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DYNAMIC POWER ALLOCATION AND ROUTING DYNAMIC POWER ALLOCATION AND ROUTING FOR TIME-VARYING WIRELESS NETWORKSFOR TIME-VARYING WIRELESS NETWORKS
Michael J. Neely, Eytan Modiano and Charles E.Rohrs
Presented by
Ruogu Li
Department of Electrical and Computer Engineering
The Ohio State University
CONTENTSCONTENTS Overview System Model and Assumptions Network Capacity Region Centralized DRPC Policy
Proof of stability Enhanced DRPC Policy Decentralized DRPC Policy Conclusion Future work
OVERVIEWOVERVIEW We consider dynamic routing and power allocation for a
wireless network with time-varying channels: The network consists of power constrained nodes; Transmission rates over the links are determined by allocated
power; Packets randomly enter the system at each node and wait in
output queues to be transmitted to their destinations; We developed a joint routing and power allocating
policy (DRPC) that stabilizes the system and provides bounded average delay.
SYSTEM MODEL AND ASSUMPTIONSSYSTEM MODEL AND ASSUMPTIONS A wireless network with nodes; Time is slotted, channel state stays the same in one slot; Multiple data stream randomly enter the system
with source and destination ; Each node can transmit data over multiple links
simultaneously; Power is assigned to links
at each node.
)(tAiji j
N
SYSTEM MODEL AND ASSUMPTIONSSYSTEM MODEL AND ASSUMPTIONS Power constraint at each node:
Transmission rate on each link is determined by a rate-power curve , where is the power matrix, and is the channel state matrix;
Channel state represents,for example, attenuationand/or noise levels; it isknown to the controllerat the beginning of eachtime slot;
totala
akkak PtP
,
)(
))(),(( tStPab )(tP
)(tS
SYSTEM MODEL AND ASSUMPTIONSSYSTEM MODEL AND ASSUMPTIONS The power curve is assumed to be upper
semi-continuous in the power matrix for all states ;
The power matrix , where is the set of acceptable power allocations.
)(tP
)(tS
))(),(( tStPab
)(tP
SYSTEM MODEL AND ASSUMPTIONSSYSTEM MODEL AND ASSUMPTIONS Each node queues data according to their destinations; We classify all data flowing through the network as
belonging to a particular commodity , representing the destination node for the data;
Define as the rateoffered to commodity traffic along link ;
},...,1{ Nc
)()( tcab
c
),( ba
SYSTEM MODEL AND ASSUMPTIONSSYSTEM MODEL AND ASSUMPTIONS The input process of the network are stationary
and ergodic with rates . represents the incoming rate at node of commodity
. The matrix is the corresponding matrix with diagonal entries equal to zero.
Further assume that the second moment of is bounded every time slot by some finite maximum value regardless of past history.
)()( tA ci
ic ic )( ic NN
ic
)()( tA ci
SYSTEM MODEL AND ASSUMPTIONSSYSTEM MODEL AND ASSUMPTIONS The control decision variables are:
Power allocation, choose such that ; Routing/Scheduling, choose such that
The backlog of bits in node destined for node c is represent by (the queue length).
)(tP )(tP
)()( tcab
))(),(()()()( tStPtt ababc
cab
)()( tU ci
i
NETWORK CAPACITY REGIONNETWORK CAPACITY REGION A queueing system is said to be stable if the queue
length does not ‘blow up’ when time goes to infinity; The network capacity region is the closure f the set of
all rates matrices that can be stably supported over the network, considering all possible algorithms.
)( ic
NETWORK CAPACITY REGIONNETWORK CAPACITY REGION Example:
The capacity region will be
1
2
1
2
CENTRALIZED DRPC POLICYCENTRALIZED DRPC POLICY Dynamic Routing and Power Control (DRPC) Policy:
For all links , find commodity such that
and define
Power allocation: choose a matrix such that
Routing: define transmission rate as follows: , if and , otherwise
),( ba )(* tcab)}()({maxarg)( )()(
},...,1{
* tUtUtc cb
ca
Ncab
))()(()( ))(())((* **
tUtUtW tcb
tcaab
abab
)(tP
ba
ababP
tWtSPtP,
* )())(,(maxarg)(
0
))(),(()()(
tStPt abc
ab
)(* tcc ab 0)(* tWab
CENTRALIZED DRPC POLICYCENTRALIZED DRPC POLICY It is inspired by the maximum differential backlog
algorithms developed by Tassiulas and Ephremids; An extension of the maximum differential backlog
algorithm which maximize the throughput of a constrained network;
Thus DRPC Policy maximizes the throughput of the network.
Ref: L. Tassiulas and A. Ephremids, “Stability properites of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks,” IEEE trans. Autom. Control, vol. 37, no.12, Dec 1992
CENTRALIZED DRPC POLICYCENTRALIZED DRPC POLICY Stability of DRPC Policy
Theorem: Suppose an N-node wireless network has capacity region and rate matrix such that for some . Then, the above DPRC policy stabilize the system and guarantees bounded average congestion.
Proof of Stability of DRPC Policy Basic idea: prove the stability of the system using a Lyapunov
function. A function is a Lyapunov candidate function if it
is locally positive definite, i.e. The choice of the Lyapunov function is based on the
problem.
