Andrey Lysykh

Use of analytical methods is connected with necessity of construction of mathematical models of a web-server for strict mathematical terms. Analytical models of computing systems usually have likelihood character and are under construction based on concepts of the theories of mass service, probabilities and Markov's processes, and also methods diffusive approximations. The differential and algebraic equations can be applied also.

Unlike analytical imitating modeling removes the majority of the restrictions connected with an opportunity of reflection in models of real investigated web-server, dynamic mutual conditionality of current and subsequent events, complex interrelation between parameters and parameters of a system effectiveness, etc. Though imitating models in many cases are more labour-consuming, less laconic, than analytical, they can be as much as close to modelled system and are simple in use.

Practical use of models of a web-server in many cases assumes presence of the information on real characteristics of it's job. Such information can be received by empirical methods. The necessary information is going to by means of special means which provide measurement of the parameters describing dynamics of functioning of a web-server in modes of skilled and normal operation.

Research of processes or objects is replaced with construction of their mathematical models and research of these models at analytical modeling. Identity of the form of the equations and unambiguity of ratio between variables in the equations describing the original and model are put in a basis of a method. As the events occuring in a web-server, have casual character the likelihood mathematical models of the theory of mass service are the most suitable for their studying.

The brief review of existing approaches in modeling

Some attempts have been accomplished to construct model productivity of a web-server.

Van der Mei [4] modelled a web-server as a queuing network with serial service. The model was used to predict parameters of productivity of a server and has been approved by means of measurements and modeling

Wells and others [5] have made the analysis productivity of web-servers, using, colored Petri nets. Their model is divided into three layers where each layer models the certain aspect of system. The model has some parameters, some of them are known. Unknown parameters are certain by means of modeling.

Banga [6] investigated traps which arise during generation of synthetic working loadings of a Web-server in the test bed consisting of a small amount of clients. They have revealed restriction of the simple scheme of generation of requests which underlies modern systems of reference testing of Web-servers. Banga has offered and has estimated new strategy which excludes this problems, using a set of specially created client processes. Initial experience in use of this method for an estimation of a usual Web-server has specified that measurement of Web-server working process at overload and pulsing conditions of the traffic gives new and important understanding about a Web-server operating. His new methodology makes possible generation realistic, pulsing HTTP traffic and thus an estimation of a prominent aspect of Web-servers operating.

However, the considered models are complex enough. There is a lack of simple model which at the simplicity remains adequate. The simple model establishes less amount of parameters, according to it it's easier to estimate, while the complex model usually contains such parameters which are difficult for receiving.

Current and planned results on a subject

In this work described the model of a web-server which consists of the shared processor with the queue attached to them. The total of tasks in the system is limited. It is supposed, that process of receipt of requests to a server is Puasson whereas distribution of a service time is any. Such system name M/G/1/K*PS queue. An average service time and the maximal number of tasks - parameters which can be certain through the maximal estimation of probability. Also expressions for parameters productivity of a web-server, such as throughput, average time of the answer and probability of blocking are received.

we shall consider M/G/1/K queue with order of service - processor sharing as the given work considers as a web-server Apache - widely used multithreading server of a web (it means, that requests in him is processed by own thread or process during all cycle of a request life). Receipt of request according to Puasson distribution occurs with rate λ. The service time has the general distribution with an average x. The arrived request will be blocked, if the total tasks in system has reached predetermined value K. The task in queue receives small quantum of service and after that stops while each other task will not receive identical quantum of service in cyclic process. When the task has received demanded quantity of service, it leaves queue. Such system can be considered also as a network with queue with one node.

Function of probability of total tasks in system has is following,

(1)

where p – acting load and it is equal λx.

M/M/1/K*FCFS queue has the same function of probability. However distribution of a service time of M/M/1/K*FCFS queue should be indicative, and it's order of service should be FCFS.

The Web-server is modelled, using M/G/1/K*PS queue as shown in fig. 1. Requests comes according to Puasson process with rate λ. An average service time of each inquiry is x.The server can process simultaneously no more K requests. The inrequest will be blocked, if this number has been reached. The blocking probability designated as P_b. Therefore the rate of the blocked requests is certain as λP_b.

Figure 1 - M/G/1/K*PS web-server model

From (1) we can receive the following three parameters of productivity: average response time, throughput and blocking probability. Blocking probability P_b is equal to probability that system contains K tasks, the system is filled,

(2)

Throughput H – rate of the processed requests. H is equal to accepted requests rate When the server reaches balance,

(3)

Average response time T – expected time of task stay. Proceeding from Little law, we have

(4)

In model there are two parameters x and K. It is supposed, that average response time at the certain frequency of requests receipt can be estimated by means of measurements. Estimations x and K obtain, minimizing the sum of squares of errors of observable average response time.

Let T_i is the average response time predicted by model and T^{^}_i is the average response time measured, at intensity of requests receipt λ, i=1..m. As prospective response time T^{^} are averages from the received results, it approximately normally distributed casual value with average T and a dispersion σ²_T/n, when the number of indications n is very big. Hence, the pair parameters of model x and K can be estimated, minimizing the sum of square mistakes as follows,

(5)

As approach the prospective dispersion of response time σ^{^2}_i can be used instead of σ²_i.

Now, the problem of parameter estimation becomes a question of optimization,

(6)

Optimization can be solved by various ways, for example a method of most speedy descent, the integrated gradient, Newton and even direct search.

Conclusion

In the given work considered the problem modeling and the analysis of productivity one of the most popular web-server Apache. The essence of a problem that character of the traffic (requested data) is obviously unknown and it is necessary to simulate approximately as much as possible productivity of system on obviously unknown data. The basic purpose of researches - construction of program server model. The decision of the given problem is the model of a web-server based on M/G/1/K*PS queue. The given model repeats real parameters well enough. It's efficiency is confirmed experimentally.

Master's work isn't completed yet by the time this abstract issued. Completion of the thesis is expected till January, 2008. The Full text of the thesis and related materials can be obtained from the author or his leader after the specified date.

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