Location Service in Ad-Hoc Networks: Modeling and Analysis

Transcription

1 Location Service in Ad-Hoc Networks: Modeling and Analysis Yinhe Yu Dept. of Computer Sci. & Engg. Univ. of Minnesota, Twin Cities Guor-Huar Lu Dept. of Electrical & Computer Engg. Univ. of Minnesota, Twin Cities Zhi-Li Zhang Dept. of Computer Sci. & Engg. Univ. of Minnesota, Twin Cities Abstract Location-based routing significantly reduces the control overhead in mobile ad hoc networks (MANETs) by utiliing position information of mobile nodes in forwarding decisions. However a location service is needed before any forwarding scheme can be applied. Therefore the scalability of the location services directly affects the overall scalability of location-based routing. Recently, several location service schemes have been proposed, most of which are evaluated based on only one or two performance metrics, and under only the uniform traffic pattern. We believe that a comprehensive comparative study is needed to gain a deeper understanding of the design trade-offs in developing scalable location services. In this paper, we first present a taxonomy of existing schemes and explore the design space and tradeoffs involved. We then develop a common theoretical framework to analye five existing and representative schemes in terms of three important cost metrics location maintenance cost, location query cost, and storage cost and under different traffic patterns. Our analysis shows that the design of location services involves tradeoffs among all three cost metrics, and overlooking any of them may lead to biased conclusions. We also show that some of the schemes are more effective in exploiting localied traffic patterns, thereby more suitable for large scale MANETs, where traffic patterns are more likely to be highly localied. I. INTRODUCTION Routing scalability is a critical issue in mobile ad hoc networks (MANETs). Unlike infrastructure-based wired networks, routing in MANETs is performed by each and every node in a cooperative manner. Due to limited resources such as power, bandwidth, processing capability and storage space at the nodes as well as mobility, it is important to reduce routing overheads in MANETs, while ensuring a high rate of packet delivery. Despite many recent advances, designing scalable routing protocols that can effectively operate in an ad hoc networking environment with a large number of mobile nodes remains a challenging research problem []. Classical ad hoc network routing protocols, such as DSR [2] and AODV [3], focus their design on accommodating the mobility of nodes. In these protocols, nodes attempt to discover routes on demand by sending route query messages that flood the entire network. As the network sie grows beyond a few hundred nodes, the cost of flooding becomes prohibitive [4]. To address the scalability problem associated with the early ad hoc network protocols, location-based routing protocols have been proposed [5], [6], [7]. Location-based routing assumes the availability (e.g., via the GPS system) of geographical location information of nodes: nodes (source or intermediate nodes) make forwarding decisions based on the location of destination, e.g., by choosing the neighbor closest to the destination. Hence nodes in location-based routing do not need maintain the traditional routing tables about all potential destinations, only information about neighbors. Although location-based routing eliminates the cost associated with route discovery and maintenance for destinations, it introduces a new problem, namely, the need for location service: before a packet can be forwarded, the (source) node has to discover the location of node to which the packet is destined. Location service is a co-operative service in which nodes are both clients and servers. As a node moves, it must update its location information. When a node needs the location information about a particular destination node, it queries the location service to retrieve the location information. Therefore scalability of location service directly affects the overall scalability of location based routing in MANET. Several location service schemes have been proposed in the literature: GLS[7], SLURP[8], SLALoM[9], DLM[0], HIGH- GRADE[] and Hierarchical Grid[2] are some representative examples. Although scalability has been a main design goal in all these schemes, many questions regarding their scalability properties still remain. First, different performance metrics are employed in previous performance evaluations. For instance, the scheme in [2] focuses mainly on the location update cost (i.e., average number of packets forwarded per second in the network to update all the location servers), while in [9] and [0], both location update and location query costs are considered. In addition, different sets of parameters (e.g., mobility pattern and traffic pattern) are used in simulation evaluation of these schemes, making direct comparison of their scalability and performance difficult if not impossible. Second, apart from location query and update overheads, which are good indicators of CPU processing and power consumption of a location service, we believe that memory/storage requirement is also an important metric that should be taken into account in the design of a location service, especially when a large number of small mobile devices are involved. Finally, scalability of location services should be considered under various traffic patterns (i.e., communication patterns among nodes), in addition to the uniform pattern considered

