Optimal Power Control for Wireless
Queueing Networks
Tolga Girici and Anthony Ephremides
Abstract
Optimally controlling transmit power for
jointly minimizing:
System Model
Wireless Link
Finite buffer capacity (length= K)
Bernoulli packet arrivals (rate= λ)
Equal length packets (length= 1 bit)
Energy expenditure
Buffer overflow
Considered System
Wireless Link,
Finite-capacity buffer,
Random packet arrivals,
Choosing from two alternative power levels.
MDP
Two alternative power Levels (P1<P2)
BPSK modulation
ARQ error control
Power decisions are made at every
time slot.
PTs
)
Success probability = 1 − Q(
t 0
Overflow cost:
O ( x, u )
State Transitions
Energy Efficiency
A key concern in wireless networks,
Limited and non renewable power supplies.
d i , j (u )
Power Control, Traditional Work:
Mitigating effects of interference,
Satisfying some QoS constraints,
Bambos(96):Maximizing battery lifetime
No error control technique is specified.
Zorzi(97):Trade-off between instantaneous
performance and energy efficiency:
ARQ error control,
Gilbert-Elliott Channel,
Decreasing power in favor of increased
retransmissions can be a more energy-efficient.
Random Arrivals
Previous work generally considered variance of
interference and noise.
Random arrival of data is not considered
generally.
In a limited buffer system with random arrivals
An arrival burst can occur when queue is long.
Buffer overflow can be avoided by increasing power.
Changing Arrival rate
ÒPu , u 1,2
Ó
Ô0, u 0
Single Stage Cost:
g ( x, u ) c1 E ( x, u ) c2O ( x, u ), x ´X , u ´U
Dynamic Programming Equations:
V 0 ( x)
min {g ( x, u )}
u´U ( x )
V n 1 ( x)
min {g ( x, u ) HË d x , x ' (u )V n ( x)}
u´U ( x )
x'
Optimality of Threshold Policy
Changing Discount Factor
For 0<x<K optimal policy chooses P2 if:
c1 ( P2 P1 ) }H[z( 2 1 )(V n ( x 1) V n ( x)) (1 z)( 2 1 )(V n ( x) V n (x 1))]
Minimizing average power subject to average success
rate constraint.
Interference has a known distribution.
E ( x, u )
i 0, j 0
1 z,
Ò
Õ
z
i 0, j 1
,
Õ
Õz
K !i !0, j i
(1 z)(1 u ),
u Õ
z(1 u ),
K !i !0, j i 1
Õ
Ó
K !i !0, j i (1 z) u
1
Õ
Õ
i K, j i
1 u ,
Õ
u ,
i K, j i 1
Õ
Õ
0
| j i |!1
Ô
SINR,BER constraints (Foschini(93),Yates(95))
Energy Efficiency
Ò0, x !K
Ó
Ôz, x K
Energy Expenditure cost:
Optimal policy is of threshold type.
Motivation
Changing Power Levels
VH E x ËHt g ( xt , ut ) x ´X , u ´{0,1,2}
N0
Result
Computational Results
State, xt: Buffer occupancy in the
tth slot (xtєX={0,1,…,K})
Control: ut: Power control decision
(ut є {0,1,2})
ut is allowed only when xt=0.
Find the policy that minimizes the
discounted cost:
H[z( 2 1 )hxn (1 z)( 2 1 )hxn1 ] (1)
Here h V ( x 1) V ( x)
Theorem: For P1<P2 the below inequalities hold in every iteration n:
n
x
n
n
For x greater than or equal to some xn: hxn≤hx+1n (2)
For x<xn: c1(P2-P1)≥α(μ1- μ2)(λhxn+(1- λ) hx-1n) (3)
For all x: 0≤hxn≤c2 (4)
(2) and (3) are useful for proving that the right hand side of inequality (1) crosses
left side at most at one point.
Practical Determination of Thresholds
Let 0≤t≤K be the threshold value.
Queue state probabilities π=[π1 π2,…, πK] can be found by solving the
Markov chain below:
Find t* minimizing the cost below: d1,2(1) d2,3(1) dt-2,t-1(1) dt-1,t(1)
dK-2,K-1(2) dK-1,K(2)
t
1
t*
K
arg min{‡0t g (0,0) ˇxt g ( x,1) ˇxt g ( x,2)}
t
x 1
x t
X=1
X=2
d2,1(1)
…
X=t-1
X=t
d3,2(1)dt-1,t-2(1) dt,t-1(2)
…
X=K-1
dK-1,K-2(2)
X=K
dK,K-1(2)
Results
The optimal power control policy of
threshold type.
Optimal threshold
Increases as P1 approaches to P2,
Decreases as the arrival rate increases,
Decreases as the discount factor α
increases.