QoS Topology Control in Ad Hoc Wireless Networks

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1 QoS Topology Control in A Hoc Wireless Networks Xiaohua Jia, Deying Li Dept of Computer Science City University of Hong Kong Hong Kong, China Dingzhu Du Dept of Computer Science an Engineering University of Minesota Minneapoli USA Abstract This paper iscusses the energy efficient QoS topology control problem in a hoc wireless networks. Given a set of noes in a plane, en-to-en traffic emans an elay bouns between noe pair the problem is to fin a network topology that can meet the QoS requirements an the maximum transmitting power of noes is minimize. We consier two cases of the problem: ) the traffic emans are not splittable, an ) the traffic emans are splittable. For the former case, the problem is formulate as an integer linear programming problem. For the latter case, the problem is formulate as a mixe integer programming problem, an an optimal algorithm has been propose to solve the problem. (Abstract) Keywors a hoc wireless networks; energy management;qos provision; QoS routing; topology control (key wors) I. INTRODUCTION An a hoc wireless network is a special type of wireless networks that oes not have a wire infrastructure to support communication among the wireless noes. In multi-hop a hoc network communication between two noes that are not irect neighbors requires the relay of messages by the intermeiate noes between them. Each noe acts as a router, as well as a communication en-point. There are many moern network applications that require QoS provisions in a hoc network such as transmission of multimeia ata, real-time collaborative work, an interactive istribute applications. Extensive research has been one on QoS provisions in a hoc network such as QoS routing or amission control [-]. Most of the existing works eal with resource allocation (e.g., scheuling or buffering) or routing for QoS requests. However, the construction of a network topology that can overall meet QoS requirements has not been stuie in the literature. In multi-hop a hoc network on-line QoS provision such as en-to-en banwith an elay, are highly epenent on the network topology. Without a proper configuration of the topology, some noes in the network coul be easily over-loae an it might be impossible to fin a QoS route uring the operation of the network. The topology of an a hoc network can be controlle by some controllable parameters such as transmitting power an antenna irections. Topology control is to allow each noe in the network to aust its transmitting power (i.e., to etermine its neighbors) so that a goo network topology can be forme. An issue often associate with topology control is energy management. In a hoc wireless network each noe is usually powere by a battery equippe with it. Since the capacity of battery power is very much limite, energy consumption is a maor concern in topology control. To increase the longevity of such network an important requirement of topology control algorithms is to achieve the esire topology by using minimum energy consumption. In this paper, we stuy the energy efficient QoS topology control problem. Given a set of wireless noes in a plane an QoS requirements between noe pair our problem is to fin a network topology that can meet the QoS requirements an the maximum transmitting power of noes is minimize. The QoS requirements of our concern are traffic emans (banwith) an maximum elay bouns (in terms of hop counts) between ennoes at the application level. With the network configure in such a topology, as many as possible QoS calls can be amitte at run-time an the network life time can be prolonge. II. RELATED WORK There are some research works that have alreay been one on topology control for a hoc wireless networks. The earlier works of topology control can be foun in [, ]. In [], Hou et al. stuie the relationship between transmission range an throughput. An analytic moel was evelope to allow each noe to aust its transmitting power to reuce interference an hence achieve high throughput. In [], a istribute algorithm was evelope for each noe to aust its transmitting power to construct a reliable high-throughput topology. Minimizing energy consumption was not a concern in both works. Recently, energy efficient topology control becomes an important topic in a hoc wireless networks. Most of the works have been focuse on the construction an

