Estimation of the length of the process simulation

Abstract

The paper is devoted to the method of determining the simulation duration suficient for estimating of unknown parameters of the distribution law for given values of the relative error and the confidence level. The method is developed for nonnegative random variables, for nonnegative regenerating sequence and for nonnegative regenerating piecewise-constant process. An algorithm for simulation a piecewise constant random process is proposed, the result of which is the number of experiments or the duration of the simulation as well as the estimates of unknown parameters obtained as a result of simulation. The results of numerical experiments illustrating the application of the method for determining the simulation duration are given for a predetermined relative error and confidence level. For the examples, two distributions are chosen from the families of one-parameter and two - from two-parameter distributions. The simulation of random variables with the chosen distributions took into account the simulation duration, suficient for estimating the unknown parameters of the distribution law for given relative error and confidence level. For chosen distributions, series of experiments were performed at diferent values of the relative error and the confidence probability.

Keywords

• and phrases
• confidence
• simulation
• estimation of an unknown parameter

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1. Introduction experiments.

One way to analyze the characteristics of a random process is to simulate it, when the model of the calculated process is fully reproduced. Simulation simulation relies on Monte Carlo methods - numerical methods for solving mathematical problems by simulation random

variables. The most important technique in this case is to bring the problem to the calculation of mathematical expectations [ 1 ]. If the mathematical expectation is unknown, the problem arises of estimating it with the required accuracy. Methods are known for obtaining estimates for unknown parameters of the distribution law of a random variable on the basis of a limited number of experiments [ 2, 3 ]. In this case, it is necessary to determine the duration of the simulation, suficient to estimate the unknown parameters of the distribution law for given relative error and confidence level. The method for determining the length of the simulation, based on the central limit theorem, is proposed in [ 4 ].

The approach is also applicable to estimating the characteristics of nonnegative regenerating process both in discrete or in continuous time. The method

makes it possible to obtain estimates of the characteristics of a random process both in discrete and continuous time, i.e. the characteristics that depend on the number of jumps in states and the characteristics that depend on the time of stay in states.

The method is applicable for the case of known values of the distribution characteristics corresponding to the experiments, and for the case of unknown values of the mathematical expectation and variance, for which they are obtained from existing

In Section 2, the method is described for the case of simulation independent identically distributed random variables. Section 3 presents the simulation algorithm. In Section 4, a numerical experiment was carried out for some of the laws of distribution — the exponential distribution, the

Weibull distribution, the Rayleigh distribution, and the Pareto distribution, the characteristics of which are known [ 5 ]. For these distributions, for a number of values of the relative error, for several values of the confidence probability, a series of experiments were conducted and estimates of the mathematical expectation ~ and variance ~ were obtained as well as a suficient length of the series of experiments (the duration of the simulation). A numerical experiment illustrates the operation of the method for determining the length of simulation for a given relative error and confidence level [ 4 ].

2.

Method for determining the length of the simulation In this section the method for determining the length of the simulation is described Let n be the number of independent experiments on a nonnegative random variable X, on the basis of which we estimate ~ and ~ for the mathematical expectation m>0 according to [ 2 ]. and variance D: ~ = ∑︀ =1

, ~ = ∑︀ =1 ( − ~)2 − 1

We express the probability on the left-hand side of the inequality in terms of the normal distribution function * (): (1) (2) (3)

⎷⎸⎸ 1 = ∑︁ ( − − 1) , = ∑︁ 2 ( − − 1) . where = *

and * () is the inverse of * () meaning the value of the argument at which the normal distribution function is equal to x.

If mathematical expectation m and variance D are not known, we use their estimates = ~ and ˜ =

Thus, the required number n of experiments will be found from the inequality [ 3 ]: (4) (5) (6) (7) (8) (9) (10)

Inequality (7) is also applicable in the case when the values 1, 2, ..., form a nonnegative regenerating sequence, i.e. discrete time random process [ 6 ].

Denote = 1 + 2 + ... + and = 21 + 22 + ... + 2, then Therefore, the required number of experiments can be found from the inequality In order to avoid overflow when exponentiation of accumulated amounts, it is more convenient to use its record in the following form: where = ︁( )︁ 2 assumes a constant value on the interval [− 1, ) , 0 = 0 < 1 < 2 < . . . < − 1 < estimated by formulas ~ = , ~ = − and the required simulation duration of the T is found from the inequality The mathematical expectation and variance of this process on the interval [0,T ) are

︂( ≤ + 1 )︂ 3.

Simulation Algorithm ︂( )︂ 2

, (12) (13) (14) (15)

Let X(t) = (1(), 2(), ..., ()) be a piecewise constant process with L components, either in discrete or in continuous time.

Let nonnegative functions () (X()) and () (X()) be characteristics of the process X() in case of discrete or continuous time correspondingly with the mathematical expectations () = lim

→∞ () = lim →∞ 0 =1 1 ∫︁ 1 ∑︁ () (X()), = 1, 2, . . . , ,

() (X()) , = 1, 2, . . . , , where and are the corresponding numbers of characteristics.

