The Message Delay in Mobile Ad Hoc Networks

1. The Message Delay inMobile Ad Hoc Networks Robin Groenevelt1,2,3 Ger Koole4 Philippe Nain1 6 October, 2005 1) INRIA, project MAESTRO 2) Université de Nice – Sophia Antipolis 3) R&D Techlabs, Accenture 4) Vrije Universiteit, Amsterdam

2. Agenda Introduction & Motivation The Model • Assumptions • Poisson Meeting Times • Analytic Solution Applications • Random Waypoint • Random Direction • Random Walker

3. Mobile Ad Hoc Networks • Autonomous mobile radio devices • Nodes have a radio transmission range • Routing capabilities • Lack of fixed infrastructure Examples: • emergency situations (rescue, physical disasters) • everyday life (connectivity anytime and anywhere) • military applications • IEEE 802.11 ad hoc mode • Bluetooth

4. Study of ad hoc networks Performance measures of mobile ad hoc networks: • Throughput/capacity • Message delay 1 ,2 ,3 3 Other parameters: • Connectivity • Interference Device characteristics: radio transmission range, energy consumption, relay protocol, memory, computing capabilities [1] P. Gupta and P.R. Kumar. The capacity of wireless networks. IEEE Transactions on Information Theory, 2000. [2] M. Grossglauser and D. Tse. Mobility increases the capacity of ad-hoc wireless networks. Best paper Infocom 2001. [3] A. Gamal et al. Throughput-delay trade-off in wireless networks. Best paper Infocom 2004.

5. Message Delay Question: What is the message delay with N+1 nodes moving independently in two dimensions? (one source, one destination, and N-1 relay nodes) Let’s start with two nodes: • in a LxL square • fixed transmission radius R

6. Assumptions • Nodes move according to the same mobility model • Nodes start from steady-state • Nodes always “on” Our focus is on message delay due to the mobility (model). → Time to transmit message is small compared to time required for nodes to come within communication range of one another. → Transmission time is zero (no queuing delay) • No interference (sparse network, R<<L)

7. Inter-meeting times Observation: Let R<<L. The inter-meeting times for the random directionand the random waypointmobility models are approximately exponentially distributed. 1,2 For random walkers the tail is exponentially distributed. [1] Sharma, Mazumdar. Delay and capacity trade-off in wireless ad hoc networks with random mobility. [2] Groenevelt. PhD thesis. Stochastic Models in Mobile Ad Hoc Networks.

8. Relay protocols We consider two relay protocols: Unrestricted multicopy protocol: nodes copy the message whenever possible. Two-hop multicopy protocol*: the message gets copied only by the source node or to reach the destination in the second hop. * Not to be confused with the two-hop relay protocol.

9. Quantities of interest Define • T2 (resp. TU), the message delay under the two-hop (resp. unrestricted) multicopy protocol. • N2{1,…,N}(resp. NU{1,…,N}), the number of occurrences of the message in the network (excluding the message at the destination) at the moment the destination receives the message.

10. The model: two-hop multicopy The model: unrestricted multicopy Model the number of occurrences of the message as an absorbing Markov chain: • State i{1,…,N} represents the number of occurrences of the message in the network. • State A represents the destination node receiving (a copy of) the message.

11. Message delay Theorem: The Laplace transform of the message delay under the two-hop multicopy protocol is: and

12. Message delay Theorem: The Laplace transform of the message delay under the unrestrictedmulticopy protocol is: and

13. Expected message delay Corollary: The expected message delay under the two-hopmulticopy protocol is and under theunrestrictedmulticopy protocol it is Where γ ≈ 0.57721 is Euler’s constant.

14. Relative performance The relative performance of the two relay protocols: and Note that these are independent of λ!

15. Some remarks Remarks: • These expressions hold for all mobility models which have exponential meeting times. • Two mobility models which give the same λ also have the same message delay for both relay protocols! • Mean message delay scales with mean meeting times.

