Prepared through collaborative participation in the Communications & Networks Consortium sponsored by the U.S. Army Research LabAn SAIC Company

Authors:İbrahim Hökelek, CCNYMariusz A. Fecko, TelcordiaM. Ümit Uyar, CCNY

ANALYTICAL MODEL OF A VIRTUAL BACKBONE STABILITY IN MOBILE ENVIRONMENT

2

General Concept: Reliable Server Pooling (RSP)

Goal: Providing naming service to clients that need uninterrupted access to servers

Focus: Scalable and survivable architecture for ad hoc networks

Name Servers (NSs)

Pool Users(PUs)

Pool 1:PE11, PE12, …

Pool N: PEN1, PEN2, …

associate,register,

I am alive

associate, request, PE failure

advertise

peer discovery

Pool Elements (PEs)

3

RSP Scope

(3) (1)

PU1

NS1(PU1’s home)

PE2PE1

Pool

(4)

(5)(2) PE2 fail

NS2(PE2’s home)

(6) PE2 is dereg’ed

4

RSP across multiple domains

R R

R

BR

PE5

R

hub

PE PUR

BR

PU

ENRP

ENRP

PU

PU

Backbone Network

ENRPName Server(Endpoint Name Resolution Protocol)

R Router

PUBR Border Router PEPool: Elements Users

PU PE1

ENRP1ENRP2'

PU1 PE2

PE3

ENRP2

PE4

PUPU

(1)(3)

(5)

(2)

(4)

(7) (8)

• ENRP1 knows only PE1 and PE2• both quickly made available to PU1• PE1 can fail over from PE1 to PE2• ENRP1 may or may not contact ENRP2• if YES, then PE3, PE4, and PE5

become available to PU1 after delay

(6)

D1

D2

Pool: PE1, PE2, PE3, PE4, PE5

5

RSP across multiple domains

•Experiments show that a single flat namespace causes problems•signaling overhead due to hunting for Home NS and/or advertisements•difficulty in synchronizing among multiple NSs

•Investigating multiple domains and local name spaces•features

•one logical NS per domain (primary plus backups)•pools may span multiple domains and local name spaces•NS keeps only partial membership information for a given pool

•advantages•limited traffic: home hunt, NS advertisements, PE heartbeats, etc...•no need to synchronize NSs•quick response within local domain

•issues•load balancing among PEs may not be optimum within domain•new procedure needed for querying NSs in other domains to get a complete pool-membership information

•protocols need to be redesigned•expect to further reduce the signaling overhead

6

RSP over Virtual Backbone

Main focus: Registration and discovery services for PEs/PUs– Developed new architecture and protocols for RSP

Novel scheme is called Dynamic Survivable Resource Pooling (DSRP)

DSRP implements RSP over virtual backbone for ad hoc networks

DSRP architecture is (practically) infrastructure-less

– No fixed infrastructure; system fully distributed

– Naming system deployed on dynamically assigned VB nodes backbone nodes serve as dynamic Name Servers NSs form an overlay of nodes as a connected dominating set (CDS)

– VB is highly survivable

– Main Features of DSRP Reorganization in response to mobility, failures, and partitioning Fast response time if local name resolution possible Load balancing of pool elements provided by NSs or pool users Scalability when the network size grows

7

Analytical Model of DSRP

Motivation– Only simulations available for single-PE discovery over VB

Approach– Main end-user metric:

What is the expected delay to get service request resolved?

– Steps probability of a PE/PU (not) having an operational PNS stability of NS, i.e., expected time for NS to leave the backbone expected delay for PE/PU to find new PNS when the previous one

becomes unavailable

– Base model We adapted the discrete-time random walk model proposed by Y. Tseng

et al., “On Route Lifetime in Multihop Mobile Ad Hoc Networks” Dynamics of nodes and VB driven by random node movement Probabilistic link creation/failure models

8

(0,-4)

(0,4)

(-3,1)

(-2,1)

(-3,2)

(-4,1)

(-4,2)

(-3,0)

(-2,0)

(-1,2)

(0,2)

(-1,3)

(-2,2)

(-2,3)

(-1,1)

(0,1)

(0,-1)

(1,-1)

(0,0)

(-1,-1)

(-1,0)

(0,-2)

(1,-2)

(2,0)

(3,0)

(2,1)

(1,0)

(1,1)

(2,-1)

(3,-1)

(3,-3)

(4,-3)

(3,-2)

(2,-3)

(2,-2)

(3,-4)

(4,-4)

(4,0)

(3,1)

(0,3)

(0,-3)

(-1,-2)

(-2,-1)

(-2,-2)

(-3,-1)

(-4,0)

(-4,3)

(-3,3)

(-4,4)

(-3,4)

(-2,4)

(1,2)

(2,2)

(4,-1)

(4,-2)

(2,-4)

(1,-3)

(-1,-3)

(1,3)(-1,4)

(1,-4)

x

y

MN1

MN2

MN3

MN4

<2,0>

<2,0><4,-4>

<-4,4>

Area covered by

MANET

Available Link state

nav=2 Total number of

layers ntot=9

Unavailable Link state n=4

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Random Walk Model and Link State Changes

<x’,y’> <x,y> <x-1,y> <x-1,y-1> <x,y-2> <x+1,y-2> <x+1,y-1> <x+1,y> <x,y-1> <x+2,y-2> <x+2,y-1>