The goal is to estimate () and () with predetermined values of relative error and confidence probability . In addition the following input data should be determined: - duration of transit period before beginning of accumulating statistics; - the maximum length of the simulation in case of failure to achieve the specified accuracy; eps - infinitesimal increment in time units; - small quantity to avoid zero divide in step d.

The algorithm for simulation a piecewise constant process to obtain estimates of required characteristics is given below. the time of the next jump . quantities () and (), = 1, 2, ..., . We set = 1 and = 0.

a) Set the duration of the transit period > 0, the maximum simulation duration > and determine the initial state of the process X(t):X = X(0).

For each characteristic () of the process X(t), we zero out the quantities () and (), = 1, 2, ..., . For each characteristic () of the process X(t), we zero out the b) Set = , and then for each component of the process () determine and the corresponding component = { }, where 1 ≤ ≤ .

We determine the instant = { } of the nearest jump of the process X(t) If ≥ We change the accumulated data: , we set = + .

() := () + () (X) , () := () + ︁( ()(X) ︁) 2

, = 1, 2, . . . , , () := () + () (X) ( − ), (16) g) End of simulation.

~() = ~() = () () , = 1, 2, . . . , ,

, = 1, 2, . . . , . to the estimates during determined period .

At the end of simulation one obtain the estimates ~() for discrete time process or ~() for continuous time process as well as the number of experiments or simulation duration. In case specified accuracy was not achieved the values ~() and ~() correspond 4.

Numerical results probability =0.95 are chosen.

For the simulation, we selected the distributions of non-negative continuous random variables — the exponential distribution and the Rayleigh distribution (one-parameter), the

Weibull distribution and the Pareto distribution (two-parameter) [ 5 ]. For all experiments three values of relative errors ∈ {10− 2, 10− 3, 10− 4 } and the confidence () := () + ︁( ()(X) ︁) 2 ( − ), = 1, 2, . . . , . (17) (18) (19) (20) (21) (22) process X(t): X = X( ).

c) If < , then we set := + 1 and find the following state X of the If ≥ , go to step f (the specified accuracy is not achieved). d) If > and for some values () < or () < , then go to step b. e) If > and for some characteristic (), we have then go to step b. If for some characteristic (), we have () then go to step b.

f ) We calculate the estimates of the average values of the characteristics () () > () (︀ + 1 )︀ , , () = 1 − − with the parameter =3.

For several values of the relative error the number n of experiments necessary to achieve the required accuracy with given confidence level =0.95 is indicated, together with the corresponding estimates ~ for mathematical expectation and ~ for variance. expectation ~ and variance ~, the straight line corresponds to the exact value of the mathematical expectation ≈ 0.333 (21) and variance ≈ 0.111 (22). The number n of experiments and numerical characteristics for the exponential distribution with the parameter =3 The number n of experiments and numerical characteristics for the Weibull distribution with the parameters =1 and k =5 with the parameters =1 and k =5.

,

In Figure 3 and Figure 4 the curves correspond to the simulation estimate of the mathematical expectation ~ and variance ~, the straight line corresponds to the analytical value of the mathematical expectation = 0.9181 (24) and variance ≈ distribution (23) show the dependence of the number of experiments on the relative pending on the relative error for the exponential distribution depending on the relative error for the with the parameters =2, =3. = ︂( )︂ 2 (31)

In Figure 9 and Figure 10 the curves correspond to the estimate of the mathematical expectation ~ and variance ~, the straight line corresponds to the exact value of the mathematical expectation = 1.5 (30) and variance = 0.75 (31).

In this paper we propose an algorithm for simulation of a regeneration multicomponent process. The length of the simulation is specified by the relative errors of the process parameters. The algorithm makes it possible to obtain estimates the characteristics of the process, both in discrete and continuous time, which depend on the number of jumps in the states, and on the characteristics of the stay time in states.

The result of the algorithm is the number of experiments (the number of process jumps) or the duration of the simulation as well as the estimates of unknown parameters obtained as a result of simulation.

The numerical experiment is an illustration of the work of the method for determining the simulation duration, suficient for estimating the unknown parameters of the distribution law for given relative error and confidence level. The evaluation of the parameters obtained on the basis of a series of experiments, in the case considered mathematical expectation and variance, was compared with the exact values of these parameters, the analytical form of which for taken distributions is known. We note that the method makes it possible to determine a suficient length of a series of experiments also for the case when the values of the distribution law parameters are unknown, and estimates for these parameters are obtained on the basis of simulation [ 2 ]. This method is planned to be used in simulating random access [ 7 ] and adaptive radio access [ 8 ] for LTE networks as a development of previous research.

Acknowledgments

The publication has been prepared with the support of the “RUDN University Program 5-100” and funded by RFBR according to the research projects No. 18-07-00576, 1907-00933. This work has been developed within the framework of the COST Action CA15104, Inclusive Radio Communication Networks for 5G and beyond (IRACON).

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