A Process Algebraic Framework for Estimating the Energy Consumption in Ad-hoc Wireless Sensor Networks

Transcription

1 A Process Agebraic Framework for Estimating the Energy Consumption in Ad-hoc Wireess Sensor Networks L. Gaina, A. Marin, S. Rossi Università Ca Foscari Venezia, Itay T. Han, M. Kwiatkowska University of Oxford, United Kingdom ABSTRACT We present a framework for modeing ad-hoc Wireess Sensor Networks (WSNs) and studying both their connectivity properties and their performances in terms of energy consumption, throughput and other reevant indices. Our framework is based on a probabiistic process cacuus where system executions are driven by Markovian probabiistic scheduers, aowing us to transate process terms into discrete time Markov chains (DTMCs) and use the probabiistic mode checker PRISM to automaticay evauate/estimate the connectivity properties and the energy costs of the networks. To the best of our knowedge, this is the first work that proposes a unique framework for studying quaitative (e.g., by proving the equivaence of components or the correctness of a behaviour) and quantitative aspects of WSNs using a too that aows both exact and approximate (via Monte Caro simuation) anayses. We demonstrate our framework at work by considering different communication strategies based on gossip routing protocos, for a typica topoogy and a mobiity scenario. Categories and Subject Descriptors C.4 [Computer Systems Organization]: Performance of systems modeing techniques; C.2.1 [Computer-Communication Networks]: Network architecture and design Wireess Communication Genera Terms Theory, reiabiity Keywords Process agebra, simuation, sensor networks Work partiay supported by the MIUR Project CINA Compositionaity, Interaction, Negoziation, Autonomicity for the future ICT society. Permission to make digita or hard copies of a or part of this work for persona or cassroom use is granted without fee provided that copies are not made or distributed for profit or commercia advantage and that copies bear this notice and the fu citation on the first page. To copy otherwise, to repubish, to post on servers or to redistribute to ists, requires prior specific permission and/or a fee. Copyright 2013 ACM...$ INTRODUCTION Wireess Sensor Networks (WSNs) [2] are coections of spatiay distributed sensing devices equipped with imited computing and radio communication capabiities. They are empoyed in a variety of appications, ranging from miitary surveiance to heath care or assisted iving, and from smart cities to environmenta monitoring. A typica sensor network is characterized by a arge number of sensor nodes, which are densey depoyed, and have frequent topoogy changes due to the mobiity of its devices. Nodes communicate using wireess transmission in a specified range, with communication between nodes impemented in terms of routing protocos. A critica issue in wireess sensor networks is the imited avaiabiity of energy within the network devices. Therefore, judicious choice of routing protocos that can reduce the nodes power consumption is crucia, not ony for the performance of each singe node, but aso for the throughout of the network ifetime. In this paper we introduce a framework for the specification, modeing and automated anaysis or simuation of connectivity properties and the evauation or estimation of energy consumption in ad-hoc WSNs. The framework is based on a variant of the Probabiistic Energy-aware Broadcast Unicast and Muticast (PEBUM) cacuus introduced in [5], and supports the automatic performance evauation through (approximate) probabiistic mode checking, e.g., using the PRISM mode checker [9]. The advantages of using a process agebra rey on its compositiona nature which aows us to decompose both the mode construction and the quaitative anaysis. Concerning the quantitative anaysis we take advantage of the various kind of properties which can be specified in terms of a tempora ogic and verified using the mode checking technique. The cacuus we propose is buit around nodes, representing the sensor devices of the systems, and ocations, identifying the position ces across which each device may move inside the network. Node mobiity is governed by probabiity distributions. Instead, wireess synchronisations are controed by sequentia processes inside the nodes: each transmission broadcasts a message within a given transmission range. The semantics of our cacuus is inspired by Segaa s probabiistic automata [15] driven by scheduers to resove the nondeterministic choice among the probabiity distributions over target states. Differenty from [5], in this work we assume that nodes are not equipped with a unique identifier and they a share the same transmission frequency. These choices refect the fact

2 Networks Processes M,N ::= 0 Empty network P,Q ::= 0 Inactive process [P ] J Sensor Node ( x).p Input M 1 M 2 Parae composition w L,r.P Output [w 1 = w 2]P, Q Matching A w Recursion Tabe 1: Syntax that transmissions in ad-hoc sensor networks are directed to a geographica ocation rather than to a specific node and, due to the ow-cost hardware of sensors, ony one frequency is used at a given time [17]. Moreover, in contrast to [6, 5], in this paper we empoy Makovian probabiistic scheduers, mapping the non-deterministic choices among the different actions a system may enabe into probabiity distributions. As a consequence, the abeed transition system underying process terms is a discrete time Markov chain (DTMC), which can be used for automaticay performing a range of quaitative and quantitative anayses by means of probabiistic mode checking, e.g., using PRISM. We define a probabiistic observationa congruence in the stye of [13] to equate networks exhibiting the same connectivity behaviour. As in [5], and in contrast to [12], the notion of observabiity is associated with specific ocations in the network refecting the fact that in ad-hoc WSNs the transmissions are not addressed to specific nodes but to specific ocations. We provide a coinductive characterization of the observationa congruence based on a probabiistic