A Probabilistic Algorithm for Efficient and Robust Data Propagation in Wireless Sensor Networks 1 2 Ioannis Chatzigiannakis† , Tassos Dimitriou‡ , Sotiris Nikoletseas† and Paul Spirakis† †
Computer Technology Institute (CTI) and Patras University, P. 0. Box 1122, 261 10 Patras, Greece. e-mail: {ichatz,nikole,spirakis}@cti.gr ‡ Athens Information Technology (AIT), Athens, Greece. e-mail:
[email protected] Abstract We study the problem of data propagation in sensor networks, comprised of a large number of very small and low-cost nodes, capable of sensing, communicating and computing. The distributed co-operation of such nodes may lead to the accomplishment of large sensing tasks, having useful applications in practice. We present a new protocol for data propagation towards a control center (“sink”) that avoids flooding by probabilistically favoring certain (“close to optimal”) data transmissions. Motivated by certain applications (see [1, 15]) and also as a starting point for a rigorous analysis, we study here lattice-shaped sensor networks. We however show that this lattice shape emerges even in randomly deployed sensor networks of sufficient sensor density. Our work is inspired and builds upon the directed diffusion paradigm of [15]. This protocol is very simple to implement in sensor devices, uses only local information and operates under total absence of co-ordination between sensors. We consider a network model of randomly deployed sensors of sufficient density. As shown by a geometry analysis, the protocol is correct, since it always propagates data to the sink, under ideal network conditions (no failures). Using stochastic processes, we show that the protocol is very energy efficient. Also, when part of the network is inoperative, the protocol manages to propagate data very close to the sink, thus in this sense it is robust. We finally present and discuss large-scale simulation findings validating the analytical results.
1 Introduction, our Results and Related Work Recent dramatic developments in micro-electro-mechanical (MEMS) systems, wireless communications and digital electronics have led to the development of small in size, low-power, low-cost sensor devices. Such extremely small devices integrate sensing, data processing and communication capabilities. Examining each such device individually might appear to have small utility, however the effective distributed co-ordination of large numbers of such devices may lead to the efficient accomplishment of large sensing tasks. Large numbers of sensors can be deployed in areas of interest (such as inaccessible terrains or disaster places) and use self-organization and collaborative methods to form a sensor network. Their wide range of applications is based on the use of various sensor types (i.e. thermal, visual, seismic, acoustic, radar, magnetic, etc.) to monitor a wide variety of conditions (e.g. temperature, object presence and movement, humidity, pressure, noise levels etc.). Thus, sensor networks can be used for continuous sensing, event detection, location sensing as well as micro-sensing. Hence, sensor networks have important applications, including (a) military (like forces and equipment monitoring, battlefield surveillance, targeting, nuclear, biological and chemical attack detection), (b) environmental applications (such as fire detection, flood detection, precision agriculture), (c) health applications (like telemonitoring of human physiological data) and (d) home applications (e.g. smart environments and home automation). For a survey of wireless sensor networks see [1] and also [13, 16]. Note however that the efficient and robust realization of such large, highly-dynamic, complex, nonconventional networks is a challenging algorithmic and technological task. Features including the huge number of sensors involved, the severe power, computational and memory limitations, their dense deployment and frequent failures, pose new design and implementation aspects which are essentially different not only to distributed computing and systems approaches but also to ad-hoc networking techniques. 1 A preliminary version of this work has appeared in the 5th European Wireless Conference on Mobile and Wireless Systems beyond 3G (EW, 2004 [7]). 2 This work has been partially supported by the IST Programme of the European Union under contract numbers IST-2001-33116 (FLAGS) and IST-2004-001907 (DELIS)
1.1
Problem Description
