Performance Analysis of Location-Based Data Consistency Algorithms in Mobile Ad Hoc Networks

Transcription

1 Performance Analyss of Locaton-Based Data Consstency Algorthms n Moble Ad Hoc Networks Ing-Ray Chen, Jeffery W. Wlson Department of Computer Scence Vrgna Tech {rchen, wlsonjw}@vt.edu Frank Drscoll, Karen Rgopoulos The MITRE Corporaton {fdrscoll, karenr}@mtre.org ABSTRACT: Many applcatons of moble ad hoc networks requre real-tme data consstency among the movng nodes wthn a geographcal area of nterest to functon correctly, e.g., battlefeld command and control applcatons. Whle t s operatonally desrable to mantan data consstency among nodes wthn a large geographcal area, the tme requred to propagate state changes to all moble nodes n that geographcal area lmts ts sze. Ths paper nvestgates the noton of locaton-based data consstency n moble ad hoc networks, and analyzes the tradeoff between data consstency and tmelness of data exchange among nodes wthn a locaton-based group n a geographcal area of nterest. By utlzng a Petr net performance model, we analyze performance characterstcs of locaton-based data consstency mantenance algorthms and dentfy desgn condtons under whch the system can best tradeoff consstency for tmelness (reflectng the tme to propagate a state change) whle satsfyng the mposed data consstency requrement, when gven a set of parameters characterzng the applcaton n the underlyng moble ad hoc network envronment.. Introducton Many applcatons operate n moble ad hoc networkng envronments wth no fxed nfrastructure connectng moble nodes n the feld. Many of these applcatons requre that nodes have some degree of data consstency wthn a communty of nterest for them to functon correctly. An example s a battlefeld applcaton n whch nodes wthn a geographcal area of nterest must have consstent data structures that contan knematc and other characterstc state data that descrbes frendly, enemy, and neutral aerospace objects to satsfy the command and control functonalty requrements. Such an ad hoc envronment s characterzed by moble nodes, mult-hop routng, planned and unplanned node dsconnecton, node falure, relatvely low communcaton system throughput, and unrelable communcaton. Node moblty s a partcular challenge because moblty translates nto mult-hop network topology changes, whch are reflected n frequent packet route changes and network parttons. The physcal envronment exacerbates the challenges caused by node moblty; by nature of ther moblty, nodes can be expected to explot cultural and natural features of the physcal envronment (e.g., takng shelter n buldngs, maneuverng around hlls) that wll have a deleterous effect on communcaton. The problem we are addressng n ths paper s analyzng performance characterstcs of algorthms for mantanng locaton-based data consstency among a group of nodes n a moble ad hoc envronment for applcatons that requre locaton-based data consstency, e.g., the DoD Sngle Ar Integrated Pcture (SIAP) project that requres all moble nodes to mantan a consstent vew of tracked objects for combat mssons [2]. It s well known that the problem of reachng an agreement ( consensus ) among all nodes n asynchronous dstrbuted systems n the presence of falures s determnstcally non-solvable even f communcaton s relable and at most one peer may crash [3]. A less constraned problem, known as the group membershp consstency for mantanng a sngle agreed vew of the group membershp among all peers, was also shown to be non-solvable n asynchronous dstrbuted systems where communcaton s relable and at most one peer may crash []. The noton of locatonbased data consstency consdered n the paper does not requre a sngle agreed vew to be mantaned (for whch there s no soluton). Rather, t allows moble nodes to jon and leave locaton-based groups, allowng multple data vews to coexst n dfferent locaton-based groups as long as nodes wthn the same group have the same vew of data. Ths requrement, although less strong n data consstency, s very useful for battlefeld applcatons where data carry geographcal meanngs, e.g., trackng frend or foe objects enterng, travelng and leavng a geographcal area by all unts n the area. The general problem of group communcaton n moble ad hoc wreless networks to mantan consstent group membershp and, as an extenson, to mantan data consstency among members of a group s a relatvely

