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The present paper analyzed a model consisting of one unit with a warm standby unit where the main unit has three states: up, degraded and down.

The semi-Markov model under the regenerative method is used to construct the mathematical model for the system.

The effectiveness measures of the system are discussed such as availability, reliability, steady-state availability and mean time to system failure. The life and repair times of the system units are assumed to be discrete follow discrete Weibull distribution. Also, the parameters of the discrete Weibull distribution are assumed to be fuzzy with bell-shaped membership function. An application is introduced to show the results obtained for the system and the profit of the presented model.

Rarely papers in literature treated the topic of the discrete-time semi-Markov process using a regenerative point technique.

This paper examines the suitability of the discretizing approach of Roy and Dasgupta, based on the discrete concentration method of Roy, for determination of reliability of complex systems under Weibull set‐up. A system is said to be complex if its distribution is intractable and analytical evaluation of probability is not available. Numerical analyses on a hollow cylinder, solid shaft, hollow rectangular tube, and the power dissipated by a resistor have been worked out to examine the closeness between the discretized reliability estimators and the simulated values based on Monte Carlo method. Also, it may be observed that the proposed approach has superior performance in comparison to the discretizing approach of English et al., based on the moment equalization method of Taguchi. The mean absolute deviation under the discrete concentration approach is much less than the same under the moment equalization approach for the Weibull set‐up.

To determine the optimal software warranty period in continuous and discrete circumstances where the difference between the software testing environment and the operational environment can be characterised by an environment factor.

Software reliability models based on continuous and discrete time non‐homogeneous Poisson processes are assumed to describe the failure occurrence phenomena under both environments. Based on the idea of accelerated life testing for hardware products, the operational profile of the software is modeled, and the total expected software cost incurred in both testing and operational phases is formulated.

Under a milder condition, the optimal warranty period which minimizes the total software cost is derived analytically.

This paper introduces the operational profile of software to model the difference between the testing environment and the operational environment.

To determine the optimal spare part order‐replacement policy for any high cost single unit complex system in a discrete‐time circumstance.

The expected total discounted cost over an infinite planning horizon is taken as a criterion of optimality as it allows us to put emphasis on the present behavior of the system.

The problem under consideration is a two‐dimensional discrete optimization problem (regular ordering time and inventory time limit for the spare are decision variables) which is difficult to handle, in general. However, it is explored that the problem can be reduced to a simple one‐dimensional one and the optimal ordering time is to be determined under the two extreme situations: no replacement of the spare until the original unit fails and replacement of the spare as soon as it is delivered.

For modeling simplicity, deterministic lead time is considered for both regular and expedited orders. A more appropriate assumption would be to consider randomized lead time for both the orders.

The research provides a useful order‐replacement strategy for a single‐unit system where the failure of the unit is better measured by the number of cycles completed before failure rather than the instant of failure.

The work done in this paper carries certain values as any continuous time model for the problem under consideration can be regarded as only an approximate model.

The purpose of this paper is to analyze mathematical aspects of the q‐Weibull model and explore the influence of the parameter q.

The paper uses analytical developments with graph illustrations and an application to a practical example.

The q‐Weibull distribution function is able to reproduce the bathtub shape curve for the failure rate function with a single set of parameters. Moments of the distribution are also presented.

The generalized q‐Weibull distribution unifies various possible descriptions for the failure rate function: monotonically decreasing, monotonically increasing, unimodal and U‐shaped (bathtub) curves. It recovers the usual Weibull distribution as a particular case. It represents a unification of models usually found in reliability analysis. Q‐Weibull model has its inspiration in nonextensive statistics, used to describe complex systems with long‐range interactions and/or long‐term memory. This theoretical background may help the understanding of the underlying mechanisms for failure events in engineering problems.

Q‐Weibull model has already been introduced in the literature, but it was not realized that it is able to reproduce a bathtub curve using a unique set of parameters. The paper brings a mapping of the parameters, showing the range of the parameters that should be used for each type of curve.

The purpose of this paper is to understand the consequence of the use of mixed Weibull distribution in the cell formation problem. In reliability theory, a mixed distribution is used for more than one hazard cause, and the Weibull distribution can be used for ascendant, monotonous and descendant failure rate. Here, the authors mixed these two theme and use it in a common problem in group technology.

