At a Dental Office, patients come in at the rate of 20 per hour and, on average, are processed at the same rate. Patients wait in a queue till they are called up to the counter for registration. After registration, patients are taken to a room where they are assigned a chair where they wait till a hygienist is ready to see them. Once a hygienist sees the patient, they identify them as a routine cleaning or they identify a problem where a dentist is needed to perform major treatments. On average, it is found that there are 30 patients waiting for registration and another 20 waiting to be seen by a hygienist.
Now in dental office one X-Ray machine was introduced which allows separation of the dental rooms into two treatment areas, one for routine cleanings and another for major treatments. A dentist’s assistant sees a patient after registration, x-rays them, and assigns the patient to one of the two tracks. Patients for routine cleaning are seen by a hygienist who either cleans their teeth and releases them or determines that the dentist’s assistant has made a mistake and the patient really needs major treatment. In this case the patient is moved to the major treatment track. Patients for major treatment are seen by a hygienist who either gets a dentist for further treatment or determines they really are routine cleanings and releases them. It is determined, using the x-ray system, that there are, on average, 20 people waiting to be registered, 5 waiting to be seen by the dentist’s assistant, 10 waiting to be seen by a hygienist in the major treatment area, and 10 waiting to be seen by a hygienist in the routine cleaning area. On average, 75% of the patients are classified by the dentist’s assistant as needing routine cleanings.
Now main problem am facing is in finding that on average, how long do patients that have major treatment spend waiting in the dental office after the introduction of the x-ray system?
Note : We can Assume the dentist’s assistant makes no mistakes in identifying a patient as needing routine cleaning or major treatment.
Your argumentation is good i think, but there the problem says that on average there are 30 patients waiting for registration and another 20 waiting to be seen by a hygienist, and because the patient after the registration wait for the hygienist i guess it is a part of the process, so instead $\frac{30}{20}$ i think you must put $\frac{30+20}{20}$, and you have 2.5, so two hours and half of waiting on average.