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Abstract In many actual issues, such as call center and cloud service computing center, retrial systems have been widely applied to derive the performance measures. This study takes up a multi-server retrial system with partial servers synchronous vacation and customer geometric loss. An arriving customer finding no servers available either enters a retrial group according to first-in-first-serve discipline or balks. In the retrial group, only the first customer can retry for his service. A customer from the retrial group finding all servers unavailable decides either balks or reenters the retrial group. Partial servers may be on vacation simultaneously when they become idle at the moment of service completion and system conditions satisfy the administrative policy. This retrial system is analyzed as a quasi-birth-death process, and some general descriptors are developed. Optimization problems, including single-objective and bi-objective, are constructed to help the manager’s decision-making. Numerical results are displayed tabularly and graphically. A potential application in which the proposed model can be applied is provided. [ABSTRACT FROM AUTHOR]

Motivated by some real stochastic service systems, we study a feedback retrial queue with customer balking and unreliable servers (servers subject to breakdowns). A customer will either join a retrial orbit or balk if all servers are busy upon his arrival. After a service completion, the customer either leave the system or enter the retrial group again for additional service. The customer who is interrupted by the server failure will be served by another server if available; otherwise, he will be contacted later and treated as leaving the system. We use a quasi-birth-and-death process to analyze this system with almost all possible uncertain factors and derive major performance measures of the system. We also develop a cost function determining the optimal parameter settings of the system under the stability condition. Probabilistic Global Search Lausanne approach is employed in solving the optimization issue. The effects of parameters on performance measures of the system are examined numerically. Finally, we discuss the application of the model in the telephone medical consulting service system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the availability of two different series system configurations with imperfect coverage, where an unreliable server is responsible for maintaining the system. The primary and standby components are included in both series system configurations. The time-to-failure of the primary and standby components are exponentially distributed. When the server is repairing, it is subject to random breakdown. The time-to-breakdown of the server is assumed to have exponential distribution. The standby components will substitute for the primary components whenever the primary components fail. The recovery from standby to active is imperfect. The coverage factors for primary component failure and for standby component failure are assumed to be the same as c. The repair time of the failed components and the repair time of the breakdown server are generally distributed. Further, we implement a practical model with two different repairable system configurations. By supplementary variable technique, we derive the steady-state availability of these two different series system configurations. Finally, we compare the cost/benefit ratio between two configurations and the availability in the different pairs of the component's repair time and server's repair time distributions. [ABSTRACT FROM AUTHOR]

This investigation is concerned with an M/G/1 machine interference problem with imperfect switchover of standbys, in which an unreliable repairman maintains a group of machines. An unreliable repairman means that the repairman is typically subject to unpredictable breakdowns. The time between two consecutive breakdowns follows an exponential distribution, and recovery time of the unreliable repairman follows a general distribution. The lifetime of operating/standby machines and the repair time of failed machines are exponentially and generally distributed, respectively. Using the method of supplementary variable, the stationary probability distribution is obtained. We develop some performance measures and system reliability indices. Furthermore, the cost effectiveness maximization is also discussed. Finally, a cost model is proposed to find the optimal numbers of operating and standby substations, which minimize the average cost per unit time. [ABSTRACT FROM AUTHOR]

We studied the inferences of an availability system with reboot delay and standby switching failures in which the system consisted of two operating units and one warm standby. The system was studied under the assumption that the time-to-failure and the time-to-repair were assumed to follow an exponential and a general distribution. The reboot times are assumed to be exponentially distributed with parameter β. We constructed a consistent and asymptotically normal estimator of availability for such a repairable system. Based on this estimator, interval estimation and testing hypothesis were developed by using log it transformation. To implement the simulation inference for the system availability, we adopted two repair-time distributions—namely, lognormal and Weibull; and three types of Weibull distributions—characterized by their shape parameters—were considered. Finally, appropriate tables and figures of all simulation results have been included. [ABSTRACT FROM AUTHOR]