In this paper, the authors made a non-polynomial-hard mathematical model based on past research and solved it with an exact algorithm. The algorithm is coded and solved in GAMS to illustrate the model, and the authors use simulation. A common numerical example is solved with the model, and the results are compared.

Reliability analysis model based on the mixed Weibull distribution approach will give options to a user to select the suitable failure rate and modes for a specific situation. If the user uses the exponential or Weibull distribution instead of the mixed Weibull distribution, the calculated cost and reliability are wrong; therefore, it leads to user making wrong decisions.

The model the authors use is the one used in past research, but in the past, researchers did not use the mixed distribution for explaining failure time. Therefore, the model can be considered as a new and more complete model.

The probability distribution of major length and aspect ratio (major length/minor length) of wear debris collected from gear oil used in planetary gear drive were analysed and modelled. The paper aims to find an appropriate probability distribution model to forecast the kind of wear particles at different running hour of the machine.

Used gear oil of the planetary gear box of a slab caster was drained out and charged with a fresh oil of grade (EP-460). Six chronological oil samples were collected at different time interval between 480 and 1,992 h of machine running. The oil samples were filtered to separate wear particles, and microscopic study of wear debris was carried out at 100X magnification. Statistical modelling of wear debris distribution was done using Weibull and exponential probability distribution model. A comparison was studied among actual, Weibull and exponential probability distribution of major length and aspect ratio of wear particles.

Distribution of major length of wear particle was found to be closer to the exponential probability density function, whereas Weibull probability density function fitted better to distribution of aspect ratio of wear particle.

The potential of the developed model can be used to analyse the distribution of major length and aspect ratio of wear debris present in planetary gear box of slab caster machine.

Downtime is a process parameter that substantially impacts on the operating hours and results in production losses, thus motivating maintenance engineers to control process plants. Notwithstanding, the impacting nature of process equipment failure on the operating hours in bottling plants remains inadequately examined. In this paper, the cause-and-effect analysis was used to establish the root cause of the downtime problem and Pareto analysis employed to justify the greatest opportunities for improvement in reducing downtime and increasing reliability levels. Weibull analysis is then conducted on the industrial setting. Novel aspect ratios are proposed.

Using the Weibull failure function of machines as a principal facilitator to produce failure predictions, the downtime behaviour of a process plant was modelled and tested with practical data from a bottling process plant. This research was conducted in a Nigerian process bottling plant where historical data were examined.

The analysis of the results shows the following principal outcome: First, the machines with the highest and least downtime values are 2 and 5, respectively, with correspondingly mean values of 22.83 and 4.39 h monthly. Second, the total downtime 92.05 and 142.14 h for the observed and target downtime, with a coefficient of determination of 0.5848 was recorded. Third, as month 1 was taken as the base period (target), all the machines, except M5 had accepted performance, indicating proper preventive maintenance plan execution for the bottling process plant. Availability shows a direct relationship between the failure and uptime of the machines and the downtime impacts on production. Two machines had random failure pattern and five machines exhibited a wear-out failure pattern and probably due to old age and wear of components in the machines.

The major contribution of the paper is the Weibull modelling in a unique application to a bottling plant to avoid current practices that use reliability software that is not easily accessible.

Maintaining a high level of availability of weapon systems during battles becomes important from the point of view of winning the battle. Due to attrition factors (failure due to battle damage and unreliability) and logistic delays in the repair process, maintaining the required level of availability is difficult. In this paper, we develop a simulation model for availability of fighter aircraft considering multiple failures causing system failure and logistic delays in the repair process. The methodology is based on discrete event simulation using Monte Carlo techniques. The failure time distribution (Weibull) and the repair time distribution (exponential) for the considered subsystems of the aircraft and the logistic delay time distribution (log‐normal) for the logistic factors spares, crew and equipment were chosen with suitable parameters. The results indicate the pronounced decrease in availability (as low as less than 10 per cent in some cases) due to multiple failures and logistic delays. The results are, however, highly sensitive to a combination of reliability, maintainability and logistic delay parameters.