1. Mean Delay Analysis of Multi Level Processor Sharing Disciplines
Samuli Aalto (TKK)
Urtzi Ayesta (before CWI, now LAAS-CNRS)

2. Teletraffic application: scheduling elastic flows
Consider a bottleneck link in an IP network
loaded with elastic flows, such as file transfers using TCP
if RTTs are of the same magnitude, then approximately fair bandwidth sharing among the flows
Intuition says that
favouring short flows reduces the total number of flows, and, thus, also the mean delay at flow level
How to schedule flows and how to analyse?
Guo and Matta (2002), Feng and Misra (2003), Avrachenkov et al. (2004), Aalto et al. (2004, 2005)

3. Queueing model Assume that
flows arrive according to a Poisson process with rate l
flow size distribution is of type DHR (decreasing hazard rate) such as hyperexponential or Pareto
So, we have a M/G/1 queue at flow level
customers in this queue are flows (and not packets)
service time = the total number of packets to be sent
attained service = the number of packets sent
remaining service = the number of packets left

4. Scheduling disciplines at flow level
FB = Foreground-Background = LAS = Least Attained Service
Choose a packet from the flow with least packets sent
MLPS = Multi Level Processor Sharing
Choose a packet from a flow with less packets sent than a given threshold
PS = Processor Sharing
Without any specific scheduling policy at packet level, the elastic flows are assumed to divide the bottleneck link bandwidth evenly
Reference model: M/G/1/PS

5. Optimality results for M/G/1
Schrage (1968)
If the remaining service times are known, then
SRPT is optimal minimizing the mean delay
Yashkov (1987)
If only the attained service times are known, then
DHR implies that FB (which gives full priority to the youngest job) is optimal minimizing the mean delay
Remark: In this study we consider work-conserving (WC) and non-anticipating (NA) service disciplines p such as FB, MLPS, and PS (but not SRPT) for which only the attained service times are known

6. MLPS disciplines
Definition: MLPS discipline, Kleinrock, vol. 2 (1976)
based on the attained service times
N+1 levels defined by N thresholds a1 < … < aN
between levels, a strict priority is applied
within a level, an internal discipline (FB, PS, or FCFS) is applied a
FCFS+PS(a)

7. Earlier results: comparison to PS
Aalto et al. (2004):
Two levels with FB and PS allowed as internal disciplines
Aalto et al. (2005):
Any number of levels with FB and PS allowed as internal disciplines

8. Idea of the proof
Definition: Ux = unfinished truncated work with threshold x
sum of remaining truncated service times min{S,x} of those customers who have attained service less than x
Kleinrock (4.60)
implies that
so that

9. Mean unfinished truncated work E[Ux]
bounded Pareto file size distribution

10. Question we wanted to answer to
Given two MLPS disciplines, which one is better in the mean delay sense?

11. Locality result x a Proposition 1:
For MLPS disciplines, unfinished truncated work Ux within a level depends on the internal discipline of that level but not on the internal disciplines of the other levels x a
FCFS+PS(a)
PS+PS(a)

12. New results 1: comparison among MLPS disciplines
Theorem 1:
Any number of levels with all internal disciplines allowed
MLPS derived from MLPS’ by changing an internal discipline from PS to FB (or from FCFS to PS)
Proof based on the locality result and sample path arguments
Prove that, for all x and t,
Tedious but straightforward

13. New results 2: comparison among MLPS disciplines
Theorem 2:
Any number of levels with all internal disciplines allowed
MLPS derived from MLPS’ by splitting any FCFS level and copying the internal discipline
Proof based on the locality result and mean value arguments
Prove that, for all x,
An easy exercise since, in this case, we have explicit formulas for the mean conditional delay

14. New results 3: comparison among MLPS disciplines
Theorem 3:
Any number of levels with all internal disciplines allowed
The internal discipline of the lowest level is PS
MLPS derived from MLPS’ by splitting the lowest level and copying the internal discipline
Proof based on the locality result and mean value arguments
Prove that, for all x,
In this case, we don’t have explicit formulas for the mean conditional delay
However, by truncating the service times, this simplifies to a comparison between PS+PS and PS

15. Note related to Theorem 3
Splitting any higher PS level is still an open problem (contrary to what we thought in an earlier phase)!
Conjecture 1:
Let p = PS+PS(a) and p’ = PS+PS(a’) with a £ a’
For any x > a’,
This would imply the result of Theorem 3 for any PS level

16. Recent results
For IMRL (Increasing Mean Residual Lifetime) service times:
FB is not necessarily optimal with respect to the mean delay
a counter example where FCFS+FB and PS+FB are better than FB
in fact, there is a class of service time distributions for which FCFS+FB is optimal
If only FB and PS allowed as internal disciplines, then
any such MLPS discipline is better than PS
Note: DHR Ì IMRL