MG1MLPS 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 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

• Definition MLPS discipline, Kleinrock, vol. 2
  (1976)
• based on the attained service times Xi(t)
• N1 levels defined by N thresholds a1 lt lt 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/PS
PS
FB
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, ExpPareto

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
• ExpPareto belongs to IMRL but not DHR (for 1 lt c
  lt e)
• There is e gt 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 CHRDHR, then FB, FCFS, or FCFSFB 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

31
THE END