Stochastic behaviour of man-machine systems operating under different weather conditions

This paper presents some comments on ‘Stochastic behaviour of man-machine systems operating under different weather condition’ [1]. In addition, reliability analysis of one of the models given in [2] is performed. State probability, human operator reliability, mean time to human error (MTTHE) and probability of human error plots are shown.     

Stochastic behaviour of man-machine systems operating under different weather conditions

This paper presents two mathematical models to predict the performance of the man—machine systems under different weather conditions. The operator may be in good or poor physical condition which also affects the performance of the system. Failure rate of the system, rate of change of weather conditions as well as rate of change of physical conditions of the operator are assumed to be constant while the repair time distributions are general. Regenerative technique in MRP is applied to obtain several reliability characteristics of interest.     

Stochastic behaviour and profit function of a system with precautionary measures under abnormal weather

A repairable system under normal and abnormal weather conditions is analysed. The system can be in one of the three modes—normal, partial and total failure. Failure rates of system and rates of change of weather conditions are constant while the repair rates are arbitrary functions of time. The repair time distributions depend upon the state from which the repair starts and are invariant with the change of weather. Using regenerative point technique various reliability characteristics such as mean time to system failure, steady state availability, the probability that the repairman is busy, expected number of visits by repairman and expected profit earned by the system are obtained.     

Stochastic behaviour of a three-state complex repairable system with three types of failures

This paper deals with the operational availability of a three-state complex repairable system consisting of n distinct units arranged in series. Single service facility is available for three types of failures of the complex system. A mathematical model for the complex system under severe restrictions has been developed for exponential failures and general repairs. Laplace transforms of various state probabilities have been evaluated and availability is then obtained by the inversion process. Few graphical illustrations are also given in the end so as to explain the practical utility of the model.     

Reliability analysis of a multistate one-unit repairable system operating under a changing environment

This paper studies a multistate repairable system operating under changing environment subject to a Markov process with two states. Failure rates are constant while the repair rates are general. Regenerative technique in MRP is applied to obtain several reliability characteristics of interest to system designers.     

Stochastic behaviour of a two-unit paralleled system

This paper deals with a two-unit paralleled system with a single repair facility. Most papers on this subject assumed that at least one of the life time and repair time distribution is exponential. However the above assumption is violated by this paper. Then the various state probabilities (densities) have derived by employing the method of supplementary variables. We obtain the stochastic behaviour from these probabilities; (1) the long-run availability, (2) the idleness probability for the repair crew, and (3) the expected number (per unit time in long run) of system failures. Also comparison between the constant failure rate and non-constant failure rate is shown by numerical example.     

On the transient behaviour of a repairable system with a warm standby

In this paper, a two-unit active standby system is analysed to evaluate various measures of system performance under the assumption that the failure and repair times follow arbitrary distributions. Systems of integral equations satisfied by various state probabilities corresponding to different initial conditions have been written using the supplementary variable technique and state-space method. A particular case is analysed numerically under the assumption that the failure and repair times of both the units follow normal distributions.     

Stochastic behaviour of a maintained system with protection system

This paper deals with the analysis of a single unit system backed by a protection system (.P.S.). Both are subject to failure with exponential failure time distribution. Repair time of working unit, fault detection and inspection time of protection system are assumed to follow general distribution. The system has been investigated in detail by the help of semi-Markov process and several parameters of interest are obtained.     

Stochastic analysis of a system operating in a multiple environment

This paper deals with the analysis of a single unit system operating in a multiple environment. The system consists of three types of states—normal, abnormal and failed. Using the regenerative point technique in a Markov renewal process, several reliability characteristics of interest are obtained.     

Stochastic analysis of a repairable system with three units and two repair facilities

This paper investigates the availability characteristics and the reliability of a three-dissimilar-unit repairable system with two different repair facilities. Under some practical assumptions, we obtain the explicit expressions of the state probabilities of the system and then the explicit expressions of the following performance measures of the system: (1) the pointwise and steady-state availability; (2) the pointwise and steady-state failure frequency; (3) the pointwise and steady-state renewal frequency; and (4) the reliability and the mean time to system failure.     

