Amotz Bar-Noy, and Richard E. Ladner

1
Windows Scheduling Problems for Broadcast System

• Amotz Bar-Noy, and Richard E. Ladner
• Presented by Qiaosheng Shi

2
Outline

• Windows scheduling problem
• The optimal windows scheduling problem
• The optimal harmonic windows scheduling problem
• Perfect schedule and tree representation
• Asymptotic bounds
• The greedy algorithm
• The combination technique
• Solutions for small h
• Open problems my project plan

3
Outline

• Windows scheduling problem
• The optimal windows scheduling problem
• The optimal harmonic windows scheduling problem
• Perfect schedule and tree representation
• Asymptotic bounds
• The greedy algorithm
• The combination technique
• Solutions for small h
• Open problems my project plan

4
Windows scheduling problem

• h slotted channels, and n pages.
• Each page i has a window size wi. i1n
• window vector
• Question Is there a schedule for the n pages on
  the h slotted channels
• one page each time slot
• the gap between two consecutive appearances of
  page i is no more than wi.

The problem is based on the max metric and not
the average metric. That is, the next appearance
of a page depends only on its previous
appearance. But in the average metric, the next
appearance of a page depends on all of its
previous appearances.
5
The push systems application

• The broadcasting environment consists of
• Clients who wish to access information pages from
  broadcast channels.
• Servers who broadcast the information pages on
  channels
• Providers who supply the information pages
• Window size of each page (quality of service) is
  determined by the money providers paid to
  servers.
• The server is left with the problem minimize the
  number of channels (bandwidth) needed to
  guarantee the quality of service.

?The optimal windows scheduling problem
6
The Optimal Windows scheduling problem

• Input A set Ww1,w2,,wn of requests for
  broadcasting. A request with window wi needs to
  be broadcasted at least once in any window of wi
  time-slots.
• Output A feasible windows scheduling of W.
• Goal minimize number of channels used H(W).

H(W)1
7
The Media-on-Demand application

• Medias are broadcast based on customer demand.
• A limited number of channels.

• The goal Minimizing clients maximal waiting
  time (delay) with given bandwidth (number of
  channels).

• Assumption
• A client that wishes to watch a movie is
  listening to all the channels and is waiting
  for his movie to start.
• Clients have large enough buffer.
• Each channel transmits data at the playback rate.
• Basic broadcasting schemes
• Broadcast popular movies continuously on h
  channels.

8
The Media-on-Demand application
Staggered broadcasting Dan96 Transmit the
movie repeatedly on each of the channels.
Guaranteed delay at most 1/h.
Can we do better?
Clients buffer!
9
The Media-on-Demand application
Partition the movie into segments (or pages).
Early segments (or pages) are transmitted more
frequently.
The client can start watching the movie without
interruptions. Maximal delay 1/3.
10
The Media-on-Demand application
Why does it work?
The 1st page is transmitted in any window of one
slot.
The 2nd page is transmitted in any window of two
slots.
The 3rd page is transmitted at least once in any
window of three slots.
11
The Media-on-Demand application

• The movie is partitioned into n pages, 1,..,n.
• Necessary and sufficient condition page i is
  transmitted at least once in any window of i
  slots (i-window).
• The client has page i available on time (from his
  buffer or from the channels).
• The maximal delay one slot 1/n.
• Therefore, the goal is to maximize n for given h.

?The optimal harmonic windows scheduling problem
12
The optimal harmonic windows scheduling problem

• Given h, maximize n such that each i in 1,..,n is
  scheduled at least once in i time slots. The
  maximum n is denoted by N(h).

h3,
n9. N(3)9?
13
Perfect channel schedule

• Another definition Matrix schedule

• Channel schedule each page is scheduled on a
  single channel.
• A schedule S is called cyclic if it is an
  infinite concatenation of a finite sequence.

14
Perfect channel schedule

• Perfect channel schedule For page i, there
  exists a , page i gets one time slot
  exactly every wi time slots.
• the window size of page i in the perfect channel
  schedule.
• Perfect channel schedule is cyclic (least common
  multiple).
• Several points
• Avoid busy-waiting the client actively listen
  until its movie arrives.
• Not optimal for windows scheduling problem
• Finding an optimal perfect channel schedule is
  NP-hard in general.
• Only need to record three numbers for one page
  channel number, period length and offset.