16. Some remarks • No dimension specified • The mobility pattern is “hidden” in λ • λ depends on: - mobility pattern - surface area - transmission radius - interference

17. Applications We consider three movement patterns: 1) Random waypoint 2) Random direction 3) Random walkers

18. Movement patterns Random Waypoint 4 ,5 ,6 Select independently from each other: • waypoint uniform from convex region • speed  (vmin,vmax) • pause times [4] Bettstetter et al. Stochastic properties of the random waypoint mobility model. ACM/Kluwer Wireless Networks: special issue on modelling and analysis of wireless networks, 2004 [5] Yoon, Liu, Noble. Random waypoint considered harmful. Infocom 2003. [6] le Boudec, Vojnovic. Perfect simulation and stationarity of a class of mobility models. Best paper Infocom 2005.

19. Movement patterns Random Direction 7 Select independently from one another: • (uniform) direction • travel time At borders: wrap-around or reflection [7] Nain, Towsley, Liu, Liu. Properties of random direction models. Infocom 2005.

20. Movement patterns Random Walkers Equal probability of hopping in any direction

21. Inter-meeting times Theorem: Let R<<L. The inter-meeting time for the random direction and the random waypointmobility models is approximately exponentially distributed with parameter where E[V*] is average relative speed between two nodes and is pdf of spatial node distribution in the point * ** [*] Groenevelt. PhD thesis. Stochastic Models in Mobile Ad Hoc Networks. [**] Navidi, Camp. Stationary distributions for the random waypoint mobility model. IEEE Transactions of mobile computing, 2004.

22. Inter-meeting times Corollary: Let R<<L. The inter-meeting time for the random direction and the random waypoint mobility models is approximately exponentially distributed with parameter Here E[V*] is the average relative speed between two nodes and ω≈ 1.3683 is the Waypoint constant. If speeds of nodes is constant and equal to v, then

23. Expected message delay Expected message delay with two-hopprotocol: Expected message delay with unrestrictedprotocol:

24. Simulation settings Nodes move on a square of size 4x4 km2 (L=4 km) Different transmission radii (R=50,100,250 m) No pause times Random Waypoint: [vmin,vmax]=[4,10] km/hour Random Direction: [vmin,vmax]=[4,10] km/hour travel time ~ exp(4) Random Walker: streets 80 meters apart speed = one street/minute

25. Inter-meeting time between two nodes

26. Example: unrestricted multicopy

27. Example: Unrestricted multicopy Distribution of the number of copies (R=50,100,250m):

28. Example: two-hop multicopy

29. Example: two-hop multicopy Distribution of the number of copies (R=50,100,250m):

30. Incorporating interference Model applies if interference is not an issue. Two observations: • When the number of nodes increases: R should decrease to prevent interference from becoming excessive. • λ linear in R (for R << L)   and

31. Incorporating interference It has been shown that a constant capacity per node is achieved with the two-hop relay protocol by taking  and Remark: Let TR be the message delay under the two-hop relay protocol. We find through similar analysis 

32. Conclusions Generic model presented with 2 parameters which: • computes message delay accurately; • captures more than one mobility model; • studies the message delay due to relay strategies apart from the underlying mobility models. Different mobility models can behave in same way (hot-spot vs. uniform) Model can also be used to model epidemics

33. ??? ANY QUESTIONS ???

34. Intuitive explanation Exponential distribution finds its roots in the independence assumptions of each mobility model: • Nodes move independently of one another • Random waypoint: future locations of a node are independent of past locations of that node. • Random direction: future speeds and directions of a node are independent of past speeds and directions of that node. There is some probability q that two nodes will meet before the next change of direction. At the next change of direction the process repeats itself.

35. The derivation of λ Assume a node in position (x1,y1) moves in a straight line with speed V1. Position of the other node comes from steady-state distribution with pdf π(x,y). Look at the area A covered in Δt time:

36. The derivation of λ Probability that nodes meet given by For small r the points in π(x,y) in A can be approximated by π(x1,y1) to give Unconditioning on (x1,y1) gives

37. Future research Extensions to the model: • different relay protocols • add an expiry time to the messages • nodes which disappear • include transmission times • non-homogeneous networks(different transmission ranges and mobility models)

38. Future research Consider two mobility models with the same λ. Results coming forth from the model:  for the same relay protocols different mobility models have the same message delay  expectation is linear in the mean meeting time. Does the same hold for other relay protocols? Does the same hold for non-exponential meeting times?