Steady-state Unavailability Research Articles

Analysis ofD-policy discrete-timeGeo/G/1queue with secondJ-optional service and unreliable server

This paper is concerned with a discrete-time Geo /G / 1 queueing system with D -policy and J -optional services in which the service station may be subject to failures at random during serving the customers. All the arriving customers require the first essential service, whereas some of them may opt for a second service from the J additional services with some probability. As soon as the system becomes empty, the server will not restart the service until the sum of the service times of the waiting customers in the system reaches or exceeds some given positive integer D . Applying the total probability decomposition law, renewal theory, and probability generating function technique, the queueing indices and reliability measures are investigated simultaneously in our work. Both the probability generating function of the transient queue length distribution and the explicit formulas of the steady-state queue length distribution at time epoch n + are derived. Meanwhile, the stochastic decomposition property is presented for the proposed model. Various reliability indices, including the transient and steady-state unavailability, the expected number of breakdowns during (0+ ,n + ], and the equilibrium failure frequency, are discussed. Finally, the optimum value of D for minimizing the system cost is numerically discussed under a given cost structure.

Asymptotic optimality of RESTART estimators in highly dependable systems

We consider a wide class of models that includes the highly reliable Markovian systems (HRMS) often used to represent the evolution of multi-component systems in reliability settings. Repair times and component lifetimes are random variables that follow a general distribution, and the repair service adopts a priority repair rule based on system failure risk. Since crude simulation has proved to be inefficient for highly-dependable systems, the RESTART method is used for the estimation of steady-state unavailability and other reliability measures. In this method, a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of a rare event (e.g., a system failure) is higher. The main difficulty involved in applying this method is finding a suitable function, called the importance function, to define the regions. In this paper we introduce an importance function which, for unbalanced systems, represents a great improvement over the importance function used in previous papers. We also demonstrate the asymptotic optimality of RESTART estimators in these models. Several examples are presented to show the effectiveness of the new approach, and probabilities up to the order of 10−42 are accurately estimated with little computational effort.

Reliability indices of discrete-time Geox/G/1 queueing system with unreliable service station and multiple adaptive delayed vacations

This paper considers the discrete-time GeoX/G/1 queueing model with unreliable service station and multiple adaptive delayed vacations from the perspective of reliability research. Following problems will be discussed: 1) The probability that the server is in a “generalized busy period” at time n; 2) The probability that the service station is in failure at time n, i.e., the transient unavailability of the service station, and the steady state unavailability of the service station; 3) The expected number of service station failures during the time interval (0, n], and the steady state failure frequency of the service station; 4) The expected number of service station breakdowns in a server’s “generalized busy period”. Finally, the authors demonstrate that some common discrete-time queueing models with unreliable service station are special cases of the model discussed in this paper.

RESTART simulation of non-Markov consecutive- k-out-of- n: F repairable systems

The reliability of consecutive- k-out-of- n: F repairable systems and ( k−1)-step Markov dependence is studied. The model analyzed in this paper is more general than those of previous studies given that repair time and component lifetimes are random variables that follow a general distribution. The system has one repair service which adopts a priority repair rule based on system failure risk. Since crude simulation has proved to be inefficient for highly dependable systems, the RESTART method was used for the estimation of steady-state unavailability, MTBF and unreliability. Probabilities up to the order of 10 −16 have been accurately estimated with little computational effort. In this method, a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of a rare event (e.g., a system failure) is higher. The main difficulty for the application of this method is to find a suitable function, called the importance function, to define the regions. Given the simplicity involved in changing some model assumptions with RESTART, the importance function used in this paper could be useful for dependability estimation of many systems.

Importance Functions for RESTART Simulation of Highly-Dependable Systems

RESTART is a widely applied accelerated simulation technique that allows the evaluation of very low probabilities. In this method a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of the rare event is higher. Formulas for evaluating the optimal number of regions and retrials have been provided in previous papers. Guidelines were also provided for obtaining a suitable function, the importance function, used to define the regions. This paper provides a simple importance function that can be useful for RESTART simulation of models of many highly dependable systems. Some eXamples from the literature illustrate the application of this importance function. Steady-state unavailability of balanced systems is accurately estimated within short computational times, and also the unavailability of an unbalanced system but with much more computational effort.