2 new research area. Klljan et al. [5] ntroduced the defnton of proxmty group communcaton n whch group membershp depends on locaton. They assocated each proxmty group wth a statc or moble area of nterest wthn whch the group members should be located. They gave a sketch of usng a partton antcpator executed on every node to detect suspcous parttonng events of a proxmty group due to node and lnk falures n order to take preventve actons for consstency group membershp mantenance. However, no concrete mechansms were gven by whch the partton antcpator may be mplemented. Roman et al. [4, 7] use the dea of safe dstance for mplementng consstent group membershp wheren membershp s based on the locaton nformaton of moble nodes. The basc dea s to group nodes logcally connected wthn a safe dstance ("close enough") n a geographc area and to perform membershp-change operatons atomcally as nodes move n and out of the geographc area. Physcal connectons that are susceptble to dsconnecton are consdered as announced dsconnectons so the system can perform membershp change n one ndvsble operaton to ensure group membershp consstency. In ths way, a group expands and contracts atomcally, preservng consstent group membershp. However, ther safe dstance-based algorthm s based on the somewhat unrealstc assumpton that dsconnectons are only caused by node movements, so the algorthm breaks when dsconnectons are caused by node falures. In ths paper, we utlze the concept of parttonable group membershp for achevng locaton-based data consstency n ad hoc systems so that each member n a locaton-based group has knowledge of other members n ts group and that such knowledge would be consstent across the entre group [4, 7]. However, nstead of mantanng membershp consstency all the tme by usng the concept of safe dstance, we allow temporary nconsstency to exst durng membershp changes to tradeoff consstency for tmelness. The degree of nconsstency s bounded by the way we check and perform membershp changes as a result of nodes leavng and jonng locaton-based groups, the rate of whch can be adjusted to satsfy the data consstency requrement of the applcaton. One key desgn ssue for mantanng locaton-based data consstency wthn a geographcal communty of nterest s to determne the logcal sze of each geographcal area. The sze drectly mpacts the tme taken for achevng data consstency among members wthn a geographcal area, thus reflectng the bound on data consstency. The optmal sze s affected by envronmental condtons characterzed by moblty rate (for moble unts n and out of an area), node falure rate, membershp-change detecton rate, data change rate (e.g., for trackng objects enterng, traversng and extng the area), etc. A smaller area ncurs a lower latency for message transfer because fewer hops are requred but ncurs a hgher overhead for membershp change operatons because of a hgher rate of members leavng and jonng geographcally smaller areas. Thus when moble hosts have low moblty rates, t may dctate a smaller geographcal area. On the other hand, a larger area ncurs a hgher latency for data change operatons because of more hops n the larger geographcal area but ncurs a lower overhead for operatons assocated wth consstent group membershp mantenance due to node moblty/falure events. Thus there exsts an optmal sze, when gven a set of parameters characterzng the moble applcaton n the underlyng ad hoc network envronment. In ths paper, we am to analyze performance characterstcs of locaton-based data consstency mantenance algorthms n terms of the optmal sze that can best tradeoff consstency for tmelness (reflectng the tme to propagate a state change) whle satsfyng the mposed data consstency requrement. 2. System Model We assume a moble ad hoc network consstng of one or more peers. The network s heterogeneous, wth peers n the system havng greatly dfferent capabltes. For a battlefeld applcaton, for example, one end of the capablty spectrum s represented by large command and control nodes (moble or fxed), such as an arcraft carrer or fxed surface-to-ar mssle ste. At the other end of the spectrum we fnd human-portable devces or plot-less vehcles wth more modest command and control capabltes. Each peer has one or more communcaton devces and may have organc sensors whose data s shared wth other peers n the dstrbuted system. Addtonally, each peer has one database n whch sensor and other state data s stored. 2. Geographcal Communty of Interest The noton of locaton-based data consstency consdered n ths work s based on the concept of a geographcal communty of nterest. Ths concept allows us to move from a requrement for data consstency among all peers n the system (whch s not achevable) to a requrement for data consstency only among peers who belong to the same geographcal communty of nterest. We also note that n battlefeld applcatons, as n any other problemsolvng, team-orented applcaton, t s more mportant (n most cases) to have greater consstency wth nearby peers than wth ones farther away. In ths paper we wll use the terms geographcal communty of nterest and locaton-based group (or just group for short) nterchangeably.

