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 construction in 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 networks. 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 be deployed 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 communicating 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).

a significant technical challenge because of the constraints imposed by the characteristics of the ad hoc networks. Resources, including energy, bandwidth, 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 that need 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

1570-8705/$ - see front matter
2004 Elsevier B.V. All rights reserved. doi:10.1016/j.adhoc.2004.03.010

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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 QoS metric bandwidth is more difficult to guarantee in wireless ad hoc networks, because the wireless bandwidth is the scarce resource 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 QoS provisioning must 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 MAC layer 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 transmissions [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, several heuristics exist which provide approximate solutions 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 group applications 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 the amount 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 bandwidth-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 energy 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 as follows. 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 (bandwidthconstrained multicast tree). Section 4 derives the linear constraint formulation for Problem QoSMEM systematically in a form of Mixed Integer Linear Programming (MILP), and proves that it produces the optimal solutions. Computational results assessing the performance are given in Section 5. Section 6 summarizes our finding and points out several future research problems. 2. Related work Over the recent few years, the design of energyefficiency routing algorithms has gained increasing

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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 minimization along the multicast routes. In the following, 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 reservation in a connection-oriented communication in order to meet the QoS requirements for each individual connection. Many mechanisms have been proposed for routing QoS constrained real-time multimedia 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 performance which 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 proposed 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], and QoS routing based on bandwidth calculation [9], have been done and show good performance. Lin [7,8] has proposed QoS routing protocols specifically designed for TDMA-based ad hoc networks. It can build a QoS route from a source to destination with reserved bandwidth. The bandwidth calculation is done hop-by-hop. CEDAR is another QoS aware protocol, which uses the idea of core nodes (dominating set) of the network while determining the paths [18]. Using routes found

219

through the network core, a QoS path can be easily found. More recently, Chen et al. [19] develop an on-demand link-state multipath QoS routing protocol in a wireless mobile ad hoc network. This protocol collects link bandwidth information from source to destination in order to construct a network topology with the information of link bandwidth at the destination. The bandwidth calculation of the QoS route is determined at the destination. The multicast protocol is a primitive communication operation for sending the same message from a source node to a group of destination nodes. It is very significant for many wireless and mobile applications. There are many existing multicast protocols such as MAODV [20], CAMP [21], ODMRP [22], and DCMP [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 protocols is much greater than for best-effort multicast protocols in such networks due to the need to take bandwidth-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 for these networks. Most recent work [24–30,38] has been proposed for the problems of minimizing the energy consumption for broadcasting and multicasting 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 be NP-hard [31,32], efficient heuristic algorithm design has received much more attention. For the MEB problem, a straight greedy approach is the use of broadcast trees that consist of the best unicast paths to each individual destination from the source node (broadcast session initiator). 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

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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 presented. The BIP algorithm is similar in principle to the 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. This operation 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 performance than the greedy algorithms SPT and MST. All the algorithms mentioned above are centralized. Recently, distributed algorithms RBOP (Related Neighbourhood Graph based Broadcast Oriented 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 obtained by pruning from the minimum energy broadcast tree all transmissions that are not needed to reach the member of the multicast group.

can receive its transmission, and the received signal power varies as r
a , where r ðr > 1Þ is the distance to the sender, and a is propagation loss exponent that typically takes on a value between 2 and 4, depending on the characteristics of the communication medium. We assume that any node u 2 N can choose its transmission power level continuously up to some maximum value pumax . Therefore, any directed arc ðv; uÞ 2 A if and only if pvu 6 pvmax , where pvu presents the minimum power needed for the link from node v to node u. For the convenience of the reader, the notations introduced in this section are summarized in Appendix A. In this paper we shall develop a constraint formulation 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 bandwidth is partitioned into a set of time slots S ¼ f1; 2; . . . ; Kg which consist the data part of a frame as shown in Fig. 1. The information concerning available 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 addition, it is also used to determine whether a new connection request is allowed into the network. Let Pui ðu 2 N Þ be the power level in the slot i assigned to node u, where 0 6 Pui 6 pumax and 1 6 i 6 K. The transmission schedule TSu of node u 2 N is thus defined as the power assignment in each time slot, i.e. TSu ¼ ðPu1 ; Pu2 ; . . . ; Puk Þ. For any new traffic request, based on its current transmis-

