Node Buffer Size in
Intermittently Connected Mobile Wireless
Networks with Infrastructure Support
Tuo Yu
Shanghai Jiao Tong University, China
Outline
 Introduction
 Motivations
 Objectives
 Model and Assumption
 Percolation of Active Nodes
 Buffer Size in Mobile Network without BSs
 Buffer Size in Mobile Network with BSs
 Discussion
2
Motivation
 Under certain constraints, wireless networks are
only intermittently connectivity:
 A complete path from the source to the destination does not exist
all the time.
 Intermittently connectivity requires adequate node buffer,
even with infinite channel capacity and processing speed.
Packets have to be
stored in the node
temporarily if its next
neighbors are inactive.
3
Motivation
 Initial work has been done by Yuanzhong Xu: the lower bound
for buffer occupation in static random wireless networks is ( n )
or (1) , according to the probability for the node to be inactive.
 How about the buffer size in mobile wireless networks with
infrastructure support such as base stations ?
4
Objectives
 We focus on node buffer occupation in mobile wireless networks
with intermittent connectivity and base stations:
1.What is the relationship between the buffer requirement,
network size and the number of BSs?
2.Is it necessary to apply infrastructure support ?
5
Outline
 Introduction
 Model and Assumption
 Percolation of Active Nodes
 Buffer Size in Mobile Network without BSs
 Buffer Size in Mobile Network with BSs
 Discussion
6
Model and Assumption – I/VI
 Network Model:
 We assume that the size of the network is L  L ,with
constant node density  . Then we have L  ( n ) .
 Locations of nodes follows uniform distribution .
 Direct Links:
 Each node covers a disk shaped area with radius r.
 Two nodes have a direct link if and only if they overlap.
7
Model and Assumption – II/VI
 Mobility Model :
 Time is slotted, slot length:
 The location of each node changes with different time
slots.
 The mobility is independent and identically
distributed(i.i.d.).
 Base Station Model :
 Base Stations are fixed in the
network uniformly with number of
m . The network is divided into m
cells.
 All the BSs are connected with wire.
Model and Assumption – III/VI
 Assumption on Node Density:
 There exists a giant cluster in the network. The size of it goes to
infinite when L   .
 The giant cluster still exists in mobile networks, but it would be
various with different time slots.
9
Model and Assumption – IV/VI
 Node inactivity – Each node switch between active state
and inactive state:
 States of each nodes would be changed among time slots.
 The probability to be active is for all nodes.
 States of different nodes are i.i.d.
10
Model and Assumption – V/VI
 Traffic Pattern of Connected Nodes – Random Unicast
 Each source messages to a single destination in constant
rate.
 Transmission is in multi-hop.
 Since the transmission rate is constant, the size of one
packet transmitted in a single time slot is in the constant
order.
11
Model and Assumption – VI/VI
 Buffering
 In each hop, if the transmitter or the
receiver is inactive, the messages
should be buffered in the
transmitter until both are active.
 Assumption on Capacity and Processing speed
 Channel capacity is large enough to be viewed as infinity,
compared to the actual utilization.
 Node processing speed is also infinite, compared to the
state-switching frequency.
12
Outline
 Introduction
 Model and Assumption
 Percolation of Active Nodes
 Buffer Size in Mobile Network without BSs
 Buffer Size in Mobile Network with BSs
 Discussion
13
Percolation of Active Nodes
 Active nodes density: p
 Threshold for probability of inactivity:
pc ( ) 
c

 Supercritical Case: p  pc ( )
active giant exists in each time slot
 Subcritical Case: p  pc ( )
no active giant in each time slot
14
Outline
 Introduction
 Model and Assumption
 Percolation of Active Nodes
 Buffer size in mobile network without BSs
 Subcritical Case
 Supercritical Case
 Buffer Size in Mobile Network with BSs
 Discussion
15
Buffer size in mobile network without
BSs -Subcritical Case – I/IV
 Two-hop transmission:
One relay node is needed to transmit a single packet.
 The transmission is divided into two
steps.
 In step 1, each packet is transmitted
by the source to a close-by relay node.
 In step 2, a packet is handed off to its
destination if the relay node is close by.
Buffer size in mobile network without
BSs -Subcritical Case – II/IV
 Subcritical case:
Buffer Size : ( n)
 Delay of step1:
n 1