2 previously. Of particular importance are more localied traffic patterns, in which nodes are more likely to communicate with those nodes that are close-by than those that are far away. Such traffic patterns have proved to be prevalent among many kinds of human or computer communications environments, which we believe is also likely to hold in ad hoc network environments. In this paper we first explore the design space of location services and present a taxonomy of existing schemes. We then develop a common theoretical framework for studying their scalability properties, based on the aforementioned three cost metrics. Using this framework, we analye the scalability of five existing and representative schemes. Our analysis shows that the design of a location service involves tradeoffs among all three cost metrics and overlooking any of them may lead to biased conclusions. We also show that some of the schemes are more effective in exploiting localied traffic patterns, thereby more suitable for large scale MANETs, where traffic patterns are more likely to be highly localied. The remainder of the paper is organied as follows. In Section II we first give a brief description of the existing location service schemes, and then present a taxonomy. A comprehensive comparison of these schemes based on theoretical analysis is described in section III. Finally, we conclude the paper and discuss some future work in section IV. II. LOCATION SERVICES: A TAXONOMY Designing efficient location services has been an active area of research in packet radio and cellular networks. However most of the schemes are based on infrastructured networks with dedicated servers handling the position information [3], [4]. In MANETs, all nodes are symmetric, and there are no dedicated location servers. In other words, location services must be provided by the nodes themselves in a copperative manner. The schemes devised for packet radio and cellular networks are thus not well suited to MANETs. In this paper we only consider the schemes specifically designed for MANETs. Previous work [5] has classified location service in MANETs based on whether all or some nodes in the network act as location servers, and whether each server stores locations of all or some other nodes. This yields four combinations: all-for-all, all-for-some, some-for-all, and somefor-some. DREAM [6] is an example of all-for-all location service, in which all the nodes flood their location information to all other nodes in the network. Clearly this approach is not scalable. Some-for-some and some-for-all schemes also have problems as they put great burden on nodes selected as servers. In this paper, we focus on analying all-for-some location services, where all the nodes in the network store some information about other nodes. In particular, we will compare five examples: HIGH-GRADE, GLS, SLURP, SLALoM and DLM. To design an all-for-some location service, we need to answer the following basic questions: How does a node (with ID) A choose a set of location server(s) to store its location information? How should A update these location servers as it moves around? How does another node (with ID) B discover the appropriate location server(s) of A to retrieve A s location? The solutions proposed by various schemes mainly differ in two dimensions. The first dimension is how servers are organied, e.g., how many location servers does every node have? If each node has multiple servers, how are they distributed across the network area? Uniformly or more densely distributed around the node being served? The second dimension is the granularity of location information stored on the servers. Some schemes store exact locations of nodes, therefore require frequent updates of location information, while other schemes store only a coarse information about a node s location. In the remainder of the section, we first briefly describe five existing and representative schemes, and then present a taxonomy of location services based on the two design dimensions. A. GLS [7] Grid Location Service (GLS) divides an area containing the ad hoc network into a hierarchical grid of squares. The largest square is called the level-h square. The level-h square is then recursively divided into four level-(h ) squares until level-0 squares are reached, forming a so-called quad-tree. In each level-i square (for i > 0), node A selects three location servers, one in each level-(i ) square that A is not in. The structure of GLS is shown in figure (c). GLS selects the location servers based on the node ID in each server s service area (i.e., a quadrant). For a node C to be node A s location server, C has the smallest ID that is larger than A s ID in that quadrant, i.e., C = min{x node x is in the quadrant, ID(x) > ID(A)} We say C is closest to A in the quadrant. For all other schemes we analye, the location servers are chosen in a way that closeness is defined in terms of geographic distance to a point obtained using hash functions, instead of the distance determined in the ID space. Each node updates its location servers with its exact location after it moves a threshold distance δ. To query for a particular node A, a node B sends the query to the node that is closest to A for which B has location information, and so on. Eventually the query would reach one of A s location servers. B. SLURP [8] In SLURP, the entire network area (a square) is divided into a flat grid of squares. Node A selects its location servers by applying a hash function to A s ID and obtains the (x, y) coordinate of a point in the entire area. The square containing that point is called the home square for node A. All nodes in that square store A s exact location information. Every time node A moves to a different square, it updates its home square with new location information. For any node B wishes to communicate with node A, the same hash function is applied to node A s ID to obtain A s home square. A query packet is then forwarded to A s home square to retrieve A s location information. This is illustrated in Figure (a).

