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Placement of Web-Server Proxies with Consideration of Read and Update

Operations on the Internet

Xiaohua Jia, Deying Li, Xiaodong Hu, Weili Wu and Dingzhu Du

THE COMPUTER JOURNAL, Vol.46, No.4, 2003

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Outline

Introduction

Problem Formulation

Optimal Placement of k Proxies

Optimal Number of Proxies

(3)

Caching

Caching is a technology to alleviate traffic congestion and improve the response time of Web servers.

There are basically two types of Web caching:

client-based caching

server-based caching

(4)

Two Problems of the Placement of Proxies

1. Given proxies, find the optimal placement of the proxies in the network, such that the overall access cost(including both read and update costs) is minimized.

2. For an unconstrained number of proxies, find the optimal number of proxies and their placement, such that the overall access cost is minimized.

They formulate the problems by using the dynamic programming

method. Optimal results are obtained in polynomial time.

(5)

Outline

Problem Formulation

(6)

Notations and Definitions

The network is modeled by a connected graph

   

. For a link

     

,

   

is the distance of the link.

Web server is

.

Each node



is associated with a non-negative number

  

, which is the access frequency to

.

  

can be the number of accesses during a certain period of time, combined with the data size retrieved from the server.

Server

has an update frequency

, which is the number of

update operations (combined with the updated data size) during a

(7)

Stable Routing

The implied routing method in the Internet is the shortest path routing.

It always takes the shortest paths to access data.

If the routing is stable, the routes to all cilents as viewed by

form a shortest path tree, where the root of the tree is

.

Let



denote such a tree induced from the original network graph.

(8)

Update Models

There are basically two models for

to transmit updated data to the proxies: unicast model and multicast model.

In this paper, multicast model is used.

(9)

Notations and Definitions (Cont.)

Let

    

be the path connecting



and



in

 

. Then,

     

P

           

. Let



    

denote the first proxy that is met while going from client



to



along the tree

 

.

Suppose the hit ratio of a proxy is



. So,

 

  

of client’s requests can be served by a proxy if the proxy is met and the remaining

        

requests will be served by the server.

Here, they assume the upstream flow from a proxy has a hit ratio zero(i.e. a request will be served by the server if it is missed by a proxy).

The hit ratio is an average value of many accesses to a proxy measured over a long period of time.

They assume the hit ratios of all proxies and for all clients are the same (i.e.



).

(10)

Hit Ratio

They assume the hit ratios of all proxies and for all clients are the same

(i.e.



).

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Cost

The cost for client



to access



is

 

                            

. The total cost for all clients in

 

to access



is

P

                                  

.

Let



denote a set of proxies in the network and

       

the shortest path tree rooted from



to reach all proxies in



.

The overall cost to update the proxies is



X

             

The total cost of all clients in

 

to access



with a set of proxies



is

     

X

                                  



X

   



        

(12)

Outline

Optimal Placement of Proxies

(13)

The Meaning of Left

Let denote a subtree of



rooted at a node



,

  

.

Given



and



in



,



is said to be the left of



if there exist





and

 

such that

   

,

   

, and





and





are siblings with





being to the left of





.

(14)

Partitioning of Subtree



can be partitioned into three subgraphs by dividing point ,

 

.

(15)

Partitioning of

  

          

and



is to the left of





.

  

is partitioned into three parts:

  

,



and

 

.

(16)

Simplicitly of Notation

The cost function can be rewritten as:

     

X

                                  



X

             

X

                   

X

                  



X

   



        

For the simplcity of notation, omit the second term.

     

X

                    

X

   



        

(17)

Cost Notations of Optimal Placement of Proxies

To recursively define the placement of proxies in a tree, the server at the root is regarded as one of the proxies.

For

   

, define

                

              



                  

               



     

(or

      

) is the minimal access cost by placing



proxies in

 

(or

  

).

(18)

Dynamic Programming

When



, we always place the only proxy at root



. So,

     

P

            

.

When



, we can always find a node



,

   

and

  

, which satisfies:

(1) a proxy is placed at



(2) no proxy is placed in

  

(

  

could be empty) (3) no proxy is placed in

          

.

(19)

Dynamic Programming (cont.)

When



, assume proxies are placed in

 

,

    

. So



proxies will be placed in

  

.

Therefore, we have

                            

, where

    

P

              

.

(20)

Dynamic Programming (cont.)

    

8

> >

> >

> <

> >

> >

> : P

             

if



                        

               

if



     

8

> >

> >

> <

> >

> >

> : P

               

if



                        

                  

if



(21)

Algorithm 1

(22)

Tracing the Placement of Proxies

   

, a 2-dimensional array, records the minimal cost of by placing



proxies in it.

    

records the partitioning point of - say



- and the number of proxies placed in



.

The final result of proxy placement is derived from arrays



   

and



      

.

(23)

Time Complexity of Algoritm 1

The work of procedures

   

    

and

   

     

is to compute results in arrays

    

and

     

, respectively.

    

and

     

have

 

and



 

different entries, respectively.

It takes at most



   

time to compute an entry.

So It takes



                

time to compute.

(24)

Outline

Optimal Number of Proxies

(25)

Cost Notations of Optimal Number of Proxies

We want to find the optimal number of proxies required in the system to make the total cost minimal.

Define

     

           

      

              



Considering the optimal placement of proxies in

 

, there are always two choices:

placing no proxy in

 

(except



) or place some proxies in

 

(besides



).

Let

   

denote the total cost in

 

with no proxy placed in it and

    

the total cost with some proxies in

 

.

   

X

            

      

                         

(26)

Dynamic Programming

                 

  

X

               

                          

                    

  

X

                 

                             

(27)

Algorithm 2

(28)

Time Complexity of Algoritm 2

Procedures

   

   

and

   

    

compute arrays

   

and

    

, respectively.

   

and

    

have



and





different entries, respectively.

It takes at most



  

time to compute an entry.

So It takes



           

time to compute.

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