We focus on an important problem under a model of sensor networks that we present. More specifically, we study the problem of local detection and propagation, i.e. the local sensing of a crucial event and the energy and time efficient propagation of data reporting its realization to a control center. This center could be some human authorities responsible of taking action upon the realization of the crucial event. We use the term “sink” for this control center (other terms used in the relevant literature [8, 9] include the term “wall” which suggests the possibility of a moving, maybe along a line, control center). The protocol we present here can also be used for the more general problem of data propagation in sensor networks (see [15]). Our protocol can be applied to any network topology. However, we so far managed to analyze its performance in exactly lattice shaped networks. In fact, the lattice shape assumption allows as to make a concrete first step towards a rigorous analysis of the protocol. We believe that is important, since such analyses are missing in the related literature, where protocols are validated (both their performance and correctness) by using simulations. Moreover, this analysis extends to “approximate” lattice-shaped networks of sufficient density (i.e. networks satisfying event F , presented in the start of Sec. 5). We furthermore note that even exactly lattice-shaped networks are useful in practice in certain applications, i.e. in a pre-deployed sensor network, where sensors are put (possibly by a human or a robot) in a way that they form a 2-dimensional lattice. Note indeed that such sensor networks, deployed in a structured way, might be useful in precise agriculture applications, where humans or robots may want to deploy the sensors in a lattice structure to monitor in a rather homogenous and uniform way certain conditions in the spatial area of interest. Certainly, exact terrain monitoring in military applications may also need some sort of a grid-like shaped sensor network. Note also that Akyildiz et al in a very recent (2002) state of the art survey published in the Journal of Computer Networks ([1]) do not exclude the pre-deployment possibility. Also Intanagonwiwat, Govindan and Estrin, a group of pioneering researchers in wireless sensor networks (from the technological point of view, which by its flavor pays special attention to modelling assumption) in a recent paper appeared in ACM MOBICOM 2002 ([15]) explicitly refer to the lattice case.
1.2
Our Protocol
For the above problem we propose a new protocol which tries to minimize energy consumption by probabilistically favoring certain paths of local data transmissions towards the sink. Thus we call this protocol PFR (Probabilistic Forwarding Protocol). The basic idea behind this protocol is to avoid flooding by favoring (in a probabilistic manner) data propagation along sensors which lie “close” to the (optimal) transmission line, ES, that connects the sensor node detecting the event, E, and the sink, S (see Fig. 1). This is implemented by locally calculating the dS), whose corner point P is the sensor currently running the local protocol, having received angle φ = (EP a transmission from a nearby sensor, previously possessing the event information. If φ is equal or greater to a predetermined threshold, then p will transmit (and thus propagate the event information further). Else, it decides whether to transmit with probability equal to πφ (see Fig. 2). The way to estimate φ is explained in Section 4.1. Because of the probabilistic nature of data propagation decisions and in order to prevent the data propagation process from early failing, we initially use (for a short time period which we evaluate) a flooding mechanism that leads to a sufficiently large “front” of sensors possessing the data under propagation. When such a “front” is created, we perform probabilistic forwarding. The protocol, as shown by a geometric analysis, always propagates data to the sink, under ideal network conditions (no failures), thus it is provably correct. Using properties of stochastic processes, we show that the protocol is very energy efficient. Also, when part of the network is inoperative (which is more realistic, because sensors are prone to faults), the protocol manages to propagate data very close to the sink, thus it is robust. The above analytical results are validated by a large scale simulation we have carried out after implementing the protocol in Sec. 6. Essentially, our protocol captures the intuitive, deterministic idea “if my distance from ES is small, then send, else do not send”. We have chosen to enhance this idea by random decisions (above a threshold) to allow some local flooding to happen with small probability and thus to cope with local sensor failures.
