ANALYTICAL MODEL OF A VIRTUAL BACKBONE
STABILITY IN MOBILE ENVIRONMENT
Authors:
İbrahim Hökelek, CCNY
Mariusz A. Fecko, Telcordia
M. Ümit Uyar, CCNY
An SAIC Company
Prepared through collaborative participation in the Communications &
Networks Consortium sponsored by the U.S. Army Research Lab
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
associate, request,
peer discovery PE failure
Name Servers
advertise
(NSs)
associate,
register,
I am alive
Pool 1:
PE11, PE12, …
Pool Users
(PUs)
Pool N:
PEN1, PEN2, …
Pool Elements
(PEs)
2
RSP Scope
NS2
(2) PE2 fail
PU1
(PE2’s home)
(5)
NS1
(PU1’s home)
(3)
PE1
(1)
PE2
Pool
(4)
(6) PE2 is
dereg’ed
3
RSP across multiple domains
R
BR
ENRP
R
R
hub
R
PE
PU
Name Server
Router
Border Router
(Endpoint Name Resolution Protocol)
Pool: Elements
Backbone
Network
R
ENRP2
PU
PU
BR
(7)
(6)
D2
PU
ENRP2'
(4)
(2)
PU1
Pool: PE1, PE2, PE3, PE4, PE5
BR
PE1
ENRP1
PU
PE3
(5)
(3) (1)
Users
• 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
ENRP
D1
PU
R
PE
ENRP
PE5
PU
PE4
PU
PE2
(8)
4
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
5
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
6
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
7
y
Area
covered by
MANET
Available
Link state
nav=2
(0,3)
(-2,4)
(-1,3)
(-3,4)
(-4,4)
(-2,3)
(2,2)
x
(1,2)
Total
number of
layers ntot=9
(0,2)
(-1,2)
(-3,3)
MN2
(-2,2)
(-4,3)
(-3,2)
<2,0>
(0,1)
(-1,1)
MN1
(-2,1)
(-4,2)
(-2,0)
(-4,1)
(0,0)
(-1,0)
(-3,1)
(4,-1)
<-4,4>
(1,-1)
MN3
(-4,0)
(-1,-1)
(-2,-1)
(-3,-1)
(-1,-2)
(4,-3)
(3,-3)
(2,-3)
(1,-3)
(0,-3)
(-2,-2)
MN4
(2,-2)
(1,-2)
(0,-2)
Unavailable Link
state n=4
(3,-2)
<2,0>
<4,-4>
(0,-1)
(-3,0)
(4,-2)
(4,-4)
(3,-4)
(2,-4)
(-1,-3)
8
Random Walk Model and Link State Changes
(0,1)
D1
(-1,1)
D6
(1,0)
(0,0)
D5
(x,y+1)
<x,y>
(x-1,y+1)
D6
D2
D4
(x,y)
D5
D3
(x+1,y)
D2
<x,y>
MN1
(-1,0)
D1
(1,-1)
MN2
(x-1,y)
<x+1,y>
D3
(x+1,y-1)
D4
(x,y-1)
(0,-1)
Figure Example 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
<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

The probability distribution for a wireless link to switch from state <x,y> to state <x’,y’>
after one time unit
9
State Transition Diagram and our modifications
nav=5
ntot
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
10
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
11
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
12
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)
13
Pk-h,K
P0,K
Pk-1,K
P0,k+h
Pk,K
Pk+h,K
0
P0,0
k-h
k-1
k
k+1
k+h
K
PK,k+h
PK,k
Pk+h,0
PK,k-1
PK,0
PK,k-h

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
14
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:
dthr
Z1, Z2, …, Zm represent the link changes for 1st, 2nd, …, mth
steps and Sm represents the net link change until the mth
step
Number of
available
links
d0
0
1
2
3
4
m-1
m m+1
Time
steps
15
First Passage Time Analysis
1
0
k-h
k-1
k
k+1
dthr
 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
16
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)
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
The expected first times vs threshold
6
10
densest
dense
typical
5
10
sparse
sparsest
4
Expected time
10
3
10
2
10
1
10
0
10
0
2
4
6
8
10
threshold
12
14
16
18
20
17
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
18
Part I: Backbone Formation and Maintenance
White – Undecided nodes
Black – VB nodes (decided)
Green – non-VB nodes (decided)
5
21
2
1
41
4
5
76
7
5
31
3
51
5
6
19