3 (a) Fig.. (b) (a) flat server organiation; (b) two-level; (c) multi-level hierarchical. (c) C. SLALoM [9] One of the main drawbacks of SLURP is that the query latency grows as the network sie grows. Even if node B is relatively close to node A, node B may still need to query A s home square that is far away. To address this problem, SLALoM uses a two-level structure. The entire network is first divided into a flat grid of level- squares as in SLURP. The network is then partitioned into various level-2 squares with each level-2 square containing many level- squares. Node A selects its location servers by hashing to the (same) point in each of the level-2 squares. Node A thus has a home square in every level-2 square as in Figure (b). Instead of storing the exact location information at every home square, SLALoM uses a two-level grained location information. SLALoM defines home squares near A as the nine level- home squares closest to A, i.e., the home square in the level-2 square where A is in, plus the eight home squares in the surrounding level-2 squares. All the home squares in the network know which level-2 square A is in, and the nine home squares near A knows the exact location of A. Using this approach, as A moves around, only closer servers need to be updated frequently, whereas remote servers require only infrequent updates. To query node A, node B sends query packet to A s home square in the level-2 square B is in. If that home square is the one near A, B can retrieve A s exact location. Otherwise the servers in that home square know which level-2 square A is in, and can forward the query to the home square in the level-2 square that A is in. D. DLM [0] DLM partitions the entire network much like GLS, i.e., there are H + level of squares. The location servers are duplicated uniformly across the region, one server in every level-k square, where K is a system parameter between and H. The servers are chosen in a way similar to SLALoM, i.e., by hashing to a point in each level-k square, therefore we say DLM also uses a two-level server structure. DLM uses two addressing policies: complete and partial address. In complete address policy, all the location servers store the exact location of a node. When a node moves, it needs to update its location information at all of its location servers. This can cause a storm of update packets when the node moves at high speed. The partial address policy is introduced Fig. 2. level-3 square boundary level-3 square LSC level-2 square boundary level-2 square LSC A B level- square LSC level- square boundary level-0 square LSC level-0 square boundary Network hierarchy and location query in HIGH-GRADE. to alleviate this problem. Each location server stores location information with different granularity. For i > K, if the location server of node A is located in the same level-i square in which A resides in, the servers store only which level-(i ) square A is in. If the server is located in the same level-k square as A, the complete location information is stored. The query operation is straightforward if the complete address policy is used. Node B simply queries the nearest location server of A to obtain A s location. If the partial address policy is used, node B first queries the nearest location server of A. If the complete address of A is found, the query is complete. Otherwise the server of A indicates which level-(i ) square A is in, the query is then forwarded to A s location server in that level-(i ) square. This process continues until A s complete information is found. E. HIGH-GRADE [] HIGH-GRADE uses a similar hierarchical grid structure as GLS, with each node A resides in exactly one level-i square, 0 i H. Figure 2 illustrates such a hierarchical grid with H = 3, and the level-0,, 2 squares of node A are highlighted with different levels of shades. In HIGH-GRADE, a node A has one set of location server(s) in each level-i square it resides in. To determine the relative location of A s location servers in the H + squares, HIGH-GRADE applies H + well-known hash functions on A s ID. We call this set of hash points Location Server Points (LSPs), denoted by

4 Fig. 3. Location granularity multi-grained two-level grained single grained SLURP DLM (partial address) SLALoM DLM (complete address) HIGH-GRADE GLS flat two-level multi-level Server organiation A taxonomy of location services along two design dimensions. LSP A,i (0 i H). Figure 2 shows node A s four LSPs with different solid markers. The location servers are a set of nodes that are closest to each LSP A,i and the location information is maintained using a ranged perimeter refresh protocol that bounds the number of servers around LSPs. For more details, please refer to []. Location servers in HIGH-GRADE store multi-grained information, i.e., each level-i location server ( i H) stores the information of which level-(i ) square A is in, and only level-0 location servers store the exact location of A. When a node B wants to find A s location, it obtains sequentially the so-called level-i potential LSP (plsp B,A,i ) by applying the same hash functions to A s ID in B s level-i square. The process starts with B s level-0 square and therefore the point plsp B,A,0. If node B and node A are co-located in the same level-0 square, then plsp B,A,0 = LSP A,0, and B can retrieve A s exact location from the servers at LSP A,0. Otherwise, the nodes at plsp B,A,0 would find that they do not have a location record for node A, and will then re-forward the query to plsp B,A,. This process continues until plsp B,A,i =LSP A,i for some i, where the first location server of A is found. Since only coarse grained location information is stored on high level servers, the query is then re-forwarded to lower level LSPs sequentially, i.e., LSP A,i, LSP A,i 2,..., until LSP A,0, where the exact location of A is finally retrieved. Figure 2 shows an example of node B querying node A using the concept of LSP and plsp. Note that in the figure, we specify plsps with dashed line markers. Since multi-grained information is stored at each level of LSP, node A only needs to update it level-j servers when A moves across a level-i square