2 maintenance of a network topology with goo (or require) connectivity by using minimal power consumption. Lloy et al. gave a goo summary of the works in this type in []. They use a -tuple <M, P, O> to represent topology control problem where M represents the graph moel (either irecte or unirecte), P represents the esire graph property (e.g., -connecte or -connecte), an O represents the minimization obective (e.g., MinMax power or Min total power). The NP-completeness of this kin of problems has been analyze an several algorithms have been propose. In [], two centralize optimal algorithms were propose for creating connecte an biconnecte static networks with the obective of minimizing the maximum transmitting power for each noe. Aitionally, two istribute heuristic LINT (local information no topology) an LILT (local information link-state topology), were propose for aaptively austing noe transmitting power to maintain a connecte topology in response to topological changes. But, neither LINT nor LILT can guarantee the connectivity of the network. Li et al. propose a minimum spanning tree base topology control algorithm that achieves network connectivity with minimal power consumption []. A cone-base istribute topology control metho was evelope in [0]. Basically, each noe graually increases its transmitting power until it fins a neighbor noe in every irection (cone). As the result, the global connectivity is guarantee with minimum power for each noe. Huang et al. extene this work in [0] to the case of using irectional antennas []. Marsan et al. presente a metho in [] to optimize the topology of Bluetooth, which aims at minimizing the maximum traffic loa of noes (thus minimizing the maximum power consumption of noes). There are a lot more works on energy efficient communication in a hoc wireless network such as in [, ]. Singh et al. stuie five ifferent metrics of energy efficient routing in [], such as minimizing energy consume per packet, minimizing variance in noe power level minimizing cost per packet, an so on. Kawaia et al. propose a clustering metho for routing in non-homogeneous networks [], where noes are istribute in clusters. The goal is to choose the transmit power level, so that low power levels can be use for intra-cluster communication an high power levels for inter-clusters. In [], Wieselthier et al. stuie the problem of austing the energy power of each noe, such that the total energy cost of a broacast/multicast tree is minimize. Some heuristic algorithms were propose, namely the Broacast Incremental Power (BIP), Multicast Incremental Power (MIP) algorithm MST (minimum spanning tree), an SPT (shortest-path tree). The propose algorithms were evaluate through simulations. Wan et al. in [] presente a quantitative analysis of performances of these three heuristics. So far, there is no publishe work that consiers how to meet the overall QoS requirements through topology control. In this paper, we aress the problem of topology control that can meet the QoS requirements an the maximal power of noes in the system is minimize. III. SYSTEM MODEL AND PROBLEM SPECIFICATION We aopt the wiely use transmitting power moel for raio networks: p i = i,, where p i is the transmitting power neee for noe i to reach noe, ( ) is the istance between i an, an is a parameter typically taking a value between an. The network is moele by G = (V, E), where V is the set of n noes an E a set of unirecte eges. Each noe has a banwith capacity B, an a maximal level of transmitting power P. The banwith of a noe is share for both transmitting an receiving signals. That i the total banwith for transmitting signals plus the total banwith for receiving signals at each noe shall not excee B. Let p i enote the transmitting power of noe i. We assume that each noe can aust its power level, but not beyon some maximum power P. That i 0 p i P for i n. The connectivity between two noes epens on their transmitting power. An ege ( ) E iff p i ( ) an p ( ). From the network moel, we can see that the network topology can be controlle by the transmitting power at each noe an the topology irectly affects the QoS provisions of the network. If the topology is too ense (i.e., noes have more neighbors), there woul be more choices for routing, but the power consumption of the system woul be high. On the other han, if the topology is too loose (i.e., with less eges), there woul be less choices for routing (hence, some noes coul be overloae) an the average hop-count between en-noes woul be high. Our goal is to fin a balance topology that can meet en-users QoS requirements an has minimum energy consumption. Let λ an enote the traffic eman an the maximally allowe hop-count for noe pair ( ), respectively. Let P max = max{ p i i n}. The topology control problem of our concern can be formally efine as: given a noe set V with their location λ an for noe pair ( ), fin transmitting power p i for i n, such that all the traffic emans can be route within the hop-count boun, an Pmax is minimize. We consier two cases: ) en-to-en traffic emans are not splittable, i.e., λ for noe pair ( ) must be route on the same path from s to ; ) en-to-en traffic

3 emans are splittable, i.e., ifferent paths from s to. λ can be route on several We assume each noe can transmit signals to its neighbors in a conflict free fashion. Thu we o not consier signal interference in this paper. There are many MAC (meium access control) layer protocols [, ] or coe assignment protocols [0, ] that have been propose to avoi (or reuce) signal interference in raio transmissions. IV. Given: TOPOLOGY CONTROL WITH TRAFFICS NON- SPLITTABLE - V, set of n noes an their locations. - B, the banwith of each noe. - λ, traffic emans for each noe pair ( ). -, maximally allowe hop-count for noe pair ( ). - P, maximally allowe transmitting power of noes. Variables: x,, boolean variable x i, = if there is a link from noe i to noe ; otherwise, x, = 0. - i - s x, i s x, i,, boolean variable = if the route from s to goes through the link ( ); otherwise s x, = 0. - P max, the maximum transmitting power of noes. Optimize: - Minimize the maximum transmitting power of noes. Min P max () Constraints: - Topology constraints: x = x, i () x x ' ( ') ( ), ' () - Transmitting power constraint: P P max x - Delay constraint: i <, () x s, ( ) () ( ) - Banwith constraint: ( ) x λ x B i () +, i λ ( ) - Route constraints: if s = i x x, i = if = i i () 0 otherwise x x () - Binary constraint: x = 0, or, x = 0, or,( ) () Remarks: Constraint () ensures that each ege correspons to two irecte links. Constraint () ensures that noes have broacast ability. That i the transmission by a noe can be receive by all the noes within its transmitting range. This feature can be represente by the links in the network as: for noe if there is a link to (i.e., x = ), then there must be a link to any noe ' (i.e., x = ) when, which is constraint (). Constraint () etermines the maximum transmitting power among all noes. Constraint () ensures that the hop-count for each noe-pair oes not excee the pre-specifie boun. Constraint () ensures that the total transmission an reception of signals at a noe o not excee the banwith capacity of this noe. The first term at the right han sie of inequality () represents all the outgoing traffics at noe i (transmitting) an the secon term represents all the incoming traffics (reception). Constraints () an () ensure that the valiity of the route for each noe-pair. Since traffics are not s splittable, x, represents that the entire traffics of (