abeed bisimiarity. Finay, we define an energy-aware preorder over networks, to contrast networks having the same behaviour, but different energy costs. We use our framework for a comparative study of gossip based routing protocos for wireess ad-hoc sensor networks. We address the probem of the state space exposion using the Monte Caro simuation impemented in PRISM in terms of statistica mode checking. Specificay, we consider different scenarios obtained by varying both the protoco parameters and the network power strategies, in order to find the best soution, which reduces the power consumption whie maintaining the same connectivity. The paper is organised as foows. Section 2 presents the cacuus, its observationa semantics expressed in terms of behavioura equivaences and a characterization of it based on a notion of probabiistic bisimiarity. In Section 3 an energy-aware preorder over networks is defined: it aows us to compare the average energy cost of different networks but exhibiting the same connectivity behaviour. Finay, in Section 4, gossip based routing protocos for different scenarios are considered and we show the framework at work studying the sensitivity of the performances of these protocos to some configuration parameters. Finay, Section 5 discusses the reated bibiography and concudes the paper. 2. A CALCULUS FOR WSN We present a variant of the PEBUM cacuus presented in [5] which focuses on the main features of ad-hoc wireess sensor networks. Specificay, nodes are not equipped with a unique identifier and ony one transmission frequency is used. Syntax. We use etters for ocations, r for transmission radii, x and y for variabes. Cosed vaues contain ocations, transmission radii and any basic vaue (booeans, integers,...). Vaues incude aso variabes. We use u and v for cosed vaues and w for (open) vaues. We write ṽ, w for tupes of vaues. We write N for the set of a networks and Loc for the set of a ocations. Whie movements may be assumed to be continuous, we identify ocations as the countabe set of ces that constitute the observing areas within the network. The syntax of our cacuus is shown in Tabe 1. This is defined in a two-eve structure: the ower one for processes, the upper one for networks. Networks are coections of sensor nodes running in parae and communicating messages. As usua, 0 denotes the empty network and M 1 M 2 denotes the parae composition of two networks. We denote by i IMi the parae composition of the networks M i, for i I. [P ] J denotes a sensor node ocated at the physica ocation and executing the process P. J is the transition matrix of a discrete time Markov chain modeing node mobiity: each entry J k is the probabiity that the sensor node ocated at moves to the ocation k. Hence, k Loc J k = 1 for a ocations Loc. Static nodes inside a network are associated with the identity Markov chain, i.e., the identity matrix J = 1 for a Loc and J k = 0 for a k. Processes are sequentia and ive within the nodes: 0 is the inactive process; ( x).p is ready to isten to a transmission, whie w L,r.P is ready to transmit. In ( x).p, the variabes in x are bound with scope in P. As to the output form, the tag r represents the transmission radius of the sender, whie the tag L is used to maintain the set of physica ocations of the intended recipients: L = Loc represents a broadcast transmission, whie a finite set of ocations L denotes a muticast communication (unicast if L is a singeton). As stated in the introduction, communication protocos for ad-hoc sensor networks are usuay intended to reach a certain ocation, rather than a specific device, due to the absence of goba identifiers associated with the sensor devices. The remaining syntactic forms are standard: [w 1 = w 2]P, Q behaves as P if w 1 = w 2, and as Q otherwise. A w is the process defined via a (possiby recursive) definition A( x) def = P, with x = w where x contains a variabes appearing free in P. Probabiity distributions for networks. We denote by µ J the probabiity distribution associated with a node ocated at with transition matrix J, i.e., the function over Loc such that µ J (k) = J k for a k Loc. We wi mode the probabiistic evoution of the network according to these distributions.

3 (R-Bcast) [ ṽ L,r.P ] J i I [( xi).pi]ji i [P ] J i I [Pi{ṽ/ xi}]ji i where i I.d(, i) r and x i = ṽ (R-Move) [P ] J [P ] J µ J (R-Par) M M θ (R-Struct) N M M M θ M N M N M N θ N N θ Tabe 2: Reduction Semantics Let M be a network. We denote by M{[P ] J k/[p ] J } the network obtained by repacing with k inside the sensor node [P ] J. We aso denote by M µ J the probabiity distribution over networks induced by µ J and defined by: for a networks M, µ J M µ J(M (k) if M = M{[P ] J k/[p ] J } ) = 0 otherwise. Intuitivey, M µ J (M ) is the probabiity that the network M evoves to M due to the movement of the sensor node [P ] J. We say that M is in the support of M µ J if M µ J (M ) 0. We write M for the Dirac distribution on the network M, i.e., the probabiity distribution defined as: M (M) = 1 and M (M ) = 0 for a M M. Finay, we et θ range over {µ J J is a transition matrix and Loc} { }. Reduction semantics. The dynamics of the cacuus is specified by the probabiistic reduction reation ( ) described in Tabe 2: it takes the form M M θ denoting a transition that eaves from M and eads to a probabiity distribution M θ. As usua, reduction reies on structura congruence ( ), such that, e.g., M N N M, (M N) M M (N M ) and M 0 M. Nodes cannot be created or destroyed, and move autonomousy. Node connectivity is verified by ooking at the physica ocation and the transmission radius of the sender: a message broadcast by a node is received ony by the nodes that ie in the area deimited by the transmission radius of the sender. We presuppose a function d(, ) which returns the distance between two ocations. Rue (R-Bcast) modes the transmission of a tupe of messages ṽ by a sensor node ocated at and using a radius r. The index set I may be empty, i.e., the rue can be appied even if no nodes are ready to receive. The radius r associated with the output action denotes the