E p2
E
p1 1
Particles that particiapate in forwarding path
S
Figure 1: Thin Zone of particles
1.3
2
S
Figure 2: Angle φ and closeness to optimal line
Related Work
Our protocol is inspired by the relevant work of [15], where “directed diffusion”, an approach to attributebased data communication for wireless sensor networks is proposed. The goal of directed diffusion is to establish communication between sources and sinks. Directed diffusion consists of several elements. Data is named using attribute-valued pairs. A sensing task is disseminated throughout the sensor network as an interest for named data. This dissemination sets up gradients within the network designed to “draw” events (i.e. data matching the interest). Events start flowing towards the originators of interests along multiple paths. The sensor network reinforces one, or a small number of these paths. Our protocol is inspired by the Directed diffusion communication paradigm in [15] where it is mentioned the possibility of “multi-path delivery with probabilistic forwarding”. However, no such protocol is designed in [15]. Furthermore, we here propose and analyze a particular probabilistic multi-path delivery. A proper modification of our protocol may be used in the gradient setup and dissemination phase of Directed diffusion. A different approach for propagating information to the sink is to use existing routing techniques for mobile ad-hoc routing protocols ([22]) in sensor networks. However, although protocols for mobile ad-hoc networks take into consideration energy conservation issues, most of them are not really suitable for sensor networks. In [17] a routing protocol is presented that is suitable for sensor networks that makes greedy forwarding decisions using only information about a node’s immediate neighbors in the network topology. This approach achieves high scalability as the density of the network increases. In [14] a clustering-based protocol is given that utilizes randomized rotation of local cluster heads to evenly distribute the energy load among the sensors in the network. In [19] a new energy efficient routing protocol is introduced that does not provide periodic data monitoring (as in [14]), but instead nodes transmit data only when sudden and drastic changes are sensed by the nodes. As such, this protocol is well suited for time critical applications and compared to [14] achieves less energy consumption and response time. Furthermore, this work is related to previous research of [8, 9], where new local detection and propagation protocols are proposed, that are very energy and time efficient, as shown by a rigorous average case analysis performed in these works under certain simplifying assumptions. In particular, the Local Target Protocol LTP protocol uses local optimization to select the next-hop neighbor, while the Sleep-Awake Protocol SWP protocol introduces alternation between sleep-awake modes to save energy. Also [2] uses variable transmission range to better perform in cases of low density of particles, obstacle presence and to avoid overusing critical (close to the sink) sensors. In [21] we introduce sleep-awake mechanisms in the PFR protocol and compare it to hierarchical clustering approaches. In [10], random geometric graphs are used to analyze performance and fault-tolerance properties of sensor networks. For a detailed discussion on data propagation protocols in wireless sensor networks see also [4]. For a comparative study of the multi-path delivery PFR protocol to single path delivery (the Local Target Protocol, LTP), see [5, 6, 9]. For other important aspects of energy efficiency (such as energy balance) see [3, 12]. For a discussion of challenges in wireless sensor networks research see [24].
2 The Model Sensor networks are comprised of a vast number of ultra-small homogenous sensors, which we call “grain” particles. Each grain particle is a fully-autonomous computing and communication device, characterized mainly by its available power supply (battery) and the energy cost of computation and transmission of data. Such particles (in our model here) cannot move. Each particle is equipped with a set of monitors (sensors) for light, pressure, humidity, temperature etc. and has a broadcast (digital radio) beacon mode. As a starting point and also motivated by certain applications (precise agriculture, military applications, see also Sec. 1), we adopt here a two-dimensional (plane) lattice deployment of grain particles (see Fig. 5). We assume an n × n lattice deployment of n2 grain particles, each one of maximum transmission range R, put in (horizontal and vertical) distances δ apart. Note√that δ depends on R and also δ characterizes the density of the network. In particular we choose R = δ 2. In the sequel we use δ = 1 for simplicity. Thus each particle (excluding the particles on the boundaries of the lattice) is surrounded by 8 immediate neighbors that lie within its transmission range. There is a single point in the network area, which we call the sink S, and represents a control center where data should be propagated to. Note that, although in the basic case we assume the sink to be static, in a variation it may be allowed to move around its initial base position, to possibly get data that failed to reach it but made it close enough to it. We assume that each grain particle has the following abilities: 1. It can estimate the direction of a received transmission (e.g. via the technology of directionsensing antennae). 2. It can estimate the distance from a nearby particle that did the transmission (e.g. via estimation of the attenuation of the received signal). 3. It knows the direction towards the sink S (i.e. not the exact coordinates). This can be implemented during a set-up phase, where the (very powerful in energy) sink broadcasts the information about itself to all particles. 4. All particles have a common co-ordinates system. The above assumptions suggest a strong model, that however does not trivialize the problem; we believe that even assuming such a model, the design of our protocol is still a challenging task. Furthermore some of our assumptions are realistic (e.g. it is indeed possible to apply smart antennas in tiny sensors, e.g. see [11]) or may become realistic in the near future. Furthermore, some modelling assumptions can be relaxed by alternatives, like introducing a preprocessing phase where sensors use localization techniques by executing an underlaying protocol f to decide on fictitious virtual coordinates [23] (i.e. relax assumption 4) and thus be able to estimate directions (i.e. relax assumption 1) and distances of transmissions (i.e. relax assumption 2). In any case, our assumptions above are simplifying ones, allowing us to simplify the analysis of the protocol properties. Notice that GPS information is not needed for our protocol. Also, there is no need to know the global structure of the network (i.e. the size of the network, the positions of other particles etc.).