4 ) go through link ( ) if it is in the route from s to. The availability of banwith along the route is ensure by constraint (). The problem of QoS topology control for nonsplittable has now been formulate as an integer linear programming problem (ILP) ()-(), which is NP-har in general. There are several tools that can be employe to compute the solution to this problem. After computing out x i, for n, the transmitting power of noe i can be etermine by the istance to its furthest neighbor. We use a user-efine toolbox calle matlog ( in Matlab. to solve the problem for experiment purpose. The experimental results are presente in section.b.. V. TOPOLOGY CONTROL WITH TRAFFICS SPLITTABLE When the network is in operation, the traffics between a noe-pair may take ifferent routes ue to congestion or failures in the network. In this subsection, we consier the case that the traffic emans can be split. That i the flow going through a path is no longer an integer. A. Formulation Given: - All the parameters in the formulation of nonsplittable case remain the same. Variables: - x i, an P max remain the same as above. - f i,, variables representing the amount of traffics of noe pair ( ) that go through link ( ). Optimize: - Minimize the maximum power of noes. Min P max (0) Constraints: - Topology constraints: x = x, i () x x 'if ( ') ( ), ' () - Transmitting power constraint: P P max x i < - Banwith constraint: ( ), () f f B i () + ( ), i - Routes constraints: λ if s = i f f, i = λ if = i i () 0 otherwise, f f x,( ) () s - Variables constraints: Remarks: x f = 0, or 0,,( ) () The obective an most of the constraints are the same as the non-splittable case. Constraint () is for flow conservation along all the routes for noe pair ( ). There is no elay constraint in the above formulation, because the traffics between a noe pair can be route via several ifferent paths. The QoS topology control problem with traffics splittable has now been formulate as a mixe integer programming problem in (0) (). B. Our Solution Our problem is to fin the network topology such that all traffics can be route an the maximal noe power is minimize. In the case where traffics are splittable, our solution consists of two maor steps: ) increment noe power to connect two noes that have the shortest Eucliean istance among the unconnecte noe-pairs; ) check if the traffics can be route on the topology constructe in step. If so, the topology is foun; otherwise repeat steps an. Because traffics are splittable, the problem in step can be transforme to a variant of the multi-commoity flow problem, that i for a given network topology, to route commoities on the network such that the maximal loa of noes is minimize. Since we assume all noes have the same banwith capacity, the obective of