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 corresponding to the unidirectional wireless communication links. Each node is equipped with a single omni-directional antenna. When considering uniform propagation condition, we observe that all nodes within the communication range of a transmitting node

v

...

1

2

...

i

...

K

...

K

...

K

Pvi

u

...

1

Control phase

2

...

i

Data phase

Pxi

x

...

1

2

...

i

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.

Fig. 1. Illustration of frame structure and data transmission in a TDMA-based wireless ad hoc network.

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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 j Pui 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 at node 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 threshold c, i.e. gþ

P

a Pvi =rvu ðx;uÞ2A;x6¼v

a Þ ðPxi =rxu

P c;

ð1Þ

where rvu is the distance between nodes v and u, and g is the thermal noise at every receiver. This model is commonly known as the Physical Interference 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 all destination nodes. Multicast employs a tree structure 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 reception 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 not relay 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 node u 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-energy 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 multicast members or may not.

221

Formally, we define Ts ðN 0 ; A0 Þ to be a bandwidth-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 bandwidth requirement (B slots per frame), and the scheduling should be conflict-free.

4. Constraints formulation The definition of bandwidth-constrained multicast tree allows us to formulate the QoS-MEM Problem as an MILP (Mixed Integer Linear Programming) model. The main idea is to extract a sub-graph Ts from the original graph G, such that Ts 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 Ts 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 ; . . . ; Puk Þ be the current conflict-free transmission schedule (before the multicast 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 to u with transmission power Pvi , the values of the decision variables qvi and tvui should be preset as

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qvi ¼ Pvi and tvui ¼ 1. For those unscheduled slots, the values of variables qvi and tvui would be obtained 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 Ts ðN 0 ; A0 Þ is the optimal tree associated with this solution, i.e. Ts ðN 0 ; A0 Þ is a BCMT of G with minimum energy consumption. In the following we formulate all the constraints for the Problem QoS-MEM. 4.1. Linear constraints for RTC We want to provide a set of constraints that would guarantee that Ts ðN 0 ; A0 Þ obtained from the formulation satisfies the rooted tree property. In this graph, N 0 ¼ fu j 9ðv; uÞ 2 A0 or ðu; vÞ 2 A0 g is  its arc set, and, A0 ¼ fðv; uÞ j Zvu ¼ 1g is its arc set. It can be characterized that Ts ðN 0 ; A0 Þ 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 exactly one incoming arc, and node s has no incoming arcs. RTC (b). Ts ðN 0 ; A0 Þ does not contain cycles. The construction and interpretation of the linear constraints for these two properties are elaborated in the following theorems. Theorem 1. Ts ðN 0 ; A0 Þ is a directed graph in which node s has no incoming arcs, and each other node has exactly one incoming arc, provided Problem QoS-MEM satisfies the following constraints: X

Zvs ¼ 0;

ð2Þ

Zvu ¼ 1 8u 2 M
fsg;

ð3Þ

P P   and v:ðu;vÞ2A Zuv are Proof. Note that v:ðv;uÞ2A Zvu the in-degree and out-degree of node u in Ts , respectively. Therefore, the root node s and the other multicast members satisfy this statement directly from the constraints (2) and (3), respectively. It remains to prove that any non-multicast member in Ts supporting the multicast communications must have exactly one incoming arc. Assume u 2 N 0 is a non-multicast member in Ts , indicated by a hollow node in Fig. 2, its incoming degree must be 1 or 0 from constraint (4). If P Z  ¼ 0, from constraint (5), it follows v:ðv;uÞ2A P vu  that v:ðu;vÞ2A Zuv 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. Therefore node u has exactly one incoming arc. h P  Note that if v:ðv;uÞ2A Zvu ¼ 1 for any non-mul ticast member u in Ts , 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: X Zvu 2 f0; 1g 8u 2 N : ð6Þ v:ðv;uÞ2A

