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Maintaining a high-level reliability and efficiency without interruption are the key concerns for many real-time machining systems. Using the redundancy and repair facility features, we develop a double retrial orbit queueing model for the fault-tolerant machining system (FTMS) operating under the restriction of admission of repair jobs based on threshold policy and working vacation. The provision of primary and secondary orbits is made so that the failed units can wait there based on the facility available in case the repairman is occupied. From the orbits, the failed units retry to get the repairman free so that the repair job can be accomplished. Chapman–Kolmogorov equations for the system states of FTMS have been constructed to evaluate the transient reliability and queueing indices using the spectral technique. The sensitivity along with the relative sensitivity analysis of crucial system parameters, have facilitated. The impacts of parameter variability on the system metrics and total expected cost are examined for illustrative examples. • Reliability Modeling of Fault-tolerant system with working vacation and double retrial orbits have been studied. • Adopt the spectral expansion method to examine the transient behavior of the developed model. • Address the sensitivity analysis of the system reliability and mean time to failure. • The numerical simulations are presented to validate the computational tractability. • A real-life based example of a Digital Subscriber Line Access Multiplexer is illustrated. [ABSTRACT FROM AUTHOR]

With the increasing reliance on cloud computing as the foundational manufacturing systems with intricate dynamics, featuring multiple service areas, varying job arrival rates, diverse service requirements, and the interplay of failures and impatience, significant analytical challenges arise. Queueing networks offer a powerful stochastic modeling framework to capture such complex dynamics. This paper develops a novel, exhaustive queueing model for a finite-capacity redundant multi-server system operating in a multi-phase random environment. The proposed model uniquely integrates real-world factors, including server breakdowns and repairs, waiting servers, synchronous working vacations, and state dependent balking and reneging, into a single queueing model, representing a significant advancement in the field. Using the matrix-analytic method, we establish the steady-state solution and derive key performance metrics. Numerical experiments and sensitivity analyses elucidate the impact of system parameters on performance measures. Additionally, a cost model is formulated, enabling cost optimization analysis using direct search method and Particle Swarm Optimization (PSO) to identify efficient operating configurations. [Display omitted] • Analytical framework for finite capacity redundant multi-server queue in multi-phase environment. • Matrix analytic method calculates steady-state probabilities to assess system performance. measures. • Through sensitivity analysis, key cost variables are identified, aiding decision-making. • Optimizing the cost function by merging Direct Search Method (DSM) and Particle Swarm Optimization (PSO). [ABSTRACT FROM AUTHOR]

• We discuss reliability analysis of a fault tolerant system with imperfect coverage. • We explore system reliability, MTTF and other queuing measures. • The cost-effective ratio is evaluated to upgrade and improve availability. • We obtain optimal control parameters via a direct search approach and PSO. Due to widespread usage in many real time systems, reliability modeling and cost optimization of fault tolerance system have drawn attention of the practitioners. The fault tolerance in these systems can be provided by the support of maintenance and redundant components that help in smooth operation of the system in spite of failure of some active components. This investigation deals with the performance modeling of a fault-tolerant system consisting of a finite number of active (online) and standby components. During the switching from active to standby, the recovery procedure is performed, which may be imperfect. In case of imperfect recovery, the system reboot takes place. The maintenance of all the components is managed by a repairman (server) which is subject to failure. When the server is interrupted for rendering the service, functioning does not get stopped due to the system switch-over from perfect working to working breakdown mode. The system works even when the server is on working vacation and performs repair jobs of the failed components. The machine repair model based on Markovian process is developed to derive the transient probabilities and other performance indices of the fault tolerant system using Laplace transforms and matrix analytical method. Using the direct search strategy and particle swarm optimization, the cost-benefit analysis is done. The optimal design of the control parameters for the fault-tolerant system are presented by framing a cost-effective ratio function. The model is examined computationally by performing the numerical simulation and cost optimization. [ABSTRACT FROM AUTHOR]

Highlights • Consider a standby system with working vacations and retrial of failed components. • We compute the steady-state availability using the matrix-analytic method. • We develop the reliability function and mean-time-to-failure. • Numerical examples are used to conduct sensitivity analysis. Abstract In this paper, we consider a repairable system consisting of M primary components, S spare components, and a repairman. In cases where none of the components in the system is failed, the repairman leaves the system for multiple vacations. During a vacation period, the repairman lowers the repair rate rather than halting repairs together. The system does not include a waiting space. If a failed component finds the repairman free upon arrival, then it immediately occupies the repairman and is being repaired. If a failed component does not find a free repairman upon arrival, then it leaves the service area to join the retrial group (orbit) to try again for a repair. For this system, the matrix-analytic method is used to compute the steady-state availability. We develop the reliability function and mean-time-to-failure (MTTF) based on the Laplace transform technique. Numerical examples are given to assess the effects of system parameters on the system reliability, MTTF, and steady-state availability. [ABSTRACT FROM AUTHOR]

A single-channel finite-buffer queueing model with a general independent input stream of customers, exponential processing times and a working vacation policy is considered. Every time , when the server becomes idle, an exponentially distributed single working vacation period is being initialized, during which the processing is provided with another (slower) rate. After the completion of the vacation period, the service is being continued normally, with the original speed. Using the idea of an embedded Markov chain, the systems of Volterra-type integral equations for the time-dependent queue-size distributions, conditioned by the initial buffer state and related to each other, are built for models beginning the operation in normal and working vacation modes, separately. The solutions of the corresponding systems written for the Laplace transforms are obtained in compact forms using the linear algebraic approach. The numerical illustrative examples are attached as well. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study a MAP/PH/1 queueing model in which the server is subject to taking vacations and offering services at a lower rate during those times. The service is returned to normal rate whenever the vacation gets over or when the queue length hits a specific threshold value. This model is analyzed in steady state using matrix analytic methods. An illustrative numerical example is discussed. [Copyright &y& Elsevier]