3 The state of the moble applcaton s characterzed by the values of state varables (e.g., track objects n the example battlefeld applcaton) mantaned by peers n the system. The locaton-based data consstency requrement means that all the peers wthn the same geographcal communty of nterest wll have the same values for state varables. The area of a geographcal communty of nterest can be modeled many ways. We can dvde the terran nto geometrc shapes lke squares or hexagons. Fgure s a coverage model showng three possble rng szes for modelng a geographcal communty of nterest based on hexagons,.e., coverng, 7 and 9 hexagons, respectvely by rng 0, rng and rng 2. The sze of each geographcal communty of nterest may vary dependng on the operatng condtons. For example, f the moblty rate s low for most moble hosts, then the sze can be small to optmze the performance. Fgure : A Representaton of a Geographcal Area based on a Hexagonal Coverage Model. 2.2 Locaton-based Group Membershp and Data Consstency Algorthm We assume that each moble host has a unque host dentfer and s equpped wth locaton sensng devces such as a Global Postonng System (GPS) recever, so t can determne ts own locaton as well as reason about ts locaton relatve to the locatons of ts neghbors wthn rado range. For a geographcal communty of nterest dentfed by a group dentfer, f the cardnalty of the membershp set (contanng members that are connected n the ad hoc envronment) s not zero, the moble host wth the smallest host dentfer wll be elected as the leader. The leader broadcasts ts presence wthn the communty of nterest perodcally. Should the leader fal, the falure event wll be detected and a re-electon protocol wll be followed to select a new leader. If two or more leaders announce ther presence, the leader wth the smallest host dentfer wns and the rest wll relnqush ther roles.the group dscovery protocol s locaton-based. When a moble host moves out of a geographcal communty of nterest, t voluntarly nforms the group leader of ts departure, who n turn wll perform a membershp change operaton to exclude the host n the group. Conversely, when a moble host enters a new geographcal area of nterest, t broadcasts a hello message contanng ts locaton nformaton and host dentfer to dscover the new locaton-based group to jon. When a host, say A, receves a hello message from host B, t nforms the correspondng group leader whch n turn wll perform a membershp change operaton to nclude B n the group. If the leader receves multple messages regardng B s new membershp, t accepts the frst and gnores the rest. Each moble host also perodcally sends an update message to the leader regardng ts locaton and dentfer so the leader s aware of who are stll wthn the communty of nterest. When the leader detects that a member moble host has not sent ts update message, t assumes that the member has been dsconnected ether due to moblty or falure and wll remove the moble host from the group membershp. A moble host can always send a hello message to request for membershp renstatement f t suspects that t has been removed from the group membershp by the leader. Ths perodc mantenance event thus allows the leader to actvely gather nformaton regardng new and mssng members to mantan consstent group membershp. Wthn a geographcal communty of nterest, f there s a state change detected by any member n the group (e.g., a hostle object approachng), the member wll send a message to the leader whch n turn wll forward the message to all members n the group. A multcast tree s bult dynamcally to permt the leader to reach all members more effcently, relably, and securely Traffc Model and Performance Metrc We assume that each moble host has ts own dstnct moblty rate n and out of geographcal communtes (groups) of nterest. For example, helcopters move faster than tanks whch n turn move faster than human bengs n general. We assume that the terran s vrtually parttoned nto equal-area regons (e.g., hexagons) for ease of analyss and presentaton as shown n Fgure. Let the moblty rate of moble host be σ ι movng n and out these regons. Each moble host also has ts own dstnct falure rate. Let the falure rate of moble host be φ ι. Followng the group membershp protocol descrbed earler, let T be the tme perod between whch each moble host sends ts locaton nformaton and dentfer to the leader n an update message. The tme requred for the leader to perform a membershp change operaton depends on the sze of the geographcal area. Let µ mc (n) be the rate at whch a membershp change operaton s executed n a geographcal communty of nterest wth a

