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Pessimism Is Mostly the Best in the Expanding Ring Search for Wireless Networks Kui Wu Computer Science Dept. University of Victoria BC, Canada V8W 3P6
[email protected]
Hong-Chuan Yang Elec. & Comp. Engr. Dept. University of Victoria BC, Canada V8W 3P6
[email protected]
Abstract—Global flooding and expanding ring search are two commonly used methods in search for an interested object in multi-hop wireless networks. While global flooding is simple, it is not scalable for large networks. To take advantage of expanding ring search, a critical question must be answered: what is the best initial TTL (Time-To-Live) value for the expanding ring search? Although an accurate answer to this question depends on specific application scenarios and system configurations, via mathematical analysis and numerical validation, we illustrate that pessimistic search (i.e., search with an initial range that just covers the object in the worst case) is mostly the best for expanding ring search. Index Terms— Multi-hop Wireless Networks, Expanding Ring Search
I. I NTRODUCTION Communication of a group of mobile users without relying on a fixed infrastructure has become popular in our daily lives. In a lot of applications over wireless networks, the source node is required to know where an interested object is located in the network. Nevertheless, due to the limitation of radio transmission range, a wireless node cannot always communicate directly with every other wireless node in the network, and as such search for a desired object must be carried out over multiple radio hops. Global flooding is a simple but expensive approach to searching over multi-hop wireless networks. With global flooding, the query message is broadcast to the whole network and the nodes owning the target object will reply. This strategy is obviously not scalable for large networks, but it has been broadly used in a lot of scenarios due to its simplicity. For instance, ad hoc routing protocols like DSR [4] and AODV [5] use global flooding to search for a destination node whenever a source node wants to send data to a destination node for which it has no routing information in its routing table.
Fulu Li The Media Laboratory M.I.T. 77 Massachusetts Ave. Cambridge, MA 02139, USA
[email protected]
Another commonly used approach is to use expanding ring search, which looks for the object with a gradually increased range. The implementation of expanding ring search is generally based on the TTL (Time-To-Live) of the query message. For example, if the initial TTL value is set to , the source node will search all nodes within one radio hop. If the object is found, the search stops. Otherwise, the source node searches all nodes within two radio hops, and so on. Although expanding ring search has been proposed in several protocols [4], [5] to reduce the search overhead of global flooding, expanding ring search is not necessarily better than global flooding if the initial TTL value for expanding ring search is not properly selected. The optimal search strategy with expanding ring search has been investigated in [1]. It has been shown that if the distribution of the location of the object is known a prior, the optimal search strategy could be obtained with dynamic programming. If such a distribution is not available and the objective is to minimize the worst-case search cost, the best strategies are randomized strategies. Although the theoretical results in [1] are strong and broadly applicable, they may not be very useful in deriving simple and efficient strategies because the calculation of the optimal search strategy with dynamic programming may be costly. The authors also admitted that “unfortunately, optimal strategies cannot in general be qualitatively described without referring to specific numerical computation.” A more recent work from the same authors [2] introduces a class of randomized optimal strategies for controlled flooding search. As the authors mentioned in the paper, the construction of those optimal strategies may pose a problem in a practical setting in the sense that the search cost is only known for integer TTL values while continuous cost functions are used in the paper to derive the optimal strategies. The process of using continuous functions to derive discrete optimal strategies is “not always easy to carry through [2].”