3 The Problem Assume that a single particle, P , senses the realization of a local crucial event E. Then the propagation problem P is the following: “How can particle P , via cooperation with the rest of the grain particles, efficiently propagate information inf o(E), reporting realization of event E, to the sink S?” To minimize the energy consumption in the sensor network we wish to avoid flooding and to minimize the number of hops (directed transmissions) performed in the data propagation process, while still managing to reach the sink.
Definition 1 Let E be the position of P which senses event E. Let S be the position of the sink S. Any protocol Π solving the propagation problem P must satisfy: • Correctness. Π must guarantee that data arrives to the position S, given that the whole network exists and is operational. • Robustness. Π must guarantee that data arrives at enough points in a small interval around S, in cases where part of the network has become inoperative, without however isolating the sink. • Efficiency. If Π activates k particles during its operation then Π should have a small ratio of the number of activated over the total number of particles r = Nk . Thus r is an energy efficiency measure of Π.
4 The Probabilistic Forwarding Protocol (PFR) As already mentioned in Section 1, the basic idea of the protocol lies in probabilistically favoring transmissions towards the sink within a thin zone of particles around the line connecting the particle sensing the event E and the sink (see Fig. 1). Note that transmission along this line is energy optimal. However it is not always possible to achieve this optimality, basically because certain sensors on this direct line might be inactive, either permanently (because their energy has been exhausted) or temporarily (because these sensors might enter a sleeping mode to save energy). Further reasons include (a) physical damage of sensors, (b) deliberate removal of some of them (possibly by an adversary in military applications), (c) changes in the position of the sensors due to a variety of reasons (weather conditions, human interaction etc). and (d) physical obstacles blocking communication. The protocol evolves in two phases: Phase 1: The “Front” Creation Phase. Initially we build (by using a limited, in terms of rounds, flooding) a sufficiently large “front” of particles, in order to guarantee the survivability of the data propagation process. During this phase, each particle having received the data to be propagated, deterministically forwards them towards the sink. In particular, and for a sufficiently √ large number of steps s = 180 2, each particle broadcasts the information to all its neighbors, towards the sink. The particular selection of s is explained further in Sec. 5.1 where we analyze the correctness of the protocol. Remark that to implement this phase, and in particular to count √ the number of steps, we use a counter in each message. This counter needs at most dlog 180 2e bits. Before a particle performing a broadcast, it reduces the counter of the message by one. As long as the counter in a message is non-zero particles execute Phase 1; when this counter becomes zero, particles receiving the message switch to Phase 2. Phase 2: The Probabilistic Forwarding Phase. During this phase, each particle P possessing the information under propagation, calculates an angle φ by calling the sub-protocol “φcalculation” (see description below) and broadcasts inf o(E) to all its neighbors with probability IPf wd (or it does not propagate any data with probability 1 − IPf wd ) defined as follows: IPf wd =
(
1 if φ ≥ φthreshold φ otherwise π
where φ is the angle defined by the line EP and the line P S and φ threshold = 134o (the selection reasons of this φthreshold will become more evident in Section 5.1). In both phases, if a particle has already broadcast inf o(E) and receives it again, it ignores it. Also the PFR protocol is presented for a single event tracing. Thus no multiple paths arise and packet sizes do not increase with time. However, PFR can be applied to multiple events as well by assuming an appropriate resolution layer (or MAC) and a coding scheme that uniquely identifies events. Also in the case where a single event is sensed by more than one sensors, one may apply data aggregation techniques during Phase 1 to avoid unnecessary repetitions of messages in Phase 1.
