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![Page 1: 1 Min-Cost Live Webcast under Joint Pricing of Data, Congestion and Virtualized Servers Rui Zhu 1, Di Niu1, Baochun Li 2 1 Department of Electrical and.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649ea25503460f94ba6bdc/html5/thumbnails/1.jpg)
1
Min-Cost Live Webcast under Joint Pricing of Data, Congestion and
Virtualized ServersRui Zhu1, Di Niu1, Baochun Li2
1Department of Electrical and Computer Engineering
University of Alberta2Department of Electrical and Computer
EngineeringUniversity of Toronto
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Roadmap
Part 1 A joint pricing of data, congestion and virtualized servers
Part 2 Min-cost multicast as k-NWST
The first PTAS proposed
Part 3 Trace-driven simulations
Part 1 A joint pricing of data, congestion and virtualized servers
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Live Webcast
Problem: Large amount of data transferringSignificantly contributing to traffic congestionEngaging many server resources, etc.
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Charge end users – conventional
Monthly flat rate/ Pay-as-you-go/Both
Excessive burden on clients
Charge content/application provider
Encourage customers to use more
E.g. Telus: free six-month subscription of Rdio
Existing pricing policies
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How should webcast operators pay for the
video delivery service?
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A road pricing motivation
Distance traveled pricing
Transferring data
Congestion specific pricing
Congestion degree
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Congestion pricingCharge the webcast provider
A per-minute price rate on each link
Pricing rate ∝ bandwidth-delay product
Related with the media streaming topology
Encourage webcast operator minimize its “waiting data”
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Cost of servers
Download from
source
Recoding and resendingClient
Operation cost
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Roadmap
Part 1 A joint pricing of data, congestion and virtualized servers
Part 2 Min-cost multicast as k-NWST
The first PTAS proposed
Part 3 Trace-driven simulations
![Page 10: 1 Min-Cost Live Webcast under Joint Pricing of Data, Congestion and Virtualized Servers Rui Zhu 1, Di Niu1, Baochun Li 2 1 Department of Electrical and.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649ea25503460f94ba6bdc/html5/thumbnails/10.jpg)
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System model
SourceCDN ServersClient
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F
F
S
F
F
Objective: minimize the total cost including data transferring, congestion
and server opening
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Formulating the problem
, ,, ,
,
min e e i i i j i jx y z
e E i F i F j T
c z f y c x
( )
,
,
,
, ( )
= 1 ( )
, ( , )
,
, , {0,1} ( , , )
e ie N
i ji F
i j i
ii F
i j i e
z y i N F
x j T
x y i F j T
y k
x y z i F j T e E
,
s.t.Server congestionService congestion
![Page 13: 1 Min-Cost Live Webcast under Joint Pricing of Data, Congestion and Virtualized Servers Rui Zhu 1, Di Niu1, Baochun Li 2 1 Department of Electrical and.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649ea25503460f94ba6bdc/html5/thumbnails/13.jpg)
13
Formulating the problem
, ,, ,
,
min e e i i i j i jx y z
e E i F i F j T
c z f y c x
( )
,
,
,
, ( )
= 1 ( )
, ( , )
,
, , {0,1} ( , , )
e ie N
i ji F
i j i
ii F
i j i e
z y i N F
x j T
x y i F j T
y k
x y z i F j T e E
,
s.t.Opening cost
![Page 14: 1 Min-Cost Live Webcast under Joint Pricing of Data, Congestion and Virtualized Servers Rui Zhu 1, Di Niu1, Baochun Li 2 1 Department of Electrical and.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649ea25503460f94ba6bdc/html5/thumbnails/14.jpg)
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Formulating the problem
, ,, ,
,
min e e i i i j i jx y z
e E i F i F j T
c z f y c x
( )
,
,
,
, ( )
= 1 ( )
, ( , )
,
, , {0,1} ( , , )
e ie N
i ji F
i j i
ii F
i j i e
z y i N F
x j T
x y i F j T
y k
x y z i F j T e E
,
s.t.