4 rng sze of n (see Fgure for llustraton), ncludng the tme to rebuld a multcast tree by the leader. Smlarly the tme requred for the leader to perform a state update operaton also depends on the sze of the geographcal area. Let µ u (n) be the rate at whch the leader can propagate an update to members wthn a geographcal communty of nterest of sze n where n s the rng sze of the geographcal communty of nterest. Fnally, as a larger geographcal communty of nterest s lkely to mantan a larger set of state varables (e.g., sensor data n the example battlefeld applcaton), let δ(n) be ths sensor-update rate wth n agan the rng sze of the geographcal communty of nterest. Ths data update rate also depends on the objects to be tracked, for example, the data update rate to track a theater ballstc mssle s generally dfferent than that requred to track an arbreathng mssle or arcraft. Later n the paper we wll show how these parameters can be parameterzed (.e., be gven values) properly reflectng the desgn choce such as the sze of a geographcal area of nterest. Our performance metrcs of nterest would measure tmelness and consstency of state nformaton dstrbuted to members wthn the geographcal communty of nterest. The tmelness metrc s measured by the response tme R requred to acheve data consstency whenever there s a state change detected by any member wthn a geographcal area of nterest. On the other hand, the consstency metrc s measured by the proporton of tme the system s n a consstent state, whch can be broken up nto two measures. The frst measure PT m s the proporton of tme the group membershp s consstent, whle the second measure PT md s the proporton of tme both membershp and state data are consstent among the node members of a locatonbased group. Our goal s to satsfy the response tme requrement whle makng the consstency measures as hgh as possble. When there s a constrant n the consstency requrement, the goal s to mnmze the response tme measure by dentfyng the best geographcal communty of nterest area sze whle satsfyng the mposed consstency requrement, when gven a set of model parameters dentfed and parameterzed characterzng the operatonal condtons of the moble applcaton n ad hoc networkng envronments. 3. Performance Model In ths Secton, we develop a Stochastc Petr net (SPN) performance model to descrbe the behavor of a moble applcaton operatng under the locaton-based data consstency algorthm descrbed earler n Secton 2. Later n Secton 4 we wll utlze ths performance model to calculate the tmelness and data consstency metrcs to analyze the tradeoff between data consstency and tmelness, gven a set of parameter values characterzng a gven moble ad hoc envronment. IC µ u (n) /Τ µ mc (n) λ f (n) Fgure 2: Petr Net Model for Locaton-Based Data Consstency Algorthm. Fgure 2 shows an SPN model for descrbng the behavor of the system operatng under the locaton-based membershp and data consstency protocol wthn a geographcal area of nterest of sze n. The SPN model can be vewed as a contnuous-tme fnte state machne whch reacts to system events that occur n the system. There are 3 places n the Petr net model, wth C standng for the state n whch the system s consstent n membershp, IC standng the state n whch the system s nconsstent n membershp due to nodes movng n and out of the geographcal area of nterest, and ICf standng for the state n whch the system s nconsstent n membershp due to unannounced node falures or dsconnectons. Intally the system s n a consstent state, represented by havng a token deposted n place C. We use the place at whch the token resdes to represent the current state of the system as tme progresses, so the ntal state s C as the token s ntally placed there. Whenever there s a membershp change due to arrvals and departures of moble nodes n and out of the geographcal area of nterest of sze n wth rate µ mc (n), the system mgrates from state C and state IC. Our algorthm requres moble hosts to nform the leader of the membershp changes when they move n and out of the locaton-based group of sze n, the dscovery rate of whch s µ mc (n). After a membershp change detecton event occurs, the leader then sends a membershp update operaton to all members n the locaton-based group, the rate of whch s the same as that for the state-update operaton,.e., µ u (n). These behavors are captured by the two transtons n the upper part of the SPN model wth rates µ mc (n) and µ u (n), respectvely. Note that the token flows from state C to state IC and then to state C agan, reflectng that a membershp update event s taken C ICf

