Optimal Flow Schedule 1. Maximization of Utility
ISSN: 1693-6930
TELKOMNIKA Vol. 13, No. 2, June 2015 : 460 – 468
462
3. Optimal Flow Schedule 3.1. Maximization of Utility
For each flow c C
, we define a utility function
c
U f to be a strictly concave,
increasing function of end-to-end flow rate
c
f . The goal of utility maximization is to achieve trade-off between efficiency and fairness. We can write the network-wide optimization problem
as: max
,
c c C
U f f
F
4 Since set of
F
is convex and the objective is strictly concave, there exists a unique solution f
of the maximization problem. Corresponding y
also exist but are not necessarily unique. We can write the KKT conditions at the optimal point:
c c
c i
ij ji
ji i Src c j
j
y y
f l
5
c c
c Src c
f U
f
6 Hence intuitively we can relate
c i
to
c i
q , the number of packets that are destined to destination of flow c . As a consequence of KKT, using some elementary algebra one can
derive:
,
arg max max max
c c
i j
i ij
R R c
i j L
i
R R p
7 3.2. 802.11-compatible Scheduling
It can be found that the optimal scheduling rule 7 is an NP-hard centralized optimization problem. It should consider a more realistic, suboptimal scheduling process and we
will show how our algorithm can be applied as a distributed heuristic. A back-pressure between nodes
i
and
j
is defined as
c c
c ij
i j
z q
q
,
i
can send packets to
j
only when
c ij
z
.We call a set of feasible activation profiles
S
802.11-compatible if for all
S S
and for all
1 1
, i J
S
, there is no
2 2
, i J
S
such that
1, 2 i i
p
, or
1, i
j
p
and
2, i
j
p
in which
1 2
j J
J
. Furthermore, we will assume that the underlying scheduling process is not under our control. We can simplify the schedule to the Eq 8.
,
arg max max
c c
j i
c c
c i
ij ij
i j L
d d
c t d z p
8 Where
z
ij
0, and Multiplier
c i
d
is used to prevent the shorter flow from occupying more resources. Condition
c c
j i
d d
is used to alleviate the side-effect of extensive exploration.
c i
d is updated as
,
1 min
c c
i j
i j L
ij
d d
p
.We can get
ij
p through statistics. Flow control: The optimal flow rate at the source
c
f t can be calculated through using a primal-dual approach as in the paper [8]:
,
1 [ ]
[ ] [ ]
M c
c c
c f
src c m
f t
f t
KU f
t q
t
9
Where the notation
b a
x projects the value of x to the closest point in the interval [a,b]. We assume that
m is a fixed positive valued quantity that can be arbitrarily small, and M is at
TELKOMNIKA ISSN: 1693-6930
The Implementation of One Opportunistic Routing in Wireless Network Li Han 463
least two times of the max link rate
.
2
1 K
,
c src c
q t
is the number of packets queued for destination of flow c on source node of flow c .