On Alarm Protocol in Wireless Sensor Networks

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21 Νοε 2013 (πριν από 3 χρόνια και 27 μέρες)

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Introduction
Upper bound
Lower boud
Summary
On AlarmProtocol in
Wireless Sensor Networks
J.Cicho´n
R.Kapelko J.Lemiesz
M.Zawada
Institute of Mathematics and Computer Science
Wrocław University of Technology
Poland
ADHOC-NOW2010,August 20-22,Edmonton
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 1/15
Introduction
Upper bound
Lower boud
Summary
Alarmprotocol for Wireless Sensor Networks
1
An unknown subset of the set of n sensor nodes
tries to notify the sink about the alert situation.
2
Sensor nodes are incapable of receiving or
forwarding messages (single-hop mode).
3
The sink is a fully equipped device that is able
to communicate with the external world.
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J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 2/15
Introduction
Upper bound
Lower boud
Summary
Model description
1
Assumptions
synchronization,
time is divided into time-slots S
0
;:::;S
L
,
fixed vector of probabilities p
0
;:::;p
L
.
2
Scenario
A  f1;:::;ng is a subset of sensors which detect an alert,
for every i 2 f0;:::;Lg each sensor fromA decides to send
an alert message in the slot S
i
with probability p
i
.
3
Success achieved if in at least one time-slot exactly one
sensor transmits.
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 3/15
Introduction
Upper bound
Lower boud
Summary
Leader election algorithms
protocol
network size
times slots with probability 1 
1
f
oblivious
known
e lnf
oblivious
upper bound
logu logf
oblivious
unknown
O(min((logn)
2
+(logf )
2
;f

logn))
uniform
unknown
loglogn +o(loglogn) +O(logf )
non-uniform
unknown
loglogn +2:78logf +o(loglogn +logf )
Table from the article"A Survey on Leader Election Protocols for Radio Networks"by
Koji Nakano and Stephen Olariu,2002.
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 4/15
Introduction
Upper bound
Lower boud
Summary
Leader election algorithms
protocol
network size
times slots with probability 1 
1
f
oblivious
known
e lnf
oblivious
upper bound
logu logf
oblivious
unknown
O(min((logn)
2
+(logf )
2
;f

logn))
uniform
unknown
loglogn +o(loglogn) +O(logf )
non-uniform
unknown
loglogn +2:78logf +o(loglogn +logf )
Table from the article"A Survey on Leader Election Protocols for Radio Networks"by
Koji Nakano and Stephen Olariu,2002.
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 4/15
Introduction
Upper bound
Lower boud
Summary
Leader election algorithms
protocol
network size
times slots with probability 1 
1
f
oblivious
known
e lnf
oblivious
upper bound
proof?
oblivious
unknown
O(min((logn)
2
+(logf )
2
;f

logn))
uniform
unknown
loglogn +o(loglogn) +O(logf )
non-uniform
unknown
loglogn +2:78logf +o(loglogn +logf )
Table from the article"A Survey on Leader Election Protocols for Radio Networks"by
Koji Nakano and Stephen Olariu,2002.
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 4/15
Introduction
Upper bound
Lower boud
Summary
Problemstatement
FIX n AND f > 1.
GOAL:Find a reasonable small L and a vector of probabilities
~
p = (p
0
;:::;p
L
)
which for arbitrary nonempty subset A  f1;:::;ng guarantee a
successful transmission with a probability at least
1 
1
f
:
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 5/15
Introduction
Upper bound
Lower boud
Summary
History
Take L = dlog
2
ne +1.Consider the vector of probabilities
~
p =


1
2

0
;

1
2

1
;:::;