boundary for i j. F. Design Tradeoffs We present a taxonomy of the above schemes in Figure 3. As we can see, each scheme differs in the way the location servers are structured: flat, two-level and multi-level. The granularity of location information stored is also different. The choice of location server structure determines the number of location servers. The more location servers, the more update packets and location information copies in the network, which translates to higher location update cost and storage cost. On the other hand, a large number of location servers usually reduces the query cost and latency. Multi-grained location information affects the rate at which location servers need to be updated. If complete location information is used, all location servers need to be updated frequently. With multi-grained location information, the frequency of updating a location server is roughly proportional to the distance between the node and the server, which means remote servers only need very infrequent updates. But multigrained information increases the cost of query, since recursive query operations are required to pin-point the exact location of the destination. In designing a location service, one must have a clear goal and carefully balance these tradeoffs involved. In the next section, we will quantify these tradeoffs and show the strength and weakness of every scheme we described in terms of different scalability metrics. III. COMPARATIVE STUDY BASED ON ANALYTIC MODELS In this section, we employ a common theoretical framework to analye the scalability of the five schemes described in section II. We are interested in how well these schemes scale as ) the sie of the network, denoted by the number of nodes N, increases, and 2) the moving speed of a node, denoted as v, increases, and 3) under different traffic patterns. A. Metrics We use three metrics to evaluate the scalability of each scheme: location maintenance cost, location query cost and storage cost, which are formally defined as follows. Definition (Location Maintenance Cost): The location maintenance cost C m is defined as the number of forwarding operations each node needs to perform in a second to handle the location update/maintenance packets. It can be viewed as the cost of maintaining fresh location information on location servers in the network. Definition 2 (Location Query Cost): The location query cost C q is defined as the number of packet forwarding operations due to location queries each node needs to perform in a second. Definition 3 (Storage Cost): The storage cost C s of a location service is defined as the number of location records a node needs to store as a location server. We separate the location maintenance and query costs for two reasons. First, as we shall see, the design choices of various schemes usually involve tradeoffs between these two types of cost. Examining these two types of cost separately allows us to derive a better understanding on the consequences of the various design choices. Second, we believe that in a location service scheme, the location query cost is relatively easy to reduce by employing various caching strategies, while the location maintenance cost is not. Therefore, separating the two types of cost provides more information for one to predict the likely scenario in practice. Both location maintenance cost and location query cost are evaluated based on the forwarding load, i.e., the number of hops a packet needs to traverse during each operation (update/query). This way, a packet travels far away has a higher cost then packet sends to a nearby destination. It is

5 natural to correlate the forwarding load to CPU processing and power consumption. Also, all three metrics are defined in terms of individual node. Since all nodes are symmetric in MANET, the expected value of the metrics are the same for each node. B. Assumptions The remainder of the section are devoted to the derivation of the expected value of the three metrics as functions of N and v. We summarie our notations in Table I. Before we proceed, we first discuss the basic assumptions we make: The node density γ is constant, i.e., the area of the network A grows linearly with the number of nodes N. We also assume that γ is high enough that geographic forwarding is possible. Nodes move according to a simplified random way-point mobility model [6]. Each node picks a random point in the network and moves toward it with velocity v chosen uniformly between [0, v max ]. After the point is reached, node selects a new random point and moves on with ero pause time. A traffic pattern is the probability distribution of traffic intensities between any pair of nodes in the network. The most commonly used traffic pattern is the uniform pattern, in which the probability of initiating a packet transmission between any two nodes is the same. However, as shown in [7], [8], the capacity of ad hoc wireless network can be surprising low when the uniform traffic pattern is assumed. In fact, [7] proved that the end-toend throughput available to each individual node has a theoretical bound of O(/ N). This demonstrates the poor scalability of any MANET under the assumption of uniform traffic: as N increases, the network throughput approaches ero quickly regardless of routing protocols! Fortunately, we expect that the traffic pattern in large MANETs usually exhibits a more localied property, i.e., nodes close to each other are more likely to communicate than nodes far apart. Therefore, in our analysis, we consider a localied traffic pattern in addition to the commonly assumed uniform traffic. Note that both SLALoM and DLM both have uniformly distributed location servers across the entire network, and SLURP only has one fixed location for location servers for each node, thus different traffic patterns are unlikely to affect the performance of the schemes. In GLS and HIGH-GRADE, location servers for a node are chosen in such a way that the servers are dense near the node and sparse far away from the node. Therefore we need some way to characterie the behavior of both schemes under both uniform and localied patterns. To study the impact of different