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