1
2

L
!
1
2000:Pr[Success]  0:6 (???)
2
2001:Pr[Success]  0:3 (???)
3
2002:Pr[Success]  0:5 (???)
4
2010:Pr[Success]  0:579
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 6/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
The idea
Let us take L = dlog
2
ne +1.
Theorem 1
For arbitrary k 2 f1;:::;ng and L = dlog
2
ne +1 there are three
slots for which the probability of success is at least   0:579.
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 7/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
The idea
Let us take L = dlog
2
ne +1.
Vector of probabilities
1
2
0
1
2
1
:::
1
2
i 1
1
2
i
1
2
i +1
:::
1
2
L
Theorem 1
For arbitrary k 2 f1;:::;ng and L = dlog
2
ne +1 there are three
slots for which the probability of success is at least   0:579.
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 7/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
The idea
Let us take L = dlog
2
ne +1.
Vector of probabilities
1
2
0
1
2
1
:::
1
2
i 1
1
2
i
1
2
i +1
:::
1
2
L
Related cardinalities
2
0
2
1
:::
2
i 1
2
i
2
i +1
:::
2
L
Theorem 1
For arbitrary k 2 f1;:::;ng and L = dlog
2
ne +1 there are three
slots for which the probability of success is at least   0:579.
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 7/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
The idea
Let us take L = dlog
2
ne +1.
Vector of probabilities
1
2
0
1
2
1
:::
1
2
i 1
1
2
i
1
2
i +1
:::
1
2
L
Related cardinalities
2
0
2
1
:::
2
i 1
2
i
2
i +1
:::
2
L
Cardinality of A k
Theorem 1
For arbitrary k 2 f1;:::;ng and L = dlog
2
ne +1 there are three
slots for which the probability of success is at least   0:579.
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 7/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
The idea
Let us take L = dlog
2
ne +1.
Vector of probabilities
1
2
0
1
2
1
:::
1
2
i 1
1
2
i
1
2
i +1
:::
1
2
L
Related cardinalities
2
0
2
1
:::
2
i 1
2
i
2
i +1
:::
2
L
"
Cardinality of A 2
i 1
<k 2
i
Theorem 1
For arbitrary k 2 f1;:::;ng and L = dlog
2
ne +1 there are three
slots for which the probability of success is at least   0:579.
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 7/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
The idea
Let us take L = dlog
2
ne +1.
###
Vector of probabilities
1
2
0
1
2
1
:::
1
2
i 1
1
2
i
1
2
i +1
:::
1
2
L
Related cardinalities
2
0
2
1
:::
2
i 1
2
i
2
i +1
:::
2
L
"
Cardinality of A 2
i 1
<k 2
i
Theorem 1
For arbitrary k 2 f1;:::;ng and L = dlog
2
ne +1 there are three
slots for which the probability of success is at least   0:579.
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 7/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
Definitions
Let SCC
L;n;k
denote the event of the successful transmission.
Probability of success in a slot j is a unimodal function in k:
f
j
(k) =

k
1


1
2
j


1 
1
2
j

k1
Pr[SCC
L;n;k
] = 1 
L
Y
j =0

1 f
j
(k)

J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 8/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
Proof(sketch)
1
Fix k 2 f1;:::;ng.
2
Choose i 2 f0;:::;L 1g such that 2
i 1
< k  2
i
:
3
Consider three cases:
i = 0:Pr[SCC
L;n;1
] = 1:
i = 1
Pr[SCC
L;n;2
]  1 (1 f
1
(2))  (1 f
2
(2)) =
11
16
> 0:579:
2  i  L 1
Pr[SCC
L;n;k
]  1 (1 f
i 1
(k))  (1 f
i
(k))  (1 f
i +1
(k))
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 9/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
Sketch of proof
We prove that
Pr[SCC
L;n;k
]    0:579
showing that
f
i 1
(k) 
1
4
f
i
(k) >
1
2
e

1
2
f
i +1
(k) >
1
4
e

1
4
for each k 2 (2
i 1
;2
i
].
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 10/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
Sketch of proof
We prove that
Pr[SCC
L;n;k
]    0:579
showing that
f
i 1
(k) 
1
4
f
i
(k) >
1
2
e

1
2
f
i +1
(k) >
1
4
e

1
4
for each k 2 (2
i 1
;2
i
].
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 10/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
Sketch of proof
We prove that
Pr[SCC
L;n;k
]    0:579
showing that
f
i 1
(k) 
1
4
f
i
(k) >
1
2
e

1
2
f
i +1
(k) >
1
4
e

1
4
for each k 2 (2
i 1
;2
i
].
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 10/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
Possible improvements
L = dlog
2
ne +1.
Original vector of
probabilities:
~
p =

1
2
k

k=0;:::;L
Modified vector of
probabilities:

max

1
n
;
1
2
k

k=0;:::;L
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 11/15
Introduction
Upper bound
Lower boud
Summary
Theorem 1
Theorem 2
Repeating the sequence
We repeat the sequence m times:
1 

1 Pr[SCC
L;n
]

m
 1 
1
f
:
Then
m 

log
1
1 

1
logf:
Theorem 2
A sufficient total number of time slots required to send a
message with probability at least 1 "is equal
d1:1553  log
1
"
e  (dlog
2
ne +1) +1:
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 12/15
Introduction
Upper bound
Lower boud
Summary
Lower bound
Known results
1
Jurdzinski,Stachowiak:L =
(
logn log
1
"
log(logn log
1
"
)
)
2
our bound:L 
logn
2W
(
3e
2
f
f 1
logn
)
1
Hypothesis
For n  20 we have
L 
1
2

log
2
nlog
2
1
"
log
2
(log
2
nlog
2
1
"
)
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 13/15
Introduction
Upper bound
Lower boud
Summary
Concluding remarks
Summary
1
There exists an oblivious alarm protocol for a sensor
network which uses O(logn) time slots.
2
Each alarm protocol for sensor network requires
(
logn
loglogn
)
time slot.
3
The algorithmic gap remains to be clarified.
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 14/15
Introduction
Upper bound
Lower boud
Summary
Thank you!
J.Cicho´n
,R.Kapelko,J.Lemiesz
,M.Zawada
On Alarm Protocol in Wireless Sensor Networks 15/15