The Tooth Care Hospital provides free dental service to the patients on every Saturday morning. There are 3 dentists on duty, who are equally qualified and experienced. It takes on an average 20 minutes for a patient to get treatment and the actual time taken is known to vary approximately exponentially around this average. The patients arrive according to the Poisson distribution with an average of 6 per hour. The officer of the hospital wants to investigate the following: 
 i) The expected number of patients in the queue.
 ii) The average time that a patient spends at the clinic.
 iii) The average percentage idle time for each of the dentists.

To solve this problem, we'll use queuing theory, specifically the M/M/1 queue model, where arrivals follow a Poisson distribution, service times follow an exponential distribution, and there is a single server (dentist) serving the queue.

Given:
- Arrival rate (λ) = 6 patients per hour
- Service rate (μ) = 3 patients per hour (since there are 3 dentists and each treats one patient at a time)
- Average service time = 20 minutes = 1/3 hour

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