37th Annual IEEE Conference on Local Computer Networks
LCN 2012, Clearwater, Florida
Interconnecting Disjoint Network Segments Using a Mix of Stationary and Mobile Nodes Ahmad Abbas and Mohamed Younis Department of Computer Science and Electrical Engineering University of Maryland, Baltimore County Baltimore, Maryland, USA aabbas1,
[email protected] network that can be repositioned. To limit the cost, both in terms of resource usage and placement overhead, the objective in such a category of work is often to minimize the number of used RNs. The second category of solutions employs one or multiple mobile data collectors (MDCs) that tour the individual terminals to carry data from one terminal to another [7][8][9][10]. The motive for this solution is the lack of sufficient RNs to form a stable inter-terminal topology and/or the availability of mobile nodes in the network to perform application tasks. If sufficient MDCs are available, a minimum spanning tree of all segments may be formed and MDCs can be designated to serve on the individual links [11]. This paper considers the interconnection problem using lS RNs and lM MDCs, whose combined count (lS + lM) falls short from the number of relay nodes lRN that is necessary for forming a stable inter-terminal topology. We propose a novel algorithm that uses a Mix of Mobile and Stationary nodes for Interconnecting a set of terminals (MiMSI). MiMSI exploits the availability of RNs in order to shorten the travel path for the, lM >1, MDCs and lower the data delivery latency. The algorithm is divided into three phases. In the first phase a stable topology is formed assuming the availability of sufficient number of RNs. The RN placement problem is modeled as Steiner Minimum Tree with Steiner Points and Bounded Edge Length (SMT-MSPBEL) and one of the published heuristics is used for determine the least number of RNs and their positions. In the second phase the RNs of the first phase and the terminals are spatially grouped into lM clusters, where lM is the number of available MDCs. The RNs and/or terminals that serve as gateways between the clusters, i.e., serve on the shortest path between pairs of clusters, are kept as part of the final solution. If fewer RNs are available than MDCs, i.e., lS < lM, MiMSI extends the boundaries of some clusters in order to have some overlapping terminals to serve as gateways. The third phase is dedicated to finding the shortest MDC tour for the terminals that are part of each cluster. The unused nodes out of the lS RNs are carefully placed to further shorten the travel path of the individual MDCs and equalize their motion overhead. The simulation results confirm the effectiveness of the proposed algorithm compared to recent work in the literature. This paper is organized as follows. The next section discusses the related work. In section III, the approach is described in detail. Section IV presents the simulation results and finally Section V concludes the paper.
Abstract—In many applications need arises to connect a set of disjoint nodes or segments. Examples include repairing a partitioned network topology after the failure of multiple nodes, federating a set of standalone networks to serve an emerging event, and forming a strongly connected topology for a sparsely located data sources. Contemporary solutions for interconnecting these disjoint segments/nodes either deploy stationary relay nodes (RN) to form data paths or employ one or multiple mobile data collectors (MDCs) that pick packets from sources and transport them to destinations. The RN-based solution is preferred since it establishes permanent links as opposed to the intermittent links provided by the MDCs. In this paper we investigate the interconnection problem when the number of available RNs is insufficient for forming a stable topology and a mix of RNs and MDCs is to be used. We present an algorithm for determining where the RNs are to be placed and planning optimized travel routes for the MDCs so that the data delivery latency as well as the MDC motion overhead are minimized. The performance of the algorithm is validated through simulation. Keywords: Repairing partitioned topology, Federating multiple network segment, Relay node placement, Mobile Data Collectors.
