This work investigates a two-way communication retrial queue with synchronous working vacation and a constant retrial policy. During the idle time, a server makes an outgoing call after a random length. The service time of the incoming call and outgoing call obeys exponential distribution with different rates. If the incoming call finds all servers to be unavailable, it may or may not enter orbit. All servers immediately go on vacation simultaneously as soon as they find an empty system after the service finishes. During vacation, the servers can provide a service to those incoming calls, but this is at a lower-speed rate. The stationary probability distribution and the ergodic condition are obtained utilizing the matrix geometric technique. Some system characteristics are developed. Using MATLAB software, the variation in average orbit length, idle ratio, and the average number of servers in different server states is plotted for different values of the incoming/outgoing call rate and retrial rate. We further propose a multi-objective optimization model from which the optimal rate of outgoing calls and optimal vacation rate are explicitly obtained.

This research analyzes a multi-server retrial queue with two types of calls: working vacation and working breakdown. The incoming call may enter the retrial queue and attempt to seize a server after a random delay if all the servers are unavailable upon arrival. In its idle time, the server makes outgoing calls. All the servers take a synchronous working vacation when the system empties after regular service. The system may fail at any time due to disasters, forcing all the calls within the service area to leave the system and causing all the main servers to fail. When the main servers fail, the repair process begins immediately. The standby servers serve arriving customers at a lower level of service during the working breakdown or working vacation. For this model, we derive an explicit expression for the stationary distribution with the help of the quasi-birth-and-death process and the matrix geometric method. Further, the formulas of various system performance indices are developed. An application example is given and several numerical experiments are performed to verify the analytical results. We also perform the comparative analysis of retrial queues with/without two-way communication and two-way communication retrial queues with/without disasters. The results reveal that the proper consideration of outgoing calls to the server can reduce the average time spent in the buffer. Furthermore, a more reliable server reduces the server idle rate.

A bivariate replacement policy for a two-unit system under the shock-effect is studied. Unit 1 is subject to one of the two types of shocks. A type 1 shock causes a minor failure of unit 1, while a type 2 shock makes a catastrophic failure of unit 1. The probabilities of the shock types depend on the number of shocks since the last replacement. Each minor failure of unit 1 brings some cumulative damage to unit 2. When the cumulative damage of unit 2 reaches L, unit 2 fails, which simultaneously causes the failure of unit 1, resulting in a catastrophic failure of system. Moreover, unit 2 with cumulative damage x has a minor failure with probability $\omega \left(x\right)$ at unit 1 failure instant, and this failure is removed by a minimal repair. A more general replacement policy is considered, in which the system is replaced when the cumulative damage to unit 2 has reached l (≤ L), or the nth type 1 shock occurs, or when type 2 shock occurs or the accumulative damage to unit 2 exceeds L, whichever comes first. The expression for average cost rate is developed and the corresponding optimal policy is determined analytically and numerically.

Replacement may occur during the mission in accordance with traditional replacement policies. However, this approach may not be feasible for deactivating a mission-oriented system when a mission is initiated to carry out maintenance operations. In this context, we introduce replacement policies for \(k\)-out-of-\(n\) systems considering mission durations. In reliability engineering, \(k\)-out-of-\(n\) systems are a key class of configurations, consisting of \(n\) independent components. These systems remain operational as long as at least k components are continuously functional, with overall performance and reliability depending on the condition that a minimum of \(k\) components are active. Each unit of the system experiences two types of failures. When a type I failure occurs, a failed unit undergoes minimal repair, whereas in the case of a type II failure, the unit remains idle. The system experiences complete failure when \(n - k + 1\) units are idle, and requires corrective replacement. Initially, a basic replacement policy for mission-critical \(k\)-out-of-\(n\) systems is presented, along with a discussion on the determination of the optimal number of units. Next, we incorporate the notations of “replacement first” and “replacement last” separately. “Replacement first” refers to preventive replacement occurring at the scheduled time or when the mission is completed, whichever happens first. “Replacement last” means that preventive replacement takes place when either of the above events occurs, whichever happens last. Subsequently, the optimal policy for each model is analyzed analytically. A numerical example is then provided to illustrate the findings. Finally, the proposed policies are applied to the maintaince of the automated system for the remote monitoring of underwater sections of a gas pipeline to show how to choose a better policy in three scenarios.

Retrial queues are used extensively to model many practical problems in service systems, call centers, data centers, and computer network systems. The average waiting time is the main observable characteristic of the retrial queues. Long queues may cause negative impacts such as waste of manpower and unnecessary crowding leading to suffocation, and can even cause trouble for customers and institutions. Applying control chart technology can help managers analyze customers’ waiting times to improve the effective performance of service and attention. This paper pioneers the developing and detailed study of a waiting time control chart for a retrial queue with general service times. Two waiting time control charts, the Shewhart control chart, and a control chart using the weighted variance method are constructed in this paper. We present three cases for the Shewhart control chart in which the service time obeys special distributions, such as exponential, Erlang, and hyper-exponential distributions. The case of an exponentially distributed service time is also presented for the control chart using the weighted variance method. Based on the numerical simulations conducted herein, managers can better monitor and analyze the customers’ waiting times for their service systems and take preventive measures.

This paper investigates a multi-server retrial machine repair system that incorporates orbital search and a synchronous vacation mechanism. The system features standby units and examines two potential vacation scenarios for servers, reflecting real-world situations, such as technical staff in teaching hospitals taking periodic administrative or training vacations. We formulate a mathematical model using birth-and-death processes to establish the governing equation and propose a recursive matrix method to systematically derive the steady-state probabilities. Performance measures, including system availability, the expected number of failed units in the orbit, and expected waiting times, are derived. To address the conflicting objectives of minimizing total operating costs, minimizing expected waiting times, and maximizing system availability, we construct a tri-objective optimization problem. By implementing a multi-objective genetic algorithm, we identify a set of Pareto-optimal frontiers and reveal the explicit financial and operational trade-offs among these competing criteria. Numerical experiments and sensitivity analyses demonstrate that enhancing the automated retrial rate and managing the emergency repair rate are most critical to minimizing system downtime. Furthermore, the joint optimization of server capacities and vacation schedules effectively eliminates operational redundancy, showing that near-perfect equipment availability (up to 0.999) can be achieved with only marginal increases in cost. This research provides administrators with a robust decision-making framework to optimize technical resource management while ensuring near-perfect equipment availability in real-world environments.