1 The Random Trip Mobility Model Milan Vojnovic (Microsoft Research) Computer Lab Seminar, University of Cambridge, UK, Nov 2004 with Jean-Yves Le Boudec (EPFL) part of simulations by Santashil PalChaudhuri (Rice University)

2 Examples

3 RWP: random waypoint (Johnson and Maltz, 1996)

4 RWP on general connected domain

5 RWP on general connected domain (cont’d) called city-section (Camp et al, 2002)

6 Space graphs readily available from road-map databases Example: Houston section, from US Bureau’s TIGER database (S. PalChaudhuri et al, 2004)

7 a restricted RWP ( Blažević et al, 2004)

8 a restricted RWP (Jardosh et al, 2003) (cont’d)

9 random walk with wrapping

10 random walk with reflection

11 What do we know about these models ? RWP considered harmful by Yoon et al (IEEE Infocom 2003) –speed decay: in ns-2 simulations, average speed decays with time –fix: redefine the speed distribution (at waypoints) Avoid transience : initialize mobility state, so that mobility is in steady-state throughout a simulation ( = perfect simulation) –Partial fix for RWP by Yoon et al (ACM Mobicom 2003): initialize the speed to a sample from its time-stationary distribution –Complete fix for RWP on a rectangle by Lin et al (IEEE Infocom 2004): initialize also node position to a sample drawn from the time- stationary distribution of position

12 Problems that we study The speed decay is due to non existence of steady-state –Under what conditions there exists a steady-state ? –If exists, is it unique ? I am interested in steady-state of my mobility model –What are steady-state distributions of mobility states for my model ? I want to run perfect simulations of mobility –How do I initialize my simulation so that it is perfect, i.e. free of transients ?

13 Why do we care about transients ? Or: why do we wish to run perfect simulations of mobility ? Simulations of mobility are commonly run with initial transient The simulation traces are then truncated and initial part thrown away in order to alleviate the transience effects How do we know where to truncate ? Initial transient may last as long as a typical simulation duration ! next couple of slides …

14 On transience longevity Example: revisit the restricted RWP instance: mobile always moves speed fixed to 1.25 m/s destination vertex drawn at random paths are shortest-length between vertices pairs default initialization: mobile placed at a random vertex (as in Jardosh et al) Q: How long it takes for this probability to converge to steady-state? Consider: Prob((Path at time t) = p)

15 On transience longevity (cont’d) Transient phase lasts 1000’s of seconds Typical simulation run is of the order 1000 seconds Prob((Path at time t) = path) Initial transient lasts as long as a typical simulation duration

16 Does transience of mobility affect performance of a protocol that I study ? Numeric speed is random, uniform on 0.01 to 9.99 m/s Pause time is random, uniform on 0 to 100 seconds 50 mobiles Default initialization: t=0 is a trip transition instant, each mobile initially in move phase 20 data connections, each with packet sent rate = 1 pkt/s packet length = 512 B Performance measure: packet delivery ratio = (# of received packets) / (# of transmitted packets), over a time interval Example: DSR protocol with restricted RWP on the Houston section

17 … transience of DSR Packet delivery ratio t (sec) default initialization (non perfect mobility simulation): perfect mobility simulation:

18 Outline Definition: The Random Trip Mobility Model –many existing mobility models in one (all on these slides), and new ones –easy-to-check conditions that guarantee existence of a unique time-stationary distribution –time-stationary distributions and their properties Perfect sampling algorithm –for the broad class of random trip mobility models –novelty: requires no knowledge of geometric normalization constants when they are difficult to compute Conclusion Pointer to randomtrip tool to use with ns-2

19 The Random Trip Mobility Model (basic definitions) domain A Path P n : [0,1]  A trip duration S n M n =P n (0) M n+1 =P n+1 (0) trip start TnTn trip end T n+1 Trip selection rule: at a trip transition instant T n, choose (P n,S n )

20 Path and Trip duration (P n,S n ) Example (RWP on a convex domain*): Path: P n (u) = u M n + (1-u) M n+1, u  [0,1] Trip duration: S n = (length of P n ) / V n V n = numeric speed drawn from a given distribution *convex domain := a domain such that for any two points in the domain, the line segment between these two points lies in the domain

21 Path and Trip duration (P n,S n ) (cont’d) Example (Random Walk Models): Pick a movement direction Draw a trip duration S n Path specified by the direction and trip duration + additional rules Additional rules: wrapping reflection

22 The Random Trip Mobility Model (further definitions) The trip selection rule is driven by phases I n Phases I n is a Markov chain –Example (RWP): I n = either pause or move Mobility state: (I(t),P(t),S(t),U(t)) U(t) = fraction of time elapsed on the trip at time t

23 The Random Trip Mobility Model (assumptions) (H1)(P n,S n ) is independent of all past, conditional on (M n,I n )

24 The Random Trip Mobility Model (assumptions cont’d) (H2) Either is true: (H2a) M n+1 independent of past phases I n,I n-1, … and n, conditional on I n (renewal points) for a set of selected transitions of I n, M n+1 independent of all past, conditional on I n or (H2b) M n independent of I n and n (S n,I n+1 ) independent of all past, conditional on I n

25 Random Trip Mobility Model (assumptions cont’d) (H3)Markov chain I n is positive recurrent True, in particular, if the state space of I n is finite, and all the states communicate. Remark: (H1)-(H3) true for all examples on these slides

