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
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