Beta Innovation Center, BIC, operates a technical support center for its software clients. Now BIC has newsoftware, security protection system. However, the clients are not familiar with the implementation. If the clientshave installation and encounter problems with security protection system, they may telephone the technicalsupport center and obtain the consultation. Currently, BIC employs four staffs to give the consultation to theclients who call to contact the center. If all staffs are busy when the client call arrives, the client will hear therecorded voices stating that all staffs are serving the other clients. The call of the first client in waiting line will be connected to any staff who is first available. The client calls are distributed with Poisson distribution, withan average of $10$ calls per hour. On average, it takes $12$ minutes for the staff to answer the questions for eachclient. The service time follows an exponential distribution$\mathrm{.}3\mathrm{.}1)$ In the current system, What is the probability that there is no client in the system $(2$ points$)3\mathrm{.}2)$ On average, how many clients are there in the technical support system? On average, how long does ittake for each client to wait before start talking to the staff? $(4$ points$)3\mathrm{.}3)$ What is the probability that a client will have to wait before start talking to a staff? $(4$ points$)3\mathrm{.}4)$ Staffs at CIC are paid $250$ baht per hour and there is opportunity cost of $300$ baht per hour for a clientwhen he$/$she must wait in the waiting line. Determine the total cost of this system. $(2$ points$)3\mathrm{.}5)$ The customer service department recently receives several letter from the clients complaining about thedifficulty to get the service. The service manager and the process manager agree to improve the service. Aftercollecting historical information, $40$ percent of clients who would like to ask about the installation. The servicetime is $15$ minutes in average and standard deviation is $4$ minutes. The service time follows general distribution.BIC arranges the new telephone number for these clients and assigns one of four staffs to serve the InstallationLine for these clients. The rest of the clients would ask about the troubles in using the software which takes $10$minutes and follows exponential distribution. The remaining staffs will take care of these clients with the currenttelephone system and rename the service line as The Trouble Shooting Line. In other words, the system has$3$ staffs and there is only $1$ waiting line for this Trouble Shooting Line. On Average, what is the total number ofclients in the new system? $(6$ points$)3\mathrm{.}6)$ Compare the current system $($from question $3\mathrm{.}4)$ and the new system $($from question $3\mathrm{.}5)$ using the totalcost. Should the manager switch the operation to the new system? $(4$ points$)$