transmission radius of that communication which may depend on the energy consumption strategy adopted by the surrounding protoco. A the nodes that ie in the range of the sender (i.e., such that d(, i) r) wi receive the messages. Rue (R-Move) deas with node mobiity: a node [P ] J executing a move action wi reach a ocation with a probabiity described by the distribution µ J that depends on the Markov chain J staticay associated with the node. The remaining rues are standard. Since we are deaing with a probabiistic reduction semantics, which reduces networks into probabiity distributions, we need a way of representing the steps of each probabiistic evoution of a network. Formay, given a network M, we write M θ N if M M θ and N is in the support of M θ. Foowing [6], an execution for M is a (possiby infinite) sequence of steps M θ1 M 1 θ2 M 2... Observationa Semantics. According to a standard practice, we formaise the observationa semantics of our cacuus in terms of a notion of barb that provides the basic unit of observation [13]. As in other cacui for wireess communication, the definition of barb is naturay expressed in terms of message transmission. We denote by behave(m) = { M θ M M θ } the set of the possibe behaviours of M. In order to sove the nondeterminism in a network execution, we consider each possibe probabiistic transition M M θ as arising from a probabiistic scheduer defined as foows. Definition 1 (Scheduer). A probabiistic scheduer is a tota function F assigning to a network M a distribution φ on the set behave(m). We denote by Sched the set of a probabiistic scheduers. Given a network M and a scheduer F, we define the set of a executions starting from M and driven by F as: Exec F M = {e = M 0 p1 θ 1 M 1 p2 θ 2 M 2... M 0 M and j > 0 : M j 1 M j θj, p j = F (M j 1)( M j θj ) and M j is in the support of M j θj }. For a finite execution e = M p1 θ 1 M 1... pk θ k M k Exec F M starting from M and driven by a scheduer F we define P F M (e) = p 1 M 1 θ1 (M 1)... p k M k θk (M k ) where j k, p j = F (M j 1)( M j θj ). We denote by ast(e) the fina state of a finite execution e, by e j the prefix execution M p1 θ 1 M 1... pj θ j M j of ength j of the execution e = M p1 θ 1 M 1 pj θ j M j pj+1 θ j+1 M j+1, and by e the set of ē such that e prefix ē. We write M F M if there exists a finite execution e Exec F M with ast(e) = M. We define the probabiity space on the executions starting from a given network M as foows. Given a scheduer F, σf ied F M is the smaest sigma fied on Exec F M that contains the basic cyinders e, where e Exec F M. The probabiity measure P rob F M is the unique measure on σf ied F M such that P rob F M (e ) = PM F (e). Given a measurabe set of networks H, we denote by Exec F M (H) the set of executions starting from M and crossing a state in H. Formay Exec F M (H) = {e Exec F M ast(e j ) H for some j}. We denote the probabiity for a network M to evove into a network in H, according to the poicy given by F, as P rob F M (H) = P rob F M (Exec F M (H)). Note that the use of probabiistic scheduers aows us to mode networks as discrete time Markov chains (DTMCs). This is the resut of the appication of a two eve probabiity distribution: the reduction semantics maps a network M into a probabiity distribution in the set behave(m) whie, in

4 (Output) ṽ L,r.P ṽl,r P (Input) ( x).p ṽ P {ṽ/ x} (Then) P η P [ṽ = ṽ]p, Q η P (Ese) η P Q η Q ṽ 1 ṽ 2 [ṽ 1 = ṽ 2]P, Q η (Rec) P {ṽ/ x} Q A ṽ η P A( x) def = P Tabe 3: LTS rues for Processes turn, the probabiistic scheduer maps M into a probabiity distribution φ over the probabiity distributions in the set behave(m), giving rise to a fuy probabiistic mode. Exampe 1. Consider the network M [ ṽ L,r.P ] J consisting of a singe sensor node. The set of possibe behaviours of M is { [P ] J, [ ṽ L,r.P ] J µ J}, since the sensor node at the next step can either move or transmit. Then, for each F Sched, e Exec F N such that N N and ast(e) = M we get F (e) = φ such that there exist p 1 and p 2 with p 1 + p 2 1 and for a M N : p 1 if M [P ] J φ(m p 2 q i if M [ L, r v.p ] J k and ) = [ L, r v.p ] J µ J([ L, r v.p ] J k) = q i 0 otherwise. The notion of barb introduced beow denotes an observabe transmission with a certain probabiity according to a fixed scheduer. We first introduce a notion of strong barb: for a network M, we write M K when M [ ṽ L,r.P ] J M with = K L and fora k K, d(, k) r. Roughy, a transmission is observabe ony if at east one ocation in the set of the intended recipients is abe to receive the message. We say that a network M has a barb with probabiity p at the set K of ocations, according to the scheduer F, written M F p K, if P robf M ({M M F M K}) = p. Intuitivey, for a given network M and scheduer F, if M F p K then there is a positive probabiity that M, driven by F, performs a transmission and at east one of the intended recipients is abe to correcty isten to it. In the foowing, we introduce a notion of probabiistic observationa congruence reative to a specific set of scheduers F Sched. Since our semantics is contextua, we need to ensure that the set of scheduers we consider aows the specific networks we anayse to interact with any possibe context. Hence for a set F of scheduers we define the contextua superset F C of F, as the argest set of scheduers aowing networks to interact with any possibe context even when driven by F (see [4] for a forma definition). It hods that Sched C = Sched. Hereafter, a context C[ ] is a term with a hoe defined by the grammar: C[ ] ::= [ ] [ ] M M [ ]. Our probabiistic observationa congruence reative to a specific set of scheduers is defined as foows. Definition 2. Given a set F Sched and a reation R over networks: - R is barb preserving reative to F if MRN and M F p K for some F F C impies that there exists F F C such that N F p K. - R is reduction cosed reative to F if MRN impies that for a F F C there exists F F C such that for a casses C N /R, P rob F M (C) = P rob F N (C). - R is contextua if MRN impies that C[M] R C[N] for every context C[ ]. - Probabiistic observationa congruence reative to