Order Optimal Delay for Opportunistic SchedulingIn Multi-User Wireless Uplinks and Downlinks
Michael J. NeelyUniversity of Southern Californiahttp://www-rcf.usc.edu/~mjneely/
*Sponsored in part by NSF OCE Grant 0520324 (DIGITAL OCEAN)
1
2
N
S2(t)
SN(t) Num. Users N
Avg
. Del
ay
S1(t) {ON, OFF}
Allerton 2006
N
1
2
The System Model: N Users , 1 Server
Discrete Time System: Timeslots t = {0, 1, 2, …}
q1
Ai(t) = Arrivals to Queue i during slot t
[ i.i.d over slots , E[Ai(t)] = i ]
Qi(t) = Current Num. Packets in queue i
Uplink
user 1 user N Downlink1 2
N
Si(t) = Current Channel State ({ON, OFF})
[ i.i.d. over slots, Pr[Si(t) = ON] = qi ]
q2
qN
i(t) = Packets Transmitted over link i on slot t
N
1
2
The System Model: N Users , 1 Server
Discrete Time System: Timeslots t = {0, 1, 2, …}
q1 Uplink
user 1 user N Downlink1 2
N
q2
qN
Qi(t)Ai(t) i(t)
Qi(t+1) = max[Qi(t) - i(t), 0] + Ai(t) Scheduling Constraints: Can serve at most one “ON” link per slot:
i(t) {0,1}i=1
N i(t) 1, , i(t)=0 if Si(t)=OFF
N
1
2
q1
q2
qN
Model is central to channel-aware (“opportunistic”) scheduling.
This model is investigated in [Tassiulas, Ephremides 93]:
Results of [Tas, Eph 93]:1) Capacity Region 2) LCQ Algorithm (“Largest Connected Queue”)3) Delay Optimality for Symmetric Systems
The Capacity Region : Set of all rate vectors (1, .., N) that can be stabilized. Example: (N=2) is the set of all (1, 2) such that:
1 q1 , 1 q2
1 + 2 q1 + (1-q1)q2 1
2
N
1
2
q1
q2
qN
Model is central to channel-aware (“opportunistic”) scheduling.
This model is investigated in [Tassiulas, Ephremides 93]:
Results of [Tas, Eph 93]:1) Capacity Region 2) LCQ Algorithm (“Largest Connected Queue”)3) Delay Optimality for Symmetric Systems
The Capacity Region : Set of all rate vectors (1, .., N) that can be stabilized.
General Case for N: (1, .., N) if and only if
ii I i I
1 - (1-qi)
for each of the 2N-1 non-empty subsets I of {1, .., N}
An isolated set of delay-optimality results:
q
q
q
-Largest Connected Queue (LCQ) [Tassiulas and Ephremides 93]:
Proof uses stochastic coupling and exploits symmetry…
For Symmetric Systems:
-Rate Allocation in Gaussian Multiple Access Channels [Yeh 2001 , Yeh and Cohen 2003]
-Multi-Server Systems: [Yeh 2001 , Ganti, Modiano, Tsitsiklis 2002]
An isolated set of delay-optimality results:
q
q
q
-Largest Connected Queue (LCQ) [Tassiulas and Ephremides 93]:
Proof uses stochastic coupling and exploits symmetry…
For Symmetric Systems:
-Rate Allocation in Gaussian Multiple Access Channels [Yeh 2001 , Yeh and Cohen 2003]
-Multi-Server Systems: [Yeh 2001 , Ganti, Modiano, Tsitsiklis 2002]
The actual delay that is achieved is unknown (even for these symmetric cases)
O(N)? O( N )? O(1)?
An isolated set of delay-optimality results:
For Heavy Traffic:
The actual delay that is achieved is unknown (even for these symmetric cases)
O(N)? O( N )? O(1)?
= fraction is away from capacity region boundary
q
q
q
Shakkottai, Srikant, Stolyar 2004 1 (Heavy Traffic)
An exponential Scheduling Rule approaches delay optimality (
Related: Delay for N x N Switch Scheduling:
1 2 3 N
1
N
-[Leonardi, Mellia, Neri, Marsan 2001]: O(N/(1-)) Delay bound (MWM Sched.)-[Neely, Modiano 2004]: O(log(N)/(1-)2) Delay bound (Frame Based Sched.)
Related: Delay for N x N Switch Scheduling:
1 2 3 N
1
N
Some Interesting Queue Grouping Approaches(mainly to reduce complexity):-Mekkittikul, McKeown (1998)-Shah (2003)-Wu, Srikant (wireless, 2006)
Related: Delay for N x N Switch Scheduling:
1 2 3 N
1
N
Some Interesting Queue Grouping Approaches(mainly to reduce complexity):-Mekkittikul, McKeown (1998)-Shah (2003)-Wu, Srikant (wireless, 2006)
-Leonardi et al. (2001)
+=
Related: Delay for N x N Switch Scheduling:
1 2 3 N
1
N
Some Interesting Queue Grouping Approaches(mainly to reduce complexity):-Mekkittikul, McKeown (1998)-Shah (2003)-Wu, Srikant (wireless, 2006)
-Leonardi et al. (2001)
+=
O(1) Delay when < 1/2(half loaded)
q
q
q
What is the optimal delay (as a function of N) for the N user wireless problem with varying channels?
Time Varying Channels make analysis more complex, cannotuse same approaches as switch problems… Previous Upper and Lower Bounds: (N users)
N(1-)O( )1
(1-)O( ) E[Delay]
“Single-Queue Bound” [Neely, Modiano, Rohrs 03]
q
q
q
What is the optimal delay (as a function of N) for the N user wireless problem with varying channels?