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As such, in this paper we try to provide some simple and effective criteria in the selection of good initial TTL value for expanding ring search. Unlike [1] which tries to provide a generic framework, we limit our investigation to several commonly used strategies in expanding ring search and provide guideline in selecting the initial TTL value suitable for most scenarios with little calculation. The same problem has also been studied in [3] under the context of ad hoc routing protocols. Unlike [3], we do not consider the cache of the object and we focus on the case where certain information, such as the historical record of the object’s TTL value and the mobility pattern, is available. II. N ETWORK M ODEL AND S EARCH S TRATEGIES We study a wireless network with many nodes connected by wireless links. We assume that each node has only limited radio transmission range and thus a node may rely on multiple wireless hops to deliver a message to another node. We assume that the radio transmission range is fixed and is the same for each node. We also assume that wireless nodes are distributed randomly in a two-dimensional plane, and the expected number of nodes within an area is proportional to the size of the area. We assume that the destination node’s mobile speed is known to the source node. In this paper, we study two mobility patterns: the transit mobility model and the random mobility model. In the transit mobility model, a node randomly selects a direction and moves without changing the direction during the interested time period, a period from the time that the source node obtains the recent information about the destination to the time that the source node initiates another new search query. We call this mobility pattern transit mobility since it represents the situation when a user moves from one place to another place. In the random mobility model, a node may change direction during the interested time period. It is called random mobility model since it represents the situation that during a time period a user may have transited to several places and its current location may be randomly distributed within an area. We investigate four different search strategies for expanding ring search: pessimistic search, optimistic search, random search, and optimal search. With pessimistic search, the source node estimates the current range of the destination node and uses the farthest distance to calculate the initial TTL value. With optimistic search, the source node uses the nearest distance to calculate the initial TTL value. With random search, the source node randomly selects a value between the farthest and the nearest distance to calculate the initial TTL value. With optimal search,
Fig. 1. Expected Search Progress
the source node is required to calculate the optimal initial TTL value based on available information. III. E XPECTED S EARCH P ROGRESS For easy reference, notations used in the paper are listed in Table I. TABLE I N OTATIONS
Symbol
Description the source node the destination node at time the time when S knows the recent location of the destination the time when S initiates a new search for the destination the movement speed of the destination
the distance between and the TTL value from to
the radio transmission range the expected search progress per broadcast the average density of network nodes the initial TTL value in expanding ring search
is dense enough, we can roughly use
network If the to approximate the diameter of the search range, where is the radio transmission range. Nevertheless, if the network is not dense, such approximation may have a large error. In this section we derive an equation to calculate the expected increase of search range for each broad cast. As shown in Figure 1, is the source node and is the destination node. Assume that the distance between and is . When broadcasts the query message, all neighboring nodes, i.e., nodes within the circle centered at with radius of , will receive the message. and one of Let denote the distance between ’s neighboring nodes. It is a random variable with PDF(Probability Density Function):
3 Expected Search Progress Per Broadcast
7# 6 # 8.9;: !"$# % '& )(+*-,.,0/214 < 3-5 < 3 375 3 8 5 > (1) = @? ? BA DC Hence, the CDF (Cumuwhere lative Density Function) of is: # < < < E "$# GF 3 & < (+*-,H,H/21 A C (2) + 8 9;: = & 3 Assume that I is the neighboring node that is the clos- Fig. 2. Expected Search Progress Per Broadcast Towards the Desti est to . Let KJ denote the distance between I and . nation (Radio Range = d e m) Based on (2), the CDF of KJ is: Based on (6) and (7), we can calculate theV@expected W E "ML# search progress towards the destination after any given # J 9M < < < times of broadcasts. By calculating the value of F 3 & < (N*7,.,0/21 A QPOR : > (3) [ \ > 5 f J fJ 9M 9M X we illustrate in Figure 2 the relationship O 8 9;: = & between node density and the expected search progress 3 50
45
40
35 500
Node Density: 1 node per 10 m*m Node Density: 1 node per 100 m*m Node Density: 1 node per 400 m*m
450
400 350 300 250 200 Distance To the Destination
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where is the density of the network, i.e., the average node degree. Based on (3), the PDF of the distance be tween I and is:
per broadcast towards the destination. From the figure, it is can be seen that when the node density is high, the expected search progress per broadcast towards the destination is very close to the radio transmission range. But when the node density is lower, the expected search progress is smaller. Equations (6) and (7) indicate that the expected search progress relies on the node density as well as the distance between the current searching node and the destination. Nevertheless, the results in Figure 2 demonstrate that the expected search progress per broadcast towards the destination node does not have significant changes, once the network density is given. As such, in our later calculation, for a given network density, we approximate the expected search progress as a constant value, denoted by .
With the same method above, we can calculate the expecJ , given that J 9M Z 9M : tation of V@W
IV. S EARCH C OST W ITH THE T RANSIT M OBILITY M ODEL
< < < ! " L# % S ;(N*7,.,0/21 A # & & < < < F 3 & < (N*-,H,H/T1 A PUR : 9M > (4) 5 O 83 9;: = & Therefore, the expectation of the distance between I and is: VW 83 6 : ! "SL # (5) JYX F 8 9;: 3 Let J be the random variable representing the distance between and the i-th round relaying node closest to .