We consider the distributions of operating characteristics of an M[x]/G/1 queue under vacation policies, where the first customer of each busy period receives an exceptional service. When all the customers are served in the system exhaustively, the server deactivates and operates one of two vacation policies: (1) multiple vacation policy and (2) single vacation policy. We develop the performance measures for both systems. Finally, some numerical illustrations are also given. These two vacation models have potential applications in day-to-day life, such as post offices, banks, hospitals, etc. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the steady-state availability of a repairable system with standby switching failure. The repairable system configuration includes the primary and standby components, where an unreliable server is responsible to repair or monitor the failed ones. The time-to-failure and time-to-repair of the components follow exponential and general distribution, respectively. The server subjects to active breakdown when it is repairing. The time-to-breakdown of the server is also assumed to be exponentially distributed. When the primary components fail, the standby components replace the primary components successfully with probability 1− q . The repair time of the failed components and the repair time of the breakdown server are generally distributed. Further, we frame a practical model with three different repairable system configurations. We use supplementary variable method and integro-differential equations to obtain the steady-state availability of these three different repairable system configurations. Finally, we compare the cost/benefit ratio among the three configurations given the distribution parameters, and to the cost of the primary and standby components. [ABSTRACT FROM AUTHOR]

In this paper, we present a detailed analysis of a multi-server retrial queue with Bernoulli feedback, where the servers are subject to starting failures. Upon completion of a service, a customer would decide either to leave the system with probabilitypor to join the retrial orbit again for another service with complementary probability 1−p. We analyse this queueing system as a quasi-birth–death process. Specifically, the equilibrium condition of the system is given for the existence of the steady-state analysis. Applying the matrix-geometric method, the formulae for computing the rate matrix and stationary probabilities are obtained. We further develop the matrix-form expressions for various system performance measures. A cost model is constructed to determine the optimal number of servers, the optimal mean service rate and the optimal mean repair rate subject to the stability condition. Finally, we give a practical example to illustrate the potential applicability of this model. [ABSTRACT FROM PUBLISHER]

This paper considers the optimal management problem of a finite-capacity M/M/1/K queueing system with-policy in which the server may break down while working. When the number of customers reaches its capacityK, no further arriving customers are allowed to enter the system. Customers are allowed to system with probabilitypor the customers are still unable to enter the system with probability 1 − pas the queue length decreases to a certain threshold valueF. By applying the birth-and-death process, some important performance measures are derived. A cost model, developed to determine the optimal continuous and discrete control parameters for the-policy at a minimum cost, and sensitivity analysis are also studied. [ABSTRACT FROM PUBLISHER]

We consider a machine repair problem with queue-dependent heterogeneous repairmen. The stationary probability distribution of the number of failed machines in the system is derived. A profit model is developed to determine the optimal values of threshold to assign/add one more repairman for increasing the total service rate where the service rate is assumed adjustable. The probabilistic global search Lausanne (PGSL) method is employed to find an initial trial solution. Based on this initial solution, a direct search method is applied to obtain the optimal values of the threshold and, then, the Quasi-Newton method is implemented to adjust the corresponding service rates. Some numerical experiments are performed to justify the efficiency of the optimum approach. [ABSTRACT FROM AUTHOR]

This paper deals with the ( p , N )-policy M/G/1 queue with an unreliable server and single vacation. Immediately after all of the customers in the system are served, the server takes single vacation. As soon as N customers are accumulated in the queue, the server is activated for services with probability p or deactivated with probability (1 − p ). When the server returns from vacation and the system size exceeds N , the server begins serving the waiting customers. If the number of customers waiting in the queue is less than N when the server returns from vacation, he waits in the system until the system size reaches or exceeds N . It is assumed that the server is subject to break down according to a Poisson process and the repair time obeys a general distribution. This paper derived the system size distribution for the system described above at a stationary point of time. Various system characteristics were also developed. We then constructed a total expected cost function per unit time and applied the Tabu search method to find the minimum cost. Some numerical results are also given for illustrative purposes. [ABSTRACT FROM AUTHOR]

In this paper, we examine a repairable system with imperfect coverage, reboot delay and one repair facility. The failure times and repair times of failed units are assumed to be exponential and general distributions, respectively. As a unit fails, it can be immediately detected, located and replaced with a coverage probability c by a standby, if one is available. We use a recursive method and the supplementary variable technique to develop the steady-state probabilities of down units at an arbitrary epoch. Then, an efficient algorithm is constructed to compute the steady-state availability. We adopt two repair time distributions, namely, exponential and gamma, to illustrate the method. Finally, we also perform a sensitivity analysis of the steady-state availability with respect to the system parameters for various repair time distributions. [ABSTRACT FROM AUTHOR]