 (2 r )2 p 2
L2   (2r ) 2 p 2 n1
n
P( step1)  1  (
)
~
1

e
L2
 1 e
 (2 r )2 p 2
 const (n  )
1
Tdelay ( step1) 
 const (n  )
1
ln(
)
1  P( step1)
17
Buffer size in mobile network without
BSs -Subcritical Case – III/IV
 Delay of step2:
P( step 2) 
 (2r ) p
2
2
L
2
1
 ( )
n
1
Tdelay ( step 2) 
 ( n )
n
ln(
)
n 1
Buffer size in mobile network without
BSs -Subcritical Case – IV/IV
 Assume that in every time slot each relay node can carry only
one packet.
 The number of packets one relay node would carry in one
period:
L2   (2r )2 p 2 n1
(1  (
) )  (n)  (n)
2
L
 One period: The length of time while a relay node carries one
package.
 Since the size of one packet is constant, the average buffer
size of one node is:
SMsub  (n)  (1)  (n)
Outline
 Introduction
 Model and Assumption
 Percolation of Active Nodes
 Buffer size in mobile network without BSs
 Subcritical Case
 Supercritical Case
 Buffer Size in Mobile Network with BSs
 Discussion
20
Buffer size in mobile network without
BSs -Supercritical Case – I/V
 Multi-hop transmission
 Step1: Packets are transmitted from
source to all the relay nodes that are in the
same cluster in every time slot.
 Step2: As long as any of the relay nodes
(or the source itself) and destination node
are in the active giant, the packet will be
sent to the destination directly. Then all the
buffers will be released.
Buffer size in mobile network without
BSs -Supercritical Case – II/V
 Supercritical case: p  pc ( )
Buffer Size : (1)
 The work of Xu has proved that the number
of relay nodes nr in one time slots follows:
P(nr  N )   ( N  1)e N
where  and  are constant. The
inequality means that nr is finite and
independent from n.
 Assume that the time taken to reach step2
is Tr , then the expect number of relay
nodes at the end of transmission is
E (nT )  (nr  1)Tr
Buffer size in mobile network without
BSs -Supercritical Case – III/V
 The expect number of Tr should satisfy:
E (Tr ) 
1
1
ln(
)
1  pr
where pr donates the probability that any of the relay nodes
and the destination node are in the active giant at the same
p
time slot. According to the theory
of percolation,
pr  1  (1  pG p 2 ) nT 1
pG is the probability for one node to be in the active giant,
which is constant when  and p do not change with n.
Buffer size in mobile network without
BSs -Supercritical Case – IV/V
 The expect number of Tr should satisfy:
1

 E (Tr ) 
1
ln(
)

1  pr


2 nT 1
p

1

(1

p
p
)
 r
G

Tr
E
(
n
)

(
n

1)
T
r




This is a transcendental equation set, which will have positive
solutions of nr and Tr . Since all the coefficient of the
equations are independent from n, (nr  1)T is also independent
from n. Then we have E (nT )  (1) .
r
Buffer size in mobile network without
BSs -Supercritical Case – V/V
 Then we draw the conclusion that the buffer size needed at
supercritical case is:
S M sup er
(1)  (n)  E (nT )

 (1)
n
 This result means that at the supercritical case, the buffer
occupied at each node is at constant order.
Outline
 Introduction
 Model and Assumption
 Percolation of Active Nodes
 Buffer size in mobile network without BSs
 Buffer Size in Mobile Network with BSs
 Discussion
26
Buffer size in mobile network with
BSs – I/V
 Transmission Scheme
 This scheme can be used in
both subcritical and
supercritical case.
 Three steps are taken to finish
the transmission from u to v.
 Step1:
u→relay node→BS
 Step2:
BSs wired link
 Step3:
BS→relay node→v
27
Buffer size in mobile network with
BSs – II/V
 Transmission Scheme
 Step1:
u→relay node→BS
 The transmission follow the same
scheme as that in the network without BS
at subcritical case. The only different is
that the destination of normal node is BS.
 According to the analysis above, the sum
of the buffer size occupied in the cell
n
is (( ) 2 ) .
m
28
Buffer size in mobile network with
BSs – III/V
 Transmission Scheme
 Step2:
BSs wired link
 Once the packet reaches the BS, it is
transmitted to another BS in the cell
which includes the destination node.
 Since the number of normal nodes that
are connected to one BS is