Remark that when φ = π then P lies on the line ES and vice-versa (and always transmits). If the density of particles is appropriately large (see Sec. 5, 6), then for a line ES there is (with high probability) a sequence of points “closely surrounding ES” whose angles φ are larger than φ threshold and so that successive points are within transmission range. All such points broadcast and thus essentially they follow the line ES (see Fig. 2). P
Pprev
S
E
Figure 3: Angle φ calculation example
4.1
The φ-calculation sub-protocol
Let Pprev the particle that transmitted inf o(E) to P (see Fig. 3). −−−−→ 1. When Pprev broadcasts inf o(E), it also attaches the info |EPprev | and the direction Pprev E. 2. P estimates the direction and length of line segment Pprev P , as described in the model. −−→ 3. P now computes angle (EPd prev P ), and computes |EP | and the direction of P E (this will be used in further transmission from P ). dP E) and by subtracting it from (Pprev dP S) it finds φ. 4. P also computes angle (Pprev
Notice the following:
(i) The direction and distance from activated sensors to E is inductively propagated (i.e. P becomes P prev in the next phase). (ii) Our protocol needs only messages of length bounded by log A, where A is some measure of the size of the network area, since (because of (i) above) there is no cumulative effect on message lengths. Note that the number of steps in the forwarding phase of the protocol depends on the φ threshold of the protocol as it can be √ seen from the analysis in Section 5. For φ threshold = 134o the number of flooding steps must be at least 180 2 for correctness reasons. We can increase the φthreshold ; this will increase also the number of flooding steps. This also implies a tradeoff between energy efficiency and robustness.
5 Properties of PFR Consider a partition of the network area into small squares of a fictitious grid G (see Fig. 4). Let the length of the side of each square be l. Let the number of squares be q. The area covered is bounded by ql 2 . Assuming that we randomly throw in the area at least αq log q = N particles (where α > 0 a suitable constant), then the probability that a particular square is avoided is µ
1 1− q
¶αq log q
≤ e−α log q = q −α
So the probability that all squares get particles is at least µ
1 − q · q −α = 1 − q −(α−1) = 1 − Θ
µ
N log N
¶¶−(α−1)
We condition all the analysis on this event, call it F , of at least one particle in each square.
E
Particles
E Sensor Particles
Lattice Dissection
Sink
S
Figure 4: A Lattice Dissection G
5.1
Figure 5: A Lattice Sensor Network
The Correctness of PFR
Without loss of generality, we assume each square of the fictitious lattice G to have side length 1. Lemma 1 PFR succeeds with probability 1 in sending the information from E to S given the event F . √ Proof: In the (trivial) case where |ES| ≤ 180 2, the protocol is clearly correct due to front creation phase. Let Σ a unit square of G intersecting ES in some way (see Fig. 6). Since a particle always exists somewhere in Σ, we will only need to examine the worst case of it being in one of the corners of Σ. Consider vertex A. The line EA is always to the left of AB (since E is at end of ES). The same is true for AS (S is to the right of B 0 ). y E
A A’ y’
1 2
1
D B
1
B’
S
Figure 6: The square Σ (A0 AB 0 B) Let AD the segment from A perpendicular to ES (and D its intersection point) and let yy 0 be the line from A parallel to ES. Then, d = 180o − (yAE) d − (yd 0 AS) φ = (EAS) d + 90o − (yd 0 AS) = 90o − (yAE)
d 0 ) − (yd d − (yAE) d + (DAy 0 AS) = (yAD)
d 0 ) − (yd d − (yAE) d and ω 0 AS) and, without loss of generality, let ED < DS. c1 = (yAD) c2 = (DAy Let ω c1 > 45o , since it includes half of the 90o -angle of A in the unit square. Also Then always ω d 0 AS) = sin (ASD) sin (yd =
but AD ≤
√
AD AD < AS DS
2 (= AB) and DS ≥
0 AS) ≤ =⇒ sin (yd
√ ES ≥ 90 2 2
1 0 AS) < 1o ⇐⇒ (yd 90
(Note that here we use the elementary fact that when x < 90 o then sinx < d 0 ) = 90o by construction. Thus c2 > 89o since (DAy ω
1 90
⇐⇒ x < 1o ). Then
c1 + ω c2 > 45o + 89o = 134o φ = ω
There are two other ways to place an intersecting to ES unit square Σ, whose analysis is similar. But (a) the initial square (from E) always broadcasts due to the first protocol phase and (b) any intermediate intersecting square will be notified (by induction) and thus will broadcast. Hence, S is always notified, when the whole grid is operational.
