Question

In the manufacturing of a new Samsung Galaxy smartphone, there is a quality control step comprised of two tasks where a factory worker checks for cracks in in the glass front and fixes them (Task 1) and another worker applies a polish and protection solvent the glass front surface (Task 2). Task 1 must happen prior to task 2 in this process. The time it takes for each worker to perform their respect task are exponential random variables with average completion times for tasks 1 and 2 of 5 and 7 minutes, respectively. Suppose that one of these smartphones arrives at this process where no phones are being processed in Task 1 and two phones being processed in Task 2 - Phone A in service and Phone B waiting to be serviced. Find the following:

 (a) The probability that Phone A is still in service when the new phone moves over to task 2. 

(b) The probability that Phone B is still in the system when the new phone moves over to task 2.

(c) E[T], where T is the time that the new phone spends in the system. Hints: • Break T into two components: i. Time until the first thing happens (T1) ii. Remaining time after that (R) • Use law of total expectation to find E[R] conditioning on a partition made of up two events.