Maximizing Path Durations in Mobile Ad-Hoc Networks

Yijie Han and Richard J. LaDepartment of ECE & ISRUniversity of Maryland, College Park

CISS, Princeton UniversityMarch 22nd, 2006

Outline

Background Basic Model Setup Distributional convergence Proposed algorithm

Maximizing expected path durations NS-2 simulation results

Parameter update Conclusion & Future Directions

Background

Ad hoc network routing protocols Table-driven routing protocols (proactive)

Attempt to maintain consistent, up-to-date routing information from each node to every other node in the network.

Each node maintains one or more tables to store routing information.

Example: DSDV (Destination-Sequenced Distance-Vector), WRP (Wireless Routing Protocol), etc

On-demand routing protocol (reactive) Attempt to minimize the number of required broadcasts by

providing a path only when requested Require path/route discovery phase/mechanism Examples: AODV( Ad-hoc On-demand Distance Vector),

DSR (Dynamic Source Routing)

Motivation On-demand routing protocols in ad-hoc networks

Path recovery procedure initiated when an existing path is broken

Disruption in network service to applications Performance and overhead shaped by the distribution of

link and path durations Suggests that (expected) path duration should be taken

into account when selecting a path Reduce overhead Provide more reliable network service to applications

Requires understanding of statistical properties of path duration

Existing protocols Ad-hoc On-demand Distance Vector (AODV)

Selects the first discovered route

Dynamic Source Routing (DSR) Selects the min-hop route

Associativity Based Routing (ABR) Each node maintains “associativity” for each neighbor from

beacons Higher beacon counts = more stable links

Destination selects the path with the highest average associativity

Outline

Background Basic Model Setup Distributional convergence Proposed algorithm NS-2 simulation results

Parameter update Conclusion & Future Directions

Basic Model (for studying statistical properties of path duration) V = {1, …, I} - set of mobile nodes moving across a

domain D of R2 or R3

- location/trajectory of node i

Connectivity between nodes {0, 1}-valued reachability process

between two nodes

ij(t) = 1 – if the link (i,j) is up

ij(t) = 0 – if the link (i,j) is down

ij(t) = ji(t) – symmetric links

Basic Model

Link durations {Uij(k), k = 1, 2, ,…} and {Dij(k), k = 1, 2, …}

Uij(k) (resp. Dij(k)) – duration of k-th up (resp. down) time

Time-varying graph (V, E(t))

t

Basic Model

Basic Model

Path discovery phase Path available between s and d if a set of links

provides connectivity

May not be unique Routing algorithm selects one

Denote the set of links along the selected path by Lsd(t)

s

d

n1

n2

n3

n4

For each link - time to live or

excess life after time t

Time to live or duration of a path Path available till one of

the links goes down Path duration = amount of

time that elapses till one of the links in breaks down

Excess Life and Path Duration

Question: What does the distribution of look like? In particular, when the hop counter is large

In a large scale MANET, the number of hops is expected to be large

Outline

Background Basic Model Setup – Parametric Scenario and Difficulties Distributional convergence Proposed algorithm NS-2 simulation results

Parameter update Conclusion & Future Directions

Scaling: For each fixed n = 1, 2, …, -- set of mobile nodes -- domain across which nodes move

Stationarity: Reachability processes jointly stationary

constitutes a stationary sequence with generic marginals

- CDF of

A pair of source and destination nodes selected at time t = 0 for each n

Parametric Scenario

Define

Excess or residual life of a link

Distribution of forward recurrence time Follows from elementary renewal theory

Parametric Scenario (cont’d)

Path duration -

Explore the distributional properties of the rvsas

Parametric Scenario (cont’d)

Sources of Difficulty

1. - random set that depends on Assume is a deterministic sequence

with for convenience

Example: Fix the domain, and randomly select the locations of the

source and destination Randomly place n2 – 2 other nodes in the domain Transmission range decreases as 1/n Number of hops along the shortest path increases with n

Sources of Difficulty (cont’d)

