M.N. Gopalan

On the stochastic behaviour of a 1-server 2-unit system subject to arbitrary failure, random inspection and two failure modes

Abstract The stochastic behaviour of a single-server two-unit system subject to random inspection and two failure modes has been investigated under the supposition that the failure-time of a unit is distributed arbitrarily while all the other time distributions involved are exponential. Further, it is assumed that there is a single service facility which serves the dual purpose of inspection and repair. The system is characterized by the probabilities of its being in the “up” or “down” state. Integral equations are set up for these probabilities by identifying the system at suitable regenerative epochs. Laplace-transform technique is adopted to solve these equations. Various system parameters of vital significance, namely: 1. (1) point-wise availability of the system at instant t, 2. (2) s-expected up-time of the system in [0, t], 3. (3) s-expected inspection-time of the server in [0, t], 4. (4) s-expected repair-time of type i (i = 1,2) in [0, t], 5. (5) s-expected net gain per unit time in [0, t] and 6. (6) steady-state availability of the system have been obtained explicitly.

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.

Probabilistic analysis of a n-unit cold-standby system with general failure and repair time distributions

In this paper, a n-unit cold-standby system with a single repair facility is analysed under the assumption that the failure and repair times are arbitrarily distributed. A mathematical model is developed using semi-regenerative processes. Systems of integral equations satisfied by various states probabilities corresponding to different initial conditions are obtained. Explicit expressions for the availability, reliability and the expected busy period of the service facility are obtained. Two special eases when either the failure or repair rate is constant are discussed. A particular ease with three units is analysed numerically by assuming Weibull distribution for failure time and normal distribution for repair time.

Stochastic modelling of a two-stage transfer-line production system with end buffer and random demand

Abstract This paper examines the transient and steady-state characteristics of a two-stage transfer-line production system subject to inter-stage and end inspections and end buffer. The inter-stage and end inspections are instantaneous and the end buffer is of finite capacity, of size N. A stochastic model is developed to describe the system, and analytical expressions for the percentage utilization, idle time and blocked duration with arbitrary distributions of processing times at both stages of production, and the exponential supply and demand pattern have been obtained by using the state-space method and the regeneration point technique. A particular case is investigated and numerical results are worked out in detail.

On the transient behaviour of an r-out-of-n:G system

In this paper, an r-out-of-n:G system with a single repair facility is analysed in connection with the cost incurred per unit time. A mathematical model is developed using the concept of semi-regenerative phenomena. Systems of convolution integral equations satisfied by various state probabilities corresponding to different initial conditions are obtained. An efficient iterative technique is used to solve the system of convolution integral equations and various measures of system performance are obtained.

On the utilization analysis of a two-stage transfer line production system subject to inter-stage inspection and initial buffer

The transient and steady-state characteristics of a two-stage transfer line production system subject to inter-stage inspection and initial buffer are studied. The initial buffer has finite capacity N, and when the buffer size reaches a level n, a replenishment order for a size (N − n) is placed. The first stage of production is never starved until the buffer level reaches zero, and the second stage of production is never blocked. A stochastic model is developed to describe the system, and analytical expressions for percentage utilization, idle time and blocked duration, with arbitrary distributions of processing times at both stages of production and exponential arrival pattern to the buffer upon ordering, have been obtained by using the state-space method and regeneration point technique.

s-expected number of inspections and repairs of a single server n-unit system subject to arbitrary failure

Abstract The paper deals with a single-server two-unit system subject to random inspection and two failure modes. It is assumed that the failure time of a unit is distributed arbitrarily while all the other time distributions involved are exponential. Further, it is assumed that there is a single-service facility which serves the dual purpose of inspection and repair. At any instant t, the system is characterized by the probabilities of its being in the “up” or “down” state. Integral equations are set up for these probabilities by adopting the regeneration point technique. A cost-benefit analysis of the system is developed by obtaining various system characteristics of importance, namely, 1. 1.s-expected number of inspections in [O, t], 2. 2.s-expected number of repairs of type i (i = 1,2) in [O, t] and 3. 3.s-expected net gain per unit time in [O, t].

Analysis of consecutive k-out-of-n:F systems with single repair facility

Abstract In this paper, a consecutive k -out-of- n :F system with non-identical units and with a single repair facility is analyzed in connection with the reliability and availability of the system. The failure rates of the units are constant and the repair time is arbitrarily distributed. A mathematical model is developed using semi-regenerative phenomena and systems of convolution integral equations satisfied by various state probabilities corresponding to different initial conditions are obtained. A particular case with k = 2 and n = 3 is analyzed numerically to study the effect of various parameters on the availability and reliability of the system.

Approximate analysis of n-unit cold-standby systems

In this paper, a n-unit cold-standby system with a single repair facility is analysed using two approximate methods namely, cutting and clustering the state space. It has been assumed that the failure rate is constant and the repair time is arbitrarily distributed. A mathematical model is developed using semi-regenerative phenomena and systems of convolution integral equations satisfied by various state probabilities corresponding to different initial conditions are obtained. Explicit expressions for the expected number of failures and expected number of repair completions in an interval [0, t] are obtained. An iterative numerical method is used to solve the systems of integral equations obtained and a comparative study has been carried out between exact and approximate solutions.

Reliability analysis of a system with dependent units

Abstract The paper deals with the reliability analysis of a system consisting of n identical units (connected in parallel) subject to exponential failure distribution. An explicit expression for the mean time to system failure has been obtained.