Инфокоммуникационные технологии и системы VERIFICATION OF KOLMOGOROV EQUATION USABILITY FOR REPRODUCTION AND DEATH PROCESSES

A simulation is widely used as a basis for decision support systems. Many production systems may be considered as queuing systems if a time of processes more valuable than their physical meaning. Program models realized queuing systems are used in a planning and in optimization tar-gets. But results of program simulation are not suitable for scientific qualification works according to traditions. Analytical conclusions are made using Kolvogorovs’ equations and some models derived from the one usually. But a question about possibility of using them with widespread statistical distributions is not quite explored. In this article we investigate a possibility of using the Kolmogorov's equations on a simple reproduction and death queuing system with some distributions. Numerical data is obtained from program models realized in GPSS and AnyLogic. Theoretical results in comparison with numerical data lead us to a conclusion. The possibility is present only when all statistical distributions in the model are exponential or very close to exponential. Else average error between the theory and the model is above 60%. So far as a small experimental data typical for obser-vations in production systems does not allow to determine own statistical distribution surely, an uniform distribution is accepted as default, and Kolmogorov’s equations could not be used.


Fig. 1. Dependences between the factor and the reaction of the model with different statistical distributions of the random factor variable
Kolmogorov's equation and many models generated by such approach are one of the most common mathematical tools for the research of QS. A main idea of the method is in follow. A velocity of Markov processes state changing is determined by probabilities of neighboring states and the flows of their changes: However, it is obvious that the Kolmogorov's equations are not valid for all statistical distributions. Let consider the simplest model of "death and reproduction" as one-channel QS with a limited queue. If a loading flow  is smaller of an unloading flow  , we can get a solution of the Kolmogorov equations as the probability of an empty queue 0  We attempt to experimentally "verify" the Kolmogorov's equations using some common different simulation environments in this article. We will make a theoretical calculation of the limiting state probabilities of QS "death and reproduction" and their comparison with the results experimentally obtained by GPSS World and AnyLogic programs. As examples two schemes is considered: a multichannel system with failures (the so-called Erlang problem), and a single-channel system with a limited queue.

Multichannel QS with failures
Let us consider a system includes three channels to process of transacts. Let a medium loading flow intensity is equal λ = 0,005 s -1 , and processing (unloading) flow intensity is equal μ = 0,0025 s -1 . The system has four states: S 0 -all channels are free, S 1 -one channel is free but others are busy and so on until S 3 . Let 0 3 p p  are probabilities of the states S 0 …S 3 respectively, A state graph of the system is presented on Fig. 2.

Fig. 2. A state graph of three-channel QS with a queue
The flow of transacts leads the system from any left state to the next right one with the same intensity λ sequentially. An unload intensity leads the system from left to the next right state. The last depends on state of the sysnem (from a number of busy channels). For example, the system can turn from state S 2 to S 1 if any of channels (1 or 2) will terminate the processing. That is why a summary unload flow is equal 2μ. So, the system of Kolmogorov's equations is a follow in a limit stationary mode:

Single-channel QS with a queue
Let us consider a single-channel QS with a three cells queue and the same flow rates. The system has 5 possible states: S 0 -the channel is free, a length of the queue is 0, S 1 -the channel is busy, the queue is free, S 2 -the channel is busy and the length of the queue is 1 and so on until S 4 -both the channel and the queue are full. Probabilities of the states are 0 A state graph of such system is similar to Fig. 2 but an unload flow rate is μ at any state. So, the system of Kolmogorov's equations is a follow in a limit stationary mode:

Modeling program and results
We designed two programs to solve both tasks: one by GPSS (General Purpose Simulation System) language and second in AnyLogic modeling suite. We can create some distributions in GPSS using libraries or using tables with points of the distribution function [5]. We used different kinds of statistical distributions to determinate whether they could be coincided with result of Kolmogorov's equations solving: Some of them are presented in GPSS/AnyLogic libraries, another are realized with table functions. All results and theoretical results above are in Table 1. Distributions marked "*" is the same but with modified parameters described below. A GPSS listing for multi-channel QS with failures is the follow:    Numbers of states are stored in variables Empty and Full. They are counted by a special flow from a block "generate 1" when the model works. Probabilities are the numbers divided on their sum.
An analogue program is designed for the single-channel QS with a limited queue. All changes was only in a follow block: nak storage 3 generate (Normal(1,200,0.01)) ; a kind of distribution gate snf nak,out ; if no space in 'nak', then 'out' enter nak seize kan leave nak advance (Normal(1,400,0.01)) release kan terminate out terminate Both models are realized in AnyLogic too [12]. For example the single-channel QS with a limited queue is shown on Fig. 3. Results of modeling with different distributions are shown in Table 2.   Table 3. There are no significant differences between using inbuilt or table functions of distributions calculations, if the last set correctly. For example, a error for inbuilt exponential distribution is 1,8% as shown above, and for table function is 2,6%.

Results and conclusion
As a result Kolmogorov's equation could be used for analytic QS modeling in tow cases: 1. If a stochastic distribution is exponential. 2. If other distributions are close to the one especially. For example, in built functions GAMMA(Stream, Locate, Scale, Shape), WEIBULL(Stream, Locate, Scale, Shape) etc. a parameter 'Shape' must be set to Shape = 1.