, theparameter SINR must satisfy Eq. (1). In fact, this

is achieved by constraints (12) and (13). When the

slot i is reserved for transmission from v to u, i.e.

Fig. 4. MILP model for problem QoS-MEM.


t�vui ¼ 1, constraint (12) is the same as Eq. (1); and

when t�vui ¼ 0, it becomes redundant. h

The constant b in constraint (12) should be

large enough to make the inequality always tena-

ble when tvui ¼ 0. A possible value is given below:

b ¼ Maxðv;uÞ2A

c gþX

x:ðx;uÞ2Ax6¼v

pmaxx

raxu

0BBB@

1CCCA

8>>><>>>:

9>>>=>>>;: ð14Þ

4.3. Problem formulation

Our previous derivation on the linear con-straints can now help us to write the problem

formulation as an MILP model. This is shown in

Fig. 4, in which Zvu and tvui are integer variables; quiand Fvu are continuous variables. The number of

variables in the formulation is approximately

ð2þ KÞn2 þ Kn, and the number of constraints is

of the order of OðKn2Þ.

5. Computational experiments

After the valid problem formulation, in a staticwireless ad hoc network with no more than 20

nodes, the optimal solution can be practically ob-

tained by CPLEX [43], which is a linear, integer

and quadratic programming package using sim-

plex method and written in C language.

The performance of the QoS routing protocol is

studied with simulations. A static wireless ad hoc

network of 20 nodes is generated in an area of1000 m · 1000 m. We have only considered prop-

agation loss exponent of a ¼ 2. The maximum

transmission power is set to be 50 mW. Every

timeframe is assumed to be composed of 10 slots

0

100

200

300

400

500

1 50Network Instance

tree

pow

er p

er b

andw

idth

(m

W)

QoS-1QoS-2QoS-4


and the noise power and the minimum required

SINR are set to be )50 dBm and 15 dB, respec-

tively. One of the nodes is randomly chosen to be

the Source. Multicast groups of a specified size are

chosen randomly from the overall set of nodes.

There are three types of QoS for the offered traffic.QoS-1, QoS-2, and QoS-4 need one, two, and four

data slots in each frame, respectively. In all cases,

(i.e., for a specified multicast group size and a QoS

traffic), the experiment results are based on the

performance of 50 randomly generated networks.

Our performance metric is the tree power per

bandwidth, defined as the ratio of actual total

0

100

200

300

400


tree

pow

er p

er b

andw

idth

(m

W)

QoS-1QoS-2QoS-4

Fig. 5. Tree power per bandwidth for 20 network instances

(multicast size¼ 5, a ¼ 2).

0

100

200

300

400

500


tree

pow

er p

er b

andw

ith (

mW

)

QoS-1QoS-2QoS-4





0

100

200

300

400

500

600


tree

pow

er p

er b

andw

idth

(m

W)

QoS-1QoS-2QoS-4



power required by a multicast tree to the band-width requirement. This metric allows us to facil-

itate the comparison of energy consumption for

different QoS traffic over a wide range of network

examples. Figs. 5–8 illustrate the performance of

the different QoS traffic we have studied under

different multicast group size. The horizontal axis

is the Instance ID (between 1 and 50), and the

vertical axis is the tree power per bandwidth. Forall the network instances, one can see that QoS-1

performs better (having lower tree power per

bandwidth) than QoS-2, and QoS-2 better than

QoS-4.

1 2v

qs1

3 4

1 2s 3 4

1 2u 3 4

qs2

qs1 qs2

1 2v

qs1

3 4

1 2s 3 4

1 2u 3 4

qs2

qs1 qs3

1 2v

qs1

3 4

1 2s 3 4

1 2u 3 4

qs2

qs 3 qs 4

(a) (b) (c)

Fig. 9. An example to explore the wireless advantage property in TDMA-based networks.


This is not surprising since as the network

traffic becomes heavy, it is harder to schedule the

transmission from a node to all its downlink tree

neighbors in the same time slot without conflicts.

That is, the wireless multicast advantage propertywould not be fully exploited due to the additional

constraints for the bandwidth guarantees com-

pared to the traditional minimum energy multicast

problem. To illustrate this phenomenon, we con-

sider a simple multicast tree with a source node s

and two tree links (s; v) and (s; u), where node vand node u are destinations. There are four data

slots in a frame, and the bandwidth requirement isB ¼ 2 time slots. Fig. 9 gives three transmission

schedules under different traffic load, where the

notation � indicates an confliction that would

happen if the transmission from s is scheduled at

that slot. We observe that only the case in Fig. 9a

(light traffic) takes advantage of the wireless mul-

ticast property with less energy consumption than

other cases (heavy traffic).

6. Conclusion

In this paper we present a constraint formula-

tion for the bandwidth-constrained minimum-en-

ergy multicast problem in multihop ad hoc wireless

networks. Based on the analysis on the propertiesof multicast tree and conflict-free scheduling, the

problem can be characterized in a form of mixed

integer linear programming problem, and we

proceed to prove the correctness of this formula-

tion. To our best knowledge, these are the first

work using mixed integer linear programming to

formulate this problem. Many application sce-

narios can be solved efficiently based on the for-

mulation using branch-and-cut or cutting planes

techniques. The optimal solutions can be used toassess the performance of heuristic algorithms for

mobile networks by running them at discrete time

instances.

A major challenge is to extend our analytical

model to large-scale networks. A near optimal

solution can be found in a polynomial time using

the Lagrange relaxation and sub-gradient tech-

niques [44] based on our formulation. It is alsoimportant to develop the distributed algorithms to

cope with the dynamic topologies.

Appendix A

Notations

A an arc set corresponding to the unidirec-

tional wireless communication link

A0 the arc set of multicast tree TSðN 0;A0Þ,A0 � A

Fvu the non-negative variables, which repre-

sent the amount of flow produced by the

multicast initiator going through (v; u)FSu a set of free time slots at node u, defined as

FSu ¼ fi jPui > 0; i 2 SgG a directed graph modeling the wireless ad

hoc network

M a set of multicast members, M � N 0

N a finite node set in a two-dimensional

plane, jN j ¼ n


N 0 the node set of multicast tree TsðN 0;A0Þincluding all the multicast nodes

M � N 0 � NPui the power level in the slot i assigned to

node u, 06 Pui 6 pmaxu and 16 i6K

pmaxu the maximum power level that node v can

choose

pvu the minimum power needed for the link

from node v to node uqvi a non-negative continuous variable which

represents the transmission power of the

node u at slot irvu the distance between node v and node uS the set of data slot in a frame

S ¼ f1; 2; . . . ;KgTs a multicast tree of GðN ;AÞ rooted at a

source node sTSu the transmission schedule of node u 2 N ,

defined as the power assignment in each

time slot

tvui a binary variable which is equal to one ifnode v transmits to node u at slot i, andzero otherwise

Zvu the binary decision variables that are

equal to one if arc (v; u) exists in the sub-

graph Ts of G, and zero otherwise

a the propagation loss exponent

c the minimum signal-to-interference plus

noise ratio (SINR)g the thermal noise at every receiver

ð�Þ� an optimized solution

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qos-aware minimum energy multicast tree construction in wireless ad hoc networks

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