## 5 Application Example  
This work can be used to model a health helpline center. The health helpline (a centralized information system) provides services to people through telecommunications. It covers a wide range of health-related problems such as health consultations, medical advice, etc. People who need health advice can use this helpline. Calls from potential customers requesting service are considered arrivals. Arrival calls are answered by the operator, who is responsible for establishing communication between the server and the customer. The operator needs to record the order of the calls (corresponding to a \"buffer\"). If all the lines are busy when arriving, the arriving call has the option of leaving a message, waiting for service, and calling again after a certain amount of time (called retrial), or leaving the system (called balking). Otherwise, a chief physician (main server) or a physician assistant (standby server) serves the customer immediately. Physician assistants provide services to customers at a lower service rate only while the chief physicians are on vacation. The system may fail due to a disaster such as no network coverage or a low telephone signal status. If a disaster occurs during the customers' consultation time, the customers being consulted will end the call. As soon as the signal strength reaches its full capacity (repair), the system provides service like new again. To avoid frequent failures, it is strongly recommended to perform regular maintenance of the service facility. The process of taking an idle server out of service for maintenance is called a vacation. If an entire service facility undergoes maintenance at the same time, we refer to this phenomenon as synchronized vacations. During maintenance, physician assistants will provide services to customers, if any. On the other hand, to minimize the chief physician's free time, the operator will call the customer.  
One scenario of the health helpline service centers is given as follows. The health helpline service, consisting of four chief physicians (main servers) (i.e., \(c=4\)), can be used by people who need health advice. Suppose that the number of physician assistants (standby servers) is the same as chief physicians. Calls from customers who may request service are considered to arrive at rate \(\lambda=4\). Arriving calls can be served immediately if at least one chief physician/physician assistant is available and the service rate is \(\mu_{1}=3\); otherwise, they have two options. First, the probability of an arriving call leaving without any attempt depends on the server states (\(1-b_{b}=0.2\) during a regular busy period; \(1-b_{r}=0.4\) during a repair period; and \(1-b_{v}=0.4\) during a working vacation period). Second, it may join the buffer and must retry again, with a retrial rate of \(\
u=10\). Disasters can occur at a rate of \(\delta=0.1\) due to a lack of network coverage or poor telephone reception. We assume that when the signal strength is full at a rate \(\gamma=1\), the system will provide service like new again. In order to avoid frequent failures, regular maintenance of service facility is performed at a rate of \(\eta=0.3\). During the maintenance, the physician assistant will serve the customers at a rate of \(\mu_{w}=1\). On the other hand, in order to minimize the chief physician's free time, the operator will call the customer at a rate of \(\alpha=3\), and the service rate is \(\mu_{2}=1.5\).