2. Dependence of reachability processes

Introduces dependence in link excess lives

Asymptotic independence – dependence in link excess lives goes away asymptotically as hop distance increases

Mixing conditions

Outline

Background Basic Model Setup Distributional convergence Proposed algorithm NS-2 simulation results

Parameter update Conclusion & Future Directions

Assumptions Assumption 1: (scaling) There exists such that

where

Scaling introduced for defining limit distribution parameter

Assumption 2: For every and any given there exists an integer such that

-Interpretation: probability that a link duration is strictly positive is one

Definitions

Array of -valued rvs

for notational

convenience

Definitions Let be a sequence of real numbers

Usually increases with n

Definitions

Sufficient condition:

Define

A sufficient condition is that there exists an arbitrarily small constant > 0 such that for all and

Assumptions

Assumptions

Interpretation of Assumption 4

Implications: For sufficiently large hop count, the expected path duration can be approximated by

Distributional convergence

Outline

Background Basic Model Setup Distributional convergence Proposed algorithm NS-2 simulation results

Parameter update Conclusion & Future Directions

Proposed algorithm

Link durations seen by a node likely to depend on its own type and the types of neighbors Different nodes with different speeds and capabilities Each node maintains average link durations Can maintain a separate average for each type of neighbors Average link duration used as estimate of expected link

durations (during path discovery)

Outline

Background Basic Model Setup Distributional convergence Proposed algorithm NS-2 simulation results - AODV

Parameter update Conclusion & Future Directions

NS-2 simulation - Setup

Modified AODV routing protocol 200 nodes in 2 km x 2 km rectangular region Transmission range = 250 m Two classes of nodes

Nodes with different speed (e.g., soldiers vs. jeeps or tanks) Class 1 node speed ~ [1, 5] m/s Class 2 node speed ~ [10, 30] m/s

Varying mixture Class1:Class2 = 140:60, 160:40, and 180:20

NS-2 simulation

NS-2 simulation

NS-2 simulation

Outline

Background Basic Model Setup Distributional convergence Proposed algorithm NS-2 simulation results

Parameter update Conclusion & Future Directions

Estimation of expected path duration

Recall: For sufficiently large hop count, the expected path duration can be approximated by

Question: For finite hop counts, how good is this approximation? For back-up paths Local recovery after a link

failure

Threshold update – local recovery

Select a back-up path only if the estimated probability of being available exceeds a certain threshold Probability of being available estimated to be

Not accurate due to discrepancy in exp. parameter and collected IPD value (sum of inverses of expected link durations)

Target probability Update the threshold as follows

where is the threshold after n back-up path tries and is the indicator function of a back-up path being available

Amount of time since last update

Threshold update Define to be the indicator function of the event that a

selected backup path is available when the threshold value is and - unknown distribution of and its

mean, respectively Assume (i) is strictly increasing in , and (ii) there exists

such that

Outline

Background Basic Model Setup Distributional convergence Proposed algorithm NS-2 simulation results

Parameter update Conclusion & Future Directions

Conclusions & Future Directions

Studied the statistical properties of path durations in MANETS Showed distributional convergence with increasing hop count Relationship between link durations and path duration

Proposed an algorithm for maximizing expected durations of selected paths Stochastic approximation based algorithm for handling the

discrepancy between IPD values and exponential parameters

Plan to implement with other on-demand routing protocols Validation of assumptions Convergence speed

Proposed algorithm in AODV Each node maintains a route entry from each known dest node

Up to k paths (instead of a single path in AODV) (i) dest seq. number, (ii) next hop, (iii) hop count, and (iv) Inverse Path

Duration (IPD) IPD = sum of the inverses of average link durations reported in a

path reply message Paths ranked based on (i) seq. number, (ii) IPD value, (iii) hop count

Request message (i) src ID, seq. number, (ii) broadcast ID, (iii) dest ID and seq. number,

and (iv) hop count to the src Reply message

(i) dest ID, (ii) dest seq. number, (iii) IPD value, and (iv) hop count Either an intermediate node or dest generates a reply message

Intermediate node – copy information from its entry Dest node – initialize IPD and hop count to zero