5 sequentally followng a membershp change detecton event. Whenever there s a membershp change due to unannounced falure or dsconnecton of moble nodes wth rate λ f (n), the system mgrates from state C to state ICf. Ths event s modeled by a lower rght transton n the SPN model wth rate λ f (n). Perodcally, the leader wll collect and analyze beacon messages sent from moble members of the locaton-based group and detect f any member needs to be removed from the membershp because of unannounced falure or dsconnecton events. Consequently, any unannounced falure or dsconnecton wll be detected by the system after a perod of tme T has elapsed. Ths detecton event s modeled by the lower left transton wth a determnstc tme perod T. Afterward the token flows to place IC n whch the system performs a membershp change operaton wth rate µ u (n) agan to brng the group membershp consstent. The last event s modeled by havng the token flow from place IC to place C through a transton wth rate µ u (n) to nform all members of the membershp change. Note that data change events are not modeled n the SPN model as one can mage emanatng from each state, whenever there s a state change due to sensor detecton wth rate δ(n), the system wll mgrate to another state n whch the system wll propagate the data update from the moble user detectng the data change (through sensors presumably) to the leader and then from the leader to all moble nodes n the locaton-based group usng the multcast tree mantaned by the leader wth the rate of data propagaton beng µ u (n). To avod clutter, we do not explctly model ths behavor n the SPN model and nstead wll consder t through probablstc arguments when we later derve expressons for computng the consstency and tmelness performance metrcs. The system evolves over three states, namely, C, IC and ICf, as tme progresses. Thus, there exsts a steadystate probablty that the system can be found n one of the three states. Let P C, P IC, P ICf be the steady state probabltes of states C, IC, and ICf respectvely, whch can be obtaned by evaluatng the SPN model constructed after model parameters are parameterzed (.e., gven specfc values) characterzng envronmentand applcaton-specfc operatng condtons. Then we can calculate consstency metrcs,.e., PT m and PT md, as follows: PT m = P C () PT md = µ u (n) P C / (µ u (n) +δ(n)) (2) Equaton () above gves the proporton of tme the system s consstent n membershp, whch s exactly the same as the equlbrum probablty that the system s found n state C. Equaton (2) gves the proporton of tme the system s consstent n both data and membershp, whch s equal to the equlbrum probablty that the system s found n state C multpled wth the probablty that the system s consstent n data, gven that the system s consstent n membershp. Ths can be reasoned by consderng splttng state C nto two states C and C2 such that C s a state that s consstent n both membershp and data whle state C2 s a state that s consstent n membershp only because a data update propagaton operaton s stll takng place. If one draws a two-state model wth C and C2 such that the rate from C to C2 s δ(n) for the data-change transton (due to sensng) whle the rate from C2 to C s µ u (n) for the data-update transton (for propagatng updated data to members), then one wll see that the probablty that the system s consstent n both membershp and data,.e., n state C, gven that t s n state C, s equal to µ u (n) / (µ u (n) +δ(n)). The tmelness metrc can be calculated by the average of the response tmes obtaned n varous states weghted by ther respectve state probabltes,.e., R = (P C + P ICf )/µ u (n) + P IC (/µ u (n) + /µ u (n)) (3) Here the frst term accounts for the response tme when the system s n ether state C or state ICf, whch ncurs an average update propagaton tme of /µ u (n), whle the second term accounts for the response tme when the system s n state IC whch ncurs a watng tme of /µ u (n) to account for the extra tme requred to process the membershp change operaton before takng another /µ u (n) tme to process the data propagaton operaton by the system (leader). Here we note that whle the system s n state ICf, the leader wll only propagate data to members nconsstently snce n state ICf the system s n a state n whch the leader s not aware of the fact that the group membershp s nconsstent. Contrarly, the system s fully aware of ts membershp nconsstency n state IC, n whch case the leader s n the process of performng a membershp change operaton, so a data propagaton operaton newly arrvng must wat for the membershp operaton to execute to completon before beng processed by the leader, thus ncurrng a watng tme to the response tme. We also note that when T s small, the probablty of the system found n state ICf wll be small snce the moment the system s n state ICf t wll transt to state IC quckly n whch a membershp change operaton wll be executed to mantan membershp consstency.