5.2
The Energy Efficiency of PFR
Consider the fictitious lattice G of the network area and let the event F hold. We have (at least) one particle inside each square. Now join all “nearby” particles (see fig. 4) of each particle to it, thus by forming a new graph G0 which is “lattice-shaped” but its elementary “boxes” may not be orthogonal and may have varied length. When G0 s squares become smaller and smaller, then G0 will look like G. Thus, for reasons of analytic tractability, we will assume in the sequel that our particles form a lattice (see Fig. 5). We also assume length l = 1 in each square, for normalization purposes. Notice however that when l → 0 then “G 0 → G” and thus all our results in this Section hold for any random deployment “in the limit”. LQ Q
E
S n0
n
Figure 7: The LQ Area
Theorem 1 The energy efficiency of the PFR protocol is Θ n0 = |ES| = o(n), this is o(1).
µ³
n0 n
´2 ¶
where n0 = |ES| and n =
√ N . For
Proof: The analysis of the energy efficiency considers particles that are active but are as far as possible from ES. Thus the approximation we do are suitable for remote particles. Here we estimate an upper bound on the number of particles in an n × n (i.e. N = n × n) lattice. If k is this number then r = nk2 (0 < r ≤ 1) is the “energy efficiency ratio” of PFR. We want r to be less than 1 and as small as possible (clearly, r = 1 leads to flooding). Since, by Lemma 1, ES is always “surrounded” by active particles, we will now assume without loss of generality that ES is part of a horizontal grid line (see Fig. 7) somewhere in the middle of the lattice. Recall that |ES| = n0 particles of all the active, via PFR, particles that continue to transmit, and let Q be a set of points whose shortest distance from particles in ES is maximum. Then L Q is the locus (curve) of such points Q. The number of particles included in LQ (i.e. the area inside LQ ) is the number k. Now, if the distance of such points Q of LQ from ES is ω then k ≤ 2(n0 + 2ω)ω (see Fig. 8) and thus r ≤ 2ω(nn02+2ω) . Notice however that ω is a random variable hence we will estimate its expected value IE(ω) and the moment IE(ω 2 ) to get 2n0 4IE(ω 2 ) + 2 IE(ω) (1) IE(r) ≤ 2 n n d = φ < 30o , i.e. φ < π . We want to use the approximation |ED| ' x, Now, look at points Q : (EQS) 6 where x is a random variable depending on the particle distribution.
n0 E
S
Figure 8: The particles inside the LQ Area
Let
|ED| x
= 1 +³². Then a bound on the approximation factor ² determines a bound on the “cut-off” angle ´
φ0 since |ED| cos
φo 2
= ²x.
µ
φo (1 + ²) x cos 2
¶
µ
φo = x =⇒ cos 2
¶
1 1+²
=
o φ0 above is constant that can be appropriately chosen by the protocol implementor. We ³ ´ chose φ 0 = 30 so ξ ² = 0.035. Also, we remark here that the energy spent increases with x 0 = n0 1 − 2 in the area below x0 , where ξ = tan 75o , but decreases with x for x > x0 . This is a trade-off and one can carefully estimate the angle φ0 (i.e. ²) to minimize the energy spent. y x
Q
xo x
1
D
E
2
E
xo
0
S
x
S
y’
Figure 9: The QES Triangle
Figure 10: The Random Walk W
d d where QD ⊥ ES and D in ES. In Fig. 9, φ = φ2 − φ1 . Let φ1 = (EQD), φ2 = (DQS) We approximate φ by sin φ2 − sin φ1 (note: φ ' sin φ ≥ − sin φ1 + sin φ2 ).
