1. Scheduling Algorithms for Multi-Carrier Wireless Data Systems
Matthew Andrews and Lisa Zhang
Bell Labs, ACM MobiCom 2007
Ji Hoon Lee

2. Agenda Background The goal of this paper System model
Multi-carrier systems
MaxWeight Algorithm
The goal of this paper
System model
Objective funtions
Candidate algorithms
Stability
Extensions to general weight
Simulations

3. Background
Multi-carrier wireless data systems  Use an OFDMA physical layer:
IEEE
EV-DO Rev. C
3GPP LTE
Difference “tones” can be assigned to different users at each time.

4. Max-Weight
L. Tassiulas and A. Ephremides, “Dynamic Server Allocation to Parallel Queues with Randomly Varying Connectivity”, IEEE Trans. on Information Theory, 39(2), Mar
MaxWeight was first shown to perform well in wireless networks by Tassiulas and Ephremides.
Single-hop network with time varying connectivity.
“Serve-the-Longest-Connected-Queue” policy
Serving the longest connected queue
Stabilize the system
Maximize the throughput and minimize the delay when packet arrivals and channel states are iid Bernoulli processes for all queues.

5. Goal
How to adapt the popular algorithm known as MaxWeight to the case of multiple carriers.
Study the problem of scheduling in multi-carrier and frame-based systems.
Focus on the issue of multiple carriers.
Multiple timeslots per frame can be regarded as a special case of multiple carriers.
i.e., the frame size is one time slot.

6. System Model A single BS transmitting data
to a set of N wireless users
on a set of C carriers
x(i,c,t): whether or not carrier c is assigned to user i at time t
r(i,c,t): the channel rate of carrier c for user i at time t
Goal
to schedule the system
to choose the values of x(i,c,t) in the most advantageous way.
Assumptions
The frame size is one time slot.
We know the channel rate for all tuples (i,c,t).
The channel rate does not vary significantly over the duration of the frame.

7. Three Objective Functions (1/2)
Qsi(t): the queue size of user i at the beginning of time slot t.
Qei(t): the queue size of user i at the end of time slot t.
u(i,t): the amount of service user i receives at time t (=Σcr(i,c,t)·x(i,c,t)).
C.f. MaxWeight
Serve the user that maximize Qsi(t)·r(i,t) at each time t
A number of ways to emulate the Max-Weight algorithm in multi- carrier systems:

8. Three Objective Functions (2/2)
(1) is the simplest analogue of MaxWeight.
Assigning more service to a user than it can actually use.
May not change the stability property of MaxWeight.
(2) is offered as a natural fix.
(3) explicitly maximize the negative drift of the Lyapunov function.
Both (2) and (3) are more sentitive to maintaining small queues.

9. Lyapunov Stability The technique of Lyapunov drift: Idea
One of the most important mathematical tools.
To prove stability of queueing networks.
To develop stabilizing control algorithms.
Idea
Define a non-negative function, called a Lyapunov function, as a scalar measure of the aggregate congestion of all queues.
Network control decisions are evaluated in terms of how they affect the change in the Lyapunov function from one slot to the next.
Lyapunov drift:
Further reading:
ection=x1-84r4

10. Candidate Algorithms MaxWeight-Alg1 MaxWeight-Alg2 MaxWeight-Alg3
Optimize (1)
MaxWeight-Alg2
1/2-app. for (2)
MaxWeight-Alg3
1/3-app. for (3)
MaxWeight-Alg4
User-by-user algo.
MaxWeight-Alg5
Based on Generalized Assignment Problem
(1-1/e-ε)-app. for (2)

11. MaxWeight-Alg1 I = argmaxi Qsi·r(i,c)
Assign each carrier c to the user that maximizes Qsi(t)·r(i,c,t).
This carrier-by-carrier algorithm optimizes objective (1):
Optimizing (1) is not ideal.
It could lead to more service being allocated to a user than it is able to use.
Hence, the queue size (and packet delay) may become larger than necessary.

12. MaxWeight-Alg2 I = argmaxi Qsi·min{r(i,c), Qci}
Carrier-by-carrier algorithm.
A constant factor approximation for objective (2) (1/2– approximation):
Significantly outperform MaxWeight-Alg1.

13. MaxWeight-Alg3 I = argmaxi Qci·min{r(i,c), Qci}
Carrier-by-carrier algorithm.
A constant factor approximation for objective (3) (1/3– approximation):
Significantly outperform MaxWeight-Alg1.

14. MaxWeight-Alg4 b(i,c): assignment of bits from carrier c to user i.
β(c): a quantity that measures the best allocation so far for c.
Does not locally optimize each carrier in isolation.
Each user one by one and finds the best carrier(s) for the user.
Alternative 1/2–approximation for objective (2):
Solution is similar to the standard algo. For Knapsack problems.

15. MaxWeight-Alg5 (1/2)
Based on Generalized Assignment Problem (Fleischer06).
Given a set of bins of different sizes
Each item has a bin-dependent profit, and a bin-dependent size.
The goal is to pack the items into bins so as to maximize the profit in such a way that no bin-size is violated.
Complex and impractical for scheduling wireless systems.
L. Fleisher et al., “Tight approximation algorithms for maximum general assignment problems,” In Proc. ACM-SIAM Symposium on Discrete Algorithms, 2006.
Improved (1-1/e-ε)-approximation for (2) (Note that 1-1/e = … > 1/2).
Theoretical interest to understand what are the limits regarding the approximability of objective (2).

16. MaxWeight-Alg5 (2/2)
Λi : the set of all possible subset of carriers assigned to user i.
For S ∈ Λi,
XSi : indicate whether or not subset S is assigned to user i.

17. Stability Informally, an algorithm is said to be stable, Formally,
If it keeps the queue sizes bounded whenever this is achievable.
Formally,
Showing that Lyapunov function( ) has a negative drift when the queues are large.
The single-carrier MaxWeight has proven to be stable.
MaxWeight-Alg1 through Alg5 are stable.

18. General Definitions of Weight
Many algorithms for single-carrier systems always schedule the user that maximize Wi(t)·r(i,t) for some weight Wi(t).
Proportional Fair schedule in EV-DO Rev. 0
Wi(t) is the reciprocal of an estimate of the average service rate provided to user i.
For multi-carrier versions, we define:
Wsi: the weight of user i at the beginning of time step.
Wei: the weight of user i at the end of time step.
Wci: the weight of user i at the end of time step.

19. Algorithm Extensions First three are straightforward. MaxWeight-Alg4

20. Objective Extensions
New definitions of objective (1) and (2).

21. Simulations (1/2) Field trace Synthetic trace (3 km/h Rayleigh fading)
A constant rate arrival model
5 – 10 users
4 – 8 carriers

22. Simulations (2/2)
Over longer timescales, Alg4 differs from the performance of Alg1-3 dramatically.

23. Summary Contribution of this paper
Generalize the well-know MaxWeight algorithm to accommodate multi-carrier setting.
State the hardness of optimization problems in the multi-carrier setting.
Present simple algorithm solutions.
Present provable performance bounds.
Validate algorithms via numerical results.