5 minimizing the maximal noe loa woul lea to the optimal routing to accommoate the traffics. We first consier the QoS routing problem for a given network topology. ) QoS Routing Problem Given a network graph G an traffic emans between noe pair route these traffics in this graph, such that the maximum noe-loa in the system, enote by L max, is minimize. This problem can be formulate as the following: MinL max () f ( ) f L max λ = λ 0 if s = i if = i i otherwise f, i V () f f, i Lmax i (0) 0, 0 + ( ),( ) G Note that: ( ), f = 0, if ( ) E( ) () Function () is the obective, which is to minimize the maximum noe loa. Constraint (0) obtains the maximum noe loa in the network. When any request cannot be route ue to the isconnection of the network, it will report an error of isconnection. This is a linear programming (LP) problem. The optimal solution can be foun in polynomial time O(( E t). ), where E is the number of eges in graph G, an t is the number of noe pairs which have non-zero traffic. We use Matlab. to compute the LP problem. The next, we integrate this QoS routing algorithm with the energy efficient topology control. ) Energy Efficient QoS Topology Control Algorithm The basic iea of the algorithm is to sort all noe pairs (in fact, only the noe pairs that can be reache within the maximal transmitting power P are consiere) in ascening orer accoring to their Eucliean istance. Each time the pair of noes that have the shortest istance an have not yet ha a link between the two noes are picke an their power is increase until they can reach each other. Then, the QoS routing algorithm runs on the network to see if the requeste traffics can be all route. This operation is repeate until the QoS topology is foun, or all noes alreay reache their maximal power P (the topology that can meet the QoS requirements oes not exist in this case). Input: noe set V with their location ( ), an banwith capacity B. λ Output: power levels p for all noes in V. for noe-pair a) sort all noe-pairs with ( ) P (i<) in ascening orer accoring to ( ). b) pick up the noe-pair with closest istance but not yet connecte an increase the power to make them connecte to get a new graph G. c) run the QoS routing algorithm on G to obtain L max. If L B, then stop; otherwise repeat (b) an (c). max In step (b), it stops if all noes alreay reach power P an an error of no solution is reporte in this case. To reuce the number of times of calling the QoS routing algorithm in step (c), we use the binary search metho to fin the QoS topology, instea of aing an ege each time an running the routing algorithm. In this algorithm, the noe power is graually increase until the require topology is forme. It is not ifficult to see that the maximum noe energy neee to form the require topology is minimum, provie the number of power-levels of a noe is finite. Furthermore, the topology foun in step (b) an (c) is minimum in the sense that it has the least number of eges that are ae-in among all the possible topologies that can meet the QoS requirements. This is because the routing prouce by our QoS routing algorithm (formulate in () ()) is also optimal in the sense that the maximum noe-loa in the topology is minimum. That i given a topology, if our routing algorithm cannot route all traffic emans without letting any noe excee its banwith capacity, there is no solution on this topology (i.e., the topology nees more eges to split traffics off). Therefore, when traffic emans are splittable, our algorithm can fin the optimal solution to the energy efficient QoS topology control problem. VI. A. Simulation Setup EXPERIMENTS The simulations are conucte in a 0 0 twoimensional free-space region. The co-orinates of the noes are ranomly an uniformly istribute insie the region. All noes have the same banwith capacity B = 00. The value of in the transmitting power function is set to, i.e., p i =, for =. i

6 The set of requests R = {(, λ )} are generate by using the Poisson function (i.e., the requests originating from a noe follow the Poisson istribution). For each noe, we use the ranom Poisson function with the mean value λ= to generate a number k, which is the number of requests originating from this noe. The estinations of the k request are ranomly picke from the other noes. The traffic eman λ for a pair of noes ( ) is assigne by a ranom function of a normal istribution with variance equal to 0. λ m, where λ m is the mean value of the normal istribution function (i.e., the average banwith eman per request). B. Simulation Results an Analysis ) Topologies for non-splittable traffic versus splittable traffic The first experiment is to compare the topologies for the two cases of traffic non-splittable an splittable. Fig. shows the topology of a network with noes an requests. The noes are ranomly isperse in a 0 0 two-imensional region. The banwith capacity of noe i.e., B, is set to 00. The source, estination, an traffic eman of the requests are generate in the same way as escribe in subsection VI.A. The average traffic amount per request (i.e., λ m ) is 0, which is 0.B. The etails of the requests an the compute routing information are in Tab.. For comparison, Fig. is the topology for the case where traffics are splittable an the topology is compute by using the metho escribe in section V. The input for Fig. is the same as Fig.. Tab. shows the routing an traffic istribution in the topology of Fig.. Comparing Fig. with Fig., we can see that the topology for non-splittable case has a long istance ege (, ) in Fig., which results in a much higher P max than the splittable case. The topology for splittable case has more short ege which helps splitting the traffic off among multiple routes. Note that: Fig. an Fig. are topologies which remove the reunant ege i.e., no traffic eges. Figure. Topology of six noes an six requests for non-splittable case TABLE.. THE QoS REQUESTS AND THEIR ROUTES FOR FIGURE s λ route Figure. Topology of six noes an six requests for splittable case TABLE. THE QoS REQUESTS AND THEIR ROUTES FOR FIGURE s splitte route.0 λ

