The work is dedicated to the development of analytical model of probability estimation of reliability, productivity, quality and efficiency of functioning of the complex technical queuing system consisting of the arbitrary number of marked groups of the service devises (channels, facilities, servers) differing with reliable characteristics (parameters of refusals and restorations) of forming their composition (also of arbitrary number) marked, identical, unreliable and restorable serving channels in which for serving come in requirements with intensities depending on marking of channels. In the considered system it is supposed that the currents of refusals of serving devices and currents of coming requirements are subdued to Poisson, and the currents of restorations of refused devices and the currents of services of coming requirements—exponential laws of distribution of probabilities. A stochastic process of transfers of a system by that is Markovian process with continuous time and discrete states. Correlations linking the basic parameters and exit characteristics of the systems of the pointed out type are obtained in a view of probabilities of the system location in the given moment of time in one of the possible states.
The end of the 20th and beginning of the 21st centuries were marked with rapid quantitative and qualitative growth of computer networks. This tendency that apparently will be kept in the nearest decades is well illustrated by unprecedented growth of the Internet network sweeping all countries of the world. Local computer networks that are the basis of automation of activities of separate enterprises and companies, and distributed computer networks involving cities, regions and continents penetrate all spheres of human activities including economics, science, culture, education, industry etc. Contemporary computer networks provide the consumers with a wide set of service including e-mail, transfer of fax and voice information, the work with distant data bases in the real scale of time, service of news and other services. On the basis of computer networks the following activities can be implemented: distant education, telemedicine, teleconferences, tele-exchanges, teleshops etc.
A quick growth of the number of computer networks, successes of fiber-optic and wireless means of communication are accompanied with a continuous change of the network technologies directed to the increase of fast activity and reliability of the networks, possibility of integrated transfer of the data, voice and video-information. The changing technologies, continuous quantitative and qualitative growth of computer networks make the following top-priority tasks: development of network designing fundamentals on the basis of systematization of already known approaches and development of new methods of analysis and synthesis of computer networks [
This article is dedicated to the development of one specific but the more generalized model of analysis and synthesis of computer networks that is based on the theory of systems and networks of mass service (theory of queues). The peculiarity of this model is the use of methods of theory of reliability for accounting of the reliable characteristics of serving devices. The article represents some perfections and generalizations of previous work of the author [
Proceeding from the above said the purpose of our work is to develop research model for probability estimation of the factors of efficiency of real time, fail-proof and high-capacity systems, having equipment and time resources for maintenance of dependability and efficiency level and functioning in the conditions of influence of random factors―refusals, repairs, and random flow of random length tasks and random time of their service.
The proposed model is developed with consideration of the following assumptions:
・ Transitions of the system between different states is realized under shared effect of several random pro- cesses―processes of incoming and service of demands, as well as processes of refusals and repairs of service facilities:
・ The technical queuing system consists of
・ Serving devices are subjected to stable refusals subdued to Poisson distribution of probabilities with the parameter of distribution
・ Stochastic process of restoration of refusing devices is described by exponential laws of distribution of probabilities with the parameter of distribution
・ The group of Poisson currents of requirements enter the system to be served with the intensity
where
・ The current of services of entering requirements is described by exponential law of distribution of probabilities and its intensity
・ In each group of serving devices it is foreseen the entrance collector for those requirements that will come in those moments of time, when the number of all correctly functioning devices in the given group are busy with serving of early coming requirements;
・ In any considered moment of time the length of the queue of requirements waiting for service in each group is determined as
For description of functioning of the above mentioned system in time, special probability functions
cesses of the system between the various states in accordance with the reasons causing these transitions― refusals and restorations of serving devices or entrance and service of requirements.
By those indexes
The introduced functions are determined as follows:
with
state with
These probabilities describe the process of change of states of
state with
These probability functions describe the process of change of the state of system at permanently changing upper and lower indexes, i.e.
Rate fixing condition has the following form:
In order to determine intensities of system transitions at fixed values of upper indices
In order to determine intensities of system transitions at fixed values of lower indices
Considering the possible changes of the system in infinitely small time interval
where
Summands entering in the right part of equality (3) should be obtained by solution of the following system of difference equations which describes the process of functioning of the considered system in assumption of fixed values of lower indices
Below we define sense of the notion of probabilities in Equations (1)-(3):
quantity―
{quantity of devices: operable―
requests: general quantity―
requests: general quantity―
ble―
service} in time
From Equations (1)-(3) by going to limit at
where
Here:
Systems of Equation (6), Equation (7), Equation (9) and Equation (10) represent the systems of inhomogeneous differential equations with fixed factors.
In assumption of given starting conditions:
partial solutions of these equations can be retrieved by method of operational calculus using the following Laplace transformations:
Systems of Equations (6)-(10) in Laplace derivatives will get the following form:
where
After simple transformations, Equations (11)-(15) are transformed into inhomogeneous systems of algebraic equations which are solved quite simply with Cramer formulas relative to
where
With the help of Cramer formula the solution of Equation (20) can be written in the following form:
where
is a principal determinant of system (20), while
right part (1/s) of equation system (20) into
The solutions of the system of Equation (17) are written similarly:
where
is a principal determinant of system (17), while
right part
3 are the numbers of matrix columns.
With the help of found probabilities of states it is easy to determine different factors of dependability and efficiency of the examined system.
The results obtained in this work represent some generalizations of already known in literature results, including the results developed by the author of this article [
Severity of given mathematical judgments and transformations, taking into account the assumptions made in this work, as well as the approach to the exposition of material, may serve as proving the adequacy of the offered model. The novelty of the model and its more general nature is confirmed by the fact that in case of some assumptions about model’s parameters, we will get the new models coinciding with the already known ones in the literature.
The work results can be successfully used during the analysis and synthesis of such complex technical systems as the system of the data transfer, system of cell communication, computer and communication networks, flexible production lines, automation control systems and also other complex technical systems used both in civil and military objectives. They can be used during estimation of the given or provision of the required levels of reliability, productivity and effectiveness of functioning of complex technical systems both on the stage of operation and the stage of designing.
Ramaz R.Shamugia,11, (2015) On One Analytical Model of a Probability Estimation of Quality and Efficiency of Functioning of Complex Technical Queuing Systems. International Journal of Communications, Network and System Sciences,08,295-303. doi: 10.4236/ijcns.2015.88029