F, written = F p, is the argest symmetric reation over networks which is reduction cosed, barb preserving and contextua. Two networks are reated by = F p if they exhibit the same probabiistic (connectivity) behaviour reative to F. In the next section a bisimuation-based proof technique for = F p is deveoped in order to provide an efficient method to check whether two networks are reated by = F p. Deciding the Observationa Congruence. We express the semantics of the cacuus in terms of abeed transition systems (LTS) which are buit upon two sets of rues: one for processes and one for networks. Tabe 3 presents the LTS rues for processes. Transitions are of the form P η P, where η ranges over input and output actions: η ::= ṽ ṽ L,r. Tabe 4 presents the LTS rues for networks. Transitions are of the form M γ M θ, where M is a network and M θ is a distribution over networks. Probabiities are used to mode the mobiity of nodes. Tag γ ranges over the abes: γ ::= L!ṽ[, r]?ṽ@ R!ṽ@K τ. Rue (Snd) modes the sending of tupe ṽ to a specific set L of ocations with transmission radius r, whie rue (Rcv) modes the reception of ṽ at. Rue (Bcast) modes the broadcast message propagation: a the nodes ying within the transmission ce of the sender may receive the message, regardess of the fact that they ie in one of the ocations in L. Rue (Obs) modes the observabiity of a transmission: every transmission may be detected (and hence observed) by any recipient ying in one of the observation ocations within the transmission ce of the sender. The abe R!ṽ@K represents the transmission of the tupe ṽ of messages: the set R is the set of a the ocations receiving the message, whie its subset K contains ony the ocations where the transmission is observed. Rue (Lose) modes message oss. As usua, τ-transitions denote non-observabe actions. Rue (Move) modes node mobiity according to the probabiity distribution µ J. Finay, (Par) is standard. Based on the LTS semantics, we define a probabiistic abeed bisimiarity that is a characterisation of our probabiistic observationa congruence. It is buit upon the actions: α ::=?ṽ@ R!ṽ@K τ. We write behave(m) for the set of a possibe behaviors of M, that is behave(m) = {(α, M θ ) M α M θ }. Labeed executions arise by resoving the non-determinism

5 (Snd) [P ] J P ṽl,r P L!ṽ[,r] [P ] J (Rcv) [P ] J P ṽ P?ṽ@ [P ] J (Bcast) M L!ṽ[,r] M N?ṽ@ N M N L!ṽ[,r] M N d(, ) r (Obs) M L!ṽ[,r] M R { Loc : d(, ) r} K = R L, K M R!ṽ@K M (Lose) M L!ṽ[,r] M M M τ (Move) [P ] J τ [P ] J µ J (Par) M γ M θ M N γ M N θ Tabe 4: LTS rues for Networks of both α and M θ. As a consequence, a scheduer 1 for the abeed semantics is a function F assigning a probabiity to each pair (α, M θ ) behave(m) with a network M. We denote by LSched the set of scheduers for the LTS semantics. A abeed execution e of a network M driven by a scheduer F is a finite (or infinite) sequence of steps: M α 1 α p1 θ 1 M 2 α 1 p2 θ 2 M k 2... pk θ k M k. With abuse of notation, we define Exec F M, ast(e), e j and e as for unabeed executions. Since we are interested in weak observationa equivaences, that abstract over τ-actions, we introduce the notion of weak action as foows: = is the transitive and refexive cosure of ; τ α = denotes = = α ˆα α τ. We write = for the weak action = α if α τ, and = otherwise. We denote by Exec F M ( =, α H) the set of a executions that, starting from M, according to the scheduer F, ead to a network in the set H by performing =. α We define the probabiity of reaching a network in H from M by performing =, α according to a scheduer F as P rob F M ( =, α H) = P rob F M (Exec F M ( =, α H)). For F Sched, we denote by ˆF C LSched its contextua superset for the LTS semantics (see [4]). Definition 3. Let M and N be two networks. An equivaence reation R over networks is a probabiistic abeed bisimuation reative to a set F of scheduers, if MRN impies: for a scheduers F ˆF C there exists a scheduer F ˆF C such that for a α and for a casses C N /R: - if α?ṽ@ then P rob F M (, α C) = P rob F N ( = ˆα C); - if α =?ṽ@ then either P rob F M (, α C) = P rob F N ( =, α C) or P rob F M (, α C) = P rob F N (=, C). Probabiistic abeed bisimiarity reative to F, written F p, is the argest probabiistic abeed bisimuation reative to F over networks. Probabiistic abeed bisimiarity is a characterization of our probabiistic observationa congruence [4]. Theorem 1. M = F p N if and ony if M F p N. 1 We abuse notation and sti use F to denote a scheduer for the LTS semantics. 3. MEASURING ENERGY CONSUMPTION In this section, based on the LTS semantics, we define a preorder over networks which aows us to study the performances, in terms of energy consumption, of different networks, but exhibiting the same (or simiar) connectivity behaviour. For this purpose we associate an energy cost with abeed transitions as foows. For a transmission with radius r, et En(r) = En eec packet en + En amp packet en r 2 where En eec (nj/b) is the energy dissipated to run the transmitter circuit, whie En amp (pj/b/m 2 ) is the radio ampifier energy (see [11]). We define En(r) if M L!ṽ[,r] N Cost(M, N) = for some L, ṽ, and r 0 otherwise α For an execution e = M 1 α 0 θ1 M 2 1 θ2 M 2... α k θk M k, Cost(e) = k i=1 Cost(Mi 1, Mi). Let H be a set of networks; we denote by Paths F M (H) the set of a executions from M ending in H and driven by F which are not prefix of any other execution ending in H. More formay, Paths F M (H) = {e Exec F M (H) ast(e) H and e such that e < prefix e, e Paths F M (H)}. Now, we are ready to define the average cost of reaching a set of networks H from the initia network M according to the scheduer F. Definition 4. The average cost of reaching a set of networks H from an initia network M according to the scheduer F is Cost F e Paths M (H) = F (H)Cost(e) P M F (e) M e Paths FM (H)P M F (e). The average cost is computed by weighting the cost of each execution by its probabiity according to F and normaized by the overa probabiity of reaching H. The foowing definition provides an efficient method to perform both quaitative and quantitative anayses of mobie networks.