traffic patterns on scalability of location services, we introduce the following notations. For i = 0,..., H, let P i denote the probability that node B (the querying node) and A (the node being queried) are co-located in the same level-i square (as defined in GLS and HIGH- GRADE). We use P u i and P l i to represent P i under uniform C m C q C s v ρ i d u d q n u n q λ κ Pi u Pi l c c 2 c 3 TABLE I NOTATIONS location maintenance cost location query cost storage cost node speed average progress of each forwarding hop level-i square boundary crossing rate distance traveled by an update packet distance traveled by a query packet number of forwarding hops of an update packet number of forwarding hops of a query packet perimeter refreshing rate distance threshold in perimeter refresh prob. querying nodes in level-i square (uniform traffic) prob. querying nodes in level-i square (localied traffic) constant of random distance within a square constant of random distance between squares another constant of random distance between squares Fig. 4. Distance traveled by a node within a region and localied traffic patterns, respectively. Based on the sie of the level-i squares of a node, Pi u can be easily obtained as follows, { 3 Pi u = 4 if i H H i 4 if i = 0. H For simplicity of analysis, we consider a specific localied traffic pattern in which the probability Pi l decreases exponentially in larger and larger level-i squares. Formally, we define Pi l = 2 P i l, for i H. Given that H 0 P i l =, we can obtain that Pi l = 2 i+, for 0 i H. 2 H C. Boundary Crossing Rate For all the schemes, the location update cost is directly related to the boundary crossing rate of a moving node. A node A generates a location update packet when it cross some square boundaries. The following Lemma gives the boundary crossing rate. Lemma (Boundary Crossing Rate): The square boundary crossing rate of a node A is ρ 0 πv 2R

6 where v is the moving speed of the node A, and R is the side length of a level-0 square. The boundary crossing rate ρ i for a level-i square in multilevel hierarchical structure is ρ i ρ 0, for 0 i H, 2i Proof: In [8], the author showed that we can approximate ρ 0 with the crossing rate of a node in a circular area with diameter R. As illustrated in figure 4, when a node enters a region, it travels some distance R cos θ before exiting. θ is the angle between the velocity vector v and the tangent at the point where node enters the region. Hence, the average distance traveled by a node within a region is Therefore we get 2 π π/2 0 ρ 0 R cos θdθ = 2R π v 2R/π πv 2R Observe that a boundary crossing is either a vertical or horiontal boundary crossing in multi-level structure. In either case, a level-i boundary is also a level-(i ) boundary, while every other (vertical/horiontal) level-(i ) boundary is a level-i boundary. Therefore, we have ρ i = 2 ρ i for i H. D. HIGH-GRADE As described in Section II, HIGH-GRADE uses a multilevel hierarchical structure. A node A selects a LSP in each level-i square, 0 i H. Node A only updates its level-j servers when A moves across a level-i square boundary for i j. To query the location of a node A, the query packet sent by a node B will be forwarded to the plscs in node B s level-i square, starting with i = 0. If a record of node A is not found at a plsc, the query packet is forwarded further to to a higher level (level-(i + )) plsc until a record of node A is found. After that, the query is forwarded to lower and lower level of LSPs and eventually reaching a level-0 server of A. We prove the following theorem for HIGH-GRADE. Theorem 2: For HIGH-GRADE, the expected location maintenance cost E(C m ) and the expected location query cost E(C q ) are: E(C m ) = O(v log N); { O( N) for uniform traffic pattern E(C q ) = O(log N) for localied traffic pattern Proof: To compute the location maintenance cost C m, we first consider the expected distance between node A and its level-i LSP, denoted as E(d u i ) ( u for update), and the average number of hops a update packet takes from node A to A s level-i LSP, denoted as E(n u i ). Since a uniform random hash function is used to obtain its level-i LSP, we can view c Fig. 5. Three constants. Left: c random distance between a pair of nodes in a unit square. Right: c 2 random distance between a pair of nodes in two unit squares adjoined on a side; c 3 random distance between a pair of nodes in two unit squares adjoined on a corner. d u i as the distance between the two random points in a level-i square, as shown in Figure 5 (left pane). Therefore, E(d u i ) = 2 i R (x x 2 ) 2 + (y y 2 ) 2 dx dy dx 2 dy = c 2 i R i R Denote as the average progress for each forwarding hop, can be viewed as a function of the radio transmission range r t and the node density γ [8]. Since we assume both r t and γ c2 c3 = c 2i R. are constant, so is. Thus we have E(n u i ) = E(du i ) In HIGH-GRADE, C m is a combination of two costs: C m, the cost of updating all H + LSPs, and C m2, the cost of perimeter refreshing operations to maintain A s location information around its LSCs []. Therefore, E(C m ) = ρ i E(n u i ) = πvc H v H 2 We hold R, the side length of a level-0 square, as constant. Then H is proportional to log A. Since A N, H log N, E(C m ) = O(v log N). For E(C m2 ), let λ be the perimeter refresh rate. The number of nodes around the perimeter is bounded by πκ γ, where κ is the distance threshold. Since there are H + LSPs which need to be refreshed, we have E(C m2 ) = (H + ) λ πκ2 γ = O(log N). Combining the two, we have E(C m ) = E(C m ) + E(C m2 ) = O(v log N). For location query cost, denote E(n q i ) as the expected forwarding hops traveled by a query packet from node B to node A s level-0 server when nodes A and B are co-located in the same level-i square. As described previously, each query packet is forwarded sequentially from plsp 0, plsp,..., up to plsp i (i.e., LSP i ), then to LSP i until LSP 0 is reached. We can view each step as the distance between two random points in a level-j square, with j first increases from 0 to i then decreases from i to. Therefore, i E(n q i ) = E(d u j ) E(d u j + ) = 2 j=0 i j=0 c 2 j R j=i c R = (2 i+2 3) c R.