I. INTRODUCTION Recent years have witnessed a massive growth in the use of networked devices in civil, scientific and military applications. Some applications scenarios require networking a set of disjoint terminals within relatively short time duration and without extensive infrastructure planning. For example, wireless sensor networks (WSNs) often serve in inhospitable environments which make nodes susceptible to failure, e.g., due to detonation of explosives in a battle field, natural calamities in forests, etc. [1]. The failure may involve multiple nodes and cause the WSN to be divided into disjoint segments. Restoring connectivity among these segments would be essential for the network to resume full operation. Another scenario is when multiple data sources or standalone networks are to be federated in order to aggregate their services for serving a certain application mission such as search and rescue, military situational awareness, criminal hunting, etc. Interconnecting a set of disjoint terminals has lately received increased attention from the research community. Published solutions can be classified into categories. The first involves the deployment of stationary relay nodes (RNs) in order to form a strongly connected inter-terminal and inter-RN topology and establish stable communication paths between every pair for terminals [2][3][4][5]. Relays can be new nodes that are to be externally provided or existing nodes in the
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IDM-kMDC and FeSMoR form stable inter-terminal topology and then convert some of the collocated RNs to an MDC in order to cope with the RN availability constraint. As mentioned, MiMSI is considering a more difficult problem since the number of mobile nodes is fixed. In Section IV, we compare the performance of MiMSI to that of IDM-kMDC and FeSMoR through simulation.
II. RELATED WORK As pointed out in the previous section, published schemes for interconnecting a set of disjoint terminals falls into two categories; namely forming stable inter-terminal topology through RN placement and establishing intermittent links by touring the terminals using MDCs. This section discusses related work in these two categories.
III. ESTABLISHING CONNECTIVITY USING A MIX OF MOBILE AND STATIONARY NODES
A. Relay Node Placement Careful positioning of nodes has been pursued as a means for shaping the network topology to meet certain performance objectives [3]. Given their cost, minimizing the number of required RNs to achieve the placement objective has been an integral goal. When connectivity is the main objective, the placement problem becomes equivalent to finding a Steiner Minimum Tree with Minimal Steiner Points and Bounded Edge-Length (SMT-MSPBEL), which is shown to be NPHard by Lin and Xue [6]. Therefore, heuristics have been pursued to find approximate or sub-optimal solutions. Published work on RN placement can be classified into three categories. The first considers unconstrained setups and tries to just establish connectivity between end points [4][5][12], or to restore lost connectivity in a partitioned network [2][13][14]. In the second category either additional performance objectives are targeted [15][16][17], or higher degree of connectivity is to be achieved [18][19][20]. In the third category a constrained version of the RN placement problem is considered [21][22], where RNs can only be placed at a set of pre-determined spots. However, unlike MiMSI, all these approaches employ only stationary relays and assume that unconstrained supply of relay nodes.
MiMSI opts to connect a set of N terminals using a mix of lS stationary relays nodes and lM >1 mobile data collectors. All lS + lM nodes are assumed to have the same communication range R. MiMSI operates in three phases. The first phase generates a stable topology that is used as a baseline for optimization while meeting the node count and mix constraints. The second phase identifies some of the RNs in the baseline topology that would optimally serve the network by minimizing the travel overhead for the mobile nodes, and consequently the data delivery delay among the terminals. Finally the third phase defines tours for the mobile nodes that involve the least travel distances. These phases are described in details in the balance of this section. A. Forming Stable Inter-Terminal Topology In the first phase MiMSI forms a stable connected interterminal topology assuming the availability of sufficient number of RNs. The topology will be used to guide the solution by identifying positions for some, or all, of the lS RNs in the final interconnected system. As pointed out earlier the RN placement problem is modeled as Steiner Minimum Tree with Steiner Points and Bounded Edge Length (SMTMSPBEL), which is NP-hard. There are numerous polynomial time heuristics for solving the SMT-MSPBEL problem in the literature. To avoid confusion, we shall refer to the nonterminal nodes in the SMT-MSPBEL as Steiner points in order to distinguish them from RNs that will be placed as part of the MiMSI solution. Among the possible choices, MiMSI employs FeSTA [2], which is of one of the recently published heuristics for the SMT-MSPBEL problems, and which has been shown to yield superior performance in terms of the RN count and the level of connectivity. The level of connectivity is typically measured in terms of the average node degree in the formed SMTMSPBEL. As will be later explained, MiMSI groups the Steiner points and terminals in the SMT-MSPBEL into clusters based on proximity. Having high node degree in the SMT-MSPBEL serves MiMSI well since there will be more gateways between the clusters which enhances the solution achieved by MiMSI. We use FeSTA in validating MiMSI, as we explain in Section IV. It should be noted that MiMSI can work other SMT-MSPBEL heuristics as well.