6 Thus a small T mproves membershp and data consstency whle compromsng the response tme performance metrc, and vce versa, and there exsts a tradeoff between the consstency metrcs (as gven by Equatons () ad (2)) and the tmelness metrc (as gven by Equaton (3)). 4. Analyss 4. Parameterzaton Consder a moble ad hoc network modeled by a hexagonal network coverage model as llustrated n Fgure wth the center regon n rng 0. Also consder a locaton-based group wth a geographcal area of nterest of sze n coverng rng 0 through rng n-. For a moble node, say, node, n the area, let λ n be the outward moblty rate of moble node to go out of rng n nto rng n+ and µ n be the nward moblty rate of the moble node to go out of rng n nto rng n-. The specfc values of λ n and µ n for moble node depend on the semantcs of the moble applcatons and the moblty model of the moble node. As an example, consder the node follows a random walk moblty model. It can be shown that [6] when a moble node s n rng n, the probabltes of the moble node wth random walk movng outward to rng n+, movng nward to rng n-, and stayng wthn rng n, upon a movement out of a hexagon regon, denoted by P omove, P move and P smove, respectvely, are gven by: P P P omove move smove f n = 0 + otherwse 0 f n = 0 otherwse 0 f n = 0 = otherwse 3 Let σ ι represent the user moblty rate of moble node movng across hexagonal areas. Agan let λ n be the outward moblty rate of moble node to go out of rng n nto rng n+ and µ n be the nward moblty rate of the moble node to go out of rng n nto rng n-. Then, σ f n = 0 n λ + σ otherwse 0 f n = 0 n µ σ otherwse (4) (5) Now consder that the moble ad hoc network s populated wth moble nodes wth an average densty of M users per hexagonal area located at rng. For the unform densty case, all M s are the same, say, equal to M 0. The more reasonable case s that there are more moble nodes close to the center of the geographcal area (snce they are nterested n the area and are members of the locaton-based group) and less nodes as we move further away from the center of the geographcal area. Ths nhomogeneous densty dstrbuton can be modeled by a populaton functon wth an exponental decay behavor. Let M 0 be the densty of the center hexagon n a geographcal area of nterest, then M s gven by: M M 0 = (6) b Here b s the populaton decay parameter whose magntude represents how fast the populaton densty decays as we move away from the center of attenton n the geographcal area, wth the specal case b= beng the unform densty case. Snce a geographcal area of nterest of sze n contans 3n 2 3n + hexagons, so there are (3(n+) 2 3(n+) + ) (3n 2 3n + ) = 6n hexagons n rng n, wth n>0. For example, rng 0 contans, rng contans 6 and rng 2 contans 2 hexagons, and so on. Snce only nodes n rng n movng nward to rng n- and nodes n rng n- movng outward to rng n wll trgger a locaton-based membershp change operaton, the overall rate at whch all the moble nodes wll trgger a membershp change for a geographcal area of nterest of sze n (consstng of rng 0 to rng n-) due to moblty, defned as µ,mc (n), s gven by: µ ( n ) = 6( n ) M 6 µ mc n n n λ + nm (7) n Here the frst term accounts for the rate at whch moble nodes move out of the geographcal area of nterest of sze n, and the second term accounts for the rate at whch moble nodes move nto the area, both trggerng a membershp change operaton. Note that we have dropped the subscrpt from λ n and µ n to refer to the fact we have consdered all moble nodes n Equaton (6). Snce a geographcal area of nterest of sze n on average wll contan M 0 +6M +2M (n-)M n- moble nodes, the rate at whch moble users wthn a geographcal area of nterest of sze n fal or dsconnect unannounced, λ f (n), s gven by: n f ( n) ( M 0 λ = φ + 6jM j ) (8) j=