)( ic )( ic0
RRV n :
00)(,0)( xx0 VV
CENTRALIZED DRPC POLICYCENTRALIZED DRPC POLICY The proof in the paper is very complicated; We consider a similar simpler case using the same
approach; Single base station sends out data to N users; Data arrive at base station with rate , same
assumption for the arriving process;nn tA ])[(E
B.S.
User 1
User 2
User 3
User N
][1 t
][2 t
][3 t
][tN
][1 tA][1 tU
][2 tA
][tAN][tU N
CENTRALIZED DRPC POLICYCENTRALIZED DRPC POLICY Constraint for the base station , ‘power
constraint’ with linear rate power curve, denote the set of feasible by ;
Control variable: choose ; The arrive rates satisfy ,
where is the capacity region;
Policy: choose
n
n t][
B.S.
User 1
User 2
User 3
User N
][1 t
][2 t
][3 t
][tN
][1 tA][1 tU
][2 tA
][tAN][tU N
][t ][tR
][t
R )( n
T
tT
tT 1
][1
lim RR
N
nnn
ttr
trtUt1][][
][][maxarg][R
CENTRALIZED DRPC POLICYCENTRALIZED DRPC POLICY Queue evolution: Choose the Lyapunov function
Thus the Lyapunov drift is given by
])[][][(]1[ ttAtUtU nnnn
][2
1][
1
2 tUtLN
nnU
]][|][]1[[E][ UtUtLtLt UU
]|))(]1[([E2
1 22 UtUtUn
nn
nnnnn UUAU ]|)[(E
2
1 22
CENTRALIZED DRPC POLICYCENTRALIZED DRPC POLICY Notice that ; Thus
From the assumption of , we know that
22)( xx
n
nnnnn UAAUt ]|)()(2[E2
1][ 2
n
nnn
nnn UAUUAU ]|)[(E2
1])|[E]|[(E 2
n second moment, C
n
nnn
nn UUUC ]|[E
R )( n
n
nnn
nn UUU ]|[E)(
CENTRALIZED DRPC POLICYCENTRALIZED DRPC POLICY Thus we get
which is an simplified version of (21) in the paper; Sum over 1 through T-1 and take expectation on
both sides, we get
From the non negativity of the Lyapunov function,
n
nUCt ][
][t
1
0
]][[E]]0[][[ET
t nnUU tUTCLTL
]]0[[E1
]][[E1 1
0U
T
t nn L
TCtU
T
CENTRALIZED DRPC POLICYCENTRALIZED DRPC POLICY Taking the limit of the above inequality
Thus we proved the stability of the system under our policy;
Using Little’s Law we can get the bound on delay; The proof in the paper is an extension of this simple
case.
/]][[E
1suplim
1
0
CNtUT
T
t nn
T
ENHANCED DRPC POLICYENHANCED DRPC POLICY Potential problem of DRPC Policy:
When the network is lightly loaded, very little information is contained in the backlog values;
Packets may wander in the network, resulting long delays; Solution:
Adding a restricted set of desirable routes; But restricting the routes may be harmful in time varying
channels; Enhanced DRPC Algorithm is introduced to solve this
problem.
ENHANCED DRPC POLICYENHANCED DRPC POLICY Basic idea: implementing a ‘bias’ in the DRPC Policy so
that in low loading situations, nodes are inclined to route packets in the direction of their destinations;
Define
and define as the maximizer of ; Power allocation and routing is done as before;
))(())(()()( cb
cb
cb
ca
ca
ca
cab VtUVtUtW *abc )()( tW c
ab
ENHANCED DRPC POLICYENHANCED DRPC POLICY The parameters can be chosen as scaled hop count
estimates between nodes and , so that, in the absence of backlog information, data is routed to reduce the remaining distance to the destination;
The values are any weights for prioritizing commodity service in node ;
It can be shown that this enhanced DRPC Policy can stabilize the system for any and .
caV
ca
0caV0ca
a c
ca
DECENTRALIZED DRPC POLICYDECENTRALIZED DRPC POLICY The DRPC Policy is a centralized control; Hard to implement in reality; The authors provided a simple decentralized
approximation without proof; Nodes have current neighbors; The current neighbors of a node is defined as the set
of the nodes to which node can currently transmit and receive.
ii
DECENTRALIZED DRPC POLICYDECENTRALIZED DRPC POLICY The Decentralized DRPC Policy
At the beginning of each time slot, nodes randomly decide to transmit with probability . All transmitting nodes send a control signal of power where is globally known;
Define as the set of all transmitting nodes. Each node measures its total resulting interference and send this quantity over a control channel to all neighbors;
Each transmitting user decides to transmit using full power to the single neighbor who maximizes
qtotalP
Q
Qi totalibb PI
ab
)1log(*
totalabbb
totalabab PIN
PW
CONCLUSIONCONCLUSION We have formulated a general power allocation problem
for a multinode wireless network with time-varying channels and adaptive transmission rates;
The network capacity region was established; A DRPC algorithm is developed and shown to stabilize
the network whenever the arrival rate matrix is within the capacity region.
FUTURE WORKFUTURE WORK The DRPC policy is based on maximum backlog
differential algorithm which tries to maximize the throughput of the network, but other network control metrics such as minimizing the delay are not considered.
In the policy, we need to find
which is not a trivial problem. A straightforward exhaustive search may not work for large networks. Many works have been done on this, for example, using greedy algorithm.
ba
ababP
tWtSPtP,
* )())(,(maxarg)(