Example 1. A generic example of a 4-node network 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-directed arc ð1; 4Þ indicates that only node 1 can reach node 4 since node 4 may not have enough power to

n3

v:ðv;uÞ2A

X

n2 nk

v:ðv;uÞ2A

X

n1

Zvu 6 1

8u 2 N
M;

ð4Þ

v:ðv;uÞ2A

X v:ðu;vÞ2A

Zuv 6 ðn
1Þ

X v:ðv;uÞ2A

Zvu

8u 2 N
M:

ð5Þ

(a)

(b)

(c)

Fig. 2. Illustration of constraints: (a) any non-multicast member in Ts must have exactly one incoming arc; (b) a connected component of Ts may be a simple cycle and (c) a cycle with subtree leaving out of it. (Solid nodes indicate multicast members, and hollow nodes indicate non-multicast members.)

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

1

2

3

4 3

1 2

3 1

2

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;

223

Assume that the nodes (n1 ; n2 ; . . . ; nk ; nkþ1 ¼ n1 ), k > 1, form a simple cycle in Ts . Then from constraint (2), node s will never be included in such a cycle. Constraint in (8) implies that Fvu could be positive if and only if ðv; uÞ 2 A0 . Letting Fn1 n2 ¼ f , then from the P constraints in (7) it follows that  Fnr nrþ1 ¼ Fn1 n2
r
1 i¼1 Zni niþ1 for r ¼ 1; . . . ; k. Each node nr (r ¼ 1; . . . ; k) is in A0 as stated in the   assumption, Pr
1 i.e., Znr nrþ1 ¼ 1. Therefore Fnr nrþ1 ¼ Fn1 n2
i¼1 Zni niþ1 ¼ f
ðr
1Þ for r ¼ 1; . . . ; k. After substituting Fnk n1 ¼ f
ðk
1Þ P into con straint (7), for u ¼ n , we obtain 1 v2N Fvn1
P  F ¼ f
ðk
1Þ
f ¼ 1
k < 0. On the v2N n1 v P P P  other hand, v2N Fvn 1
v2N Fn1 v ¼ v2N Zvn P 0 1 from constraints (6). Thus the constraints in (7) are violated, and therefore simple cycles are not possible in Ts . Similar reasoning shows that the topology in Fig. 2c also violates the constraints in (7), and therefore Ts 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:

3Z14 P 0:

F12 þ F32
F21
F23 ¼ Z12 þ Z32 ; We shall see in Theorem 2 that the introduction of variable Fvu is to help to prevent loops in Ts , and this variable only represents fictitious flow produced by the multicast initiator s going through arc (v; u). Theorem 2. TS ðN 0 ; A0 Þ does not contain cycles, if Problem QoS-MEM satisfies constraint (2)–(4) and the following constraints: X X X Fvu
Fuv ¼ Zvu 8u 2 N
fsg; v:ðv;uÞ2A

v:ðu;vÞ2A

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 :

v:ðv;uÞ2A

ð7Þ Zvu 6 Fvu 6 ðn
1ÞZvu

F13 þ F23
F32 ¼ Z13 þ Z23 ; F14 ¼ Z14 ;

8u 2 N
fsg; ðv; uÞ 2 A: ð8Þ

Proof. From the constraints in (2)–(4), it follows that the only connected components in Ts 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 topologies are not feasible for Problem QoS-MEM.

