M/G/1/MLPS Queue Mean Delay Analysis

1. M/G/1/MLPS QueueMean Delay Analysis Samuli Aalto (TKK) in cooperation with Urtzi Ayesta (LAAS-CNRS) and Eeva Nyberg-Oksanen

2. Outline • Introduction • DHR service times • IMRL service times • Ongoing work

3. Application: bandwidth sharing in IP networks • 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 (that is, the average file transfer time) • How to service flows and how to analyse? • Guo and Matta (2002), Feng and Misra (2003), Avrachenkov et al. (2004), Aalto et al. (2004,2005,2006)

4. Queueing model • Assume that • flows arrive according to a Poisson process with rate l • flow size distribution is ”heavier” than exponential, 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 already sent • remaining service = the number of packets still left

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 is optimalminimizing the mean delay • Remark: 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. Service 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

7. FCFS+PS(a) a MLPS disciplines • Definition: MLPS discipline, Kleinrock, vol. 2 (1976) • based on the attained service times Xi(t) • 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 Xi(t) PS FCFS t

8. Our objective • We compare MLPS disciplines in terms of the mean delay • MLPS vs PS • MLPS vs MLPS • We assume that service times are • DHR • IMRL • Note: • DHR Ì IMRL

9. References • K. Avrachenkov, U. Ayesta, P. Brown and E. Nyberg (2004) • IEEE INFOCOM • S. Aalto, U. Ayesta and E. Nyberg-Oksanen (2004) • ACM SIGMETRICS – PERFORMANCE • S. Aalto, U. Ayesta and E. Nyberg-Oksanen (2005) • Operations Research Letters 33 • S. Aalto and U. Ayesta (2006a) • IEEE INFOCOM • S. Aalto and U. Ayesta (2006b) • to appear in Journal of Applied Probability • S. Aalto (2006) • to appear in Mathematical Methods of Operations Research

10. Outline • Introduction • DHR service times • IMRL service times • Ongoing work

11. DHR service times • Service time distribution: • Density function: • Hazard rate: • Definition: • Service time distribution belongs to class DHR (Decreasing Hazard Rate) if h(x) is decreasing • Examples: • Pareto (taking values from 0 on) and hyperexponential

12. Unfinished truncated work Ux • Customers with attained service Xi(t) less than x: • Unfinished truncated work with truncation threshold x: • Unfinished work:

13. Optimality of FB • Aalto et al. (2004): • FB minimizes the unfinished truncated work for any x and t in each sample path Xi(t) Ux(t) FCFS FB s s x x t t

14. Idea of the mean delay comparison • Kleinrock (1976): • For all WC & NA service disciplines p • so that (by applying integration by parts) • Thus,

15. MLPS vs 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 FB FB/PS PS FB/PS FB/PS

16. Mean unfinished truncated work bounded Pareto service time distribution

17. MLPS vs MLPS: changing internal disciplines • Aalto and Ayesta (2006a): • 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) MLPS MLPS’ FB/PS PS/FCFS

18. MLPS vs MLPS: splitting FCFS levels • Aalto and Ayesta (2006a): • Any number of levels with all internal disciplines allowed • MLPS derived from MLPS’ by splitting any FCFS level and copying the internal discipline MLPS MLPS’ FCFS FCFS FCFS

19. MLPS vs MLPS: splitting PS levels • Aalto and Ayesta (2006a): • 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 • Splitting any higher PS level is still an open problem (contrary to what we thought in an earlier phase)! MLPS MLPS’ PS PS PS

20. Outline • Introduction • DHR service times • IMRL service times • Ongoing work

21. IMRL service times • Service time distribution: • H-function: • Mean residual lifetime: • Definition: • Service time distribution belongs to class IMRL (Increasing Mean Residual Lifetime) if H(x) is decreasing • Examples: • all DHR service time distributions, Exp+Pareto

22. Level-x workload • Customers with attained service less than x: • Unfinished truncated work with truncation threshold x: • Level-x workload: • Workload = unfinished work:

23. Non-optimality of FB • Aalto and Ayesta (2006b): • FB does not minimize the level-x workload Xi(t) Vx(t) FCFS FB not optimal FB s s x x t t

24. Idea of the mean delay comparison • Righter et al. (1990): • For all WC & NA service disciplines p • so that (by applying integration by parts) • Thus,

25. MLPS vs PS • Aalto (2006): • Any number of levels with FB and PS allowed as internal disciplines • Aalto and Ayesta (2006b): • Any number of levels with FB and PS allowed as internal disciplines FB/PS PS FB/PS FB/PS

26. Mean level-x workload bounded Pareto service time distribution

27. Non-optimality of FB • Aalto and Ayesta (2006b): • FB does not necessarily minimize the mean delay for IMRL service times • Counter-example: • Exp+Pareto belongs to IMRL but not DHR (for 1 < c < e): • There is e> 0such that FB FB FCFS

28. Outline • Introduction • DHR service times • IMRL service times • Ongoing work

29. Gittins index • Gittins (1989): • J-function • Gittins index for a customer with attained service a: • Gittins discipline serves the customer with highest index • Gittins discipline minimizes the mean delay in M/G/1 (among NA disciplines): • If DHR, then FB optimal • If NBUE, then FCFS optimal • If CHR+DHR, then FB, FCFS, or FCFS+FB optimal

30. Bandwidth sharing networks • Bandwidth sharing network • multiple links shared by elastic flows with different routes • necessary stability conditions: for all links l, • Verloop et al. (2005): • global SRPT instable in the network case, that is necessary stability conditions are not sufficient • However, local SRPT (applied to each route separately) is stable and reduces the mean delay in an optimal way

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