26 When a time-stationary distribution of mobility state exists and is unique ? Theorem: Under (H1)-(H3), a random trip mobility model has a time-stationary distribution, if and only if the mean trip duration sampled at trip transition instants, E 0 (S 0 ), is finite. Whenever it exists, a time-stationary distribution is unique. Proof: shows that (I n,P n,S n ) has a unique stationary distribution verifies conditions of Slivnyak’s inverse construction

27 When the conditions fail ? Example: RWP as was implemented in ns-2 At trip endpoints, numeric speed is independent of trip distance => Numeric speed is uniformly distributed on an interval (0,vmax] => Found and called “harmful” by Yoon et al (IEEE Infocom 2003) The theorem tells us that for this RWP, no steady-state exists Renders many simulations results unreliable

28 What is time-stationary distribution of mobility state ? Theorem: Assume (I(t),P(t),S(t),U(t)) has a unique time-stationary distribution (provided by our previous theorem). The time-stationary distribution of (I(t),P(t),S(t),U(t)) is U(0) is independent of (I(0),P(0),S(0)) and uniform on [0,1] Proof: Palm inversion formula. Prob 0 (I 0 = i) E 0 (S 0 | I 0 = i)

29 What is Palm inversion formula ? A mean-value formula of Palm calculus ( = a set of results for stationary point processes) Palm inversion formula relates time-stationary distribution and event-stationary distribution ( = as seen at instants of a point process) Holds in general for a stationary point process, not only for renewal processes as assumed in previous work

30 Knowing Palm inversion formula, the rest is easy Time-stationary distribution of phase:

31 Knowing Palm inversion formula, the rest is easy (cont’d) Intermediate step: Time-stationary distribution of (phase, trip duration, and trip elapsed time), conditional on phase:

32 RWP time-stationary distributions Theorem: Under the time-stationary distribution: Conditionally on the phase I(t)=(l,l’,r,move) 1.Numerical speed is independent of path and position; speed density = 2.dP(P(t)(0)=m 0,P(t)(1)=m 1 )=K ll’ d(m 0,m 1 ) 3.Given (P(t)(0) =m 0,P(t)(1) =m 1 ), position X(t) uniform on the segment [m 0,m 1 ] Conditionally on the phase I(t)=(l,l’,r,pause), 1.Position and remaining pause time are independent 2.Position is uniform in A 3.Density of the remaining pause time = Remark: the independency property in item 1 previously only conjectured

33 Perfect sampling Goal: draw a sample from the time-stationary distribution (provided it exists) of the mobility state (I(t),P(t),S(t),U(t)) Recall: normalization constants

34 Perfect sampling (cont’d) For i specifying move phase, and numerical speed and distance on a trip independent: for RWP-like models this is a geometric constant For RWP with domain = rectangle, the geometric constant is average distance between two points on a rectangle (known in closed-form by Ghosh (1951)) Such geometric const. are known for some elementary domains: They are in general difficult to compute, if not impossible

35 Rejection sampling lemma For perfect sampling, we do not need to know geometric constants, when they are difficult to compute We want to sample a random vector (J,Y) on a space ( J,R d ) with density Suppose we know a factorization where g i (.) is a probability density

36 Rejection sampling lemma (cont’d) “Twist” the distribution of J as follows The sample is drawn from the given density of (J,Y)

37 Perfect sampling for restricted RWP with one sub-domain A 1 = average distance between two random points on a domain A 1 = bound on distance between any two points in A 1 The general case with an arbitrary number of sub-domains is in principle similar, but with description complexity

38 What do I gain with this perfect sampling algorithm ? When geometric constants are unknown, we may estimate them by Monte Carlo –This may be time consuming The proposed perfect sampling algorithm needs only: a bound on any possible trip distance, under a given phase In many cases these bounds are easy to compute Example (the restricted RWP):

39 Illustration: Perfect samples of positions for some of our examples Restricted RWPs: RWP on a non convex domain:

40 Perfect sampling for random walk models By definition, for RWP models, we know distributions of the mobility state at trip transition instants For random walk models we need first to find these distributions Theorems: For random walk with wrapping, if M 0 is uniformly distributed on A, so is M n for any n>0. The same holds for random walk with reflection. Proof: By periodicity of the wrapping and reflection mappings.

41 Perfect sampling for random walk models (cont’d) For RW with wrapping: Similar result obtained for RW with reflection

42 Conclusion Proposed: the Random Trip Mobility Model –contains many existing and new mobility models in one Gave conditions for the Random Trip Mobility Model that guarantee existence and uniqueness of a time-stationary distribution Proposed a perfect sampling algorithm to sample mobility state from its time-stationary distribution (whenever exists) –The sampling algorithm is for a broad set of the random trip mobility models –The sampling algorithm does not require knowing normalization constants when they are difficult to compute – a bound on trip distance suffices –The sampling algorithm is implemented to use with ns-2, which enables to run perfect simulations of mobility

43 Conclusion (cont’d) By-products: Demonstrated that transience for some mobility models may last as long as a typical simulation duration --- a compelling reason to run perfect simulations of mobility Proved that in steady-state of RWP models, node position and numerical speed are independent --- previously conjectured Showed new distribution invariance properties for random walk models with wrapping and reflection, which yield perfect sampling algorithm for these models

44 Pointers The Random Trip Mobility Model described in: “Perfect Simulation and Stationarity of a Class of Mobility Models,” J.-Y. Le Boudec and V. IEEE Infocom 2005 (to appear) available as EPFL Technical Report IC/2004/59:

45 Additional pointers Web Page: The Random Trip Mobility Model On this web page: Download: randomtrip –ns-2 code of random trip, with perfect simulation (by S. PalChaudhuri, Rice University)