Reliability and availability analysis of standby systems with working vacations and retrial of failed components
Introduction
Retrial queues have attracted considerable attention, due to their wide applicability in telephone switching systems, computer systems, and telecommunication networks. Retrial queues are characterized by the feature that arriving customers who find the server busy may enter a retrial group (orbit) for a randomly determined duration of time to try their requests again. For more detail on retrial queues, please refer to [37], [8], [2], [1], [27], [12], and the references therein. More recently, Chang et al. [3] analyzed a multi-server retrial queue with server breakdowns, feedback, and impatient customers.
Comprehensive reviews of vacation queues can be found in [7], [29], [11], [31]. For a cold standby repairable system with multiple vacations, Yuan and Xu [41] determined an optimal replacement policy to minimize the long-run expected cost per unit time. Yuan [40] employed the discrete transforms and Runge–Kutta methods to analyze a k-out-of-n:G repairable system with redundant dependency and repairman's multiple vacations. Yu et al. [39] considered a geometric process repair model with spare device procurement and one repairman who may take multiple vacations. They not only derived steady-state reliability measures, but also found the optimal maintenance policy. Servi and Finn [26] introduced a variant vacation policy referred to as a working vacation for use in the analysis of a WDM optical access network. In [26], they examined M/M/1 queues with working vacations, in which the service rate of the server is varied rather than completely stopping during vacation periods. In recent years, a growing number of researchers have focused on queueing models with working vacations, including Li and Tian [19], Wang et al. [32], Zhang and Hou [42], Gao and Yao [9], Yang and Wu [35], Luo et al. [25], and Sun et al. [28]. Lin and Ke [22] performed stationary analysis of the M/M/R queue with single working vacation. They also were concerned with the problem of minimizing the expected cost per unit time. Do [4] analyzed the M/M/1 retrial queue with working vacations using the quasi-birth-and-death process. Li et al. [20] considered the discrete-time Geo/Geo/1 retrial queue with working vacations, where the server returns from a vacation at a service completion epoch when there is at least one customer in the orbit. Tao et al. [30] obtained the stationary probability distribution for the M/M/1 retrial queue with collisions of customers, working vacations and vacation interruption under N-policy. Liu and Song [24] applied the matrix-geometric method to investigate the discrete-time Geo/Geo/1 retrial queue with working vacations, where retrial customers are assumed to be non-persistent. Do et al. [5] obtained the stationary probability distribution for the M/M/1 retrial queue with working vacations and negative customers. Liu et al. [23] used the matrix-analytic method to study a coldstandby repairable system with multiple working vacations and vacation interruption, where the vacation process of the repairman is described by a Markovian arrival process. Li et al. [21] applied the probability generating function method to find the stationary probability distribution for the M/M/1 retrial queue with working vacations and vacation interruption under classical retrial policy. Recently, Do et al. [6] have studied equilibrium and socially optimal balking strategies in an M/M/1 retrial queue with working vacations and constant retrial policy.
Reliability and availability play important roles in repairable systems. These have been explored extensively in the cloud computing systems, manufacturing systems, computer and communication networks, and industrial systems. Standby redundancy is a common technique used to improve system reliability and availability. Standbys are generally divided into three categories in accordance with their failure rates: cold standby, hot standby, and warm standby. Several studies have addressed repairable systems with standbys (see, e.g., [17], [33], [36]). Yen et al. [38] analyzed the system reliability of a repairable system with warm standbys and a control policy, in which the service station is unreliable and subject to working breakdowns. Kuo and Ke [15] and Lee [18] studied the steady-state availability of a repairable system with an unreliable server, where standbys are subject to switching failures. More recently, Ke et al. [10] employed the supplementary variable method to investigate an M/G/1 machine repair model with an unreliable repairman and standby switching failures. However, relatively little work has gone into repairable systems with retrials. Krishnamoorthy and Ushakumari [14] first introduced retrials within a repairable system obtaining system reliability from a k-out-of-n system with retrial of failed units. Krishnamoorthy et al. [13] provided detailed analysis of a k-out-of-n system, in which the server deals with external customers in a retrial queue. Kuo et al. [16] investigated a repairable system with mixed standby components and retrial of failed components. Our paper differs from that of Kuo et al. [16], and the main differences are that (1) our system does not consider mixed standby components; (2) the retrial policy in our system is the constant retrial policy; (3) the retrial policy in [16] is the classical retrial policy; and (4) Kuo et al. [16] are not concerned with working vacations. Recently, Yang and Chang [34] presented a mathematical model of a machine repair problem with constant retrial policy to evaluate the machine availability and operative efficiency.
Queueing systems with retrial customers and working vacations have been widely studied; however, no reports have been published from the perspective of reliability. In this paper, we focus on reliability-based measures of standby systems with retrials and working vacations. The paper is organized as follows. In the following section, we present a detailed description of the proposed repairable system. In Section 3, we conduct the availability analysis and develop the steady-state availability using probability analysis. In Section 4, we derive the reliability function and mean-time-to-failure (MTTF) using the Laplace transforms. In Section 5, we provide numerical results illustrating the effects of various system parameters on the reliability indices of the system and steady-state availability. In Section 6, we draw conclusions and outline suggestions for future work.
Section snippets
Assumptions
We consider a repairable system with a single repairman and identical components, where M components are operating independently and in parallel, and S components are spare. The system and components may fail, and the failed component is repairable. We present a schematic illustration of the proposed repairable system in Fig. 1 and make the following assumptions:
- •
At time , all components are new and operational. Each of the operating components and each of the spare components fail in
Availability analysis
We attempt to evaluate steady-state availability of the repairable system using Markov model. The matrix-analytic method is applied for computing the stationary distribution in a recursive manner. Let N(t) be the number of failed components in orbit at time t and Y(t) denote the state of the repairman at time t, which is defined as
Reliability analysis
We present the reliability function and MTTF of the repairable system with standbys, working vacations, and retrial of failed components. The state-transition-rate diagram for the reliability model with standbys, working vacations, and retrial of failed components is shown in Fig. 3.
Numerical examples
In this section, we present numerical examples to explore the effects of various system parameters on RY(t), MTTF, and the steady-state availability Av. We fixed the number of operating components at and the minimum number of operating components at . The system parameters are set as follows: , , , , , , and . The numerical results in Fig. 4 illustrate the effects of various system parameters on RY(t). In Fig. 4(a) and (b), we can see that system
Conclusions
In this paper, we provided the reliability and availability analysis of a repairable system with standbys, working vacations, and retrial of failed components. The matrix-analytic method was used to compute steady-state availability. We applied the Laplace transforms to obtain the reliability function and MTTF. Sensitivity analysis revealed that system reliability, MTTF, and steady-state availability can be improved significantly by increasing the number of spare components and repair rates of
Acknowledgment
This research was partially supported by the Ministry of Science and Technology of Taiwan under grant no. MOST 105-2221-E-141-005.
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