APPROXIMATE ANALYSIS AND MEASUREMENT OF EQUIVALENT MODEL FOR TREE CONNECTION OF WEB SERVER SYSTEMS

Chung-Ping Chen*, Ying-Wen Bai** and Yin-Sheng Li** Graduate Institute of Applied Science and Engineering, Fu Jen Catholic University* Department of Electronic Engineering, National Taipei University of Technology*

Department of Electronic Engineering, Fu Jen Catholic University** 510 Chung Cheng Rd , Hsinchuang , Taipei County 24205

Taiwan, Republic of China cpchen@ntut.edu.tw, bai@ee.fju.edu.tw, sam6813@ms27.hinet.net

ABSTRACT In this paper we propose a method to estimate the system performance before constructing the tree connection of a Web server system, which means all nodes have exactly zero or two childes and haven’t loops; we aim at a serial-parallel connection by carrying on simulation, analysis equation and measurement. We analyze the arrival rate with respect to the service rates where there are three kinds of relation - greater than, equal to and less than - which requires 27 cases or regions to represent the complex model of a complicated system using equivalence. For the physical verification we install a Web server system represented by a serial-parallel queue. We use measurement software to measure the system response time of the Web page access, and we find that the equivalent service rate of the actual measurement by linear regression estimate. The equivalent service rate is 54 requests/sec by the specific experimental measurement, and at 48.21 requests/sec we compute the difference between the two service rates as 10.85%. In the actual measurement, if we use a Web page data size of 25Kbytes, measure the system response time and compute the equivalent system response time, the difference between them is 46.12% KEY WORDS Equivalent Queue, Queuing Network, Web Server System, Service Rate, Arrival Rate 1. Introduction Lately an increasing number of Internet applications can be noticed and the Web services are booming. The development of various network service techniques has caused users to depend on the Website of their particular application, and therefore there is a rapidly increasing need for services born by the network servers [1-2]. In this paper we derive equivalent equations to represent the serial-parallel queue which models the Web server system. An earlier research paper, aiming at a low blocking rate, has obtained the equivalent service rate of

the serial equivalent queue system. The equivalent parallel queuing system is still uninvestigated. However, the methods of that paper can be extended to model the serial-parallel queues [3-5]. We thus develop the equations to represent the serial-parallel queues for a tree connection of the network for the serial-parallel Web server system. We also use mathematics tool software, network simulation software and network measurement software to verify the differences between analysis, simulation and measurement. We also consider the various blocking probabilities which extend from the basic case to other cases with different relations between service rates and arrival rates of the Web server system. This paper is organized as follows. In Section 2 we derive the equations of the serial-parallel queues to represent the tree connection of a Web server system. In Section 3 we provide the simulation to obtain the system performance that serial-parallel queues. In Section 4 we install the actual Web server system and obtain the physical measurements of the system performance. In Section 5 we provide linear regression and find the differences between analysis, simulation and measurement. In Section 6 we provide summary and conclusions. 2. The Derived Equations for the Performance of Serial-Parallel Queues Serial-parallel queues can be basic connection models to represent simple computer networks. If the amount of error between the model and the physical network is limited, the expansion of the basic serial-parallel queues can model a tree connection with a set of system parameters. We use the system definition and the model parameters as shown in Table 1.

590-132 91

Table 1 System definition and model parameters

Parameters Description Definition or Unit

λ Web request rate Requests/sec

1P Probability of parallel first queue

2P Probability of parallel second queue

sμ Service rate of serial queue Requests/sec

1Pμ Service rate of parallel first queue Requests/sec

2Pμ Service rate of parallel second queue Requests/sec

eqμ Service rate of equivalent for serial-parallel network Requests/sec

sB Buffer size of serial first queue Bytes

1PB Buffer size of parallel first queue Bytes

2PB Buffer size of parallel second queue Bytes

TotalB Buffer size total of serial-parallel queues Bytes

( )sE T System response time of serial queue msec

1( )pE T System response time of parallel first

queue msec

2( )pE T System response time of parallel

second queue msec

( )eqE T System response time of equivalent for serial-parallel queues msec

We use the serial-parallel queues as a basic component to simplify a tree connection, and we aim at equations of equivalent serial-parallel queues based on the serial-parallel equivalent model shown in Fig.1 [3-4,6].