B. Mobile Data Collectors Mobility has been exploited for data delivery in sparse mobile ad hoc networks (MANETs), delay tolerant networks, and fragmented sensor networks. In these networks, a mobile node plays one of two roles: a data collector that tours the sensors and carries their readings to user nodes, or a base-station that consumes the data. For example, Shah et al. [23] have introduced data MULEs that move randomly and carry data in a sparse sensor network. Meanwhile, Shen et al. [24] use the MDCs as access points in order to connect nodes in isolated networks through airborne units or satellites. MDCs have also served in linking disjoint batches of nodes [7][8][9]. Almasaeid, and Kamal [7][8] focus on studying the delay effect of using mobile relays. On the other hand, Message Ferries have been introduced in [9] to efficiently deliver data in sparse MANETs using deterministic movement. Seah et al. [25], exploit the use of underwater unmanned vehicles (UUVs) to repair link breakages due to fading and ambient noise. UUVs also ferry the data from the isolated sensors when the network gets partitioned. IDM-kMDC [11] and FeSMoR [26] are two recent approaches that exploit the mobility of some relays to deal with the limited RN count. In essence these two approaches have some similarity with MiMSI . However, MiMSI tackles a more constrained version of the problem where only few nodes can move and the rest are stationary. Like MiMSI,
B. Positioning Stationary Relays The goal of the second phase of MiMSI is to determine a set candidate positions to all or some of the available lS stationary relays and assign a set of terminals to each mobile node to serve. To achieve this goal, MiMSI groups the nodes (Steiner
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points and terminals) in the baseline SMT-MSPBEL, formed in the first phase, based on proximity. The number of clusters equals the number of available MDCs, lM. The rationale behind the proximity-based clustering is to group a set of terminals that are close to each other so that an MDC can tour them with the least travel distance overhead. The question that is worth addressing is why the clustering is not performed without forming a baseline SMT-MSPBEL. The main advantage of the baseline SMT-MSPBEL is to point out possible gateways between the clusters, something that would not be easy to determine if the clustering is performed without knowing the baseline SMT-MSPBEL. The Steiner points corresponding to these gateways will be populated with RNs, if they are fewer or equal to lS, as we explain below. There are many proximity-based clustering algorithms in the literature, e.g., the k-means, which can be applied for this step. Upon forming the clusters, MiMSI identify the gateway nodes that connect two or more clusters. This is done by forming a minimum spanning tree (mst) on the set of Steiner points and terminals, and selecting the shortest edges that connect pairs of the lM clusters. The end points of these edges may be terminals or Steiner points. Unselected Steiner points in the baseline SMT-MSPBEL are not considered any further. Depending on the relationship between the number of gateway Steiner point “g” and lS, the available RNs are assigned as follow:
(a)
(b)
Figure 1: Overlapping node in inter-cluster link to reduce the number of RNs (a) before overlapping (b) after overlapping. C. Determining the MDC Tours In the last phase of MiMSI, the travel paths of MDCs are determined. Each of the lM clusters contains a subset of the N terminals and possibly one or multiple RNs that serve as gateways to other clusters. These terminals and RNs are considered stops on the MDC tour for the cluster. Two cases can be identified based on the number of stops “STP”, as follows: STP = 2: If the distance between the two stops is ≤ 2R, where R is the communication range of a node, the MDC will not move and it will stay stationary in the middle point between the two stops, as shown in Figure 2-a. Otherwise, we find the line connecting the two stops as shown in Figure 2-b Then we find the intersection points, α and β , of this line and the two circles that represent the range of the nodes at the two stops. MiMSI uses the points, α and β , as the data collection points and the MDC will move in straight line forward and backward between them.
