Delia Montoro-Cazorla

A deteriorating two-system with two repair modes and sojourn times phase-type distributed

We study a two-unit cold standby system in steady-state. The online unit goes through a finite number of stages of successive degradation preceding the failure. The units are reparable, there is a repairman and two types of maintenance are considered, preventive and corrective. The preventive repair aims to improve the degradation of a unit being operative. The corrective repair is necessary when the unit fails. We will assume that the preventive repair will be interrupted in favour of a corrective repair in order to increase the availability of the system. The random operational and repair times follow phase-type distributions. For this system, the stationary probability vector, the replacement times, and the involved costs are calculated. An optimisation problem is illustrated by a numerical example. In this, the optimal degradation stage for the preventive repair of the online unit is determined by taking into account the system availability and the incurred costs.

Transient analysis of a repairable system, using phase-type distributions and geometric processes

The transient behavior of a system with operational and repair times distributed following phase-type distributions is studied. These times are alternated in the evolution of the system, and they form 2 separate geometric processes. The stationary study of this system when the repair times form a renewal process has been made . This paper also considers that operational times are partitioned into two well-distinguished classes successively occupied: good, and preventive. An algorithmic approach is performed to determine the transition probabilities for the Markov process which governs the system, and other performance measures beyond those in are calculated in a well-structured form. The results are applied to a numerical example, and the transient quantities are compared with the ones obtained in the stationary case. The computational implementation of the mathematical expressions formulated are performed using the Matlab program.

Reliability of a system under two types of failures using a Markovian arrival process

We consider a system subject to external and internal failures. The operational time has a phase-type distribution (PH-distribution). Failures arrive following a Markovian arrival process (MAP). Some failures require the replacement of the system, and others a minimal repair. This model extends previous papers with arrivals governed by PH-renewal processes.

Replacement policy in a system under shocks following a Markovian arrival process

We present a system subject to shocks that arrive following a Markovian arrival process. The system is minimally repaired. It is replaced when a certain number of shocks arrive. A general model where the replacements are governed by a discrete phase-type distribution is studied. For this system, the Markov process governing the system is constructed, and the interarrival times between replacements and the number of replacements are calculated. A special case of this system is when it can stand a prefixed number of shocks. For this new system, the same performance measures are calculated. The systems are considered in transient and stationary regime.

A multiple system governed by a quasi-birth-and-death process

The system we consider comprises n units, of which one has to operate for the system to work. The other units are in repair, in cold standby, or waiting for repair. Only the working unit can fail. The operational and repair times follow phase-type distributions. Upon failure, it is replaced by a standby unit and goes to the repair facility. There is only one repairman. When one unit operates the system is up and when all the units are in repair or waiting for repair, the system is down. This system is governed by a finite quasi-birth-and-death process. The stationary probability vector and useful performance measures in reliability, such as the availability and the rate of occurrence of failures are explicitly calculated. This model extends other previously considered in the literature. The case with an infinite number of units in cold standby is also studied. Computational implementation of the results is performed via a numerical example, and the different systems considered are compared from the reliability measures determined.

A multiple warm standby system with operational and repair times following phase-type distributions

Abstract We study a warm standby n -unit system. The system functions as long as there is one operative unit. When the unit online fails, a unit in standby becomes the new unit online, if any. When a unit fails it goes to repair. There is a repairman. The units are repaired following the arrival order. The unit online has a lifetime governed by a phase-time distribution. The repair times follow a phase-type distribution. The warm standby units have lifetimes exponentially distributed. All the other times are negligible. This system extends many others of frequent use in the literature. We show that this system is governed by a level-dependent quasi-birth-and-death process (LDQBD process). The availability, rate of occurrence of failures and other magnitudes of interest are calculated. The mathematical expressions are algorithmically and computationally implemented, using the Matlab programme.

Replacement times and costs in a degrading system with several types of failure: The case of phase-type holding times

A reliability system submitted to external and internal failures, that can be repairable or non-repairable, with degradation levels, and with sojourn times phase-type distributed, is considered. Repair is not as good as new, and the repair of internal failure follows policy N, that is, after N completed repairs the system is replaced by a new one to the following failure, repairable or not. For this system, a Markov model is constructed, and the stationary probability vector is calculated. It is shown that the distribution of the time between two consecutive replacements follows a phase-type distribution, whose representation is determined. The costs of these periods are calculated. An optimization problem involving the costs, the availability, and the number of internal repairs is illustrated by a numerical example.

A shock and wear system under environmental conditions subject to internal failures, repair, and replacement

A system in a random environment is considered. The influence of the external conditions is governed by a Markovian arrival process. The internal structure of failure and repair are governed by phase-type distributions. The maintenance is performed by policy N. Under these assumptions, the Markov process governing the system is constructed, and it is studied in transient and stationary regime, calculating the availability and the rate of occurrence of failures. The renewal process due to the replacements of the system is studied, and expressions for the number of replacements and for the number of repairs between replacements are calculated. This paper extends other previously published since it incorporates a shock arrival process with dependence among the interarrival times and the renewal process associated to the replacements. A numerical application illustrates the calculations.

Survival Probabilities for Shock and Wear Models Governed by Phase-Type Distributions

Abstract A device submitted to shocks arriving randomly and causing damage is considered. The interarrival times follow continuous phase-type distributions. Lifetimes between shocks are affected by the number of cumulated shocks and they follow continuous phase-type distributions. Every shock can be fatal or not, with a probability that follows a discrete phase-type distribution. The device can support a maximum of N shocks. We calculate the distribution of the lifetime of the device in terms of the counting process of the number of shocks, and illustrate the calculations by means of a numerical application. Computational aspects are introduced. This model extends other previously considered in the literature.

TRANSIENT ANALYSIS OF A MULTI-COMPONENT SYSTEM MODELED BY A GENERAL MARKOV PROCESS

An M-unit system in dynamic environment with operational and repair times following phase-type distributions and incorporating geometrical processes is considered. A general Markov process with vectorial states is the appropriate structure for modeling this system. A transient analysis is performed for this complex system and the transition probabilities are calculated. Some performance measures of general interest in the study of systems are obtained using an algorithmic approach, and applied to G-out-of-M systems. A numerical example is presented and the transient performance measures are calculated and compared with the stationary ones. This paper extends previous reliability systems, that can be considered as particular cases of this one. Throughout the paper, the mathematical expressions are given by algorithmic methods, that emphasized the utility of phase-type distributions in the analysis of lifetime data.