Optimal solution is a tree
Each client belongs to one server
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The data costThe total data transferred per unit time is proportional to the total number of selected edges
Given the video bit rate r, the total data transferred is
Since nr is a constant, this cost can be incorporated into the server opening cost
ii F
y r nr
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Unfortunately, it is a hard problem.
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Let’s start by ignoring the opening costThen, fi=0 for all relay servers.
Only congestion cost are considered.
Equivalent with an very famous hard problem, Steiner Tree. (NP-hard, even within 1.0105)
M. Chlebik, J. Chlebikova.The Steiner Tree problem on graphs: Inapproximability results. Theoretical Computer Science, 2008
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If we don’t consider the inter-server connection
18
Case 1: No cost for inter-server connections.
Case 2: No inter-server connections are permitted.
In both case, they are equivalent with Uncapacitated Facility Location problem, another NP-hard problem.
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No server number constraint?
Well, it is called Node-Weighted Steiner Tree problem (NWST).
![Page 20: 1 Min-Cost Live Webcast under Joint Pricing of Data, Congestion and Virtualized Servers Rui Zhu 1, Di Niu1, Baochun Li 2 1 Department of Electrical and.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649ea25503460f94ba6bdc/html5/thumbnails/20.jpg)
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NWST – Existing Results
NP-hard to approximate withinC.Lund, M. YannakakisOn the hardness of approximating minimization problems. Journal of the ACM, 1994
Currently best known ratio:S. Guha, S. Khuller.Improved methods for approximating node weighted Steiner trees and connected dominating sets. Information and Computation, 1999
(1 ) ln n
1.35ln n
![Page 21: 1 Min-Cost Live Webcast under Joint Pricing of Data, Congestion and Virtualized Servers Rui Zhu 1, Di Niu1, Baochun Li 2 1 Department of Electrical and.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649ea25503460f94ba6bdc/html5/thumbnails/21.jpg)
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The linear relaxation, ,
, ,,
min e e i i i j i jx y z
e E i F i F j T
c z f y c x
( )
,
,
,
, ( )
= 1 ( )
, ( , )
,
, , 0 ( , , )
e ie N
i ji F
i j i
ii F
i j i e
z y i N F
x j T
x y i F j T
y k
x y z i F j T e E
,
s.t.
![Page 22: 1 Min-Cost Live Webcast under Joint Pricing of Data, Congestion and Virtualized Servers Rui Zhu 1, Di Niu1, Baochun Li 2 1 Department of Electrical and.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649ea25503460f94ba6bdc/html5/thumbnails/22.jpg)
22
Original problem
A PTAS for k-NWST
, ,,
min (x, y, z)
s.t. ,
e e i i i j i je E i F i F j T
ii F
f c z f y c x
y k
, ,,
(x, y, z, ) ( )e e i i i j i j ie E i F i F j T i F
L c z f y c x y k
The Lagrangian relaxation
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Lagrange multiplier λ as opening cost: fi’ := fi
+ λ
Subroutine Algorithm1:
A PTAS for NWST with additional opening cost
, ,,
, ,,
(x, y, z, ) ( )
( )
e e i i i j i j ie E i F i F j T i F
e e i i i j i je E i F i F j T
L c z f y c x y k
c z f y c x k
( , , )P c f A G1P. Klein, R. Ravi.A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithm, 1995
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A PTAS for our problemSearching for proper Lagrange multiplier λ 1
1 1
2 2
( )
( )
P
P
A
A
1 1 2 2k k k
Convex combination of P1 and P2
2
If μ2>1/2, output P2. Otherwise, select some nodes in P2 and add them in P1
3
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Step 1: find proper λ
• For sufficiently large λ, the opening cost dominates
• For sufficiently small λ, the cost depends on congestion, making more to open
• The binary search can find two trees near the server constraint