2( )pE T

( )eqE T

1( )pE T

( )sE T

1 2(1 )P P= −

2Pλ

eqμ

2Pμ

1Pλ

sB

1PB

2PB

1Pμ

TotalB

Fig. 1 The equivalent model of serial-parallel queues

According to the following assumptions we analyze the system response time of the serial-parallel queues [4, 5].

The request in the system will not classify grade and priority.

The total number of requests is unlimited in the system.

Each arrival that requests to get into the system from the external transport is presented by Poisson distribution.

The request can leave the queue. The service rate of requests presents an exponential distribution.

Each request is a queue service, and all service

sequences are first in first out. The blocking nodes are used to block entering. The arriving probabilities of the two parallel queues are equal.

Arrival rate λ and three nodal point service rates in the serial-parallel queues present the correlation between different ranges. When the service rate is greater than the arrival rate, the system is at low utilization, and the system response time of the system is ( ) 1/( )n nE t μ λ= − [4, 6]. When the service rate is equal to the arrival rate, the system response time of the system is ( ) /(2 )n n nE t B μ= . When the service rate is less than the arrival rate, the system response time of the system is ( ) /( )n n nE t B μ= . Overall there are three serial-parallel queues in the model, and each queue contains an individual service rate. The service rate in relation to the arrival rate of each queue is greater than, equal to or less than, therefore there are 27 case working regions of the serial-parallel queues as shown in Fig. 2.

1 4 72 5 8

3 6 9

10 13 1611 14 17

12 15 18

19 22 2520 23 26

21 24 27sμ

1Pμ

2Pμ

sλ μ>

sλ μ<

sλ μ=

1 2P Pμ μ> 1 2P Pμ μ=

1 2P Pμ μ<

Fig. 2 The working regions of the serial-parallel queues

The following describes the cases of the serial-parallel queues. In Case 1 the service rate of the serial queue is greater than the arrival rate, the first service rate of the parallel queue is greater than the arrival rate, and the second service rate of the parallel queue is also greater than the arrival rate, therefore all three queues have low blocking rates. The total system response time as shown in Eq. (5) is the sum of the system response times of the individual queues; its mathematics equation is deduced as follows.

Case 1: sμ λ> ,1 1P Pμ λ> ,

2 2P Pμ λ> The system response time of the serial queue is

s s( ) 1/( - )E T μ λ= (1) The system response time of the first parallel queue is

92

1 1P 1 P 1( ) /( )E T P Pμ λ= − (2)

The system response time of the second parallel queue is

2 2P 2 P 2( ) /( )E T P Pμ λ= − (3) The system response time of the equivalent serial-parallel queues is ( ) 1/( - )eq eqE T μ λ= (4) To begin with, we assume that the system is linear, later we will verify this by measurement [5]. We add up the system response time of each part and find the total system response time of the serial-parallel queues:

1 2s P P( ) ( ) ( ) ( )eqE T E T E T E T+ + = (5) We plug Eq. (1), (2), (3) and (4) into Eq. (5) and obtain

1 2

1 2

s 1 p 2 P

1 1+- - -eq

P PP Pμ λ μ λ μ λ μ λ

+ =−

(6)

By rearranging Eq. (6) we obtain the system response time and the service rate equation of the equivalent serial-parallel queues:

1 2 2 1

1 2

1 2 1 2 2 1

1 2

( )( ) ( )( ) ( )( )( )

( )( )( )P P s P s P

eqs P P

P P P P P PE T

P Pμ λ μ λ μ λ μ λ μ λ μ λ

μ λ μ λ μ λ− − + − − + − −

=− − −

(7)

1 2

1 2 2 1

1 2

1 2 1 2 2 1

( )( )( )( )( ) ( )[ ( ) ( )]

s P Peq

P P s P P

P PP P P P P P

μ λ μ λ μ λμ λ

μ λ μ λ μ λ μ λ μ λ− − −

= +− − + − − + −

(8)

The other cases of the equivalent serial-parallel queues are shown in Table 2.