7 ) Topologies versus traffic loa This experiment shows how the topology changes as the increase traffic emans. The network topology is ictate by the traffic emans once noe locations are given. Fig. shows the topology changes as the increase of λ m. For the clarity of rawing, only 0 noes are use in this experiment. When λ m is relatively small (to B), the topology oes not change much (as seen from Fig.a an Fig.b). When λ m reaches a certain threshol ( λ m = 0.B), the number of eges in a topology increases in a fast spee (which can also be evience in Fig.). The topology in Fig. is almost saturate ue to the high traffic loa in the network. It has been observe that about 0% of the cases have no solution (i.e., no such a topology that can meet the QoS emans) when reaches 0.B. The thir experiment is to analyze the generate topologies. In this experiment, 0 noes are istribute in the 0 0 region to form an a hoc network. Fig. show how the noe-egree, noe-loa, an noetransmitting-power change, respectively, as the changes of λ m. From Fig. -, the following observations can be mae: (c) 0 0 () 0 Figure. The network topology for (a) λ m = 0.0B; b) λ m = 0.B; c) 0.B ) λ m = 0.B. (a) 0 ) Noe egrees increase slowly when λ m is small (in fact, the topology has little change before reaches 0.0B), as shown in Fig.. When is greater than 0.0B, it is observe the noe egrees increase quickly. After λ m reaches 0.B, it starts to have no solution case an the simulation results become unstable ue to the rop out of the no solution samples. ) The QoS routing algorithm is very effective in loa-balancing. From Fig., we can see maximum noe loa, L max, becomes closer to the average noe-loa, L avg, when the loa in the network gets heavier, an the increase of L max slows own quickly an becomes quite stable after reaches 0.B. (b)

8 max avg min noe loa noe egree B 0.0B 0.0B 0.0B 0.B 0.B 0.B Figure. Noe-egrees versus 0.0B 0.0B 0.B λ m max avg min 0.B 0.B 0.B Figure. Noe-loa versus λ m 0.B 0.B 0.B 0.B 0.B VII. CONCLUSIONS We have iscusse the energy efficient QoS topology control problem. This is the first time in the literature that topology control is stuie regaring to QoS provisions. Both cases of traffic splittable an nonsplittable have been consiere. For the former case, we consiere both banwith an elay boun as QoS requirements. The problem has been formulate as an integer linear programming problem. For the latter case, we consiere only the banwith requirement, an the problem has been formulate as a mixe integer programming problem. A polynomial time algorithm has been propose to compute the optimal solution. The problem iscusse is a static configuration problem. The traffic emans are assume to be known in prior. By configuring a goo QoS topology, QoS requests can be best serve in the system (i.e., less requests will be blocke). However, ue to the ynamics an the unpreictability of network traffic a QoS request can still be blocke no matter how goo the topology is. In a ynamic environment where noes are mobile an traffics are ynamic, the propose topology control algorithm can be run perioically to keep the topology optimal in the sense that it balances the noe energy consumption an, at the same time, meets users QoS requirements. noe transmission power B 0.0B 0.0B max avg min 0.B 0.B 0.B 0.B Figure. Noe-transmitting-power versus 0.B λ m ) Fig. shows that our topology control metho is very effective in balancing the uses of noe transmitting power. We can see the maximal noe power, P max, is constantly quite close to the average noe power, P avg. The shows that the topology generate by our metho uses short eges an avois using long istance eges. This feature of balancing the use of noe power extens greatly the lifetime of the network. 0.B ACKNOWLEDGMENT This work is supporte by a grant from the Research Grants Council of Hong Kong uner grant No. CityU 0/0E. REFERENCES [] C. Zhu an M. S. Corson, QoS routing for mobile a hoc networks, IEEE INFOCOM 0. [] C.R. Lin an J.S. Liu, QoS routing in a hoc wireless networks, IEEE Journal on Selecte Areas in Communication vol, no., August, pp. -. [] S. Chen an Klara Nahrstet, Distribute quality-ofservice routing in a hoc networks, IEEE Journal on Selecte Areas in Communication vol., no., August, pp.-0. [] C.R. Lin, Amission control in time-slotte multihop mobile networks, IEEE Journal on Selecte Areas in Communication vol., no. 0, Oct 00, pp. -. [] L. Hu, Topology control for multihop packet raio networks, IEEE Trans. On Communication vol., no. 0,, pp. -. [] T. Hou an Victor O.K. L Transmission range control in multihop packet raio networks, IEEE Trans on Communication vol., no., Jan, pp.-. [] E. L. Lloy, R. Liu, M. V. Marathe, R. Ramanathan an S. S. Rav Algorithmic aspects of topology control problems for a hoc networks, ACM MobiHoc 0.

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