Optimal Delay for Opportunistic Schedu Multi-User Wireless Uplinks and Downlin Michael J. Neely University of Southern California http://www-rcf.usc.edu/~mjneely/ ored in part by NSF OCE Grant 0520324 (DIGITAL OCE 1 2 N S 2 (t) S N (t) Num. Users N Avg. Delay S 1 (t) {ON, OFF} Allerton 2006
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Order Optimal Delay for Opportunistic Scheduling In Multi-User Wireless Uplinks and Downlinks Michael J. Neely University of Southern California mjneely
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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)
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