6 Figure 1: Topoogy of the Static Network (SN) Figure 2: Topoogy of the Mobie Network (MN) Definition 5. Let H be a countabe set of sets of networks and et F LSched a set of scheduers. We write N F H M, if N F p M and, for a scheduers F LSched and for a H H, there exists a scheduer F LSched such that Cost F N (H) Cost F M (H). 4. STUDYING GOSSIP PROTOCOLS Gossip protocos are a famiy of communication protocos inspired by the way that gossiping disseminates information in socia networks. A gossip protoco is a variant of the fooding agorithm, where each node forwards a message with some probabiity to reduce the overhead of the routing protocos. Gossiping based routing protocos are commony used in arge-scae networks (see, e.g., [3, 10, 7]) to reduce the number of retransmissions and the energy cost. In this section we show that our framework is suitabe for providing an integrated automatic anaysis of the gossip strategy in terms of both connectivity maintenance and energy consumption. In particuar, we assume that, when a node receives a message, it forwards it with a fixed probabiity psend and discards it with probabiity 1 psend. Common vaues for psend ranges from 0.6 to 0.8: it is shown that, in practica scenarios, these vaues provide a reduction of more than 30% of the forwarding transmissions without deteriorating the network connectivity [7]. Sensor networks are usuay characterised by a arge number of sma devices, densey distributed in the network area, sensing the environment and forwarding data. Here we consider two different network configurations on a rectanguar area of m. We assume omnidirectiona antenna and a fixed transmission power for each sensor node, which covers circuar areas with a radius of 10m. In the foowing, we denote by [P i] J the sensor node i ocated at, executing the process P i and moving according to the transition matrix J. We study the behaviour of the networks by varying the vaue of the parameter psend. The first network we consider consists of 50 static nodes, eveny distributed within the network area (see Figure 1). Node mobiity is characterised by the identity matrix I. In our tests, we consider a fixed receiver [P 50] I 50, whie the sender node s ocation varies in the set {12, 23, 35, 37, 44}, in order to study how the connectivity behaviour of the network changes, depending on the distance between the sender node and the receiver. The network is expressed by the term: M j def = 50 i=1 [Pi]I i, with j {12, 23, 35, 37, 44}, and def P i = (x i). x i {50},10.P i, i {j, 50}, def P j = x j {50},10.P j, P 50 def = (x 50).P 50, modeing the communication between [P j] I j and [P 50] I 50. The second configuration consists of 25 mobie sensor nodes again eveny distributed within the network area. Each sensor node can move between two adjacent ocations, modeing the instabiity caused by, e.g., environmenta conditions (see Figure 2). The probabiity distribution associated with node mobiity can be captured by the transition matrix J such that: J (+5) = J (+5) = ε {5, 15, 25, 35, 45}, and J (+1) = J (+1) = ε for a the other odd ocations in the network area, and J = 1 ε for a the ocations, with 0 < ε < 1. Notice that the choice of ε and the definition of the scheduer aow us to mode the reative speed between movements and transmissions. Henceforth we assume that ε = 0.8. This network is expressed by the term: N h 25 i=1 [Pi]J (2i 1), with h {12, 17, 22}, where P i (x i). x i {45,50},10.P i, i {h, 25}, P h x h {45,50},10.P h and P 25 (x 25).P 25, modeing the communication between [P h ] J (2h 1) and [P 25] J z, where z {45, 50} is the set of ocations where we expect to find P 25. We mode severa different gossip strategies by varying the vaue of psend in the interva [ ]. In particuar, for each vaue of psend we assume a set F psend of scheduers such that, at each step, the probabiity for each node to perform a synchronisation or a movement is the same. Moreover, we do not consider message oss due to ink faiure or other environmenta causes: a message can be ost ony when a node discards it, consistenty with the protoco. The anaysis is performed using the PRISM mode checker [8] (see the Appendix for detais). The first step of our methodoogy consists in transating the process agebraic definition of our networks into the anguage supported by PRISM. This can be achieved in a purey agorithmic way. In genera the exact anayses of rea WSNs modes is unfeasibe due to the exposion of the state space of the mode. For this reason, we choose to perform an exact anaysis to study probems of equivaences or performances in case of sma components, e.g., to repace a network s node with a functionay equivaent one that has better performances in terms of throughput or energy consumption. Conversey, when studying the overa properties of wide WSNs we appy the approximate mode checking (aso known as statisti-

