1. Achieving Delay Rate-function Optimality in OFDM Downlink with Time-correlated Channels
Joint work with Bo Ji, Kannan Srinivasan, and Ness Shroff
Zhenzhi Qian
The Ohio State University
April

2. Challenges in Wireless Networks
Users have stringent delay requirements
Video calls, online gaming cannot tolerate large delay
Nearly half of the tasks are delay-sensitive

3. Challenges in Wireless Networks
Need to deal with much larger data traffic
>4 Billion mobile users by 2018
12X Mobile data traffic between 2012 and 2018

4. Opportunities in Multi-channel System
Key design mechanisms to boost throughput for 3G/4G wireless networks may actually hurt delay
Opportunistic scheduling: Higher priority for users having good channel NOT for users experience large delay
Schedule users in sub-bands with favorable channel will achieve high throughput without hurting delay

5. OFDM Downlink A time-slotted multi-queue multi-server system
Large numbers of sub-carriers ( channels/servers)
Large numbers of mobile users ( queues)
BS maintains a queue/buffer for each user
Perfect channel state information at BS
Channel estimation via pilot signal
Time-varying connectivity
User 1
Channel Feedback
User 2
Base Station

6. System Model Scheduling Problem Packet arrives at each queue :
At each time-slot , BS needs to determine:
Allocate which channel/server to transmit data for which user/queue
Constraints
Only “ON” channels can be used for transmission
One channel can only be allocate to one user
Channel 1
User 1
Channel 2
User 2
Channel 3
User 3

7. Objective Minimize delay
Given delay threshold , minimize the delay violation probability:
where is the Head-Of-Line delay of queue in time-slot
Maximize the delay rate-function (decay rate) of the delay violation probability in large deviation sense

8. Related works
Queue-length-based metric [Bodas & Shakkottai & Ying & Srikant ’09,’10,’11, Bodas & Javidi ’11]
Optimality results established only for restricted arrival processes (i.i.d. not only across users but also in time)
Good queue-length performance does NOT necessarily imply good delay performance
Target delay rate-function [Sharma & Lin ’11, Ji & Gupta & Lin & Shroff ’14, ’15]
The class of OPF policies is delay-optimal for restricted channel model (i.i.d. not only across users but also in time). In reality, channels in OFDM systems are usually time-correlated

9. Time-correlated Channel
ON/OFF channel
Channel rate is either 0 or 1
Alternating renewal process
“ON” period: The number of time-slots between the last time the channel was “OFF” and the next time-slot it becomes “OFF” again
The length of ith “ON” period is i.i.d. and independent of “OFF” period
Generalization of a two-state Markov chain
OFF ON
Is the class of OPF policies still delay-optimal even
when the channels are time-correlated?

10. Delay-optimality: Intuition
Single-queue single-server with fixed capacity
First-Come First-Serve policy
Serve oldest packets first
Delay optimal
Multi-queue multi-server with full-connectivity
An analogous single-user single-server system
A policy that serves the n oldest packets first, is delay-optimal S1 S2 S3 7 5 6 3 2 1 4 7 6 5 4 3 2 1
Rate 3

11. Delay-optimality: Intuition
Multi-queue multi-server system with time-varying connectivity
Policy that schedules the n oldest packets first is delay-optimal
Not feasible due to partial connectivity
Consider the policy that serve oldest packets for the largest possible value of
All policies satisfy this condition form the class of Oldest Packets First (OPF) policies
Does this mean that policies that satisfy this condition are good from the point of view of minimizing delay? S1 S2 S3 7 6 5 3 2 1 4
Largest k=2

12. Main Results
[ Theorem 1 ] The class of OPF policies achieve optimal delay rate-function performance under the general time-correlated channel model if the following two conditions hold:
Any vector of finite channel states satisfies non -negative correlation condition
“OFF” period D is geometrically distributed
[ Theorem 2 ] Under negatively correlated Markovian channel model, the class of OPF policies can achieve a delay rate-function that is no smaller than p10/log(1-p01) fraction of the optimal value.
This brings us to our main results that the class of OPF policies are indeed good from the point of view of minimizing delay

13. Optimality condition A &B
Non-negative correlation condition [Dubhashi & Ranjan ’96]
The conditional expectation is non- decreasing in each for any disjoint index set
Generalization of the non-negative correlation for two variables
More likely to stay in its current state rather than make transitions
What kind of channel models satisfy these conditions?
Positively correlated Markovian model ( ) )
And even more since the “ON” period could be very general

14. Proof Sketch An upper bound on the delay rate-function
Consider three types events
These events account for sluggish service and bursty arrivals
Take , which is the tightest upper bound constrained by these events

15. Proof Sketch (cont.) Achievable Rate-function of OPF policies
A combined version of Frame Based Scheduling (FBS) with appropriately chosen parameter and perfect-matching policy is shown to be delay-optimal in i.i.d. channel
Given the history information on channel connectivity, we are able to find a frame/perfect-matching with high probability in each time-slot
(Sample-path dominance) Given the same settings, any OPF policy will serve every packet that the FBS/perfect-matching policy has served
We have optimality result if

16. Simulation Setup 10-user system with running time 107 slots
i.i.d. arrival
Channel settings
i.i.d. and Markovian channels (positively and negatively correlated)
Same expected rate
Scheduling policy
Delay weighted matching (DWM, an OPF policy) [Sharma & Lin ’11]
What is the impact of time-correlation on delay
performance by applying the same DWM policy?

17. Numerical Result I More positively correlatedlarger delay
Larger service variation
Smaller loadLarger delay gap
The effect of channel correlation is increasing

18. Numerical Result II More negatively correlatedsmaller delay
Extreme case: ONOFFON sequence, deterministic
Indicate some space for delay improvement under negatively correlated channels

19. Future Work Multi-rate channel model Minimize lateness
The current technical approach fails since the sample-path dominance does not hold
A lexicographically-optimal algorithm that makes the HOL delays most balanced should work well
Introduce a new trade-off between maximizing instantaneous throughput and balancing delays
Minimize lateness
Set a soft deadline for each user’s packets, minimize the lateness between the deadline and the actual completion time
Earliest Due Date (EDD) first algorithm should have good delay performance

20. Q&A
Thanks

21. Assumptions Assumption 1 Assumption 2 Assumption 3 for any and
The arrival process are i.i.d. across all users
Given any and , there exists a positive function
such that:
Assumption 3
The sum of “ON” and “OFF” periods is aperiodic
Finite expectation: and

22. Example 1 “Last packet” problem
Assume one packet is the last packet of a certain user flow
All other queues have plenty of packets buffered and incoming
The HOL delay of queue 2 is the largest among all queues
This queue may never be allocated due to its short queue-length S1 S2 S3 7 15 3 6 5 1 2 Q1 Q2 Q3 S1 S2 S3 7 15 3 6 5 1 2

23. Example 2 Trade-off in multi-rate model
Each channel can serve at most packets from the same queue
Channel rate vs. delay
Whether to serve the queue which has the highest rate
Or to serve the queue with largest delay
Difficulty: Never know the resulting maximum delay! S1 S2 S3 8 15 3 5 1 2 S1 S2 S3 8 1 9 6 2
Rate=2
Rate=2
Rate=2