4.2. Linear constraints for BWC The bandwidth constraints reflect the conditions that bandwidth allocated on each link of the multicast tree should be conflict-free and meet the bandwidth requirement, which can be characterized as follows. BWC (a). Along each wireless link (v; u) in the  optimal multicast tree Ts (Zvu ¼ 1), the transmissions for this multicast session are scheduled in the set of free  time slots fi j tvui ¼ 1; i 2 FSv g with

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cardinality of BP(the bandwidth  requirement), i.e. i2FSv tvui ¼ B. For  non-multicast link (Zvu ¼ 0), no additional should be reserved, P bandwidth  i.e. i2FSv tvui ¼ 0. BWC (b). If the current transmission schedule (before the multicast request) TSu ¼ ðPu1 ; Pu2 ; . . . ; PuK Þ is conflict-free, then the transmission schedule TSu0 ¼ ðqu1 ; qu2 ; . . . ; quK Þ 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. Ts ðN 0 ; A0 Þ satisfies the bandwidth requirement, if the formulation of Problem QoSMEM includes the constraints (9) and (10): tvui 6 Zvu 8ðv; uÞ 2 A 8i 2 FSv ; X X X tvui ¼ B  Zvu 8u 2 N : v:ðv;uÞ2A i2FSv

ð9Þ ð10Þ

ðv;uÞ2A

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 either with reservations established earlier or within the multicast traffic itself allowed into the network. Theorem 4. The new transmission schedule (qu1 ; qu2 ; . . . ; quk ) is conflict-free, if the formulation of Problem QoS-MEM includes constraints (11)–(13), where b is a relatively large number. X

X

tuvi 6 ðn
1Þ 1


v:ðu;vÞ2A

! tvui

v:ðv;uÞ2A

8i 2 S 8u 2 N ; 0

ð11Þ 1

X qxi C B qvi B gþ C
c B a C P bðtvui
1Þ a rxu @ A rvu x:ðx;uÞ2A x6¼v

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.  tvui ¼ 1 ði 2 FSv Þ, only if (v; u) is included in the  multicast tree Ts , i.e. Zvu ¼ 1. This is equivalent to constraint (9). P  Recall that v:ðv;uÞ2A Zvu is the in-degree of node  u in Ts and could only be 1 or arc (v; u) P0. For any    in Ts , i.e. Zvu ¼ 1, we have y:ðy;uÞ2A Zyu ¼ Zvu ¼1  and Zxu ¼ 0 for any ðx; uÞ 2 A and x 6¼ v, resulting  in txui ¼ 0 ði 2 FSx Þ from constraint (9). After    substituting the values of Zvu ; Zxu , and txui into constraint (10), we obtain X i2FSv

 tvui ¼

X

X

x:ðx;uÞ2A i2FSx x6¼v

¼B

X

 txui þ

X

 tvui ¼

i2FSv

X X

 txui

x:ðx;uÞ2A i2FSx

  Zxu ¼ B  Zvu ¼ B:

ðx;uÞ2A  ¼ 0, then Similarly, we can verify that if Zvu P  t ¼ 0. h i2FSv vui

8i 2 S 8ðv; uÞ 2 A; 0 6 qui 6 pumax

8i 2 S 8u 2 N :

ð12Þ ð13Þ

Proof. Note that ftxyi jðx; yÞ 2 A and y ¼ ug and ftxyi jðx; 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 P only receive a single  transmission at a time, i.e. v:ðv;uÞ2A tvui 2 f0; 1g, and cannot transmit and P receive simultaneously, P   i.e. if v:ðv;uÞ2A tvui ¼ 1 then v:ðu;vÞ2A tuvi ¼ 0. This is equivalent to constraint (11). We also note that if P  t ¼ 0 for any node u, constraints (11) v:ðv;uÞ2A vui becomes redundant since its simultaneous receivers 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, the parameter 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.