Abstract: This paper treats an M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Whenever the system becomes empty at a service completion instant, the server goes for a single working vacation. In the working vacation, a customer is served at a lower speed, and if there are customers in the queue at the instant of a service completion, the server is resumed to a regular busy period with probability p (i.e., the vacation is interrupted) or continues the vacation with probability . Using the matrix analytic method, we obtain the distribution for the stationary queue length at departure epochs. The joint distribution for the stationary queue length and service status at the arbitrary epoch is also obtained by using supplementary variable technique. We also develop a variety of stationary performance measures for this system and give a conditional stochastic decomposition result. Finally, several numerical examples are presented. [Copyright &y& Elsevier]

Abstract: Consider a GI/M/1 queue with start-up period and single working vacation. When the system is in a closed state, an arriving customer leading to a start-up period, after the start-up period, the system becomes a normal service state. And during the working vacation period, if there are customers at a service completion instant, the vacation can be interrupted and the server will come back to the normal working level with probability p (0⩽ p ⩽1) or continue the vacation with probability 1− p. Meanwhile, if there is no customer when a vacation ends, the system is closed. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length at both arrival epochs and arbitrary epochs, the waiting time and sojourn time. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a finite buffer size discrete-time multiple working vacation queue with changeover time. Employing the supplementary variable and embedded Markov chain techniques, we derive the steady state system length distributions at different time epochs. Based on the various system length distributions, the blocking probability, probability mass function of sojourn time and other performance measures along with some numerical examples have been discussed. Then, we use the parabolic method to search the optimum value of the service rate in working vacation period under a given cost structure. [ABSTRACT FROM AUTHOR]

Abstract: This paper analyzes the -policy M/M/1/K queueing system with working vacation and an exponential startup time. The -policy deals with the issue of controlling arrivals to a queueing system, and the server requires a startup time before allowing customers to enter the system. For the queueing systems with working vacation, the server can still provide service to customers rather than completely stop the service during a vacation period. The matrix-analytic method is applied to develop the steady-state probabilities, and then obtain several system characteristics. We construct the expected cost function and formulate an optimization problem to find the minimum cost. The direct search method and Quasi-Newton method are implemented to determine the optimal system capacity , the optimal threshold and the optimal service rates at the minimum cost. A sensitivity analysis is conducted to investigate the effect of changes in the system parameters on the expected cost function. Finally, numerical examples are provided for illustration purpose. [Copyright &y& Elsevier]

Abstract: We consider a discrete-time single-server queueing model where arrivals are governed by a discrete Markovian arrival process (DMAP), which captures both burstiness and correlation in the interarrival times, and the service times and the vacation duration times are assumed to have a general phase-type distributions. The vacation policy is that of a working vacation policy where the server serves the customers at a lower rate during the vacation period as compared to the rate during the normal busy period. Various performance measures of this queueing system like the stationary queue length distribution, waiting time distribution and the distribution of regular busy period are derived. Through numerical experiments, certain insights are presented based on a comparison of the considered model with an equivalent model with independent arrivals, and the effect of the parameters on the performance measures of this model are analyzed. [Copyright &y& Elsevier]

Abstract: This paper deals with a single server working vacation queueing model with multiple types of server breakdowns. In a working vacations queueing model, the server works at a different rate instead of being completely idle during the vacation period; the arrival rate varies according to the server’s status. It is assumed that the server is subject to interruption due to multiple types of breakdowns and is sent immediately for repair. Each type of breakdown requires a finite random number of stages of repair. The life time of the server and the repair time of each phase are assumed to be exponentially distributed. We propose a matrix–geometric approach for computing the stationary queue length distribution. Various performance indices namely the expected length of busy period, the expected length of working vacation period, the mean waiting time and average delay, etc. are established. In order to validate the analytical approach, by taking illustration, we compute numerical results. The sensitivity analysis is also performed to explore the effect of different parameters. [Copyright &y& Elsevier]

Abstract: This paper studies the M/M/1 machine repair problem with working vacation in which the server works with different repair rates rather than completely terminating the repair during a vacation period. We assume that the server begins the working vacation when the system is empty. The failure times, repair times, and vacation times are all assumed to be exponentially distributed. We use the MAPLE software to compute steady-state probabilities and several system performance measures. A cost model is derived to determine the optimal values of the number of operating machines and two different repair rates simultaneously, and maintain the system availability at a certain level. We use the direct search method and Newton’s method for unconstrained optimization to repeatedly find the global minimum value until the system availability constraint is satisfied. Some numerical examples are provided to illustrate Newton’s method. [Copyright &y& Elsevier]

Abstract: In this paper, we study a discrete time Geom/Geom/1 queue with multiple working vacations. Using the quasi birth and death chain and matrix-geometric solution method, we give distributions for the number of customers in system and the waiting time of a customer and their stochastic decomposition structures, and obtain distributions of the additional number of customers and additional delay. Furthermore, we derive the formulae of expected regular busy period and expected busy cycle. Finally, by numerical examples, we analyze the effect of the parameters on the expected queue length and sojourn time. [Copyright &y& Elsevier]