Note sin φ1 '
|ED| |ED| ' |QE| x
and
=⇒ sin φ ' sin φ2 − sin φ1 '
sin φ2 '
|DS| x
n0 |ES| = x x
n0 d < π and thus for such points Q. We note that the analysis is very similar when ( QES) Thus πφ ' πx 2 φ = φ 1 + φ2 . Let M be the stochastic process that represents the vertical (to ES) distance of active points from ES, and W be the random walk on the vertical line yy 0 (see fig. 10) such that, when the walk is at distance x ≥ x0 n0 n0 from ES then it (a) goes to x + 1 with probability πx or (b) goes to x − 1 with probability 1 − πx and never n ξ o 0 goes below x0 , where x0 is the “30 -distance” i.e. x = n2 . o o 0 n
Clearly W dominates M i.e. IPM x ≥ x1 ≤ IPW x ≥ x1 , ∀ x1 > x0 . Furthermore, W is dominated by the continuous time “discouraged arrivals” birth-death process W 0 (for x ≥ x0 ) where the rate of going from x to x + 1 in W 0 is αx = n0x/π and the rate of returning to x − 1 is n0 2 = 1 − πξ = β. 1 − πx 0
We know from [18] that for ∆x = x − x0 (x > x0 )
n0 α = ³ β π 1−
2 πξ
IE(ω) ≤ IEM (x) ≤ x0 +
α β
IEW 0 (∆x) = Thus, IEW 0 (x) = x0 + αβ , hence by domination
n
Also from [18] the process W 0 is a Poisson one and IP ∆x = k of ∆x is σ 2 = α/β (again) i.e. for ω = x0 + ∆x ³
IEW 0 (ω 2 ) = IEW 0 (x0 + ∆x)2
o
=
´
(2) (α/β)k k!
e−(α/β) . From this, the variance
´
= x20 + 2x0 IEW 0 (∆x) + IEW 0 (∆x2 ) ³ ´ α = x20 + 2 x0 + σ 2 + IE2 (∆x) β µ ¶2 α α α 2 + = x 0 + 2 x0 + β β β 3n20 α n0 ξ α IEW 0 (ω ) = + 2 + + 4 β 2 β α n0 n ´ = 0 where = ³ β τ π 1 − π2ξ 2
So
µ
2 τ = π 1− πξ
and
thus
IE
W0
2
(ω ) =
n20
µ
¶
= π−
2 ξ
3 ξ 1 + + 2 4 τ τ
¶
+
µ ¶2
α β
n0 τ
(3)
and by domination IE(ω 2 ) ≤ IEM (ω 2 ) ≤ IEW 0 (ω 2 ). So, finally 4 IE(r) ≤
·
³
n20 43
+
ξ τ
+
1 τ2
n2
´
+
n0 τ
which proves the theorem.
5.3
¸
+
n20 ³ 2´ ξ+ 2 n τ
(4)
The Robustness of PFR
We now consider the robustness of our protocol, in the sense that PFR must guarantee that data arrives at enough points in a small interval around S, in cases where part of the network has become inoperative, without however isolating the sink. Q’
y
y’ 2
1
x
E
D
Figure 11: The Q0 ES Triangle
S
Lemma 2 PFR manages to propagate the crucial data across lines parallel to ES, and of constant distance, with fixed nonzero probability (not depending on n, |ES|). Proof: Here we consider particles very near the line ES. Thus the approximations that we do are suitable for nearby particles. Let Q0 an active particle at vertical distance x from ES and D ∈ ES such that Q0 D ⊥ ES. d d0 y 0 ). Then φ = 180o −(φ1 +φ2 ), 0 E), φ = (SQ Let yy 0 k ES, drawn from Q0 (see fig. 11) and φ1 = (yQ 2 2 i.e. πφ = 1 − φ1 +φ π . Since φ1 , φ2 are small (for small x) we use the approximation φ1 ' sin φ1 and φ2 ' sin φ2 φ1 + φ2 ' sin φ1 + sin φ2 =
x x + 0 0 EQ QS
d0 S), thus EQ0 ≤ n0 and Q0 S ≤ n0 . Hence, Since φ > 900 , ES is the biggest edge of triangle (EQ
sin φ1 + sin φ2 ≤
and
1−
2x n0
2x sin φ1 + sin φ2 ≥ 1− π πn0
(for small x)
Without loss of generality, assume ES is part of a horizontal grid line. Here we study the case in which some of the particles on ES or very near ES (i.e. at angles > 134 o ) are not operating. Consider now a horizontal line ` in the grid, at distance x from ES. Clearly, some particles of ` near E will be activated during the initial broadcasting phase (see Phase 1). Let A be the event that all the particles of ` (starting from p) will be activated, until a vertical line from S is reached. Then à ! n0 2x 2x IP(A) ≥ 1 − ' e− π πn0
6 Simulation Results We validate the theoretical results and investigate the asymptotic behavior of the PFR protocol by conducting a set of large scale simulations. Our implementation follows closely the protocol description of Section 4 and is based on C++ and the Library of Efficient Data types and Algorithms (LEDA) [20]. We start by examining the energy efficiency of PFR by measuring the ratio r of the number of activated particles over the total number of particles r = nk2 . We used lattice shaped sensor fields comprised of extremely large number of sensors (n ∈ [1000, 8000]) that √ are put in (horizontal and vertical) distances δ = 5m apart and each particle’s broadcast range R = δ 2. In each case, we repeated the simulations for at least 100 times to get good average results (as indicated by the smoothness of curves). In the first set of simulations, we drop a fixed number of particles (n = 8000) and then position E and S in a way such that ||ES|| ∈ [1000R, 4000R]. Figure 12 depicts the ratio r for various ||ES||. In the second set of simulations we work in a different way: we drop sensor fields comprised of different number of total particles (n ∈ 1000, 4000) and then position E and S so that their distance is always fixed to ||ES|| = 1000R. Fig. 13 depicts the ratio r for different n. ³