Analysis of an unreliable-server retrial queue with customer's feedback and impatience is presented. Truncated classical and constant retrial policies are taken into account. This system is analyzed as a process of quasi-birth-and-death (QBD). The quasi-progression algorithm is applied to compute the rate matrix of QBD model. A recursive solver algorithm for computing the stationary probabilities is also developed. To make the investigated system viable economically, a cost function is developed to decide the optimum values of servers, mean service rate and mean repair rate. Quasi-Newton method, pattern search method and Nelder–Mead simplex direct search method are employed to implement the optimization tasks. Under optimum operating conditions, numerical results are provided for a comparison of retrial policies. We also give a potential application to illustrate the system's applicability. [ABSTRACT FROM AUTHOR]

In this paper, we consider a single-server retrial queue with constant retrial rate and batch arrivals, in which the unreliable server has the option to take an additional vacation after the first essential vacation. Customers arriving at the system according to a compound Poisson process are served immediately as long as the server is available; otherwise, they enter a retrial orbit and form a single queue. When the orbit becomes empty, the server leaves for the first essential vacation, after which he may either remain idle within the system or take one of J optional vacations. When the server is busy, he is subject to random breakdowns and repairs. It is assumed that the service times, repair times and customer retrial times are arbitrarily distributed. The implementation of the supplementary variable technique makes it possible to derive the probability generating functions of the system size distribution at a random epoch. We also develop a variety of system performance measures as well as two reliability indices. Finally, we deal with the problem of cost optimization and provide a number of numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider a retrial and repairable multi-component system with mixed warm and cold standby components. It is assumed that the failure times of primary (operating) and warm standby components follow exponential distributions. When a component fails, it is sent to a service station with a single server (repairman) and no waiting space. The failed component is repaired if the server is idle and it has to enter an orbit if the server is busy. The failed component in the orbit will try to get the repair service again after an exponentially distributed random time period. The repair time also has an exponential distribution. The mean time-to-failure, MTTF , and the steady-state availability, A T (∞), are derived in this retrial and repairable system. Using a numerical example, we compare the systems with and without retrials in terms of the cost/benefit ratios. Sensitivity analysis for the mean time-to-failure and the steady-state availability are investigated as well. [ABSTRACT FROM AUTHOR]

• A multi-criteria replacement policy under cumulative damage models is discussed. • The mean cist rate is formulated. • The optimal replacement policy is presented theoretically. • Numerical examples are provided. This paper investigates an extended replacement policy for a two-unit system subject to shocks and cumulative damage. As a shock occurs, the system suffers two types of effects: type I shock-effect is rectified by a minimal repair and type II shock-effect is removed by a corrective replacement. The probability of type II shock depends on the number of shocks from the last replacement. Each type I shock causes a minor failure of unit 1 and some damage to unit 2. These damages to unit 2 are additive and the system fails when the total damage exceeds a critical level L. A replacement policy (T, m, l) is considered in which the system is preventively replaced at age T or at the m th type I shock or at the time which the total damage to unit 2 exceeds a pre-specified level l ; and is correctively replaced either at the first type II shock or when the total damage to unit 2 exceeds a failure level L , whichever occurs first. The optimal policy that minimizes the expected cost rate is determined analytically. The proposed model extends several existing results. Our replacement policy is more general, flexible, and applicable in the real situations. [ABSTRACT FROM AUTHOR]

• A multi-criteria replacement policy under cumulative damage models is discussed. • The concept of replacement last is analyzed in detail. • The optimal replacement policy is presented theoretically. • An algorithm for searching the optimal replacement policy is presented. • Numerical examples are provided. −In this study, we discuss a preventive replacement policy under cumulative damage models that include the concept of replacement last. The system is subject to two types of shocks (represented by type 1 shock and type 2 shock) whose probabilities depend on age. Each type 1 shock leads to an amount of additive damage to the system and also causes a minor failure of the system, which can be rectified by a minimal repair. The system will have a catastrophic failure as long as the accumulated damage exceeds a failure level threshold L. Meanwhile, type 2 shock will cause catastrophic failure of the system. The system is replaced preventively at age τ or the n -th type 1 shock or the time instant when the accumulated damage exceeds a pre-specified level k (but less than L), whichever occurs last, and is replaced correctively at the first type 2 shock or the time which the accumulated damage exceeds L , whichever occurs first. The average cost rate is developed and the optimal replacement policy is discussed analytically and computed numerically. [ABSTRACT FROM AUTHOR]