1. Connection-Level Scheduling in Wireless Networks Using Only MAC-Layer Information
Javad Ghaderi, Tianxiong Ji and R. Srikant
Coordinated Science Laboratory
and
Department of Electrical and Computer Engineering

2. Outline Background and Description of Problem
Centralized scheduling algorithm with only MAC layer information
Distributed implementation using CSMA.

3. Outline Background and Description of Problem
Centralized scheduling algorithm with only MAC layer information
Distributed implementation using CSMA.

4. Wireless Networks: Interference Model
A collection of links with interference constraints
Users may not be able to transmit simultaneously due to interference.
Feasible schedule: a collection of links that can be activated simultaneously
Medium Access Protocol (MAC): determines which users are to transmit at each time instant.
Must satisfy interference constraints
Achieve efficient use of resources
Performance metrics: throughput and delay

5. Wireless Networks: Traffic Model
Files/Connections arrive at link “l” according to some stochastic process with rate λl .
File/Connection size ~ mixture of Geometric distributions
There are K file types, each type is geometrically distributed with different mean. A file arrived at links “l” can belong to type i with probability pli , i=1, 2, .., K.
Motivation: Heavy-tail-distribution of file sizes in Internet
Most of files are short but most of bytes are generated by long files.
By controlling probabilities pli , for a fixed average file size, we can generate distributions with such properties.

6. MaxWeight: Static Connections
Connections are not time varying. Packets of a connection arrive at a link according to some stochastic process.
Selects a feasible schedule which maximizes the sum of weights of the selected links
Weights are functions of queue lengths

7. MaxWeight: Dynamic Connections
In presence of file/connection arrivals and departures
Weight = a function of the total number of packets of all files at link
Problem:
Links with small queues may starve for long periods of time.
Severe unfairness if file sizes can vary widely (short files may starve for long periods of time)

8. Background and Description of Problem
Centralized Scheduling Algorithm with Only MAC-Layer Information
Distributed implementation using CSMA.

9. Layered Network Model Scheduler is implemented as part of MAC layer.
The TCP layer controls packet arrivals into the MAC layer
Window flow control protocol, window size ≥ 1
Questions:
Is MaxWeight with only MAC layer information still throughput optimal?
Can it resolve the unfairness issue (better delay performance)?

10. Network State Scheduling algorithm:
Which links to activate at each time?
Which MAC layer packets to serve at an active link?
Decisions should be made only based on MAC layer information.
Network State:
For each file: MAC-layer queue, congestion window size, transport-layer flow indicator (whether there are still packets of file at Transport layer or not)
Additional information required for MAC-layer service discipline (FIFO, random, …), e.g., ordering of MAC-layer packets.
Congestion window dynamics determined by the state
Network Markov Chain is well-defined.

11. MaxWeight with MAC layer Information
Weight of link “l” is a function of its MAC queue length (with modification when queue is zero):
Our scheduling protocol:
At each time t, the Log-Max-Weight algorithm finds a schedule s which maximizes

12. Throughput Optimality
Main Result: The scheduling algorithm maximizes throughput, independent of congestion control algorithm and (non-idling) service discipline.
Proof Sketch:
The max weight algorithm with weight log(ql) is throughput optimal, where ql is the expected number of packets waiting at link l. (Note: ql is a function of network state.)
The difference between the link weight using ql and the weight using the MAC queue length qlmac is small.
Therefore, using the MAC queue length is sufficient to ensure throughput optimality

13. Throughput Optimality: Proof Sketch
nl: number of files at link l
 : max mean file size
Because
Taking log()

14. Simulations: File Transfer Delays
Only assumption on congestion window sizes used by the transport layer is that they are greater than zero and upper bounded
1-hop Interference model
Mixture of Geometric distributions: mean 2 with probability 5/6 and mean 100 with probability 1/6.
Simulations for different traffic intensities
MAC-layer packets are removed in a FIFO order.

15. Simulations: File Transfer Delays
Delay performance of short files
Delay performance of long files
Short file : files which are less than half of the mean file size.

16. Simulations: File Transfer Delays
A large total queue length at a link does not imply a large number of files at that link.
Regular MaxWeight chooses a link with large queue length but containing only a few files.
MAC scheduler yields better delay performance compared to regular MaxWeight.
1-bit additional information to identify short files for short-file-first MAC service discipline: delay performance will be even better.

17. Background and Description of Problem
Centralized Scheduling Algorithm with Only MAC-Layer Information
Distributed Implementation Using CSMA.

18. Distributed Implementation
Previous result holds with weight functions of form
where g(.) is an arbitrary increasing function.
Use CSMA using such weight functions:
Each time slot (t) is divided into two parts
Control
Data
choose a decision schedule m(t)
choose a data transmission schedule x(t)

19. Distributed Implementation
Step 1: Generating Decision Schedule m(t)
Each link “i” transmits an INTENT message with probability ai in the control slot t.
Those links that transmit INTENT messages and do not hear any INTENT messages from the neighboring links consist a decision schedule m(t).
Step 2: Generating Transmission Schedule x(t)
For any link i in m(t) do
If no links in its conflict set C(i) were active in the previous data slot, link i will decide to become
active with probability pi: xi(t)=1
inactive with probability : xi(t)=0
Else, link i will remain inactive: xi(t)=0
Step 3: Data Transmission: In the data slot, use x(t) as the transmission schedule.

20. Distributed Implementation: Example
Current schedule: x(t)={1, 5}
Select a decision schedule: m={3, 5, 6}
Allowed decisions for links in m:
link 3: x3=0 (no choice)
Link 5: x5=0 (w.p. 1-p5)
link 6: x6=1 (w.p. p6)
Other links’ states unchanged
New schedule: x(t+1)={1, 6} 5 2 7 1 4 6 3

21. Distributed Implementation
But queues are actually changing with time, so weights are time-varying.
ensures that weights change slowly enough such that the Markov chain remains close to its equilibrium distribution.
x(t) evolves as a Discrete-Time Markov Chain.
If the weights do not change with time, the steady-state probability of using schedule x has the following product-form: By letting , we have

22. Distributed Implementation
Hence, distributed algorithm chooses schedule x with probability where
Distributed algorithm chooses the MaxWeight schedule with high probability.
Combined with previous result CSMA using only MAC layer information is throughput optimal where
(No time scale separation assumption)
g(.) can be arbitrarily slow increasing , so for large range of practical queue lengths.

23. Conclusions Scheduling Using MAC layer information
Scheduler is implemented as part of MAC Layer
Can resolve the unfairness issue (better delay)
Can be throughput optimal if the link’s weight is chosen as a log-wise function of its MAC queue.
Such a weight function can be used in distributed implementation using CSMA (still throughput optimal with no time scale separation assumption)
Under investigation: Extension to Multihop, general file size distributions