Tuning Message Size in Opportunistic Mobile Networks

arXiv:1001.3439v1 [cs.NI] 20 Jan 2010 T uning Message Size in Oppor tunistic Mobile Netwo rks John Whitbec k Thales Communi cations and UPMC P aris Univ ersitas V ania Conan Thales Communi cations Marcelo Dias de Amorim UPMC P aris Universitas ABSTRA CT W e describe a new mod el for stu dying in termittently co n- nected mo bile net w orks, based on Marko vian random tem- p oral graphs, that capt ures the influ ence of message size, maximum tolerated delay and link stabilit y on the deliver y ratio. Categories and Subject Descriptors C.2.1 [ Netw ork Arc hitecture and Design ]: Store and F orward Netw o rks; C. 4 [ Perfor mance of System s ]: Mod- eling T echniques General T e rms Theory , R eli abilit y Keyw ords Dela y T oleran t Netw orks, Random T emp oral Graphs, Mes- sage Size, Delivery Ratio 1. INTR O DUCTION The topology of a real-lif e netw ork of mobile h andheld devices evolv es o ver time as links come up and d own. Suc- cessiv e snapshots of the evolving connectivit y graph yields a temp or al gr aph , a time-indexed sequ ence of traditional static graphs. These present a number of new interesting metrics. F or example, there might exist a space-time path b etw een tw o vertices even if th ere never ex ists an end to end p ath b et ween them at any given moment. Since such temp oral graphs appear naturally when analyzing connectivity traces in which n odes p eriodically scan for neighbors, their theo- retical study is imp ortan t for und erstanding the und erlying netw ork dy namics. Modeling temp oral netw orks using random graphs is a relativ ely unexp lored field. Simple sequences of indep en- dent regular random graphs are u sed in [1] to analyze the diameter of opp ortunistic mobile netw orks. The notion of c onne ctivity over tim e is explored in [2 ] b u t looses any in- formation ab out the order in which contact opp ortu nities app ear. In this pap er, we improv e up on previous work, by captur- ing the strong real-life correlation b etw een the connectivity graphs at times t and t + 1. Since we will be examining Copyri ght is held by the autho r/o wner(s). MobiHel d’09, August 17, 2009, Barcelona , Spain. A CM 978-1-60558-444 -7/09/08 . T able 1: Model parameters N Number of nodes d Maximum dela y τ Time step r Av erage link lifetime α Pac ket size λ fraction of time a link is do wn how t he mess age size influences the delivery ratio, it is cru- cial to mo del link stabilit y , i.e., the p robab ility that a link up at time t remains so at time t + 1. In order to capture these correlations, we propose a Marko vian t emporal graph mod el. 2. MODEL W e consider temporal graphs of N mobile no des that evol ve in discrete time. The time step τ is equal to the shortest con t act or inter-con tact time. In a real-life trace, τ will b e equal to th e sampling perio d. The only differences b et ween successiv e t ime steps will b e whic h links are up and whic h are down. They can come up or go down at the b e- ginning of each time step, but the top ology then remains static until th e n ext time step. Eac h of the p oten tial N ( N − 1) 2 links is considered ind epen- dent and is mo deled as a tw o-state ( ↑ or ↓ ) Marko v chain. The evolution of the entire connectivity graph can also b e described as a Mark o v chain on th e tensor pro duct of th e state spaces of all links. W e note r the av erage num ber of time steps th at a link sp en ds in t h e ↑ state, and λ the fraction of time that a link sp en ds in the ↓ state. In a sense, r measures the evolution sp eed of the netw ork’s top olo gy while λ is related to its density . The av erage link lifetime is by defin itio n r τ while the a vera ge nod e degree is N − 1 1+ λ . When up, all links share th e same capacity φ and thus can transp ort the same quantit y φτ of information during one time step. W e will refer to φτ as the link size . Mess age size is equal to αφτ where α can b e grea ter or smaller than 1. By abuse of language, we will refer to α as the message size. F or example, a message of size 2 ( α = 2) will only be able to trav erse link s that last for more than 2 time step s, whereas a message of size 0 . 5, will b e able to p erform tw o hops during eac h time step. The message size thus defin ed is numerical ly prop ortional to τ . Small v alues of τ mean that the net work top ology’s c h ar- acteristic ev olution time is short an d thus only small amounts of information may b e t ransmitted o ver a link d u ring one time step. F urthermore we supp ose th an a mobile appli- cation can only tolerate a given delay in message deliv ery . W e note d the maximum dela y b eyond whic h a d eliv ery is 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 Delivery ratio Pac ket size ( α ) N = 20 r = 2 . 