M/G/1 - compulsory test

1. Consider the expected remaining service time we derived in class: R_s=lambda/2 * E[X^2]. How does this expression look like if the service time is Exponential with parameter mu? How does this result relates to the memoryless property?

2. To work with M/G/1 queues one needs to be able to quickly find mean value, second moment, variance, and Laplace transform of probability distributions. Find these for the following service times:
a) The service time is the sum of two exponentially distributed random variables, one with a mean of 3 seconds and the other with a mean of 1 second.
b) Deterministic service time, always 5 minutes.
c) Uniformly distributed service time with values between 1 and 5 minutes. 
You can find a small collection of Laplace transforms here. Feel free to use other resources as well.