7 The tme for a leader to propagate a state-update operaton or a membershp change operaton to all members n the geographcal area depends on f the propagaton method s broadcast-based or multcastbased. Suppose that we adopt multcast-based for the sake of securty. Then the propagaton tme depends on the number of members n the group and the way the leader bulds the multcast tree to reach all members. Assume a perfect balance tree. Then on average t takes log n 2 ( M 6 ) 0 + jm hops to reach all members and the j j= communcaton tme per hop s τ dependng on the underlyng communcaton technology deployed n the ad hoc network. Consequently, the rate at whch the leader performs a state-update operaton to all members n the locaton-based group of sze n, µ u (n), s gven by: µ u ( n) = n τ log [ M + 6 jm ] 2 0 j= Equatons (5), (6), (7), (8) and (9) thus parameterze model parameters µ mc (n), λ f (n) and µ u (n) once we are gven values of basc parameters M 0 and b (densty of moble nodes), σ (moblty rate per node), φ (falure rate per moble node) and τ (communcaton delay per hop) characterzng the network and applcaton operatng condtons. 4.2 Numercal Data Here we present numercal data obtaned from evaluatng the Petr net model developed usng SPNP [8] based on Equaton (), (2) and (3) to show desgn tradeoffs between the tmelness (R) and consstency metrcs (PT m and PT md ) obtaned, as a result of applyng our locatonbased data consstency algorthm n moble ad hoc networkng envronments. The set of parameters characterzng the moble applcaton n the underlyng ad hoc network envronment s gven by τ=, M 0 =2, b=4, δ(n)=0.0, σ=0.05, φ=0.00, T=5. These parameters are normalzed wth respect to τ= (hop-by-hop delay) for ease of presenataton, e.g., φ=0.00 means that the falure rate on average s once per 000τ, and T=5 means that the perodc check s about once every 5τ perod. We wll analyze the effects of some of these parameters n the paper by changng ther values to observe ther mpacts on R, PT m and PT md obtaned. Fgures 3 and 4 show the consstency metrcs PT m and PT md verse the sze of geographcal area of a locaton-based group. Unlke the tmelness metrc whch monotoncally ncreases wth n, we observe that there exsts an optmal n, say, n opt, at whch the consstency metrc s maxmzed. For example, j (9) when b= or 2, n opt =2, when b=4, n opt =4 and when b=8 or 6, n opt,=3 n both Fgures 3 and 4. Membershp Consstency (PTm) Membershp and Data Consstency (PTmd) Geographcal Area Sze (n) Fgure 3: Membershp Consstency (PT m ) vs. Geographcal Area Sze (n) Geographcal Area Sze (n) Fgure 4: Membershp and Data Consstency (PT md ) vs. Geographcal Area Sze (n). The reason that an optmal area sze exsts for maxmzng membershp and data consstency s that membershp nconsstency s attrbuted to the system beng n state IC due to moblty events for nodes n and out of the group, and also n state ICf due to falure events for member nodes. The rate of node falure events s drectly proportonal to the number of member nodes n the locaton-based group. Thus, as n ncreases, more falure events are lkely to occur as there are more member nodes n the group. On the other hand, the rate at whch moblty events occur due to nodes movng nto and out of the geographcal area s not necessarly proportonal to n. For the nhomogeneous populaton model defned by Equaton (5), the rate of membershp changes nduced by user moblty actually decreases as n ncreases when b>2, because there are fewer nodes resdng at the outer hexagons (due to exponental populaton decay) as we move away from the center hexagon of the locaton-based group, so most nodes n the group are lkely to be contaned wthn the area when n s large. These two effects counterbalance each other, thus resultng n an b=6 b=8 b=4 b=2 b= b=6 b=8 b=4 b=2 b=

