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
Ying-Wen Bai
Department of Electronic Engineering,
Fu Jen Catholic University,
Taipei, Taiwan, 242, R.O.C.
[email protected]
Yu-Nien Yang
Department of Electronic Engineering,
Fu Jen Catholic University,
Taipei, Taiwan, 242, R.O.C.
Abstract
In this paper, we propose a formula that can be used
in the extensive range, and not merely applied to the
Web cluster, so long as the serial connection system
which is close to the linear system has a supposition
property.
In this paper, we propose an equivalent model in a
serial queue for representing the serial connection of
the load balancer and a Web server of the Web cluster.
We have set up an experimental Web cluster for doing
some performance measurements. Moreover, we
compare either the simulation results or the
measurement results for the mean system response
time of the serial queues forming the equivalent model
which in turn has the supposition property derived by
the use of two subsystems.
1. Introduction
With the development of the computer networks,
the load of a Web server is increasing. Usually, it is
hard to predict the network transmission time.
However, we can estimate the Web site time based on
the arrival rate of the Web requests and service rate of
the Web servers.
As a traditional single server has not been able to
give prompt service, a variety of a Web cluster now
provides one of the solutions [1]. Some research shows
the use of both analytic modeling and the simulation
method to investigate a system’s performance [2]. The
system architecture of a Web cluster is as shown in Fig.
1.
2. The System Model for a Web Cluster
System
In the real Web cluster system, all Web requests
will be sent to the load balancer first. The load
balancer will redirect the requests to each server. We
propose a queuing model for a Web cluster with
balancing architecture as shown in Fig. 2.
ӳ
Ӵ
Ӵ
˄
Ӵ
˅
Ӵ
́
.
.
.
˷
n
Figure 2. A queuing model for a Web cluster with a load balancer.
Under such architecture, we introduce the mimic
concept that an equivalent queue can represent a
couple of serial queues. In this paper, we will
investigate the equivalent formula in serial queues as
shown in Fig. 3. The first step is using this queuing
theory is to verify the correctness of the equivalent
formula [3].
P1
P2
E1(T)
E2(T)
Peq
Eeq(T)
Figure 3. The equivalent model of serial queues
Figure 1. A system architecture of a Web cluster
First, we assume that the system is close to a linear
system, so we can obtain (1) for the mean response
time. Later, we will verify this assumption by both the
simulation and the experimental measurement.
Proceedings of the 13th Annual IEEE International Symposium and Workshop on Engineering of Computer Based Systems (ECBS’06)
0-7695-2546-6/06 $20.00 © 2006
IEEE
P
1
1
O
P
2
1
O
(1)
1
P
eq
O
By using the above algebraic procedure, we will
obtain the equivalent service rate as shown in (2).
P 1P 2 O 2
(2)
P
eq
P
P
1
2
2 O
3. The Simulation of the Equivalent Serial
Queues
We can see that the equivalent response time is quite
close to the mean response time of the Web cluster that
we measured. We compute the maximum error which
is 10.37% between both of the results from the
measurement and analysis.
Furthermore, we use the approximate relationship to
get the general form for n stages of serial servers. By
use algebraic procedure, we obtain the equivalent
service rate as shown in (3).
(3)
1
P
n
¦
K
1
P
K
O
1
O
We obtain the mean system response time in
proportion to the stage numbers of the serial system as
shown in Fig. 5.
350
4. The Comparison of the System
Performance with respect to Analysis,
Simulation and Measurement
In order to verify the equivalent model in serial
queues, we have set up a real Web cluster, and we use
Webstress to measure the system performance [5].
In a real Web cluster, except for the load balancer
and Web server, the system still includes the
transmission time. In order to reduce the effect of the
transmission time, we use the high-speed Ethernet. In
order to adjust the service time, we will use ASP
grammar to let the Web server perform multiplication
operations. In this way, we can neglect the
transmission time which is less than 1ms. Therefore,
the balancer and the Web server can be modeled by
using two serial queues. Fig. 4 shows the measurement
of the mean system response time of the real Web
cluster and the computation of that by use of the
equivalent formula.
eq
The system response time
In order to verify the correctness of the equivalent
formula in serial queues, we will use the QNAT tool to
show the characteristics of the queuing model [4].
We compute the equivalent service rate up to the
third decimal point in this simulation. The computation
error is according to the number of effective digits.
300
250
Arrival rate = 30 (Requests/s)
Arrival rate = 50 (Requests/s)
Arrival rate = 70 (Requests/s)
200
150
100
50
0
0
2
4
6
8
The number of servers : "n"
10
Figure 5. The relationship between the number of servers “n” and
the mean system response time
5. Conclusion
At a low reject rate, we propose an equivalent
queue in serial queues to represent the serial
connection of the load balancer and a Web server of
the Web cluster system. The system response time of
the Web cluster system calculated from the equivalent
model is quite close to the system response time from
the experiment measurements.
6. References
[1]
[2]
[3]
*Κ
P 1 =100, P 2 =50
**Κ
P 1 =100, P 2 =70
***Κ
P 1 =100, P 2 =90
[4]
Unit: Requests/s
Figure 4. The measurement of the system response time of the real
Web cluster
[5]
Cardellini, V., Colajanni, M., Yu, P.S., “Dynamic load
balancing on Web-server systems”, IEEE Internet Computing,
Volume 3, Issue 3, May-June 1999 pp. 28 – 39.
E. V. Carrera and Ricardo Bianchini, “Evaluating ClusterBased Network Servers”, Proceedings of the Ninth
International Symposium on High-Performance Distributed
Computing, 2000, pp. 63-70.
Thomas G. Robertazzi, “Computer Networks and Systems”,
Springer, 2000, pp 19 – 47, 101-108.
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, pp. 341-347.
http://www.paessler.com/webstres
Proceedings of the 13th Annual IEEE International Symposium and Workshop on Engineering of Computer Based Systems (ECBS’06)
0-7695-2546-6/06 $20.00 © 2006
IEEE