7 ca mode checking), that reies on a Monte Caro simuation of the underying DTMC. As a consequence, PRISM wi compute estimates of the desired indices rather then resuts, whose precision is controed by means of confidence interva specifications (absoute width and confidence). This approach is suitabe for most of practica purposes. When simuation is adopted, the estimates are obtained by samping, i.e., generating a arge number of random paths through the process underying the mode, hence avoiding the generation of whoe DTMC. In our case studies we assume that the sender node keeps retransmitting the same packet unti the destination node receives it. The outcomes of this study aow us to determine the expected number of retransmissions of the same packet that are needed to reach the intended recipient or, in more detai, which is the number of retransmissions needed in order to reach the destination with a probabiity higher than a given threshod. Our goa is the comparison between the different network configurations, according to the definition of energy aware preorder introduced in Section 3. The foowing proposition is important for the termination condition of the simuations. It states that, by varying the sender ocation, the packet eventuay reaches the intended recipient. Proposition 1. (i) F psend, psend [ ] and j 1, j 2 {12, 23, 35, 37, 44} M j1 Fpsend p M j2. (ii) F psend, psend [ ] and h 1, h 2 {12, 17, 22} N h1 Fpsend p N h2. Proof. The proof can be formay done by observing that the sender keeps retransmitting the packet and that there is a non-nu probabiity for a packet to reach the destination given any WSN configuration. Nevertheess, we can use the PRISM mode checker in order to automaticay verify the bisimuation among the different networks within a certain confidence. Usuay, since this patform supports severa tempora ogics, this can be done by constructing characteristic formuas for bisimuation. Since in our case the movements and the forwarding transmissions are sient actions whie the ony observabe action is the transmission of the packet to the ocation 50 (resp. {45, 50} for the mobie network), and our bisimuation does not take into account sient actions, we can simpy verify that the probabiity for [P 50] I 50 ([P 25] J 45) to receive the message is aways 1 (with the specified confidence). We use the PRISM P operator (for reachabiity properties). In particuar we verify P=? [F goa] i.e., which is the probabiity to eventuay perform a transmission at ocation 50 ({45, 50}). goa is the formua indicating that P 50 (P 25) has correcty received the message, and F means that the goa state wi be eventuay reached in a finite number of steps. For each network the probabiity to eventuay reach the successfu state resuts to be 1, where the confidence interva width is 0.01 based on 95% confidence eve. Once we proved that the networks we are considering have the same connectivity, we are ready to compare their energy costs, by changing the vaue of psend and the distance among the sender and the receiver. Using the PRISM mode Figure 3: (SN) Fraction of nodes reached by a transmission Figure 4: (MN) Fraction of nodes reached by a transmission checker, we expoit the possibiity of defining reward measures to compute the energy cost function defined in Section 3. Assuming that the energy spent for each transmission is fixed and that a the nodes have the same physica characteristics, we simpy count the number of transmissions rather than summing their energy cost. The cost function is expressed in terms of a Probabiistic Computation Tree Logic (PCTL) formua in the PRISM property specification anguage, augmented with rewards (or costs), which are rea-vaued quantities associated with states and/or transitions (see the Appendix for more detais). Specificay, we verify the formua R =? [F goa], which expresses the cumuative expected energy cost to compete the communication. Vaidation of the simuator. We have vaidated our simuations with those proposed in [7]. Specificay, the same kind of estimates shown in [7] can be computed by simuating our modes in PRISM. For the static and mobie networks described above, we show the estimates of the fractions of nodes that are expected to be reached by a transmission in Figure 3 and 4, respectivey. Simuation of static Networks. The estimates for the static network are shown in Figure 5. The simuations have been performed with an average of experiments, a maximum confidence interva width of 1% of the estimated measure based on 95% of confidence. The pots show how the distance between sender and receiver criticay infuences the energy performance of the agorithm. For a distance arger then 30m we have a monotonic decreasing pot showing that, for arge distances, the gossip protoco can cause energy waste. Using the stan-

8 Figure 5: (SN) Expected energy cost Figure 7: (MN) Expected energy cost Figure 6: (SN) Expected number of transmissions for a successfu communication Figure 8: (MN) Expected number of transmissions for a successfu communication dard fooding strategy (psend = 1.0) a the cases converge to 49, because each node wi forward the message exacty one time. We can verify that there exists a preorder among different network configurations within the confidence of the simuation. Proposition 2. psend {0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 1.0} and j 1 < j 2 {12, 23, 35, 37, 44}: M j1 F psend H M j2 where H is the set of network configurations where the communication has been successfuy competed. Figure 6 shows the expected number of retransmissions that the sender node must perform before the communication is successfuy competed. Notice that the smaer is the vaue of psend, the higher is the probabiity that the message is ost during the path, forcing a new transmission (for the sake of simpicity we don t mode the acknowedges, but we assume that the sender node wi wait for an acknowedge unti a timeout occurs, then it wi transmit again); hence, even if a sma vaue of psend reduces