Table 2 The 27 working regions or cases of the serial-parallel queues

1 1P Pμ λ>

1 1P Pμ λ= 1 1P Pμ λ<

2 2P Pμ λ> Case 1 Case 4 Case 7

2 2P Pμ λ= Case 2 Case 5 Case 8 sμ λ>

2 2P Pμ λ< Case 3 Case 6 Case 9

2 2P Pμ λ> Case 10 Case 13 Case 16 2 2P Pμ λ= Case 11 Case 14 Case 17 sμ λ=

2 2P Pμ λ< Case 12 Case 15 Case 18 2 2P Pμ λ> Case 19 Case 22 Case 25 2 2P Pμ λ= Case 20 Case 23 Case 26 sμ λ<

2 2P Pμ λ< Case 21 Case 24 Case 27

There are 27 kinds of cases in Table 2, use the case that the thick solid line crid gets up among them, their bottlenecks are same, therefore system response time and equivalence service rate generating are also similar, so, there are 15 kinds of equations of conditions, as Table 3 show. The equivalent system response time equation and the equivalent service rate of the serial-parallel queues are shown in Table 3.

Table 3 The equivalent system response time equation and the

equivalent service rate of the serial-parallel queues Case 1

1 2 2 1

1 2

1 2 1 2 2 1

1 2

( )( ) ( )( ) ( )( )( )

( )( )( )P P s P s P

eqs P P

P P P P P PE T

P Pμ λ μ λ μ λ μ λ μ λ μ λ

μ λ μ λ μ λ− − + − − + − −

=− − −

(7)

1 2

1 2 2 1

1 2

1 2 1 2 2 1

( )( )( )( )( ) ( )[ ( ) ( )]

s P Peq

P P s P P

P PP P P P P P

μ λ μ λ μ λμ λ

μ λ μ λ μ λ μ λ μ λ− − −

= +− − + − − + −

(8)

Case 2 2 2

2

22 ( )( )

2 ( )P P s

eqP s

P BE T

μ μ λμ μ λ+ −

=−

2 2

2 22

( )2 ( )

P P seq

P P s

BP B

μ μ λμ

μ μ λ−

=+ −

Case 3 = Case 6

2 2

2

P 2 P s

P s

( )( )

( )P B

Eeq Tμ μ λ

μ μ λ+ −

=−

2 2

2 2

P P s

P 2 P s

( )( )eq

BP B

μ μ λμ

μ μ λ−

=+ −

Case 4 = Case 5

1 1

1

12 ( )( )

2 ( )P P s

eqP s

PBE T

μ μ λμ μ λ+ −

=−

1 1

1 11

( )2 ( )

P P seq

P P s

BPB

μ μ λμ

μ μ λ−

=+ −

Case 7 = Case 8 = Case 9

1 1

1

1 ( )( )

( )P P s

eqP s

PBE T

μ μ λμ μ λ+ −

=−

1 1

1 11

( )( )

P P seq

P P s

BPB

μ μ λμ

μ μ λ−

=+ −

Case 10

1

1

1 1

1

( ) 2( )

2 ( )s P s

eqs P

B P PE T

Pμ λ μμ μ λ− +

=−

1

1

1

1 1

( )( ) 2

s s Peq

s P s

B PB P Pμ μ λ

μμ λ μ

−=

− +

Case 11

2 2

2

2( )2

P s s Peq

s P

B P BE T

μ μμ μ+

=

2 2

2 22

( )s P Peq

P s P

Bs BBs P B

μ μμ

μ μ+

=+

Case 12 = Case 15

2 2

2

22( )

2P S S P

eqS P

B P BE T

μ μμ μ+

=

2 2

2 22

2 ( )2

s P S Peq

P S S P

B BB P B

μ μμ

μ μ+

=+

Case 13 = Case 14

1 1

1

1( )2

s P s Peq

s P

B PBE T

μ μμ μ+

=

1 1

1 11

( )s P s Peq

s P s P

B BB PBμ μ

μμ μ

+=

+

Case 16 = Case 17 = case 18

1 1

1

12( )

2s P s P

eqs P

B PBE T

μ μμ μ+

=

1 1

1 11

2 ( )2

s P s Peq

s P s P

B BB PBμ μ

μμ μ

+=

+

Case 19

1

1

1 1

1

( )( )

( )s P s

eqs P

B P PE T

Pμ λ μμ μ λ

− +=

−

1

1

1

1 1

( )( )

s s Peq

s P s

B PB P Pμ μ λ

μμ λ μ

−=

− +

Case 20

2 2

2

22( )