7 For the uniform traffic pattern, the expected query cost is, E(C q ) = E(n q i ) P u i (2 i+2 3) c R 3 4 H i 2 H+ c R = O( N). For the localied traffic pattern, E(C q ) = E(n q i ) P l i (2 i+2 3) c R 2 c R H = O(log N). 2 i+ 2 H Theorem 3: The expected value of the storage cost E(C s ) for HIGH-GRADE is: E(C s ) = O(log N). Proof: The average number of records a node stores is the total number of records stored in the network divided by the total number of nodes. Each node stores its location information at H + LSP s. Since the average number of servers at each LSP is bound by πκ2 γ, we have E. GLS E(C s ) = 2 πκ N (H + ) γ N = O(log N) The GLS scheme uses a similar multilevel hierarchical structure as HIGH-GRADE. A node A selects three location servers in each level-i square, one in each level-(i ) squares that node A is not in, as shown in Figure (c). An important difference between GLS and HIGH-GRADE is that GLS stores the exact location information on every server. Therefore, to ensure freshness of location information and to reduce the query failure rate, all location servers of node A need to be updated periodically. The update period is set as the expected time a node moves a distance of δ, namely δ/v When a node B wants to find the location of A, it sends a query packet towards a node C, the node closest to A in the ID space that B knows of. C does the same, re-sending the query packet to C 2, the node closest to A that C has a record of and so forth, until a location server of A is found. Assuming nodes are relatively static during the lifetime of a packet, the GLS scheme guarantees that in i steps, the location server will be reached, where i is the level of the minimal common square A and B are co-located. In addition, in each of the i steps, the source and destination of the re-sent packet in that step are within a level-j square, where j decrease gradually from i to. We now prove the following theorem about GLS. Theorem 4: For GLS, E(C m ) = O(v N); { O( N) for uniform traffic pattern E(C q ) = O(log N) for localied traffic pattern ; E(C s ) = O(log N). Proof: We first consider the location maintenance cost metric C m. Since in GLS, a node updates all its servers with the same period, no location maintenance packet (as in the perimeter refreshing of HIGH-GRADE) is needed. C m is solely due to location updates in the GLS scheme. Consider the expected distances the three update packets traveled to update the three locations servers at the level-i square, denoted E(d u i ). We have E(d u i ) = (2c 2 + c 3 ) 2 i R, where 2 i R is the side length of a level-i square, c 2 and c 3 are two constant factors representing the average random distance between two points in two neighboring squares, as shown in Figure 5 (right pane). Obviously, we have c 2 5, and c Since updates are sent out at a rate of v δ, we have E(C m ) = v δ i= (2 c 2 + c 3 ) 2 i R = v δ (2c 2 + c 3 ) R (2H+ 2) = O(v N). Next we consider the location query cost C q. Based on the location query procedure described above, the expected location query cost when A and B are co-located in a level-i square is E(n q i ) = i = j=0 i j=0 E(d u i ) c 2 j R = (2 i+ ) c R. For the uniform traffic pattern, E(C q ) = E(n q i ) P u i (2 i+ ) c R 3 4 H i 2 H c R = O( N).

8 and for the localied traffic pattern, E(C q ) = E(n q i ) P i l c R H = O(log N). Finally, the storage cost is, F. DLM (2 i+ ) c R E(C s ) = N 3H N 2 i+ = O(log N). 2 H In our analysis we will focus on DLM partial address option. As described in Section II, DLM uses multi-grained location information similar to HIGH-GRADE. A major difference is that in DLM the location servers of a node A are distributed uniformly across the network: one server in every level-k square, where K is a network parameter chosen between and H. The choice of value for K involves a tradeoff between the location maintenance cost and the location query cost of the DLM scheme. Intuitive, when K is small, there is a location server for node A in every low level squares. Therefore, the location query cost is low as any node B can find a nearby server of A. However, the location update cost is high, since many servers need to be updated across the network. When K is large, the reverse is true. We show that the performance metrics are functions of the parameter K, as in the following theorem. Theorem 5: The expected location maintenance cost and the expected location query cost of DLM are: E(C m ) = O(v(2 K + 2 H K ) + 2 2K ) E(C q ) = O(2 2K ). independent of traffic patterns. Proof: The location update cost of DLM (denoted by C m ) consists of two parts. First, when a node A moves out of its level-0 square, it needs to update only its closest location server (the server within its level-k square). In addition, when a node A makes a level-(i + K) square boundary crossing (i 0), it needs to update all its servers within its level-(i + K + ) square. We next derive the cost of updating all the servers within the level-(i + K) square. The DLM paper [0] did not discuss how to perform the updates. A naive approach would be to send a unicast message to each server. However, a better approach is to build a multicast tree-like structure connecting all the servers within the level-(i+k) square, and send update messages along the tree. Since there are 4 i servers in a level- (i + K) square, the tree will have 4 i edges. It is easy to see that each edge will have the geographical length of 2 K R. Therefore, the total cost of updating all the servers within the level-(i + K) square is 2 K R (4 i ). Using the same analysis as in HIGH-GRADE, the level-j square boundary crossing rate is ρ j = 2 j location update cost of DLM is E(C m ) = ρ c 2 K R + H πv 2R. Therefore, the P i+k ρ 2K R (4 i ) = 2 K c πv + πv 2 (2H + ) (c 2 K + 2 H ) πv Note that H is the number of levels above level K, i.e., H = H K, H is the total number of levels in the network. In DLM, a location server of node A is chosen by hashing the ID of node A to one level-0 square in the level-k grid. If that level-0 square is void (i.e., no node in the level-0 square), a backup search process is performed to search the entire level-k grid area in a certain order. In addition, as the current location server moves, it needs to periodically search at least a partial area of the level-k grid to ensure that it is still the closest node to the hashed position of node A; otherwise it needs to pass the location information of node A to the closer node, which now serve as the new location servers of A. In the worse case, the current server needs to search all the 2 K 2 K level-0 squares periodically to ensure the correctness of location server selection. Therefore, another part of location maintenance cost (denoted by C m2 ) is E(C m2 ) = λ m R (2K 2 K ) = O(2 2K ), where λ m is the rate of the periodical check by the current location server. Note that in the derivation, we assume the backup search process follows a tree like sequence similar to those discussed in the last subsection. Therefore, E(C m ) = O(v(2 K + 2 H K ) + 2 2K ) The location query process of DLM is simple. To find the location of node A, a node B will query the location server of A within B s level-k square. However, because of the backup search scheme, in the worst case node B may need to search all the 2 K 2 K level-0 squares to find out node A s location server. Therefore, the location query cost of DLM is E(C q ) = R (2K 2 K ) = O(2 2K ), We note that the sum of the location maintenance and query costs are minimied when K = H 3. In that case, E(C m ) = O(v 3 N) E(C q ) = O( 3 N).

