June 3, 2015Windows Scheduling Problems for Broadcast System 1 Amotz Bar-Noy, and Richard E. Ladner...

April 18, 2023 Windows Scheduling Problems for Broadcast System

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Windows Scheduling Problems for Broadcast

System

Amotz Bar-Noy, and Richard E. Ladner

Presented by Qiaosheng Shi


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Outline

Windows scheduling problem The optimal windows scheduling problem The optimal harmonic windows scheduling

problem

Perfect schedule and tree representationAsymptotic boundsThe greedy algorithmThe combination techniqueSolutions for small hOpen problems & my project plan


3

Outline


problem



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Windows scheduling problem

h slotted channels, and n pages. Each page i has a window size wi. i:1…n

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.

nwwW ,...,1

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.


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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


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Input: A set W={w1,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).

Example: Input: W={2,4,5} Output: one channel

4 2 5 2 4 2 425 252 …

There is at least one transmission of in any window of 5 time-slots

5There is at least one transmission of in any window of 4 time-slots

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The Optimal Windows scheduling problem

H(W)=1


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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).

AssumptionA 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 schemesBroadcast popular movies continuously on h channels.

The Media-on-Demand application


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Staggered broadcasting [Dan96]: Transmit the movie repeatedly on each of the channels. Guaranteed delay: at most 1/h.


Can we do better?Client’s buffer!


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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.

arrive watch & buffer

1 32 (3 pages)

Each time-slot has length 1/3.

0 1/3 2/3 1 4/3 5/3 2

C1: 11111 1

2 2 2 333C2:

…

…

arrive watch & buffer


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Why does it work?

The 1st page is transmitted in any window of one slot.

C1: 11111 1

2 2 2 333C2:

…

…

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.



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•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


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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).

The optimal harmonic windows scheduling problem

6 6

2 2 22 224 4 45 5 5

3 33 37 79 98 8

1 1 1 1 1 1 1 1 1 1 1 1 …

…

…

C1

C2

C3

1 1 1 …C1

Examples: h=1, n=1, N(1)=1

C1 11111 1

2 2 2 333C2

…

…

h=2, n=3. N(2)=3

h=3, n=9. N(3)=9?


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Perfect channel 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.

Another definition: Matrix schedule

...),(...)1,(

...),1(...)1,1(

thChC

tCC

)1,(...),(

)1,1(...),1(

i

i

wthCthC

wtCtC


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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.

ii ww '


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Tree representation

1

2

4 5

3

6 7 8 9

6 6

2 2 22 224 4 45 5 5

3 33 37 79 98 8

1 1 1 1 1 1 1 1 1 1 1 1 …

…

…

C1

C2

C3

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

0 1 2 3 4 5 6 7 8 9 10 11Tree is simple


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Tree representation

1

2

4 5

3

6 7 8 9 One ordered tree per channel Leaves represent the pages

1

0 00

100 2

10 1 0 01

)(xo

Offset )()()()( xpxoxx

The period of each page is the product of the degrees of the nodes on the path from the root to its corresponding leaf.

)(xp

21*20)'( x

53*12)( x


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Tree representation

Page A B C D

Period 2 6 6 6

Offset 0 1 3 5

0 1 2 3 4 5


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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


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

Tree representation

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

A B C D1 D2 D3 A D4 D5 D6 D7 B A D8 D9

D10 D11 C A D12 D13 B D14 D15 A D16 D17 D18 D19 D20 … …

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

Can we always get the perfect channel schedule from an ordered tree?

A A A

A A

BB

BC

C D1 D2 D3 D4 D5 D6 D7 D8 D9

D10 D11 D12 D13 D14 D15 D16 D17 D18 D19 D20

Degree of root must divide the periods 6, 10, 15


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Asymptotic bounds for H(W)

Page i requires at least a fraction of a channel

iw1

n

i iwWh

1

1)(

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

nwwW ,...,1 )()( WhWH

Lower bound For any window vector

Minimum number of channels needed to schedule window vector W

, N(h)


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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) )()( WhWH

First upper bound (round the window sizes down to the nearest power of 2): For any window vector W, there exists a perfect schedule that uses no more than channels.

)(2 Wh

)(2122

1)'(11

WhwWhn

i i

n

ivi

122 ii vi

v wi

v wi

122

1


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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.

iw ivu20iv

1u

)()( WhWH

))(ln()( WheWh


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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 k=1, round window size down to closest to

get a schedule with at most channels. If k>1, 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

iv2 )(2 Wh

xuxWu 2,

iw iw uw ivi 2 u

uu v 2

iw ui Ww ui Ww

uW


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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.

iwiw

)( uWh uW

rWrW

rW

iwiw


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Upper bound for H(W) --- example

Let W=<2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19> At least 3 channels to schedule the windows in W (2<h

(W)<3) k=1: W’ = <2,2,4,4,4,4,8,8,8,8,8,8,8,8,16,16,16,16> k=2, x=3: We get the following 3 vectors

12,12,12,12,6,6,3'3W 15,14,13,12,7,6,33W

10,10,5'5W

19,18,17,16,9,8,4,24W 16,16,16,16,8,8,4,2'4W

11,10,55W

9,8,4,2,15,14,13,12,7,6,3

19,18,17,16,11,10,5rW 3

4

6

4

5

1

3

5

16,16,16,16,8,8,4rW

K=1


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

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

1

)1(

1

)( 21

1

kk

kk

e

ekrkr

for k=1

for k>1

The function r is monotonic increasing function whose exist and is approximately 4.6412

)(lim krk

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

kk eWhe 1

krWkhWh k 1


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

Theorem: Every window vector W, with h(W)>1, 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.

WheWh ln 3595.7lim

kre

k

)(/))(ln(1 WhWhO


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Bounds on N(h)

Lower bound of H(W): )()( WhWH

56147.057721.0 ec

hcehN

Upper bound of H(W):

))(ln()()( WheWhWH

e

h

e

h

he

e

he

ehN

93669.7

Given h channels, maximize n such that each page i is scheduled at least once in any consecutive i slots


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Outline


problem


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June 3, 2015Windows Scheduling Problems for Broadcast System 1 Amotz Bar-Noy, and Richard E. Ladner...

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