the forwarding exposion, it may increase the number of repications. Simuation of networks with mobiity. Figure 7 shows the estimates of the expected energy cost for a successfu transmission in the WSN with mobiity. The mobiity of the nodes criticay increases the size of the state space, hence the obtained resuts have wider confidence intervas (ranging from 5% to 10% of the measure) than those obtained with the static network simuation, based on 95% of confidence. However, the resuts are very simiar to the previous case: for distances arger than 25m the gossip protoco causes a very high energy waste. Aso in this case we can verify that there exists a preorder among different network configurations: Proposition 3. psend {0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 1.0} and h 1 < h 2 {12, 17, 22}: N h1 F psend H N h2 where H is the set of network configurations where the communications has been successfuy competed. Figure 8 shows the average number of retransmissions that the sender must perform before the communication has successfuy competed. It is worth of notice that with respect to the static network, in this case the distance between the sender and the receiver has a great impact on the protoco performances. 5. RELATED WORK AND CONCLUSIONS A arge amount of research on sensor networks has been recenty reported in the ast decade. From the energy consumption viewpoint, severa papers address the probem of studying the energy consumption for specific communication protocos. For instance, in [18] the authors define a Markov reward process (see, e.g., [14]) modeing some protocos for point to point reiabe transmissions. A steady-state quantitative anaysis is then obtained, and from this the average performance indices are computed. In [1], Bernardo et a. present a methodoogy to predict the impact of power

9 management techniques on system functionaity and performance. In [16], the authors define a set of metrics to anayse the energy consumption which are then estimated through simuation, and show how some modifications in the protocos can improve their efficiency. In [7], gossip protocos running on WSNs are studied but the authors deveop an ad hoc simuator to estimate their performances. Conversey, in our setting, a genera purpose too, e.g., PRISM, can be used since the performance indices or properties to be evauated (or estimated) can be formay specified according to a rigorous ogic. Moreover, with respect to a the above mentioned contributions, the mode we propose here aims at providing a common framework for automaticay perform both quaitative and quantitative anayses. Indeed PRISM can be used for different purposes. The energy preorder defined in Section 3 can be efficienty decided for sma networks components using mode checking methods and hence one may decide to repace a node with another which is behaviouray equivaent but ess energy consuming; conversey, when the compexity of the process underying the mode makes exact anayses unfeasibe, approximate (or statistica) mode checking can be empoyed. This corresponds to the weknown Monte Caro simuation; using the tempora ogic impemented in the too, one can verify a proposition within a certain eve of confidence (e.g., the network equipped with a certain protoco is connected with a confidence of 99.9%). To the best of our knowedge such a quaitative and quantitative approach supported by the same too represents a novety in the study of WSNs. 6. REFERENCES [1] A. Acquaviva, A. Adini, M. Bernardo, A. Bogioo, E. Bontà, and E. Lattanzi. A methodoogy based on forma methods for predicting the impact of dynamic power management. In Forma Methods for Mobie Computing, voume 3465 of ncs, pages Springer Berin / Heideberg, [2] I. Akyidiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci. Wireess Sensor Networks: A Survey. Computer Networks, 38(4): , [3] A. Dimakis, A. Sarwate,, and M. Wainwright. Geographic Gossip: Efficient Aggregation for Sensor Networks. In Proc. of the 5th internationa conference on Information processing in sensor networks, pages ACM, [4] L. Gaina. Forma Modes for Quaitative and Quantitative Anaysis of Mobie Ad Hoc and Sensor Networks. PhD thesis, DAIS, University Ca Foscari of Venice, [5] L. Gaina, S. Hamadou, A. Marin, and S. Rossi. A probabiistic energy-aware mode for mobie ad-hoc networks. In Proc. of the 18th Internationa Conference on Anaytica and Stochastic Modeing Techniques and Appications (ASMTA 11), voume 6751 of LNCS, pages Springer-Verag, [6] J. Goubaut-Larrecq, C. Paamidessi, and A. Troina. A probabiistic appied pi-cacuus. In Proc. of the 5th Asian Symposium on Programming Languages and Systems (APLAS 07), voume 4807/2009 of LNCS, pages Springer-Verag, [7] Z. Haas, J. Hapern, and L. Li. Gossip-based Ad Hoc Routing. IEEE/ACM Trans. Netw., 14(3): , [8] A. Hinton, M. Kwiatkowska, G. Norman, and D. Parker. Prism: A too for automatic verification of probabiistic systems. In Proc. of the 12th Internationa Conference on Toos and Agorithms for the Construction and Anaysis of Systems (TACAS 06), voume 3920 of LNCS, pages Springer-Verag, [9] M. Z. Kwiatkowska, G. Norman, and D. Parker. Prism 4.0: Verification of probabiistic rea-time systems. In G. Gopaakrishnan and S. Qadeer, editors, CAV, voume 6806 of Lecture Notes in Computer Science, pages Springer, [10] P. Kyasanur, R. Choudhury, and I. Gupta. Smart Gossip: An Adaptive Gossip-based Broadcasting Service for Sensor Networks. In