Failure Transition Distance-Based Importance Sampling Schemes for theSimulation of Repairable Fault-Tolerant Computer Systems

Markov models are often used to evaluate dependability attributes of fault-tolerant computer systems. The use in practice of Markov models is, however, hampered by the well-known state space explosion problem. Simulation alleviates the problem. For Markov models of repairable fault-tolerant systems, standard simulation of dependability measures tends to be expensive due to the rarity of the system failure event. Importance sampling can speed up the simulation. This paper develops two importance sampling schemes, called failure transition distance biasing & balanced failure transition distance biasing, which exploit the failure transition distance concept in an attempt to improve the efficiency of two other schemes, failure biasing & and balanced failure biasing. The schemes require the computation of the so-called failure transition distances, and procedures to perform those computations are developed. The presentation is tied to a previously proposed measure-specific simulation method for the steady-state unavailability. An optimization method of the parameters of the importance sampling schemes is also developed. For the simulation of the steady-state unavailability, failure transition distance biasing has (as failure biasing) the bounded relative error property for balanced fault-tolerant systems & balanced failure transition distance biasing has (as balanced failure biasing) the bounded relative error property for both balanced & unbalanced fault-tolerant systems. It is proved that, for balanced fault-tolerant systems, both failure transition distance biasing & balanced failure transition distance biasing can indeed improve the efficiency of failure biasing & balanced failure biasing. In addition, numerical experiments seem to indicate that, for unbalanced fault-tolerant systems, balanced failure transition distance biasing can also improve the efficiency of balanced failure biasing. The application of the failure transition distance-based importance sampling schemes is, however, limited to systems not having too many minimal failure covers, or, at least, not having too many minimal failure covers of small cardinality. A minimal failure cover is a minimal bag of failure bags such that the failure of its components implies the failure of the system; a failure bag is any non-empty bag of component classes which can fail simultaneously.

Fine grained software degradation models for optimal rejuvenation policies

In this paper, we address the problem of preventive maintenance of operational software systems, an approach recently proposed to counteract the phenomenon of software “aging”. We consider the so-called “software rejuvenation” technique [Software rejuvenation: analysis, module and applications, in: Proceedings of the 25th International Symposium on Fault-Tolerance Computing (FTCS-25), Pasadena, CA, USA, June 1995], which consists in periodically stopping the software system and then restarting it in a “robust” state after a proper maintenance, and we propose a methodology for the quantitative analysis of rejuvenation strategies. The methodology is based on a fine grained model that assumes that it is possible to identify the current degradation level of the system by monitoring an observable quantity, so that the rejuvenation strategy can be tuned on the measured degradation. Based on this methodology, we present two different strategies that allow to decide whether and when to rejuvenate, and we exploit the theory of renewal processes with reward to estimate the steady-state unavailability of the software system, which is used to define an optimality criterion that allows us to evaluate the proper rejuvenation intervals. The methodology and the rejuvenation strategies are demonstrated by applying them to a real-world case study, arising in the area of database maintenance for data archiving, and to a hypothetical setting used to assess the sensitivity of the technique to various degradation processes.

Large-capacity strictly nonblocking optical cross-connects based on microelectrooptomechanical systems (MEOMS) switch matrices: reliability performance analysis

The reliability performance of 128/spl times/ 128 optical cross-connects (OXCs) based on microelectrooptomechanical systems (MEOMS) switch matrices is considered. First, we compare a strictly nonblocking wavelength selective switch with a strictly nonblocking three-stage Clos architecture. The probability of maintenance of free operation has been investigated for both OXC structures. We present our calculation results for such commonly used reliability measures as mean time between failures (MTBF), mean downtime (MDT) per gear, and steady-state unavailability. It is shown that the reliability performances of the considered OXCs are far from that requested. In this paper, we also investigate possibilities of improving the reliability performance of the considered OXCs by introducing shared redundancy of the MEOMS matrices. We propose two different protection schemes: one for the wavelength selective switch and another for the three-stage Clos architecture. It is shown that the proposed protection schemes significantly improve the reliability performance for both cases. Finally, we compare the performance of the all-optical version of the OXC based on MEOMS matrices with the optoelectronic version of the OXC based on electronic cross-point switch matrices. It is shown that from a reliability viewpoint, the optical cross-connect based on MEOMS matrices is better than that with electrical cross-point switches. The influence of capacity expansion on the system reliability is discussed.

A Highly Efficient Monte Carlo Method for Assessment of System Reliability Based on a Markov Model

SYNOPTIC ABSTRACTThis paper proposes a Monte Carlo procedure for evaluating the reliability of a system consisting of independent components with two states, failed and working. This procedure replaces the failure distribution of each individual binary component with a two-state continuous time Markov chain. The transition rates for the process are selected such that the steady-state unavailability index equals the component unreliability. The Markov chain for each component is simulated over a chosen time interval and a simulated value of the system uptime is obtained therefrom using the system logic. The system reliability is estimated as the ratio of the system uptime to the length of the simulation time horizon. An example is given on the application of this procedure to the evaluation of a reliability index extensively used in the electric power industry. This example as well as others on several simple systems show that the proposed procedure is more efficient than crude Monte Carlo in that it requires fewer random numbers on the average than the latter to obtain the same level of precision. The relative efficiency of this procedure increases with increasing value of system reliability.

