$)(25$ points$)-$ develop a model

The pharmacy at Arnold Palmer Hospital receives, on average, $14$ requests for prescriptions filling per hour, according to a Poisson process.

These prescriptions are generated by doctors all over the hospital, and they use the Medications Information system to prepare the prescription. Once the prescription is completed by the doctor, it is placed in a virtual queue in the information system, which is then access by the $1$ pharmacist working the pharmacy.

It takes the pharmacist a mean service time of $4$ minutes to fill each prescription, according to an exponential distribution, which means the rate of service is $15$ prescriptions per hour.

The hospital is in the process of adding $30$ clinics, which could lead to an increase in the usage of the pharmacy services. Doctors at these new clinics will ask patients if they want their prescriptions fill at the hospital or not. It has been estimated that the rate of prescriptions submitted will increase to $25$ per hour; thus, the hospital is considering hiring a second pharmacist.

a$)(10$ points$)$ For the current situation, identify the system using Kendall's notation, and compute probability of the pharmacist being idle, the average number of prescription waiting to be filled, How long it takes to fill a prescription from the moment they are placed in the information system, and utilization of the pharmacist.

b$)(10$ points$)$ For the situation where the new clinics are active, identify the system using Kendall's notation, and compute probability of the pharmacist being idle, the average number of prescription waiting to be filled, How long it takes to fill a prescription from the moment they are placed in the information system, and utilization of the pharmacist.

c$)(5$ points$)$ Compare both scenarios, and discuss whether the hospital should offer the pharmacy services for the new clinics. BE THOROUGH in your reasoning.