g = lS: This is the simplest case for which an RN is positioned at every selected Steiner point to serve as an inter-cluster gateway. g < lS: In this case we populate all “g” gateway Steiner points with stationary relays. The remaining (g - lS) RNs will used for reducing the length of the travel paths for MDCs as we explain when discussing the third phase. g > lS: This case is tricky since the number of available RNs is insufficient to cover all picked Steiner points. Two issues arise. First, these Steiner points are used to connect the clusters. Hence, excluding any of them will prevent a pair of clusters from having a stable inter-cluster link. Second, the travel path for the MDCs that serve the affected clusters ought to be extended. To address the first issue we extend the cluster boundaries for the uncovered gateways so that the clusters overlap, i.e., share at least one terminal or Steiner point. Obviously, this approach will grow the tour of the MDC that serves such an extended cluster. Therefore, the Steiner points that are not covered will be picked based on the effect on the tour length of the MDCs serving the respective clusters. Since the tour will be determined in the third phase of MiMSI, we just use the sum of the mst edge weights (distance) between the terminals in a cluster as an indication of the tour length. In general, the growth in the tour length will not exceed twice the distance between the gateway nodes; one of which is not populated and the second serves as an overlapped node. The handling of this scenario is illustrated in Figure 1.
(a)
(b)
Figure 2: Illustration of finding data collection points for 2 stops, n1 and n2 when the distance between them is, (a) ≤ 2R, and (b) > 2R. STP > 2: In this case a tour is defined so that the MDC becomes in the range of all stops, i.e., the terminals and populated RNs in the clusters, in order to collect and deliver data. The first step is to determine the convex hull for the STP stops. Then we calculate the center of mass for the stops on the convex hull and identify the data collection point for these stops by drawing a line from the center of mass to each stop. The data collection points are the intersections between these lines and the circles of radius R that represent the communication range of each convex stop. The formed path is considered the initial MDC tour. Figure 3 shows an illustration. The tour can be further optimized. For a two-edge path { , } between three stops x, y and z, if the line cut across the circle of radius R that centered at y, the path { , } can be replaced by . This optimization is illustrated in Figure 4.
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(a)
Figure 3: Illustrating how to identify the data collection points for more than two MDC stops.
(b)
Figure 5: (a) MDC edge within the communication range of non-convex nodes, (b) Optimization of MDC edge for nonconvex nodes. Figure 4: Illustrating how to optimize the MDC route for convex nodes. Next, we determine the data collection points for the non-convex stops. This is done by finding the nearest edge in convex hull to each non-convex stop. Then we check whether an edge “e” cut across the circle of radius R centered at a non-convex stop “i”, i.e., the MDC will be with the communication range of “i” while traveling on “e”. Figure 5-a shows an example. If so, the stop “i” will be considered as covered. Otherwise, a data collection point is define by finding the intersection point “q” of the perpendicular line from “i” on “e” and the circles of radius R that centered at “i”. The edge “e” will be then be replaced by two edges connecting at “q” as shown in Figure 5-b
Figure 6: Illustrating the selection of the corner of the polygon to be optimized. (2) Pnew is located on the bisector of θmin at a distance R from P i. Proof: Consider the illustration in Figure 6. In order for the perimeter of the polygon to shrink, Pnew obviously must be located inside the polygon, i.e., in an inward direction with respect to Pi. Assume the angle at Pi is θ and the line divides the angle into and
. Obviously, θ < π, and thus
and
are acute angles. Assume that the reduction in the perimeter is Q, which corresponds to the combined reduction in the lines and , as shown in Figure 6. In other words, Q = d1 + d2