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

225

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

 tvui ¼ 1, constraint (12) is the same as Eq. (1); and  when tvui ¼ 0, it becomes redundant. h

The constant b in constraint (12) should be large enough to make the inequality always tenable when tvui ¼ 0. A possible value is given below: 8 0 19 > > > > > > < B max C= X px C B ð14Þ b ¼ Max cBg þ C : a A> @ ðv;uÞ2A> rxu > > > > x:ðx;uÞ2A ; : x6¼v

4.3. Problem formulation Our previous derivation on the linear constraints 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; qui and Fvu are continuous variables. The number of

variables in the formulation is approximately ð2 þ KÞn2 þ Kn, and the number of constraints is of the order of OðKn2 Þ.

5. Computational experiments After the valid problem formulation, in a static wireless ad hoc network with no more than 20 nodes, the optimal solution can be practically obtained by CPLEX [43], which is a linear, integer and quadratic programming package using simplex 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 of 1000 m · 1000 m. We have only considered propagation 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

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and the noise power and the minimum required SINR are set to be )50 dB m and 15 dB, respectively. 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

500 tree power per bandwidth (mW)

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QoS-1 QoS-2 QoS-4

400

300

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0

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Fig. 7. Tree power per bandwidth for 20 network instances (multicast size ¼ 15, a ¼ 2). QoS-1 QoS-2 QoS-4

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Fig. 5. Tree power per bandwidth for 20 network instances (multicast size ¼ 5, a ¼ 2).

tree power per bandwidth (mW)

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500 400 300 200 100 0

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Fig. 8. Tree power per bandwidth for 20 network instances (multicast size ¼ 20, a ¼ 2).

tree power per bandwith (mW)

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0

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50

Fig. 6. Tree power per bandwidth for 20 network instances (multicast size ¼ 10, a ¼ 2).

power required by a multicast tree to the bandwidth requirement. This metric allows us to facilitate 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. For all 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.

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1

v

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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 property would not be fully exploited due to the additional constraints for the bandwidth guarantees compared to the traditional minimum energy multicast problem. To illustrate this phenomenon, we consider a simple multicast tree with a source node s and two tree links (s; v) and (s; u), where node v and node u are destinations. There are four data slots in a frame, and the bandwidth requirement is B 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 multicast property with less energy consumption than other cases (heavy traffic).

6. Conclusion In this paper we present a constraint formulation for the bandwidth-constrained minimum-energy multicast problem in multihop ad hoc wireless networks. Based on the analysis on the properties of 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 formulation. To our best knowledge, these are the first

work using mixed integer linear programming to formulate this problem. Many application scenarios can be solved efficiently based on the formulation using branch-and-cut or cutting planes techniques. The optimal solutions can be used to assess 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 techniques [44] based on our formulation. It is also important to develop the distributed algorithms to cope with the dynamic topologies.

Appendix A Notations A an arc set corresponding to the unidirectional wireless communication link A0 the arc set of multicast tree TS ðN 0 ; A0 Þ, A0  A Fvu the non-negative variables, which represent 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 j Pui > 0; i 2 Sg G 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

228

N0

Pui pumax pvu qvi

rvu S Ts TSu

tvui

Zvu

a c g ðÞ

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the node set of multicast tree Ts ðN 0 ; A0 Þ including all the multicast nodes M  N0  N the power level in the slot i assigned to node u, 0 6 Pui 6 pumax and 1 6 i 6 K the maximum power level that node v can choose the minimum power needed for the link from node v to node u a non-negative continuous variable which represents the transmission power of the node u at slot i the distance between node v and node u the set of data slot in a frame S ¼ f1; 2; . . . ; Kg a multicast tree of GðN ; AÞ rooted at a source node s the transmission schedule of node u 2 N , defined as the power assignment in each time slot a binary variable which is equal to one if node v transmits to node u at slot i, and zero otherwise the binary decision variables that are equal to one if arc (v; u) exists in the subgraph Ts of G, and zero otherwise the propagation loss exponent the minimum signal-to-interference plus noise ratio (SINR) the thermal noise at every receiver an optimized solution

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