15
Tree representation
0 1 2 3 4 5 6 7 8
9 10 11
Tree is simple
Page 1 2 3 4 5 6 7 8 9
Channel 1 2 3 2 2 3 3 3 3
Period 1 2 3 4 4 6 6 6 6
Offset 0 0 0 1 3 1 2 4 5
16
Tree representation

• One ordered tree per channel

• Leaves represent the pages

17
Tree representation
0 1 2 3 4 5
Page A B C D
Period 2 6 6 6
Offset 0 1 3 5
18
Tree representation
Page A B C D E F G H
Period 6 6 12 12 12 12 12 4
Offset 0 2 4 10 1 5 9 3
19
Tree representation
Can we always get the perfect channel schedule
from an ordered tree?

• If all leaves are distinct in forest, the
  corresponding schedule is perfect channel
  schedule.

Can we always construct an ordered tree for a
perfect channel schedule?

• However, there exist perfect channel schedule
  that cannot be embedded in a tree.

Degree of root must divide the periods 6, 10, 15
20
Asymptotic bounds for H(W)
, N(h)
Minimum number of channels needed to schedule
window vector W

• Upper bound
• It is achieved by constructing a perfect channel
  schedule.

21
Upper bound for H(W) --- simple case

• Window sizes are all powers of 2.
• Lemma there exists a perfect schedule that uses
  exactly channels. (the first
  lemma)

22
Upper bound for H(W) --- simple case

• All the window sizes are powers of 2 multiplied
  by some number u.
• Lemma If all the are of the form for
  some and , then there exists a
  perfect schedule that uses exactly
  channels. (the 2nd lemma)
• Construct an algorithm that for given window
  vector W creates perfect schedules with about
  channels.

23
Upper bound for H(W) --- the algorithm

• The algorithm use two parameters k and x that are
  optimized to obtain the best bound.
• k the depth of the recursion
• x is optimized for each value of k.
• If k1, round window size down to closest to
  get a schedule with at most channels.
• If kgt1, partition the window vector W into x
  vectors denoted by .
• is rounded down to maximal such that
  , is an odd number and
  for some u. Then
• The set of such that is
  denoted by

24
Upper bound for H(W) --- the algorithm

• channels needed to schedule all
  windows in
• Some windows scheduled into non-fully used
  channels. The set of all these windows is denoted
  by
• If x is larger, then is closer to .
  However, is too big.
• If x is smaller, then is small. But is
  too small compared to
• For each k, find the best value for x.

25
Upper bound for H(W) --- example

• Let Wlt2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18
  ,19gt
• At least 3 channels to schedule the windows in W
  (2lth(W)lt3)
• k1 W lt2,2,4,4,4,4,8,8,8,8,8,8,8,8,16,16,16,16
  gt
• k2, x3 We get the following 3 vectors

?4
6
4
5
26
Upper bound for H(W) --- major lemma

• Define r as mapping from the positive integer to
  the reals by

for k1
for kgt1

• For window vector W and positive integers k if
  then there exists a
  perfect schedule with number of channels bounded
  above by

27
Upper bound for H(W) --- major lemma

• Theorem Every window vector W, with h(W)gt1, has
  perfect schedule using number of channels bounded
  above by , where
• Theorem For any window vector W, there exists an
  algorithm for the optimal windows scheduling
  problem yielding a solution that is within a
  factor of of the
  optimal solution.

28
Bounds on N(h)
Given h channels, maximize n such that each page
i is scheduled at least once in any consecutive i
slots

• Lower bound of H(W) ?

• Upper bound of H(W)
  ?

29
Outline

• Windows scheduling problem
• The optimal windows scheduling problem
• The optimal harmonic windows scheduling problem
• Perfect schedule and tree representation
• Asymptotic bounds
• The greedy algorithm
• The combination technique
• Solutions for small h
• Open problems my project plan