In case g < lS, some RNs will be available for optimizing the tour of the individual MDCs, as pointed out when discussing the second phase of MiMSI. The optimization simply tries to place an RN in an inner point to the tour while being in range of a convex stop so that the travel path is shortened. The following theorem provides guidelines for determining where an RN can be placed. Theorem 1: Consider a polygon that has n corners point {P1, P2, ..., Pn}. Replacing one of the points Pi with a new point Pnew that is at most R units away from Pi, will achieve the maximum reduction in the perimeter of the polygon if: (1) Pi is the corner with the smallest angle θmin in the polygon, and
From Figure 6, ≈ and ≈
Q ≈ +
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Q ≈ ( +
)
sin( − ) − sin = 0
(1) To maximize Q, we should set d to the largest possible value, which equals R, and split θ such that ( +
) is maximal, which is equivalent to minimizing and
. Since = +
, minimizing
and
would imply we should pick the smallest angle
and thus Pi is the corner with the smallest angle θmin in the polygon. (2) = +
and thus
= - Substituting for
and d
⇒ sin( − ) = sin
2 Based on theorem 1, the corner with the smallest angle θmin on the MDC tour is to be selected and the RN will be placed at a distance R on the bisector of θmin towards the center of the cluster. Clusters will be optimized according to the tour length of their MDCs, with preference given to the MDC with the longest travel path. RNs will be allocated incrementally so that the tour length for all MDCs is equalized in order to balance the load. It should be noted that the tour of a MDC gets adjusted after the deployment of an RN and the length of the new tour is considered in the next iteration. In other words, a cluster may be allocated multiple RNs in a row if the tour of its MDC continues to be the longest among all clusters. ⇒ − = ⟹ =
Q ≈ ( +cos ( − )) To find that maximizes Q, let’s differentiate both sides with respect to = (−1) sin + (−1)(−1) sin( − )$ = sin( − ) − sin $ To find the maximum Q, set
%& %' (
D. Illustrative Example and Pseudo Code The various steps of MiMSI are explained using a detailed example in Figure 7. The disjoint set of terminals, shown in
= 0 and thus,
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
Figure 7: Detailed example to illustrate MiMSI’s execution.
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Figure 7-a, are to be interconnected using 3 MDCs and 2 RNs. First a SMT-MSPBEL is formed assuming the availability of sufficient stationary relays. The SMT-MSPBEL is shown in Figure 7-b, which involves a total of 18 Steiner points. A terminal and a Steiner point are denoted in Figure 7 by a square and a circle, respectively. Then, the terminals and the Steiner points are grouped to three clusters since we have 3 MDCs. The clusters are color-coded and annotated in Figure 7-c .The gateways that connect pairs of clusters are circled in Figure 7-d.Basically the gateways are 3 Steiner points and one terminal. Since there are only 2 available RNs for the interconnection process, only 2 out of the 3 Steiner points can be populated RNs and one of the clusters has be expanded to overlap with a neighboring cluster. Since the link between clusters “A” and “B” is longer than that between “B” and “C”, in Figure 7-e cluster “C” has been expanded and the circled Steiner point serves in both clusters “B” and “C”. The rationale for selecting cluster “C” for expansion is that it involves a shorter tour than cluster “B” and the load on the corresponding MDCs will be more balanced than when cluster “B” is expanded. Now the gateways and the terminals are considered for forming the MDC tours. In Figure 7-f, the convex hull is formed for each cluster. Also the center of mass is identified for each cluster and the collection points for each convex node are determined. As shown in Figure 7-g, one of the collection points associated with a convex node is removed because it is covered by the edge formed by other collecting points. After that, we consider the non-convex nodes. All of them are covered by the MDCs routes as shown in Figure 7-h. The final results which include the MDCs routes and RNs locations are shown in Figure 7-i. Figure 8 provides a high level pseudo code for MiMSI.