9 The storage cost metric for DLM is given in the following theorem. Theorem 6: The expected storage cost of DLM is E(C s ) = O(2 2(H K) ). when K = H 3, E(C s ) = O( 3 N 2 ). Proof: In DLM, a node A has a location server in each level-k square. This means that all the nodes in a level-k grid will host the location information for all the nodes in the entire network. Therefore, the storage cost for each node is E(C s ) = (2H R)(2 H R) γ (2 K R)(2 K R) γ = 2 2(H K) = O(N 2/3 ). E(C s ) = N R R γ N = O() We next analye the storage cost for SLALoM. In SLALoM, every K K level- squares (a level-2 square) contains the location information of all N nodes in the network. Assuming K = Θ(N /3 ), we can calculate the number of nodes in a level-2 squares area is (N /3 )(N /3 ) N N 2/3 A. Since these Θ(N 2/3 ) nodes store the location information of all N nodes, we have E(C s ) N N 2/3 = O(N /3 ) G. SLURP and SLALoM In [8] and [9], the authors gave the upper bound of the expected value of the location maintenance cost and location query cost for SLURP and SLALoM. The analysis of these studies are based on the same assumptions (e.g., γ is held constant, etc.) except that only the uniform traffic pattern is used. However, the results of their analysis still hold in the localied traffic pattern, i.e., the results do not depend on which traffic pattern is assumed for the SLURP and SLALoM schemes. We duplicate their results in the following two theorems. Theorem 7 (Woo and Singh): For SLURP, E(C m ) = O(v N) E(C q ) = O( N). Theorem 8 (Cheng et al.): For SLALoM, E(C m ) = O(v 3 N) E(C q ) = O( 3 N). Note that in SLALoM, the location maintenance and query costs are actually functions of K, the sie of the level-2 squares. The results in Theorem 8 is based on K = Θ(N /3 ), in which case E(C m ) + E(C q ) is minimied over K. Finally, our last theorem of our study is on the storage cost of SLURP and SLALoM. Theorem 9: For SLURP, E(C s ) = O(). For SLALoM, E(C s ) = O( 3 N). Proof: To analye the storage cost of SLURP, we note that in SLURP, the location information of a node A is stored in each node of the level-0 square containing the hashed point of A. Assuming the node density is γ, we have H. Summary We summarie the three performance metrics of the five location service schemes in Table II, and make several observations about the results. The results of HIGH-GRADE and GLS are very similar, which is not surprising, as their designs exhibit the most similarity. However HIGH-GRADE outperforms GLS in terms of location maintenance cost while having the same asymptotic costs for query and storage. The saving in the location update cost in HIGH-GRADE is mostly due to the use of multigrained location information, which avoids frequent and costly updates to remote servers. We also note that the query cost of both schemes are reduced when the traffic pattern is localied. This shows the advantage of having a multi-level server structure, with servers denser near the node and sparser farther away. In terms of the total overheads, both schemes have the same asymptotic results O(v N) if the uniform traffic pattern is used (obtained by summing the first two rows for both schemes). But if the localied traffic pattern is assumed, HIGH-GRADE has a better asymptotic cost, O(v log N) versus O(v N) for GLS. The reduction in the location update cost in HIGH-GRADE has several advantages. First, the location update cost is usually the dominating cost in location services and cannot be avoided. Second, as mentioned previously, the location query cost can be reduced by using various caching techniques. Any improvement from caching has direct impact on the overall cost for HIGH-GRADE, whereas caching does not offer much benefit in GLS since the total cost is dominated by the location update cost. We note that both DLM and SLALoM improve the overall location maintenance/query cost over SLURP by using a twolevel server structure and multi-grained location information. However this improvement comes at the cost of increasing storage cost, which is another important metric for scalability.