2P s P s

eqs P

B P BE T

μ μμ μ+

=

2 2

2 22

2 ( )2

s P s Peq

P s P s

B BB P B

μ μμ

μ μ+

=+

Case 21 = Case 24

2 2

2

2( ) P s P seq

s P

B P BE T

μ μμ μ+

=

2 2

2 22

( )s P s Peq

P s P s

B BB P B

μ μμ

μ μ+

=+

Case 22 = Case 23

1 1

1

12( )

2P s s P

eqs P

B PBE T

μ μμ μ+

=

1 1

1 11

2 ( )2

s P s Peq

P s s P

B BB PB

μ μμ

μ μ+

=+

Case 25 = Case 26 = Case 27

1 1

1

1( ) P s s Peq

s P

B PBE T

μ μμ μ+

=

1 1

1 11

( )s P s Peq

P s s P

B BB PB

μ μμ

μ μ+

=+

3. Simulation for a Single Equivalent Queue To verify the analysis of the serial-parallel queues we use the software simulation tool, Queuing Network Analysis Tool (QNAT) [7]. Based on Fig. 3 we determine the arrival rate and the service rate, each ranging from 1 to 100 requests/sec, and obtain the simulation results shown in Figure 4, in which we find that, whenμ λ> , the system response time is very low, less than 1 ms, when μ λ= , the system response time starts rising, reaching in to the 600 ms, but when μ λ< , the system response time can reach 12×104ms. Whenμ λ> , at a low blocking probability, the system response time is small, this is acceptable to customers.

Fig. 3 A single equivalent queue

93

2040

6080

100

2040

6080

100

2468

1012

x 104

Arrival Rate (requests/sec)Service Rate (requests/sec)

Syste

m R

espo

nse

Tim

e (m

s)

Fig. 4 The simulation results of a single equivalent queue

We use the result of simulation, Fig 4, is a performance compare of the reference is gone to by the measurement of the network server system performance. The response time of our measurement single server and serial - parallel server, uses Webserver Stress Tool, will this result again and the compare of the linear regression of the as a result making of the Fig4, take minimum error value as the service rate contrasted by it. 4. Performance Measurement for the Serial- parallel Web Server System To verify the viewpoint in estimating the performance, we install the physical Web server system with serial parallel architecture by using the measurement software tool, Webserver Stress Tool, and some related techniques to carry out the measurement [8,9]. We install two kinds of operation modes which represent the “different” and the “same” domains for the operation of the serial-parallel connection of the Web server system. Table 4 shows the installation by Windows Advanced Server 2000 with SP4 and Webstress.

Table 4 The specifications for the physical installation of the measurement system

The performance measurement includes the system response times of the switch, the Web server and the network. To understand the individual components of the system response time we use two kinds of measurement method, as shown in Fig. 5.

Win2003 Web serverpc3

Win2003 Web serverpc2

Win2003 Web serverpc1

Win2000

Win2003 Web serverpc3

Win2003 Web serverpc2

Win2003 Web serverpc1

Win2000

Fig. 5 The performance measurement of the serial-parallel connection

for different domains Figure 6 shows the actual measurement architecture. When the number of users and the Web size product are over 3Mbytes, the measurement shows that the system response time of the whole network will change randomly due to the loading of the operating system. When the size of the Web page is 25Kbytes and the number of users is from 1 to 100, the curve of the system response time is predictable, therefore we take 25Kbytes of Web size as the base to carry out the measurement with the purpose of understanding the transmission time and the delay time of the network and the bottleneck of the network performance. We use ASP syntax for the measurement transmission time of the network and take into account the multiplication loop together with the 25Kbytes of data. When the Web server receives a user request, it provides the multiplication operation and then delivers a certain amount of Web data. We carry out two part measurements, beginning with one using the ASP Web page and the other by determining the multiplication loop at 10000 times and the number of user requests at 1-100. The measurement results show high, medium and low blocking probability of the serial-parallel queues and try to find out the system response time without considering the network transmission time.

