In a printing shop, print requests arrive randomly and independently at an average rate of 20 per hour and are placed in a queue according to arrival times. Suppose that the time it takes to process each request is exponentially distributed and that the print times for different jobs are independent.

(a) Calculate the probability that this printer will be idle for the next 5 minutes if the queue is currently empty.

(b) A particular printer in this shop is capable of printing five pages per minute. The average request results in ten printed pages of poster.

(i) Calculate the probability that the next request will take more than 2 minutes to process.

(ii) There is currently one job in the queue and it has been active for the last 5 minutes. Calculate the probability that this job will still be active 1 minute from now.

(iii) State the property of the distribution that you can see from the result in (ii).

(iv) There are currently five jobs in the queue - one active and four waiting to be processed. A new customer submit a job to this printer and it becomes the sixth job in the queue. Calculate the probability that he will have to wait more than 5 minutes for the printer to begin processing his job. Assume that the time between consecutive jobs in the queue is negligible.

(v) Name the alternative distribution of the random variable that you are considering in (iv).