Mean Delay Analysis of Multi Level Processor Sharing Disciplines

1. Mean Delay Analysis of Multi Level Processor SharingDisciplines 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 optimalminimizing 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 optimalminimizing 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. FCFS+PS(a) a MLPS disciplines • Definition: MLPS discipline, Kleinrock, vol. 2 (1976) • based on the attained service times • N+1levels 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

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. FCFS+PS(a) PS+PS(a) x a Locality result • 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

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