2
n

(2
r
)
p
  (1)
 
2 
m

n/m 




 Then the bandwidth needed would be
constant, so there is no need to take the
limit of bandwidth into account.
29
Buffer size in mobile network with
BSs – IV/V
 Transmission Scheme
 Step3:
BS→relay node→v
 Step 3 is actually reverse to step 1. So
we directly reach the conclusion that the
sum of the buffer size occupied in the cell
n 2
is (( ) ) .
m
30
Buffer size in mobile network with
BSs – V/V
 Occupied Buffer Size :   n 
m
S MB
  n 2  n 2 
    
m m 
n



  
n


m
2



m


 This result implies that the average
occupied buffer size in the network is in
inverse proportion to the number of BSs.
31
Outline
 Introduction
 Model and Assumption
 Percolation of Active Nodes
 Buffer Size in Mobile Network without BSs
 Buffer Size in Mobile Network with BSs
 Discussion
 Feasibility of Base Stations
 Base Stations with Changeable Radius
 Other Achievements
32
Discussion -Feasibility of Base
Stations
 Mobile network without BSs:
 Subcritical:
( n)
 Supercritical:  1
 Mobile network with BSs:
  mn 
 The conclusion show us that in subcritical case, the application of
BSs will decrease the buffer size needed. If the number of BSs is
in the same order of n, the buffer size will be in the constant
order, which is a good result.
 However, in supercritical case it seems that there is no need to
apply BSs to cut down occupied buffer size, because the
existence of active giant ensures that the buffer size in need is in
the constant order.
33
Outline
 Introduction
 Model and Assumption
 Percolation of Active Nodes
 Buffer Size in Mobile Network without BSs
 Buffer Size in Mobile Network with BSs
 Discussion
 Feasibility of Base Stations
 Base Stations with Changeable Radius
 Other Achievements
34
Discussion -Base Stations with
Changeable Radius
-I/II
 In the analysis above, we assume that the radius of BSs is r,
which is the same as that of normal nodes.
 What if the radius of BSs is changeable?
 Assume that the radius of BSs is R . To take the limit of
bandwidth into account, the maximum number of nodes that are
connected to a BS is limited to k .
 The analysis is similar to that above. The result is :
S MBR


n

  n / m   ( R  r ) 2   (2r ) 2  m 1  


   ln( 1n/m )  1   p 2


 n / mk  
n / m   ( R  r )2




1
ln( n/nm/ mk )

35
Discussion -Base Stations with
Changeable Radius
-II/II
SMBR  

1
ln( n/nm/ mk )

 According to the result, when k is in the constant order, the
occupied buffer size is still  n , which is independent of R.
m
 When k is in the order of
• When R  
R
2
, the result is S MBR
  , the buffer size in need is S
n
m


1
    n/ m  
 ln n/m ( Rr )2  

 
MBR
  1 .
• When R   1 , the buffer size in need is still SMBR    mn  .
36
Outline
 Introduction
 Model and Assumption
 Percolation of Active Nodes
 Buffer Size in Mobile Network without BSs
 Buffer Size in Mobile Network with BSs
 Discussion
 Feasibility of Base Stations
 Base Stations with Changeable Radius
 Other Achievements
37
Discussion
-Other Achievements
Subcritical Case
Static Wireless Networks
(Yuanzhong Xu)

 n
Static Wireless Networks
with BSs

 
Mobile Wireless Networks
Mobile Wireless Networks
with BSs
Mobile Wireless Networks
with BSs(changeable
radius)(channel limit: k)
Supercritical Case
 1

n
m
 
n
m
 n
 1
  mn 
  mn 


1
ln( n/nm/ mk )



1
ln( n/nm/ mk )

38
Thank you !
The Calculation of Buffer Size in Static
Networks with BSs
-I/II
Buffer in Intermittently Connected Network Presentation
40
The Calculation of Buffer Size in Static
Networks with BSs
-II/II
Buffer in Intermittently Connected Network Presentation
41
The Distribution of Occupied Buffer Size
Near Base Station
Buffer in Intermittently Connected Network Presentation
42