As shown in Figure 3, let denote the source node and J [ J 9M \ 9M X the destination node at time Y . Since with the transit #6 : mobility model the destination node moves in an arbitrary GF 3]_^ # a ! " L " L # J \ [ J 9M \ 9M C direction and does not change its direction during the time ]b ]`^ 3]`^ 9;: period from to , the location of the destination node (6) at time is uniformly distributed on the circle centered ! " L " L # is similar to (4):
h i ji , where is the The calculate of at with the radius g ]b ]`^ . to denote this circle. movement speed. We use k ! " L " L # J c [ J Z 9M- Assume that the initial TTL value in expanding ring < 9M < < ]Qb ]_^ search is . Assume that the distance between and m l , where is the expected search progress M (N*7,.,H/T1 A 9M is & & 9M < < per broadcast towards the destination as discussed in Sec< F 3] # & < (+*-,.,0/21 A 9M PUR : 9M > tion III. As discussed in the previous section, we can ap5 + = proximate the expected search progress as a fixed value & 9M 3]_^ 9;: (7) once the node density is given.
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The expected total number of message broadcasts is:
8 ^Nv-w o n qpDr o r Atsu+x s w Dp r 6 x o r 6 z > (8) `y zy where p{r is the probability that the source node can find the destination using as the initial TTL value, and o r = Q < node is the cost of this search in terms of the8 total number of broadcasts. pSr can be calculated by | , 0+ as shown in Figure R 3. where } is the angle of ~jI pDr 6 is the probability that the destination can be found if the TTL value becomes A but cannot if the TTL value o 6 is A . r z is the search cost when the TTL value is A . Next, we need to derive a general expression to calcu6 and o r 6 where
. lating p{r to denote the circle centered at with the We use k m Ac . For simplicity, we assume radius 4 because otherwise either is very close to that or the destination node deviates from its previous location too far. In the first case the search is too trivial, and in the latter case, the value of may not be very helpful in
making a right decision. By calculation, the two intersection points of the cir and the circle k , denoted as and J as cle k shown in Figure 3, are
< < < A > 7 A < < < U < C & &
With expanding ring search, if the source node cannot find the destination node with the current TTL value, the TTL value will be increased by , or equivalently, will 6 J` 6 denote be increased by . Let A . Let k the circle centered at with the radius equal to . The J and the circle two intersection points of the circle k k , denoted as o and o J as shown in Figure 3, are
o } is the angle of ~ < } o N( *7,.,H/T1 3-
in Equation (8) with Equations (9) and (10) respectively, we can calculate the cost of expanding ring search with the initial TTL value set to . We can replace with +A 3 and 2 3 to calculate the expected total number of broadcasts with the pessimistic search strategy and the optimistic search strategy, respectively. For the random search strategy, the expected cost could be calculated by replacing with a random integer value uniformly distributed between 3 and A 3 . In our late numerical calculation in Section VI, we treat Equa Q . Based on tion (8) as a function of , denoted as Proposition 2.1 in [6], we can calculate the expected cost of the random search as:
VW
86 x w Q C m X r y 8 s & 9 w s The optimal initial TTL value can be obtained with the
dynamic programming method introduced in [1]. For this, we need to obtain the tail distribution of the object location. This distribution can be obtained from Equation (9). Note that the calculation of the optimal initial TTL value with the dynamic programming method [1] is non-trivial and costly.
< < < THE R ANDOM M OBILITY 6 A > 6 A < < < 6 < V. C S EARCH C OST W ITH M ODEL & & As shown in Figure 3, let denote the source node and the destination node at time Y . With the ranBy symmetry, we only consider the intersection points dom mobility model, we have assumed that the location o above the -axis, labeled as and as shown in Figure 3. of the destination node at time % is uniformly disThe probability that the destination node cannot be found tributed within the circle centered at with the radius with the TTL value of AhD but can be found with the -{@ where is the movement speed. We use TTL value of A is k to denote this circle. Assume that the source node sets the initial TTL value in expanding ring search as . Assume that the distance pDr 6 } = C m l (9) between and is , where is the expected
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the pessimistic search strategy and the optimistic search strategy, respectively. By replacing with a random inte ger uniformly distributed between ¥ 3 and ¤A 3 , we can obtain the expected total number of broadcasts with the random search strategy. The optimal initial TTL value can be obtained with the dynamic programming method introduced in [1], since the tail distribution of6 the r.object 6 ? ? 3 3 8 9 . location can be obtained from p{r VI. N UMERICAL R ESULTS
Fig. 3. Illustration of Expanding Ring Search
search progress per broadcast towards the destination as discussed in Section III. With the similar method as in Section IV, we can calculate the expected total number of message broadcasts with Formula (8) if the initial TTL 6
will be value is set as . But the calculation of pSr different. In the following, we elaborate their calculation. Since the destination is uniformly