Server 1 (loop.ASP)

Server 2 (2.ASP)

Server 3 (1.ASP)

Client

request

requestrequestrequest

response

response

2.ASP1.ASP

Server 1 (loop.ASP)

Server 2 (2.ASP)

Server 3 (1.ASP)

Client

request

requestrequestrequest

response

response

2.ASP1.ASP

Fig. 6 The performance measurement for the same domain serial-parallel

Web server system

CPU RAM O.S. Web Tools

Client 3.2GHz 2G Windows Server 2000 SP4

WebStress Tools

Server1 1GHz 512M Windows Server 2003

Server2 1GHz 512M Windows Server 2003

Server3 1GHz 512M Windows Server 2003

IIS 5.0

Switch

Network LAN 100M bits/sec

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5. Linear Regression for the Simulation and the Performance Measurement of the Serial-parallel Web Server System To learn the accuracy of the simulation and the measurement we use the linear regression method, with actual measurement results in respect to the simulation results. We model the approximation solution of the linear equation to obtain the service rate of the computer system. In the linear regression method, the variable “ A ” represents the simulation value of the system response time, with an arrival rate of 1-100 requests/sec, while the variable “ B ” represents the measurement value of the system response time, with the same arrival rate. We assume Ax B= as a power approximation solution, assume y a bx= + as an approximation solution, and we assume that 2y a bx cx= + + , and plug it into Ax B= . We multiply both sides with TA , get T TA Ax A B= , move TA A to the other side and so get 1( )T Tx A A A B−= . Figure 7 shows a curve, therefore, when we bring back this “ x ” as the approximate value solution 2y a bx cx= + + , we will get every simulation value and every measurement value. Again in accordance with this relation, we obtain the difference between the simulation and the measurement at the same service rate. We solve the approximation solution 2y a bx cx= + + and get 0a = , 0.5587b = − and

193.6013c = . We plug all values into the simulation curve, bring back the “ x ” in the linear regression equation ( 20.5587 193.6013y x x= − + ), compute the “ y ” value, find the value of “ y ” and subtract all actual measurement values, divide the actual measurement and obtain the average error. We solve the equation for the average error, while obtaining different service rates at the minimum value of the error of the service rates of the simulation and the actual measurement. Fig. 7 shows the single equivalent service rate to be about 55 requests/sec from the actual measurement with the error at 2.7011%. As according to the above-mentioned procedure, as shown in Fig. 7 the approximation solution is 2y a bx cx= + + , we then get 0a = ,

0.7806b = and 413.4354c = . From Fig. 7 we learn 0a = that can also predict the equivalent service rate is

about 54 requests/sec based on the set of system parameters of the serial-parallel server with the error ratio at about 5.8662%. Due to the physical restriction on the actual measurement, we install and set up two computers of a tree connection by the allocation probability

1 2 0.5P P= = . The equivalent service rates of the computers are

1 255s P Pμ μ μ= = = requests/sec by the linear regression

method. For analysis we use the equivalent service rate of 54 requests/sec. Assuming a number of users from 1 to 100 when we measure, the arrival rate λ is 1 to 100 requests/sec. Now we bring the parameters together in,

Case 1 ( sμ λ> , 1 1P Pμ λ> ,

2 2P Pμ λ> ):

1 2

1 2 2 1

1 2

1 2 1 2 2 1

( )( )( )( )( ) ( )[ ( ) ( )]

s P Peq

P P s P P

P PP P P P P P

μ λ μ λ μ λμ λ

μ λ μ λ μ λ μ λ μ λ− − −

= +− − + − − + − ,

Case 10 ( sμ λ= ,1 1P Pμ λ> ,

2 2P Pμ λ> ):

1

1

1

1 1

( )( ) 2

s s Peq

s P s

B PB P P

μ μ λμ

μ λ μ−

=− +

and Case 19 ( sμ λ< ,1 1P Pμ λ> ,

2 2P Pμ λ> ):

1

1

1

1 1

( )( )

s s Peq

s P s

B PB P Pμ μ λ

μμ λ μ

−=

− +

We obtain 100 values eqμ and use the error equation 100

1100%n n

n n

M FError

F=

−= ×∑ . M is the equivalent service rate

value eqμ , F is the equivalent service rate ( 54eqμ = requests/sec) of the serial-parallel Web server system by the actual measurement, and we obtain an error ratio of about 10.84%. We again use the arrival rate 1λ = to 100 requests/sec, the service rate

1 255s P Pμ μ μ= = = requests/sec,

the distribution probability 1 0.5P = , 2 0.5P = and obtain the

equivalent system response time as follows.