Mobie Adhoc and Sensor Systems (MASS), 2006 IEEE Internationa Conference on, pages , [11] T. V. Madhav and N. Sarma. Maximizing network ifetime through varying transmission radii with energy efficient custer routing agorithm in wireess sensor networks. Internationa Journa of Information and Eectronics Engineering, 2(2): , [12] M. Merro. An observationa theory for mobie ad hoc networks. Information and Computation, 207(2): , [13] R. Miner and D. Sangiorgi. Barbed bisimuation. In Proc. of Internationa Cooquium on Automata, Languages and Programming (ICALP 92), voume 623 of LNCS, pages Springer-Verag, [14] S. M. Ross. Stochastic Processes. John Wiey & Sons, 2nd edition, [15] R. Segaa and N. Lynch. Probabiistic simuations for probabiistic processes. In Proc. of the 5th Internationa Conference on Concurrency Theory (CONCUR 94), voume 836 of LNCS, pages Springer-Verag, [16] S. Singh, M. Woo, and C. Raghavendra. Power-aware routing in mobie ad hoc networks. In Proc. of the 4th annua ACM/IEEE Internationa Conference on Mobie Computing and Networking (MobiCom 98), pages ACM, [17] G. Zhou, C. Huang, T. Yan, T. He, J. A. Stankovic, and T. F. Abdezaher. Mmsn: Muti-frequency media access contro for wireess sensor networks. In In IEEE INFOCOM, page 7, [18] M. Zorzi and R. Rao. Error contro and energy consumption in communications for nomadic computing. IEEE Transactions on Computers, 46(3): , APPENDIX PRISM is a probabiistic mode checker, a too for forma modeing and anaysis of systems that exhibit random or probabiistic behaviour. It supports a wide range of probabiistic modes, such as Markov decision processes (MDPs), discrete-time Markov chains (DTMCs) and continuous-time Markov chains (CTMCs), and these modes are described using the PRISM anguage. Moreover, PRISM provides a specification anguage to specify rewards and quantitative properties and it supports the automated anaysis of these properties with respect to the probabiistic modes.

10 modue P8 steps8 : [0.. 2] init 2; 8 : [ ] init 15; [] (8 = 15) 0.8 : (8 = 20) : (8 = 15); [] (8 = 20) 0.8 : (8 = 15) : (8 = 20); //beginning of a new round [round] no one sending (steps8 = 2); //beginning of a new round //transmission //[c8] (steps8 = 1) & (conn38 conn78 conn810 conn813) (steps8 = 0); //receives [c3] (steps8 = 2)& (conn38) psend : (steps8 = 1) + (1 psend) : (steps8 = 0); [c3] (steps8! = 2)!(conn38) (steps8 = steps8) [c7] (steps8 = 2)&(conn78) psend : (steps8 = 1) + (1 psend) : (steps8 = 0); [c7] (steps8! = 2)!(conn78) (steps8 = steps8) [c10] (steps8 = 2)&(conn810) psend : (steps8 = 1) + (1 psend) : (steps8 = 0); [c10] (steps8! = 2)!(conn810) (steps8 = steps8) [c13] (steps8 = 2)&(conn813) psend : (steps8 = 1) + (1 psend) : (steps8 = 0); [c13] (steps8! = 2)!(conn813) (steps8 = steps8) endmodue Tabe 5: The PRISM modue for a node Modeing the network. We transate process terms into DTMCs in PRISM in order to automaticay evauate the performance, in terms of energy, of a sensor network using a basic gossip protoco for the information exchange among the sensor nodes. In the PRISM anguage we define a constant psend, ranging from 0.6 to 1, and we study the different modes that are generated by the different vaues assigned to psend. Each node [P i] J i is transated into a corresponding modue Pi, which is associated with two variabes: steps i: contros the sequentiaity of the process executed by the sensor node. In particuar, steps i = 2 means that the node is ready to receive, steps i = 1 means that the node is ready to transmit, and steps i = 0 means that the node has competed a transmission. i: is the variabe containing the actua ocation of the sensor node. Tabe 5 shows the impementation of a singe node. Each transition of the PRISM mode corresponds to a transition of the abeed transition system underying the PEBUM network term: the unabeed commands mode the possibe node s movements, whie the abeed commands are used to mode synchronisations. Each action tagged with [ci] represents a transmission from the source sensor node P i. Each node starts its transmission (steps8 = 1) ony if there is at east one of the neighbours ready to receive it (conn38 conn78 conn810 conn813). This is a standard strategy for gossip protocos. However, since the neighbour nodes wi forward the message ony with a certain probabiity, the presence of at east one receiver inside the sender node s area does not ensure the competeness of the communication. A node receives the packet ony if it is inside the transmission area of the sender node, and forwards it with probabiity psend: [c13] (steps8 = 2)&(conn813) psend : (steps8 = 1) +(1 psend) : (steps8 = 0); Property Specification. PRISM provides a property specification anguage which supports various tempora ogics as we as extensions for rewards (or costs). Rewards can be associated to states or to transitions. The cumuative rewards in the PRISM anguage are expressed as: rewards reward_name [transition] condition : vaue; condition : vaue; endrewards In particuar, [transition] condition : vaue associates a reward vaue to each transition tagged with [transition] when the condition condition is true, whie condition : vaue associates a reward vaue to each state where the condition condition is true. In the PRISM specification anguage, R =?[F goa state] denotes the cumuative expected reward to eventuay reach goa state. We use PRISM to find the vaue of psend which minimises the costs of communications.