Obzherin / Boyko Semi-Markov Models

Control of Restorable Systems with Latent Failures
1. Auflage 2015
ISBN: 978-0-12-802486-7
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 212 Seiten

Featuring previously unpublished results, Semi-Markov Models: Control of Restorable Systems with Latent Failures describes valuable methodology which can be used by readers to build mathematical models of a wide class of systems for various applications. In particular, this information can be applied to build models of reliability, queuing systems, and technical control. Beginning with a brief introduction to the area, the book covers semi-Markov models for different control strategies in one-component systems, defining their stationary characteristics of reliability and efficiency, and utilizing the method of asymptotic phase enlargement developed by V.S. Korolyuk and A.F. Turbin. The work then explores semi-Markov models of latent failures control in two-component systems. Building on these results, solutions are provided for the problems of optimal periodicity of control execution. Finally, the book presents a comparative analysis of analytical and imitational modeling of some one- and two-component systems, before discussing practical applications of the results - Reflects the possibility and effectiveness of this method of modeling systems, such as phase merging algorithms developed by V.S. Korolyuk, A.F. Turbin, A.V. Swishchuk, little covered elsewhere - Focuses on possible applications to engineering control systems

Autoren/Hrsg.

Obzherin, Yuriy E

Boyko, Elena G

Leseproben

Chapter 1 Preliminaries
Abstract
In present chapter, discussed preliminaries will be used afterwards. In Section 1.1, strategies and characteristics of technical control are considered. Section 1.2 covers necessary results from renewal theory. In Section 1.3, preliminaries on semi-Markov processes with arbitrary phase space and algorithm of semi-Markov model building are discussed. Keywords
control latent failure renewal process semi-Markov process stationary characteristics Chapter Outline 1.1 Strategies and Characteristics of Technical Control 1 1.2 Preliminaries on Renewal Theory 2 1.3 Preliminaries on Semi-Markov Processes with Arbitrary Phase Space of States 5 1.1. Strategies and characteristics of technical control
Automatic checkout systems consist of the object, engineering devices, programs, and operator, which enable to carry out automatic control. Control strategy usually means the rule defining the choice of checkout means with regard to the system controlled. There exist efficiency control and preventive control [6]. Efficiency control is checkout of the production capability to fulfil functions under parameters, determined by manuals. Preventive control is a technical checkout for detection and prevention of defects or flaws. In the monograph, efficiency control is investigated. Efficiency control is devided into ideal and nonideal [4]. Under ideal efficiency control, all the failures are detected immediately and reliably [4]. Under nonideal efficiency control, latent failures and automatic checkout system failures take place [4]. In Figure 1.1, a general scheme of efficiency control execution with the help of automatic checkout systems is presented. One of the control characteristics is its periodicity. The control periodicity is time period between two successive checkout processes, executed by certain control instrument [6]. Figure 1.1 General scheme of efficiency control execution According to the object, continuous, periodical, and casual kinds of control are singled out. Under continuous control, the information on parameters is received constantly, while under periodical control it happens at certain time intervals. Casual control is carried out at random time intervals [6]. Casual control includes single control. The latter is executed, for instance, before the use of stored system, in case the system reliability is ensured by the storage measures. In the monograph, periodic control with full efficiency restoration is investigated. In Sections 2.1, 2.4, 3.3–3.5 and Sections 2.2, 2.3, 3.1, 3.2, system efficiency control with component deactivation and without deactivation while control execution are considered, respectively. 1.2. Preliminaries on renewal theory
In the present section, renewal theory review is made [1–3, 7]. The information is mainly given according to monograph [3]. Renewal theory origins from the simplest restoration model: after each failure the system is restored immediately. Definition [3]. Renewal process is a sequence n;n=1 of non-negative independent random variables (RVs) with the same distribution function (DF) F(t). RVs k=?n=1kan,k=1,T0=0 are called renewal moments. Renewal process is often defined as a sequence of RV k as well. The counting renewal process, defined with the help of renewal moments, is of particular interest (t)=maxk:Tk=t. For each moment t, the value of N(t) determines the random number or renewal moments in , t. Renewal function H(t), defining the mean of renewal moments in , t, plays fundamental role in renewal theory: (t)=EN(t)=?k=18kPN(t)=k=?k=18F*(k)(t), (1.1) where *(k)(t) is a k-fold convolution of DF (t), F*(1)(t)=F(t). The function is a solution of the integral renewal equation: (t)=F(t)+?0tH(t-x)dF(x). Renewal function derivative (t)=H'(t)=?k=18f*(k)(t) is called renewal density. It satisfies the following integral equation: (t)=f(t)+?0th(t-x)f(x)dx, (1.2) where f(t) is the density of DF F(t). The following formulas are true: 0tF¯(s)h(t-s)ds+F¯(t)=1,??0tH(t-s)F¯(s)ds+?0tF¯(s)ds=t. (1.3) Under sufficiently small t, the value (t)?t approximately equals the probability of renewal moment appearance in ,t+?t. Explicit form of functions (t) for some renewal processes can be found in [3, 7]. The functions: ~(t)=1+H(t),?H^(t)=1+H(t),t>0,0,t=0. (1.4) will be applied as well. The function ^(t), having a unit jump at t = 0, is used to write down expressions in form of Stieltjes integral. The following theorems, describing asymptotic behavior (under ?+8) of the function H(t), take place. Renewal theorem (elementary) [3]. For any distribution law of F(t) t?+8H(t)t=1µ, (1.5) where =Ea is expectation of RV a. Renewal theorem (key) [3]. If F(t) is not an arithmetic distribution, and g(t) is integrable nonincreasing in 0,+8) function, then t?+8?0tg(t-x)dH(x)=1µ?08g(x)dx. (1.6) The process of direct residual time t=tN(t)+1-t,t=0, Vt being residual time to failure by t, is connected with the renewal process n;n=1. Vt is a homogeneous Markov process with phase state 0,+8). DF (t,x)=PVt=x of the direct residual time is defined by the formula [3]: (t,x)=F(t+x)-?0tF¯(t+x-s)dH(s), (1.7) and corresponding distribution density is: (t,x)=f(t+x)+?0tf(t+x-u)h(u)du. (1.8) The expectation of the direct residual time equals: (Vt)=Ea(1+H(t))-t. (1.9) In the renewal process, the restoration of failed system is considered to be negligible in comparison with operating time. This assumption does not take place in practice. That is why the following system renewal process is considered [3]. For the first time, the system fails in a random time period 1 and is restored in random time 1. The restored system operates for 2 time, then it fails, and is restored in 2, and so on. Time moments 1=a1, T2=a1+ß1+a2,…, of the system failure are called failure moments or moments of 0-renewation, and the moments 1=a1+ß1, 2=a1+ß1+a2+ß2,…, of the restoration end are restoration moments (or 1-renewations). Definition [3]. If n;n=1 and n;n=1 are two sequences of independent similarly distributed RVs, the sequence an,ßn);n=1, as well as Tn,Sn);n=1, is called an alternating renewal process. Alternating renewal process can be equivalently given by (t),t=0 with the help of (t)=0,?if?t?[Tk,Sk),1,?otherwise. According to the definition, the process (t) determines system states at the moment : (t)=1 for system up-state at , and (t)=0 for restoration at . Let us denote by (0)(t) the random number of 0-renewations, and by...