1. M/G/1/MLPS Queue Mean 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 optimal minimizing the mean delay
Yashkov (1987)
If only the attained service times are known, then
DHR implies that FB is optimal minimizing 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. MLPS disciplines Xi(t) a t
Definition: MLPS discipline, Kleinrock, vol. 2 (1976)
based on the attained service times Xi(t)
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
Xi(t) a
FCFS+PS(a) 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)
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 Xi(t) Ux(t) s s x x t t 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 Xi(t) Vx(t) s s x x t t 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):
FB/PS PS
FB/PS
FB/PS

26. bounded Pareto service time distribution
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 > 0 such 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
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

31. THE END