´2
In both figures the dashed line depicts the theoretical result of Theorem 1, i.e. an upper bound r 0 = nn0 on the ratio of activated particles, where ||ES|| = n0 . The above figures indeed validate our analytical results, since in each case the efficiency ratio r exhibits a quadratic behavior with respect to ||ES|| (in the first case) and n (in the second). Furthermore, notice that our analysis is tight enough, since the curve of the simulation measurements is very close to the curve of the analytic result (i.e the upper bound is pretty close to the measured ratio). We then investigate the robustness of our protocol, in the case where some particles (permanently) fail. In particular, the failure probability is taken “large” for particles close to ES line (i.e. f c = {0.4, 0.45, 0.5}). The particles that are located further away from ES line, fail with smaller failure probability (i.e. f r ∈
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Figure 12: Energy efficiency measure r for different ||ES|| when n = 8000.
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Figure 14: Robustness for ||ES|| = 1000R and n = 3000 [0.01, 0.7]). The different failure probabilities for “close” and “remote” (to ES line) particles captures the fact that the particles tend to consume high (or low) energy in each case respectively, due to the fact that “close” to ES particles transmit more frequently than “remote” ones. To evaluate the robustness of the protocol we measure in the case of failure to reach the sink, the distance D from it. Figure 14 shows that our protocol is very robust. In particular, even for large failure probabilities as high as 0.5, the protocol successfully propagates data to the sink. As the failure probability increases (i.e. f r > 0.4) the protocol manages to get close enough to the sink (i.e. in the worst case the final position of the propagation is 1400m, while ||ES|| = 7000m). We note that proximity of final propagation position to the sink also increases with failure probability of “close” particles (f c ). Note that the proximity of the final position to the sink seems to exhibit a certain threshold behavior (in the case of values studied, this threshold is around fr = 0.5). This threshold behavior is probably due to the stochastic process evolution and we intend to also evaluate it analytically. A possible intuitive explanation of this behavior is that when particles lying further away from the optimal transmissions line are more or less operational (i.e. fr < 0.5) then, regardless of the failure of particles on the optimal line, information messages rearch close to the sink (i.e. the distance D is small); when however the failure probability of particles (fr ) becomes larger, then it becomes impossible to get the messages to the
sink since no sufficient number of particles to propagate data are operational (and thus distance D becomes very large).
7 Conclusions and Future Work We presented here a new protocol for information propagation in sensor networks. This protocol can be used either directly to solve a local crucial event detection and propagation problem or as part of a more general information dissemination paradigm (such as in [15]). PFR is based only on local information (and thus it is necessary a probabilistic one) and succeeds to efficiently propagate information to the sink without flooding the network (although each transmitting particle broadcasts around it), by probabilistically favoring certain close to optimal “paths” towards the sink and also by avoiding any control messages. Our analysis shows that the protocol is very efficient, in terms of energy consumption, while achieving high success rates even in the case where part of the network fails. We have implemented our protocol and conducted extensive simulations on networks of large size to validate the analysis and also investigate other important performance measures, such as proximity of final position to the sink when a significant part of the network becomes inoperative. We plan to extend our simulation results by studying the performance of our protocol in more general topologies and for additional network parameters. We believe that the protocol indeed would work in random topologies with appropriate modifications, such as a different threshold angle in the probabilistic choice taking into account the network density. Also, to comparatively evaluate PFR against other protocols in the recent literature.
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