S ← Steiner points on SMT-MSPBEL for terminal set T Ω ← {S, T} C [1,.., lM] ← Proximity-based clustering of T mst(V, E) ←Minimum Spanning tree for Ω CL ← Set of gateways links e(x,y) such that e(x,y) ∈E && x ∈ Ci and y ∈ Cj, && i ≠ j CG ← Set of gateway Steiner points x such that e(x,y) ∈ CL && x ∈ S and y ∈ Ω IF |CG| < lS // need to extend some clusters mst_tour(Ci) ← MDC Travel path (Ci) CLsort ← Sort C [1,.., lM] in ascendant order of edge length FOR i = 1 to |CG|- lS ei(x,y) = CLsort[i] Costx = mst_tour(Ci) | x ∈ Ci Costy = mst_tour(Cj) | y ∈ Cj IF Costx < Costy Make y element of Ci and remove x from CG ELSE Make x element of Cj and remove y from CG END FOR END IF Position RN at each gateway Steiner points in CG FOR i= 1 to lM // 3rd phase of MiMSI MDC Travel path (Ci) END FOR Find MDC Travel path (Tc) Form the convex hull H for the nodes in Tc Calculate the center of mass M of the node in H Calculate the collection points for H using M and the communication range R Eliminate unnecessary collection points for nodes on H Determine the collection points for non-convex nodes Optimize all collecting points RETURN
IV. PERFORMANCE EVALUATION The performance of the MiMSI algorithm is validated through simulation. This section discusses the simulation environment, performance metrics and simulation results. A. Experimental Setup and Performance Metrics In the simulation, terminals are randomly placed to a 1500 m × 1500 m area. The number of terminals is varied from 5 to 15. The transmission range of stationary relays and MDCs, R, is fixed at 100 m. For simulation, the total number of available nodes, RNs and MDCs, is set to the number of Steiner points to form SMT-MSPBEL multiplied by a fraction “φ”. The ratio of number of MDCs “lM”, to RNs, “lS”, varies between 0.5 and 0.7. The performance of the algorithm under varying number of nodes, MDCs and RNs, is studied by repeating the experiment for different φ values. The reason is that, for different topologies the terminals can be scattered too far or too close to each other and this affects the number of inserted Steiner points. Hence, varying total number of nodes will not lead us to a fair conclusion. Instead, the ratio φ is varied for different topologies to handle such a condition. The following performance metrics are considered for the simulation:
Figure 8: High level pseudo code for the MiMSI algorithm •
•
•
Total Tour Length: This is the sum of distances traveled by all MDCs. This metric gauges the inflicted overhead due to mobility. Average Tour Length: This metric again reflects the overhead due to mobility and factors in the number of engaged MDCs. Maximum Tour Length: This metric shows the longest distance that an MDC has to travel in a network. This is important in analyzing the maximum delay incurred in the network. In addition, this metric indicates the load that one of the MDCs will experience, which would affect the rate of energy depletion and the node lifetime
B. Performance Results We compare the performance of MiMSI to that of IDMkMDC [11] and FeSMoR [26]. IDM-kMDC first forms a minimum spanning tree and assumes that MDCs are deployed
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on all edges. Tours are merged if the number of available MDCs is less than the number of edges, i.e., N-1, where N is the number of terminals. Meanwhile, FeSMoR opts to optimize the date delivery delay. It forms SMT-MSPBEL and replaces subset RNs close to the network periphery with MDCs in order to limit the effect of link unavailability on the data delivery delay while an MDC is in motion. Each simulation experiment is repeated for 50 different topologies and the average is reported. It is observed that with a 95% confidence interval, our results stayed within 6% - 12% of the sample mean. Figure 9 shows the observed total tour lengths results for MiMSI and the baseline approaches. The results show that MiMSI outperforms both IDM-kMDC and FeSMOR under the various values of φ, which again reflects how resources are available relative to the case of forming stable inter-terminal topology. This performance is expected since MiMSI clusters the terminals based on proximity and allocates stationary relays to clusters based on the tour length of the MDC that serve them. All these features contribute to shortening the
tour length of the individual MDCs. Figure 10 indicates that MiMSI does a good job balancing the load among the available MDCs by yielding superior performance in terms the maximum distance that an MDC has to travel. The maximum distance grows very slightly with the increase in the number of segments due to the proximity based clustering of the terminals. For the same reason, Figure 11 shows that MiMSI sustains it performance advantage over IDM-kMDC and FeSMoR with respect to the average tour length. Acknowledgement: This work is supported by the National Science Foundation, award # CNS 1018171. V. CONCLUSION Interconnecting a set of terminals is a requirement in many networking applications. The terminals may be data centers or disjoint network segments that should be connected to serve an emerging event or task. Examples scenarios include repairing a partitioned network topology after the failure of multiple nodes and federating a set of standalone data sources
Figure 9: The total distance to be traveled by all MDCs for MiMSI and the baseline approaches for various values of φ
Figure 10: Maximum travel distance overhead experienced by an MDCs for MiMSI in comparison to the baseline approaches for various values of φ
Figure 11: The average distance that an MDC travel under MiMSI and the baseline approaches for various values of φ
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