Some characteristics of a man-machine system operating subject to different weather conditions

This paper deals with some characteristics of a single unit of a man-machine system operating under different weather conditions. It is assumed that the failure, repair and change of weather conditions (normal-stormy) are stochastically independent random variables, each having an arbitrary distribution. The system is analysed by the semi-Markov process technique. Some reliability measures of interest to system designers as well as operations managers have been obtained. Explicit expressions for the Laplace-Stieltjes transforms of the distribution function of the first passage time, mean time to system failure, pointwise availability and steady-state availability of the system are obtained. Several important results have been derived as particular cases.     

Reliability and MTTF evaluation of a repairable complex system under waiting

This paper deals with the reliability and mean time to failure (MTTF) evaluation of a complex system under waiting incorporating the concept of hardware failure and human error. Failure rates of the complex system follow exponential time distributions, whereas repair follows a general repair time distribution. Laplace transforms of various state probabilities have been evaluated and then reliability is obtained by the inversion process. A formula for variance of time to failure has also been developed. A particular case is also given to highlight some important results. Moreover, various plots have been sketched at the end.     

Stochastic behaviour of a two-unit redundant system with switchover time

There is a two-unit standby redundant system. Standby is kept in cold state. Whenever one unit is operating and the other is in the waiting standby state, switching is initiated on the latter after a random time. If the operative unit fails before the time to initiate switching action on the waiting standby unit, the system fails and the failed unit immediately undergoes repair type 2 along with the switching. However, if the operative unit fails when the other unit is as standby (after switchover), the failed unit undergoes repair type 1 and the unit as standby takes over the operation. All the distributions are arbitrary except failure-time, which is exponential.The system is defined with the help of states which generate a pseudo semi-Markov process. Abundant use of results from the theory of SMP has been made to obtain a large number of parameters which measure reliability characteristics of the system viz. MTSF, steady-state availability, expected number of visits to a state, conditional transition probabilities, first passage time distributions, expected profit rate, etc. Numerical examples are included to illustrate the results.     

Stochastic behaviour of a standby redundant system with three modes

This paper discusses the stochastic behaviour of a two-unit cold standby redundant system in which each unit works in three different modes-normal, partial failure and total failure. Failure-time distributions of units are exponential, whereas repair-time distributions are arbitrary. Explicit expressions for the Laplace-Stieltjes transforms of the distribution function of the first passage time, mean time to system failure and steady state availability of the system are obtained. A few particular cases are discussed.     

Reliability analysis of a two state repairable parallel redundant system under human failure

In this paper, the investigations have been carried out for the MTTF and reliability analysis of a repairable two-unit redundant electronic equipment having two states under human failures. The two-unit repairable parallel redundant system suffers two types of failures; viz; unit failure and human failure. Human failure brings the system to a complete failure stage. There is only one server who is always available. Laplace transforms of the probabilities of the complex system being in up and down states have been derived and have been inverted to obtain time dependent probabilities. Two graphs have also been given in the end.     

Optimal allocation for a repairable system

A system is composed of n series subsystems. Each subsystem consists of some same type of parallel or cold stand-by components. These components are repairable. We discuss the optimal allocation of this system under various constraints, and give some procedures for solution of this problem.     

Long-run availability of a repairable parallel system

We analyze the long-run availability of a two-unit parallel system sustained by a cold standby unit and attended by two identical repairmen. The system satisfies the usual conditions (i.i.d. random variables, perfect repair, instantaneous and perfect switch, queueing). Each operative unit has a constant failure rate and a general repair time distribution. We use Hokstad's supplementary variable method to construct a system of simultaneous partial differential equations. We present a first-order numerical scheme and an iterative algorithm for the solution of the equations. A particular but important numerical example (Weibull repair), illustrated by computer-plotted graphs, motivates the proposed approximation.     

Cost analysis of a three-state repairable redundant complex system under various modes of failures

In this paper investigations have been carried out for the availability and mean time to failure analysis of a three unit repairable electronic equipment having three states; viz; good, degraded and failed under critical human errors. The three states three units repairable electronic equipment suffers two types of failures; viz; unit failure and failure due to critical human errors. Entire system can fail due to critical human errors. The failure and repair times for the system follow exponential and general distributions respectively. Laplace transforms of the probabilities of the complex system being in various states are obtained along with steady state behaviour of the equipment. A numerical example has also been appended to highlight the important results. Three graphs have also been given in the end. There is only one repair facility, which is availed only when the system is in either degraded or failed state due to unit failure.