Steady-state reliability of repairable systems by combined probability and possibility assumptions

We study the steady-state reliability behavior of a repairable one-unit system which can be described by the alternating process of failures and repairs. We assume that the failure behavior is possibilistic and the repair behavior is probabilistic, and vice versa. Two approaches for computing the steady-state availability and unavailability as the measures of the system effectiveness are proposed and compared. The limit theorems related to the stationary behavior of possibility distribution functions are proved. The applicability of the considered approaches to the steady-state reliability analysis is discussed. An example of a manufacturing system with the human operator illustrates one of the approaches.

On the steady state unavailability of standby systems

In this paper we study the steady state unavailability of standby systems comprising n identical components of which n − 1 are normally operating and one is in standby, with one repair facility. The components are assumed to have constant failure rates, but arbitrary repair time distribution. Attention is given to the case that the mean time to repair MTTR is relatively small compared to the mean time to failure MTTF, which is the most common situation in practice. Simple approximation formulae are presented and compared with the standard Markov expressions. It is demonstrated that the Markov expressions produce relatively large errors when the coefficient of variation of the repair time distribution is not close to 1. Some easily computed error bounds of the approximation formulae are established.

Availability analysis in a cable transmission system with repair restrictions

Single-system generating capacity adequacy analysis normally does not include recognition of transmission constraints or availabilities. These considerations are usually included in multiarea adequacy studies and in single system studies involving remote generating facilities connected to the system through either high-voltage AC or DC transmission. The availability of the transmission link can be critical to the overall system adequacy and to the benefit associated with the remote generation facility. The authors present a technique for modelling a transmission facility with repair restrictions. The example chosen is an underwater direct-current cable installation. The system is analyzed using a Markov model, and results are presented on time-dependent and steady-state unavailability, as well as on the common mode failure of trenches. >

An availability analysis of a k-out-of- N:G redundant system with dependant failure rates and common-cause failures

This paper presents an availability analysis of a k-out-of- N:G redundant system with dependant failure rates, common-cause failures and r repair facilities. The failure rates of the components increase as the number of components failed increases, while the repair rates are constants. The failed system is repaired to ‘as good as new’. Common-cause failure is not considered in Model I. In Model II the common-cause failures are involved. Steady-state probabilities and steady-state availability and steady-state unavailability are derived.

A Conditional Probability Approach to Reliability with Common-Cause Failures

This paper presents a conditional probability method to evaluate the reliability or availability of a system whose components failures can be s-independent or have a common-cause. Different failure types can be in each cut-set. By applying pivotal decomposition, the chain rule of conditional probability, and the recursive rule of probability of a union of events, I obtain the steady-state system unavailability. This approach requires the failure & repair rates of each component due to s-independent causes, the occurrence rate of each common-cause failure, and the mean duration of each common-cause. Hence it is practical and useful. The general model in the adverse environment assumes s-dependency and faces the dilemma of how to estimate the transition rates of each component.

Steady state unavailability of a [formula omitted] system with total repair

The steady state unavailability of the k- out-of- n system is evaluated from special Markov models assuming constant transition rates, total repair from every system state with failed components and a common transition into the totally failed state due to common mode failures. The repair time distribution is modelled individually for each system state with failed components using an Erlang distribution of appropriate order. The general formulae of the solution are derived. An exact solution is also given for the limiting case of fixed repair times. Simple approximate formulae are given for application to important special cases.

On Availability of a Series system with Imperfect Detectors

An n-unit series system with exponential distributions for life-times and repair-time has been considered. Each unit is equipped with a detector to detect failures. Detectors are subject to two failure modes: viz. (i) instantaneous failure i.e. it fails at the time of need when a unit fails; (ii) gradual failure i.e. it fails and gives false alarm for system failure. Steady-state availability of the system is obtained by studying the underlying system equations. Behavior of steady-state unavailability has also been studied analytically.

A semi-regenerative process of two-unit warm standby system

This paper deals with two identical units warm standby system; a failure of operating unit can be detected at any time but a failure of standby unit can not be done until a system is inspected. We are able to look upon the stochastic behavior of our model as that of semi-regenerative process. The pointwise unavailability and the steady state unavailability of the system are derived by using the limit theorem of semi-regenerativeprocess. Further, we shall discuss the optimum inspection period minimizing the steady state unavailability. A numerical example is presented.

General Formulas for Calculating the Steady-State Frequency of System Failure

This paper develops two general formulas for directly converting the expression for steady-state availability or unavailability into frequency of system failure. Using the Aggarwal-Misra-Gupta fast algorithm, formulas for calculating the availability and failure-frequency of a s-coherent system are derived.