8 optmal area sze that maxmzes the membershp and data consstency metrcs. Membershp Consstency (PTm) Geographcal Area Sze (n) Fgure 5: Effect of T on Membershp Consstency (PT m ) and Optmal n opt. At a large group sze (.e., a large n) we can expect that practcally there wll be lttle moblty-nduced membershp changes snce all moble nodes would be reasonably contaned wthn the area most of the tme f b>2. Most membershp change operatons ncurred n ths case would be due to node falures whose rate ncreases as n ncreases. Fgure 5 shows the effect of T on the optmal sze n opt for membershp consstency. (The graph for the effect of T on the optmal sze n opt for both membershp and data consstency exhbts the same trend and s not shown to avod clutter.) We see that for the same operatonal condton (b=4 s chosen as the example), as T decreases n opt ncreases, e.g., n opt goes from to 4 as T goes from 0τ to τ, because wth a smaller T, membershp changes due to node falures can be performed more rapdly, thus favorng a larger area for whch membershp changes are mostly due to node falures. T= T=2 T=5 T=0 T=20 response tme metrc. A more frequent perodc detecton actvty (.e., a smaller T) degrades the response tme metrc more because more tme wll be spent by the leader to do membershp mantenance nduced by node falures, thus causng any concurrent state-change operaton to be delayed. Isolatng out n=3 as a case study, Fgure 6 shows that the adverse effect of T on R s especally pronounced when the falure rate (φ) s hgh at whch the leader must perform falure-nduced membershp change operatons very frequently n order to mantan membershp consstency, thus causng a hgh delay n the response tme per state-change operaton. Whether we should select a short T and a large area sze n to yeld hgh consstency at the expense of tmelness, or vce versa, depends on the applcaton s QoS requrements n consstency and tmelness. References [] T.D. Chandra, V. Hadzlacos, S. Toueg and B. Charron- Bost, On the mpossblty of group membershp, 5 th ACM Symposum on Prncples of Dstrbuted Computng, New York, USA, 996, pp [2] J. Dahmann and J.W. Wlson, MDA and HLA: Applyng standards to development, ntegraton and test of the sngle ar pcture ntegrated archtecture behavor model, Fall Smulaton Interoperablty Workshop (SIW), Orlando, Florda, [3] M.J. Fsher, N.A. Lynch and M.S. Paterson, Impossblty of dstrbuted consensus wth one faulty process, Journal of the ACM, Vol. 32, Aprl 985, pp [4] Q. Huang, C. Julen and G.-C. Roman, Relyng on safe dstance to acheve strong parttonable group membershp n ad hoc networks, IEEE Transactons on Moble Computng, Vol. 3, No. 3, Response Tme (R) Faluer Rate (φ ) per Moble Node T= T=2 T=5 T=0 n =3 T=20 [5] M.-O. Klljan, R. Cunnngham, R. Meer, L. Mazare and V. Cahll, Towards group communcaton for moble partcpants, ACM st Internatonal Workshop on Prncple of Moble Computng, Newport, Rhode Island, USA. [6] Y.B. Ln, L.F. Chang and A. Noerpel, Modelng Herarchcal Mcrocell and Macrocell PCS Archtecture, ICC, June 995. Fgure 6: Pronounced Adverse Effect of T on R at Hgh Falure Rate (φ). Wth the above analyss, we know that we could acheve reasonably hgh membershp consstency wth a large geographcal area sze n, and a small T. Unfortunately, both a large n and a small T adversely degrade the [7] G.-C. Roman, A. Huang and A. Hazem, Consstent group membershp n ad hoc networks, IEEE Internatonal Conference on Software Engneerng, 200, pp [8] Stochastc Petr Net Package (SPNP) User s Manual, Verson 6, Department of Electrcal and Computer Engneerng, Duke Unversty, 999.