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Step 2: Convex combination
• Convex combination of P1 and P2
1 1 1
2 2 2
( ) ( ) ( )
( ) ( ) ( )OPEN D
OPEN D
C X C X k k OPT
C X C X k k OPT
where is the total opening cost
is the total congestion cost
( )OPENC X
( )DC X
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Step 3: Merge P1 and P2
![Page 28: 1 Min-Cost Live Webcast under Joint Pricing of Data, Congestion and Virtualized Servers Rui Zhu 1, Di Niu1, Baochun Li 2 1 Department of Electrical and.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649ea25503460f94ba6bdc/html5/thumbnails/28.jpg)
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Target: select k-k1 nodes from P2
P1
P2
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Double edges of P2
P1
P2
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Find the Euler tour and shortcut to tour
P1
P2
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Find the Euler tour and shortcut to tour
P1
P2Average cost:
Then, we have:1 2 1( ) / ( )k k k k
12 2
2 1
2 2 2
( ) ( ')
2( ( ) ( ))
2 ( ( ) ( ))
OPEN D
OPEN D
OPEN D
C X C P
k kC X C P
k k
C X C P
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Connect P1 to the cheapest path of tour
P1
P2
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The total server cost
1 1
1 1 1
2 2 2
( ( , ')) ( ) ( )
( ') ( ')
( ( , ')) 2 ( ( ) ( ))
2 ( ( ) ( ))
D OPEN D
OPEN D
D OPEN D
OPEN D
C PATH S X C X C P
C X C P
C PATH S X C X C P
C X C P
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The upper bound for total cost
Since , we have1( ( , ))DC PATH P S OPT
1 1 1 2 2 2
1
1 2
( ) ( )
2 ( ( ) ( )) 2 ( ( ) ( ))
( ( , ))
2 2
2
(2 1)
OPEN D
OPEN D OPEN D
DS
C X C X
C X C X C X C X
C PATH P S
OPT OPT OPT
OPT OPT
OPT
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Conclusion (Approximation Ratio)Our PTAS can approximate k-NWST with a ratio of
2 1.35ln 1 2.7 ln 1.n n
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Roadmap
Part 1 A joint pricing of data, congestion and virtualized servers
Part 2 Min-cost multicast as k-NWST
The first PTAS proposed
Part 3 Trace-driven simulations
![Page 37: 1 Min-Cost Live Webcast under Joint Pricing of Data, Congestion and Virtualized Servers Rui Zhu 1, Di Niu1, Baochun Li 2 1 Department of Electrical and.](https://reader035.fdocuments.us/reader035/viewer/2022070410/56649ea25503460f94ba6bdc/html5/thumbnails/37.jpg)
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Inter-server and server-client delay traces
Traces collected from PlanetLab and from the Seattle projectMonitor the RTTs among 8 Planet nodes for a 15-day periodMonitor the RTTs from the 8 Planet nodes to 19 Seattle nodes
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Opening cost assignment
The opening costs (including data) for CDN edge nodes are from pricing policy by Amazon Web Service (Amazon CloudFront)
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Baseline Algorithm
Randomly chooses a subset of servers to openWith no inter-server connectionsConnects each client to its closet server.
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12345678 0
0.5
1
1.5
2
2.5
3 Total Cost Congestion Cost Opening CostPe
rform
ance
Rati
o
The cost computed by our algorithm
Number of Servers
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12345678 0
0.5
1
1.5
2
2.5
3 Total Cost Congestion Cost Opening CostPe
rform
ance
Rati
o
The cost computed by baseline algorithm
Number of Servers
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ConclusionsA joint pricing policy of data, congestion and virtual servers for live webcasting application providers
Model the Min-cost multicast and provide the first PTAS for it
Future work:
Only routing are considered, how about using network coding?
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Thank you
Rui Zhu
Department of Electrical and Computer Engineering
University of Toronto