0 λ = 10 . 0 d = 10 d = 4 Figure 1: Influ ence of message size on delivery probability for different v alues of maximum d ela y ( d ). Each v alue of d corresponds to tw o lines: its u pper and lo w er b ounds. considered to hav e fail ed. 3. RESUL TS Using th is model, we can derive th e deliv ery ratio of a message using epidemic routin g for α ≤ 1, as well as upp er and low er b ounds when α > 1. T o b e successful, the delive ry has to o ccu r without exceeding the maxim um allo w ed dela y . Epidemic routing is useful for theoretical purp oses, since its delivery ratio is also that of the optimal single-cop y time- space routing proto col. Message size. ( Fig . 1) Messages larger than the link size see th eir deliv ery probability severel y d egra ded, though this is somewhat mitigated by longer maximum delays. On the other hand , messages smaller than the link size are able of making several h ops in a single t ime step. This is a great adv antage when the time constraints are p articularl y tigh t ( d = 4 in Fig. 1), but barely has any effect when the time constrain ts are lo oser. This also highligh ts t he influence of nod e mobilit y . Indeed, since the actual message size is p ro- p ortional to τ , high no de mobility (i.e. small τ ) make s the actual link size smaller and thus further constrai ns p ossible message size. F urth ermore, t he gain achiev ed by using small messages is b ounded b ecause because it hits the p erformance limit of epidemic routing. Indeed, the b est p ossible epidemic dif- fusion of a message will, at eac h time step, infect a whole connected component if at least one of its nodes is infected. A small enough pack et can sp read sufficiently quickly to ac hieve this, and thus even smaller pack ets bring n o p er- formance gain ( d = 4 in Fig. 1). Number of no des. (Fig. 2a) The delivery ratio tends to 1 as N increases. Indeed, for a giv en source/destination pair, each new node is a new potential rela y in the epidemic dissemination and thus can only help the delivery ratio. Average link lifetime. ( Fig . 2b) S h orter a vera ge link life- times make for a more dyn amic netw ork top olog y . Ind eed smaller va lues of r mak e for shorter contact and inter-con tact times and increases contact opp ortunities. Small messages ( α ≤ 1) take adv antage of this and their delivery ratio in- creases as r decreases. On th e other hand, excessiv e link in- stabilit y d riv es the delivery ratio for larger messages ( α > 1) to 0, b ecause few er links last longer than one time step. Average no de degree. (Fig. 2c) Greater connectivity in- creases the delivery probability . The sharp slop e of the curve when α ≥ 1 is reminiscent of p ercolation in random graphs 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 Delivery ratio N α = 1 α = 1/2 α = 2 (a) Number of no des 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 Delivery ratio r α = 1 α = 1/2 α = 2 (b) Average link lifetime 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 Delivery ratio ( N − 1) / (1 + λ ) α = 1 α = 1/2 α = 2 (c) Average no de degree 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 Delivery ratio d α = 1 α = 1/2 α = 2 (d) Maximum d ela y Figure 2: Influence of mod el parameters on the deliv ery ra- tio. When unsp ecified, N = 20, r = 2, λ = 10 and d = 5. when the av erage no de degree hits 1. Maximum D ela y . (Fig. 2d) A ll else b eing equal, there is clearly a threshold va lue b ey ond whic h almost all messages are delive red. This can b e linke d to t he space-time diameter of the underly in g top olog y [1]. Exp eriment al resul ts. W e measured the deliv ery ratio ac hieved by v arious message sizes by replaying a real-lif e wireless mobility trace (R ollernet [3 ]). A lthough our mo del ma y not b e quantitativ ely comparable to real-life traces due to unw anted small-w orld prop erties, it accurately p redicts the relations b etw een delivery ratio, maxim um dela y and message size. 4. CONCLUSION In this pap er, w e proposed a new mo del of random tem- p oral graphs that, for the first time, captures the correlation b et ween su ccess ive conn ectiv it y graph, and provides insights on the interactio n betw een no de mobility , maximum dela y and message size. In particular, we hav e sho wn t hat, given a certain maximum dela y and no de mobilit y , message size has a ma jor impact on th e delivery ratio. These results, which w e repro duced b y repla ying a real -life connectivity trace, should b e taken into consideration when designing and im- plementing services for mobile handhelds. 5. REFERE NCES [1] A. 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