Probability 6/36 2/36 2/36 1/36 2/36 2/36 2/36 2/36 1/36 2/36

(0,0)

(1,0)

(0,1)

(-1,0)

(-1,1)

(0,-1)

(1,-1)

D1

D2

D3

D4

D5

D6 (x,y)

(x+1,y)

(x,y+1)

(x-1,y)

(x-1,y+1)

(x,y-1)

(x+1,y-1)

D1

D2

D3

D4

D5

D6

<x,y>

<x+1,y>

<x,y>

MN1 MN2

<x’,y’> <x+1,y+1> <x,y+1> <x+2,y> <x,y+2> <x-1,y+2> <x-1,y+1> <x-2,y+2> <x-2,y+1> <x-2,y>

Probability 2/36 2/36 1/36 1/36 2/36 2/36 1/36 2/36 1/36

Figure Example link state changes

The probability distribution for a wireless link to switch from state <x,y> to state <x’,y’> after one time unit

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State Transition Diagram and our modifications

NOTE: taken from the Tseng’s paper

They consider only available links

Extending the number of layers to cover all area (all available and unavailable links)

Bouncing back from the highest layer M represents state transition matrix obtained using the state transition diagram

ntotnav=5

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Analytical Model

VB behavior with respect to link changes

– VB nodes are determined dynamically when the network topology changes

– Preference given to a node with the highest degree, i.e., the number of available links

– We approximate this behavior by considering the threshold number of available links

– We are interested in expected times to cross the threshold

Mi,j represents the probability to transit from the ith state to jth state

Suppose that a wireless link is in state i at initial. Pa(i) and Pu(i) denote the probabilities that the link will be available and unavailable in the next time unit, respectively

j= 0, 1, 2, …., sa represent available link states

j= sa+1, sa+2, …., sT represent unavailable link states

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Analytical Model

Assume there are N mobile nodes in the network

Consider only a particular node. There are K=N-1 possible bidirectional links from this node to all other nodes

Assume k available links for this node, there are Ku=K-k unavailable links

Let Pdap(k,l) denote the probability that l of k available links will disappear and Pap(Ku,l+1) denote the probability that l+1 of Ku unavailable links will appear in one time unit

If we use the steady state values of the state transition matrix Mi,j , then Pa(i) will be same for all inner link states i and Pu(i) will be same for all outer link states i

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Analytical Model

Then Equations 3 and 4 will be simplified as follows:

Given that there are k available links, Pk,k+1 denotes the probability that there will be k+1 available links in the next time unit

If we generalize the above formula for Pk,k+h where h can be negative or positive (all possible number of link changes)

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0 k-h k-1 k k+1 k+h K

A new Markov chain obtained using the stationary distribution of the state transition matrix M. Here, a state represents the number of available links for a node

P is the corresponding state transition matrix

P0,K

PK,0

Pk-h,K

Pk+h,K

Pk,K

P0,k+h

PK,k-h

PK,k-1

PK,k+h

PK,k

Pk-1,K

Pk+h,0

P0,0

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Analytical Model

πk denotes the steady state probabilities of the P matrix. Let a random variable Z denote the number of link changes in one time unit. The probability distribution of Z can be calculated as follow:

Number of available links

Time steps

dthr

0 m-1 m+1

d0

m1 2 3 4

Z1, Z2, …, Zm represent the link changes for 1st, 2nd, …, mth

steps and Sm represents the net link change until the mth step

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First Passage Time Analysis

The number of transitions going from one state to another for the first time

We combined states equal to or greater than dthr into a single state dthr

We modified the transition probabilities: only the dashed lines are modified

The expected first times going from k to dthr, given that there are k (k < dthr) available links at initial, using the above Markov chain

0 k-h k-1 k k+1 dthr

1

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Numerical Results

N: number of nodes, ntot: total number of layers, nav: number of layers representing available links

ntot determines the size of the geographic area for the fixed cell size

For numerical results, N=106, nav=5, d0=0 and dthr varied

Network types in terms of its density: sparsest (ntot=40), sparse (ntot=30), typical (ntot=20), dense (ntot=15), and densest (ntot=10)

0 2 4 6 8 10 12 14 16 18 2010

0

101

102

103

104

105

106

The expected first times vs threshold

threshold

Exp

ecte

d tim

e

densest

dense

typical

sparse

sparsest

Network Mean N E. Time

Sparsest 2.07 ~2

Sparse 3.66 ~2

Typical 8.33 ~2

Dense 15.05 ~2

Densest 34.90 ~2

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Conclusion and Future Work

The mobility part of DSRP has been modeled analytically

Future Work:

Finding one unit time for different cell size and mobile node speed distributions

Combining this analysis with backbone formation and maintenance algorithms to find the expected time that an NS will remain an NS and the expected time that a non-NS will be an NS

Finally, developing an analytical model for DSRP using the above expected times together with a service discovery model

Application to other schemes depending on link stability– Routing– Bandwidth-estimation algorithms

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Part I: Backbone Formation and Maintenance

5

51

2

53

4

6

721

1

31

41

51 6

5

76

White – Undecided nodesBlack – VB nodes (decided) Green – non-VB nodes (decided)