Our Results: (part 1) If scheduling doesn’t consider queue backlog(such as stationary randomized scheduling) then:
1) E[Delay] is at least linear in N
2) Uniform Poisson Traffic: E[Delay] > N2rN(1-)
rN = 1-(1-q)N
(max possible output rate)
What is the optimal delay (as a function of N) for the N user wireless problem with varying channels?
Our Results: (part 2) For any such that < 1
Av. Delay log(1/(1-))(1-)
O( ) Independent of N
Holds for Symmetric Systems and a large class of Asymmetric ones
q
q
q
rN = 1-(1-q)N
(max possible output rate)
What is the optimal delay (as a function of N) for the N user wireless problem with varying channels?
Our Results: (part 2) For any such that < 1
Av. Delay log(1/(1-))(1-)
O( ) Independent of N
We use a form of queue grouping together with Lyapunov driftAnd statistical multiplexing
q
q
q
rN = 1-(1-q)N
(max possible output rate)
Intuition about Queue Grouping:
q
q
q
N user System, Uniform Poisson inputs:
rN = 1-(1-q)N
(max possible output rate)
Compare to a single-queue system with Pr[ON] = q
Pr[serve]=q
Can show that any work conserving scheduling policy in multi-queue system yields delay that is stochastically smaller than single-queue system. Leads An easy upper bound on delay…
(GI/GI/1 queue)
Intuition about Queue Grouping:
q
q
q
N user System, Uniform Poisson inputs:
rN = 1-(1-q)N
(max possible output rate)
Compare to a single-queue system with Pr[ON] = q
Pr[serve]=q
Single Queue Upper Bound on Avg. Delay:
(GI/GI/1 queue)
Poisson Bernoulli
E[Delay] = 1 - tot/2 q - tot
Only works fortot < q(i.e., < where = q/rN)
1
(1-/)O( )=
Queue Grouping Approach: Form K Groups: {G1, G2, …, GK}
1
2
N
M1
M1+1
G1
G2
GK
Qsum, k(t) = Qi(t)i Gk
G1
G2
GK
Qsum, k(t) = Qi(t)i Gk
sum, k = i i Gk
The Largest Connected Group (LCG) Algorithm: Every slot t, observe the queue backlogs and channel states, and select the group k {1, …, K} that maximizes 1k(t)Qsum, k(t).Then serve any non-empty connected queue in thatgroup (breaking ties arbitrarily).
Define: 1k(t) = {1 , if group Gk has at least one non-empty connected queue. 0 , else
G1
G2
GK
q1
qN
q21 qmin, 1
2 qmin, 2
K qmin, K
sum, 1
sum, 2
sum, N
Define: K = Capacity region of the K-queue System
sum, k= i i Gk
qmin, k= min {qi}i Gk
Theorem: If there is an > 0 such that:
(sum, 1 + , sum, 2 + sum, K + K
Actual N-queue System Comparison K-queue System
Then LCG stabilizes the system and yields average delay:
G1
G2
GK
q1
qN
q21 qmin, 1
2 qmin, 2
K qmin, K
sum, 1
sum, 2
sum, N
Define: K = Capacity region of the K-queue System
sum, k= i i Gk
qmin, k= min {qi}i Gk
Theorem: If there is an > 0 such that:
(sum, 1 + , sum, 2 + sum, K + K
Actual N-queue System Comparison K-queue System
If arrivals are independent and Poisson, then we have:
Theorem: If there is an > 0 such that:
(sum, 1 + , sum, 2 + sum, K + K
If arrivals are independent and Poisson, then we have:
Proof Concept: Use the following Lyapunov function:
1) LCG comes within additive constant of minimizing: (Lyapunov drift)
2) (tricky part) Prove there exists another algorithm that yields:
(h() linear)
Application to Symmetric Systems: rN = 1-(1-q)N
(max possible output rate)
q
q
qQN-1(t)
QN(t)
Q1(t)
Q2(t)
q
For any loading such that 0 < < 1:
log(2/(1-))log(1/(1-q))
Choose K = For simplicity assume N = MK (K groups of equal size M)
Then = rN(1-)/(2K) , … Plug this into the theorem…
tot = rN
Application to Symmetric Systems: rN = 1-(1-q)N
(max possible output rate)
For any loading such that 0 < < 1:
log(2/(1-))log(1/(1-q))
Choose K = For simplicity assume N = MK (K groups of equal size M)
tot = rN
E[W] 2K
rN(1-) = log(1/(1-))(1-)
O( )Then LCG =>
q
q
qQN-1(t)
QN(t)
Q1(t)
Q2(t)
q
Application to Asymmetric Systems:
tot = rmax
i=1
N
(1-qi)rmax = 1 -
(max possible output rate)
q1
q2
qN-1QN-1(t)
QN(t)
Q1(t)
Q2(t)
qN
tot = 1 + … + N
Form variable length groups by iteratively packing individualstreams until total rate of the group exceeds tot/N.
Then: sum, k < tot/N + max for all groups k
Application to Asymmetric Systems:
For any loading such that 0 < < 1:
log(2/(1-))log(1/(1-qmin))
Choose K =
tot = rmax
E[W] log(1/(1-))(1-)
O( )For any N K, LCG =>>
i=1
N
(1-qi)rmax = 1 -
(max possible output rate)
Assume max < (1-)rmax/(3K)
q1
q2
qN-1QN-1(t)
QN(t)
Q1(t)
Q2(t)
qN
Conclusions:
Order-Optimal Delay for Opportunistic Scheduling in a Multi-User System (N users)
-Backlog-unaware scheduling: Delay grows at least linear with N
-Backlog-aware scheduling: It is possible to achieve O(1) delay (independent of N)
-The first explicit bound for optimal delay in this setting
-Queue Grouping is a useful tool for analysis and design
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