1 2 2 1

1 2

1 2 1 2 2 1

1 2

( )( ) ( )( ) ( )( )( )

( )( )( )P P s P s P

s P P

P P P P P PE T

P Pμ λ μ λ μ λ μ λ μ λ μ λ

μ λ μ λ μ λ− − + − − + − −

=− − − ,

1

1

1 1

1

( ) 2( )

2 ( )s P s

s P

B P PE T

Pμ λ μμ μ λ− +

=−

and 1

1

1 1

1

( )( )

( )s P s

s P

B P PE T

Pμ λ μμ μ λ

− +=

− .

Based on this equation we find the system response time of each arrival rate. Again we use

100

1

100%n n

n n

M FError

F=

−= ×∑ as the error equation where M

represents the system response time and F the actual system response time by measurement, and we obtain the error ratio of the system response time as 46.12%.

The service rate of the single

server The service rate of the serial-

parallel server A1 B1 A2 B2

0 20 40 60 80 1000

500

1000

1500

2000

2500

Users

Res

pons

e Ti

me

(ms)

Measurement 53 requests/sec 54 requests/sec 55 requests/sec 56 requests/sec 57 requests/sec

eqμ =

eqμ =

eqμ =

eqμ =

eqμ =

A1

95

48 50 52 54 56 58 600

5

10

15

20

25

μ (request/sec)

Erro

r (%

)

Single queue

A2

0 20 40 60 80 1000

500

1000

1500

2000

2500

Users

Res

pons

e Ti

me

(ms)

Measurement 52 requests/sec 53 requests/sec 54 requests/sec 55 requests/sec 56 requests/sec

eqμ =

eqμ =

eqμ =

eqμ =

eqμ =

B1

48 50 52 54 56 58 604

6

8

10

12

14

16

18

20

22

μ (request/sec)

Erro

r (%

)

Serial-parallel queue

B2

Fig.7 The comparison of the service rates between the single server and the serial-parallel Web server system

6. Conclusion In this paper we propose the serial-parallel equivalent model to represent the tree connection of a Web server system. Before the implementation of the Web server system the system response time of the system must be known. By using our equivalent model for the serial-parallel connection one can estimate the equivalent service rate and the system response time of the Web server system. The equivalent system response times are those of both the simulation and the analysis equations.

Both can be used to verify the equivalent system response time before installing the system because of the limited average error of 46.12% which we have observed. Future work may include multiple serial-parallel queues with a feedback mechanism which is similar to the more complex connections of a Web server system. References [1] Danniel A. Menasce and Virgilio A. F. Almeida, Capacity Planning for Web Services, Metric, Models, and Methods, (Upper Saddle River, NJ: Prentice-Hall, 2002.) [2] Ying-Wen Bai, Chia-Yu Chen, and Yu-Nien Yang, A Two-Pass Web Document Allocation Method for Load Balance in Multiple Grouping of a Web Cluster System, ICON’04, 12th IEEE International Conference on Networks, 1, 2004, 177-181. [3] Gunter Bolch, Stefan Greiner, Hermann de Meer, and Kishor S. Trivedi, Queueing Networks and Markov Chains - Modeling and performance evaluation with computer science applications (Hoboken, NJ: Wiley-Interscience, second edition, 2006) 410-414. [4] Ying-Wen Bai, and Yu-Nien Yang, An Analysis and Measurement of the Equivalent Model of Serial Queues for a Load Balancer and a Web Server of a Web Cluster with a Low Rejection Rate. A tutorial review, Proc. IEEE Conf. on the Engineering of Computer Based Systems, , 2006, 485-486. [5] Ying-Wen Bai, and Yu-Nien Yang, An Approximate Performance Analysis and Measurement of the Equivalent Model of Parallel Queues for a Web Cluster with a Low Rejection. A tutorial review, Proc. 14th IEEE Conf. on Networks, , 2006, 1-6. [6] Thomas G. Robertazzi, Computer Networks and Systems - Queueing Theory and Performance Evaluation, (Stony Brook, NY: Springer-Verlag, third Edition, 2000.) [7] H. T. Kaur, D. Manjunath and S. K. Bose, The Queueing Network Analysis Tool (QNAT), Proceedings of International Symposium on Modeling, Analysis and simulation of Computer and Telecommunication Systems, 2000, 341-347. [8] Ying-Wen Bai and Chien-Yung Cheng, The performance estimation by queuing network models for a Web-based medical information system, CBMS’04, Proceedings of 17th IEEE Symposium on Computer-Based Medical Systems, ,2004,191-196. [9] http://www.paessler.com/webstress/

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