## 1 Introduction  
In queueing theory, retrial queues have been intensive research hotspots for a long time. These retrial queues have been widely applied in analog telephone systems, call centers, computer systems, and many problems in daily life. For an overview of retrial queues, refer to the surveys [1, 2, 3, 4]. In most studies on retrial queues, it is generally assumed that servers only serve incoming calls. However, in various service systems, especially systems with human servers, such as call centers, the servers not only need to service incoming calls but also do some private things in their free time. Researchers can read [5, 6, 7, 8, 9] to learn more about the two-way communication retrial queue approach. In real life, for a period of time, servers may be inaccessible for various reasons. The server's absence could mean that the server performs some other work, undergoes maintenance checks, or simply takes a break. General models in vacation queues can be found in [10, 11, 12, 13, 14]. A vacation occurs when a server is unable to serve the next customer for some reason. Allowing servers to take time off can make the queues more accurate and adaptable in situations where people are waiting in line. There could be a loss of profits if jobs accumulate during the vacation. To avoid this, servers will serve customers at different speeds. Servers give service at a reduced speed during the vacation period, rather than stopping service completely. Such a model plays an important role in some practical systems, such as network service, internet service, data transfer service, and mail service. In recent years, many authors have conducted extensive research on the retrial queues with working vacations and have achieved rich theoretical results, such as [15; 16; 17; 18; 19; 20]. In practice, multi-server vacation queues, in which servers go on vacation together (called synchronous vacation) or individually (asynchronous vacations) work better than single-server queues. But, due to their complexity, analyses of these systems appear to be limited. [PERSON] and [PERSON] [21] analyzed a multi-server queue with a variant working vacation. [PERSON] et al. [22] analyzed the synchronous multiple and single vacation policies in a multi-server queue. [PERSON] et al. [23] investigated a finite-capacity Markovian queue with Bernoulli feedback and impatience under a synchronous multiple-vacation policy. A multi-server repairable machine system with Bernoulli feedback, customer impatience, and multiple synchronous working vacations was proposed by [24]. [PERSON] et al. [25], which considered a multi-server retrial model with a synchronous working vacation, vacation interruption, and impatience customers under a constant retrial policy. The authors extensively investigated retrial queues where customers are removed from the system due to a disaster or catastrophe. Modeling specialized systems such as call centers, automated teller machines, and wireless communication networks may require different types of failures. Disasters are known to be caused by negative arrivals. The presence of negative customers can cause dangers to the system. The arrival of a negative customer induces one or more positive customers to leave the system immediately. It will also cause the server to crash for a short period of time and the service channel to fail. Once a server fails, it is sent for repair, and the repair period begins immediately. The repaired server is considered as good as a new one. Negative arrivals can be seen as viruses, suppressed signals, operating errors, or system and server disasters in neural and computer communication networks. A detailed overview of negative customers is presented in [26; 27; 28; 29; 30], which discussed the retrial queues with the concept of negative arrivals. In many real-life situations, it is common for servers to suffer lengthy and unpredictable failures while serving customers. This will cause the server to be unavailable for some time until it is repaired.