Analysis of reability of ICS states

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Analysis of reability of ICS states

Volotka V.S.

A model of information and communication system is presented in the form of 4 main states: serviceable and unserviceable, each of them may be both in operative and standby modes. The main object of analysis is failure and appropriate parameters: mean time between failures, availability coefficient, reliability probability, probability of survivability, an average total risk of failure probability.

Such properties of elements, blocks, networks and overall systems as reliability, durability, maintainability and persistence are related to the concept of reliability of any purposeful system [1, 2]. The failure is considered to be the central object of focusing. It is necessary to distinguish methods for ensuring reliability at the design and implementation planning stages and at the stage of operation. At the first stage high reliability is achieved due to manufacturing techniques, appropriate methods for constructing reliable modes and structures, while at the functioning stage we normally deal with ready-made structures. In this case high reliability is reached by monitoring current conditions and the state of network elements, followed by the response to the failure.

In information and communication systems (ICS) as in any complex controlled systems there are various factors that directly cause failure or disruption. Among these factors are the following:

- failing in network elements or trunks;

- network overloading, traffic mismatch;

- error actions of the operators and subscribers;

- failures of the control programs complex.

Reliability is a probabilistic characteristics, it is determined by the following parameters: - reliability probability, - mean time between failures, - availability coefficient [1, 2].

From the viewpoint of reliability ICS may be in one of the 4 states (see Fig. 1).

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Fig. 1. Probabilistic graph of the system states.

These states have the content:

- system is serviceable and it performs in the standby mode;

- system is serviceable and it performs in the operative mode;

- system is unserviceable and it performs in the standby mode;

- system is unserviceable and it performs in the operative mode.

Arcs correspond to the following transition intensities:

, - intensity of arrival and processing of information blocks (packets), respectively;

, - intensity of failures in standby and operative modes, respectively;

- intensity of system recovery after the failure.

For each probability of the states we can point out probabilities of dependencies on operation time [1]:

(1)

where is the dynamics in changes of probabilities that system is in the state .

The given states are interconnected. Thus, the probability of failure is , and the probability of system survival is . In the transition from the state into the state the physical failure takes place while in transitions form states and into the state the functional failure accompanied with the partial loss of ICS operability takes place.

Solution of the differential equation system (1) is possible using the direct and inverse Laplace transforms [1, 3]. However, obtaining originals using the inverse transforms is a rather intricate problem. Therefore, in practice the system (1) is used in a static mode when obtaining The received system of algebraic equations allows to find the relations between the average values.

Thus the mean time between failures (in the assumption =0):

.

Fig. 2 shows the graph , where is the system performance.

Fig. 2. Graph of mean time between failures.

Using mean time to repair we can determine another important value of reliability - coefficient :

, (2)

where is mean time between failures;

is the system mean time to repair after the failure. in its turn consists of mean times: diagnosing , waiting , switching , reservation . Thus:

. (3)

The graph is presented on the Fig. 3.

To calculate the stability of the system we often use a different probabilistic characteristic - the probability of system stability:

, (4)

where is probability of survivability determined as addition to damage probability .

Fig. 3. The dependence of the availability coefficient on the intensity of recovery.

The graph shows that the availability coefficient increases with the intensity of recovery.

A priori estimation of the stability and thus the reliability can be determined by the degree of risk which probability of any state is assumed with. The average size of the total risk can be determined by the following formula [3]:

, (5)

mathematical reliability differential

where is the probability determined from (1);

is the weight factor, the risk of taking any state , i=1, 2, 3, 4. The risk level (5) can be used to compare qualities of different systems.

In practice usage of the risk theory in multidimensional problems (5) seems to be rather uncertain first of all due to the subjectivism of assignment of weight factors that are usually assigned by the decision-makers. Values can be determined accurately enough on the basis of a priori data about reliability of certain network elements.

In this case it is advisable to do likewise with the solution of multi-criteria optimization. The key point in solving the problem (5) is to construct the Pareto set (PS). This set is defined as follows [7]: a point belongs to a given set, if it is impossible to find such a point at which the following inequalities are executed at least for one value of i:

for all i=1,2,…,n.

Among the well-known methods for solutions are [7]:

- method of average criterion;

- method of Germeer convolution;

- method of E-Restrictions.

The results obtained by the analysis coincide with the state logic for the system reliability; the technique itself makes it possible to obtain the parameters of changes in both stationary and dynamic modes of operation. At the same time more informative model of ICS in the dynamic mode is a system of differential equations obtained with respect to the state of individual network elements [4, 5]. Great informativeness of such a representation is defined by the fact that here we consider the dynamics of the state itself, not only its features - probabilities.

Conclusions

1. Sufficiently general mathematical model for probabilities of the infocommunication system states is presented.

2. The obtained analytical dependences of the traditional parameters of system reliability are associated with traditional parameters of queuing techniques.

References

1. L. Kleinrok, Teoriyа massovogo obsluzhivaniyа [Queueing theory]. M.: Mashinostroenie, 1979, 432 p.

2. A.M. Polovko and S.V. Turov, Osnovy teorii nadyozhnosti [Fundamentals of reliability theory]. SPb.: BHV - Petersburg, 2006, 704 p.

3. V.YU. Korolev, V.E. Bening, and S.Ya. Shorgin, Matematicheskie osnovy teorii riska [Mathematical fundamentals of risk theory]. M.: Fizmatlit, 2011, 591 p.

4. V.V. Popovskyy and V.S. Volotka, “Metody samodiagnostirovaniyа [Methods of self-diagnosis], Mezhdunarodnayа nauchno-tehnicheskayа konferentsiyа «Analiz i sintez slozhnykh sistem v prirode i tekhnike», [Analysis and synthesis of complex systems in nature and technology], Voronezh, 2013, pp. 45- 51.

5. V.V. Popovskyy and V.S. Volotka, “Metody analiza dinamicheskikh struktur telekommunikatsionnyikh system [Methods of analysis for dynamic structures of telecommunication systems],” Eastern European Scientific Journal. Kh. 2013, vol. 5/2 (65), pp. 18-22.

6. A.P. Samoilenko and D.Ye. Rud', “Integral'nayа model' nadyozhnosti funktsionirovaniyа uzla telekommunikatsionnoy seti [Integrated model of reliability of functioning for a telecommunications network node].” Telekommunikatsii, vol. 7, 2013, pp. 23-30.

7. L.I. Polishchuk, Analiz mnogokriterial'nykh еkonomiko-matematicheskikh modeley. [Multicriteria analysis of economic and mathematical models]. Novosibirsk, Nauka, 1989, 484 p.

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