10 TABLE II SUMMARY OF THREE SCALABILITY METRICS FOR THE FIVE LOCATION SERVICE SCHEMES HIGH-GRADE GLS DLM SLURP SLALoM Location maintenance cost Location query cost O(v log N) O( N) O(v N) O( N) O(v 3 N) O( 3 N) O(v N) O( N) O(v 3 N) O( 3 N) (uniform) (uniform) (both) (both) (both) O(log N) O(log N) (localied) (localied) Storage cost O(log N) O(log N) O( 3 N 2 ) O() O( 3 N) In the case of DLM, the storage cost actually dominates the overall cost. This indicates that to present a complete picture of the scalability of a location service scheme, we need to take into account all the three metrics to avoid biased conclusions. Finally, we point out that because SLALoM and DLM uniformly distribute location servers across the network, they can not take advantage of localied traffic patterns (they have the same asymptotic costs with both uniform and localied traffic patterns). This suggests that they are less suitable than HIGH-GRADE and GLS for large scale ad hoc networks, where localied traffics are more likely to be true. IV. CONCLUSIONS We have explored the design space of location services and classified previously proposed schemes. We have developed a uniform theoretical framework to compare the scalability of five existing and representative schemes. Our analysis shows that the design of scalable location services involves tradeoffs among all three cost metrics location query cost, location maintenance cost, and storage cost. We show that different design choices can significantly affect the scalability of location services, and in turn the overall scalability of locationbased routing. Therefore, in designing a location service we must carefully balance the tradeoffs involved. We believe that our comparative study will enable a deeper understanding of routing scalability in MANETs. REFERENCES [] Onur Arpacioglu, Tara Small, and Zygmunt J. Hass. Notes on scalability of wireless ad hoc networks. October, [2] David B Johnson and David A Malt. Dynamic source routing in ad hoc wireless networks. In Imielinski and Korth, editors, Mobile Computing, volume 353. Kluwer Academic Publishers, 996. [3] Charles Perkins and Eliabeth Royer. Ad-hoc on-demand distance vector routing. In Proceedings of IEEE Workshop on Mobile Computing Systems and Applications, 999. [4] Charles Perkins. Ad Hoc Networking. Addison Wesley Publishing Co., [5] Brad Karp and H. T. Kung. GPSR: greedy perimeter stateless routing for wireless networks. In Mobile Computing and Networking, pages , [6] Stefano Basagni, Imrich Chlamtac, Violet R. Syrotiuk, and Barry A. Woodward. A distance routing effect algorithm for mobility (DREAM). In Proceedings of ACM MobiCom 98, pages 76 84, 998. [7] J. Li, J. Jannotti, D. De Couto, D. Karger, and R. Morris. A scalable location service for geographic ad-hoc routing. In Proceedings of ACM MobiCom, pages 20 30, August [8] Seung-Chul M. Woo and Suresh Singh. Scalable routing protocol for ad hoc networks. Wireless Networks, 7(5):53 529, 200. [9] Christine T. Cheng, H. L. Lemberg, Sumesh J. Philip, E. van den Berg, and T. Zhang. SLALoM: A scalable location management scheme for large mobile ad-hoc networks. In Proceedings of IEEE WCNC, March [0] Y. Xue, B. Li, and K. Nahrstedt. A scalable location management scheme in mobile ad-hoc networks. In Proceedings of the IEEE Conference on Local Computer Networks (LCN 0), 200. [] Yinhe Yu, Guor-Huar Lu, and Zhi-Li Zhang. Enhancing Location Service Scalability with HIGH-GRADE, Dept. of Comp. Sci. & Eng., U of Minnesota, Technical Report TR , [2] Sumesh J. Philip and Chunming Qiao. Poster: Hierarchical grid location management for large wireless ad hoc networks. In Proceedings of ACM MobiHoc 03, Poster session, June [3] Gregory Lauer. Address servers in hierarchical networks. In Proceedings of IEEE ICC, 988. [4] Kamal K. Kasera and Ram Ramanathan. A location management protocol for hierarchically organied multihop mobile wireless networks. In Proceedings of IEEE ICUPC, 997. [5] M. Mauve, J. Widmer, and H. Hartenstein. A survey on position-based routing in mobile ad hoc networks. IEEE Network Magaine, 5(6):30 39, november 200. [6] Josh Broch, David A. Malt, David B. Johnson, Yih-Chun Hu, and Jorjeta Jetcheva. A performance comparison of multi-hop wireless ad hoc network routing protocols. In Mobile Computing and Networking, pages 85 97, 998. [7] Piyush Gupta and P. R. Kumar. The capacity of wireless networks. IEEE Transactions On Information Theory, 46(2): , March [8] Jinyang Li, Charles Blake, Douglas S. J. De Couto, Hu Imm Lee, and Robert Morris. Capacity of ad hoc wireless networks. In Mobile Computing and Networking, pages 6 69, 200.