In a single server infinite population queuing model, arrivals follow a Poisson distribution with mean $$\lambda = 4$$ per hour. The service times are exponential with mean service time equal to $$12$$ minutes. The expected length of the queue will be

At a production machine, parts arrive according to a Poisson process at the rate of $$0.35$$ parts per minute. Processing time for parts have exponential distribution with mean of $$2$$ minutes. What is the probability that a random part arrival finds that there are already $$8$$ parts in the system (in machine $$ + $$ in queue)?

On the average $$100$$ customers arrive at a place each hour, and on the average the server can process $$120$$ customers per hour. What is the proportion of time the server is idle?