I'm preparing for an exam and one of the preparation questions for the exam is the following:
Consider a computer system with TWO processors and NO waiting queue. Out of the two processors, one is faster than the other. That is, the average execution time of a job on the fast processor is 0.5 sec while that on the slow processor is 1 sec. Computational jobs arrive at this system with the average rate of 4 jobs/sec following the Poisson distribution. If both processors are idle, a job is assigned to the fast processor. Only when the fast processor is busy, a job is assigned to a slow processor. Once a job is assigned to a processor, it must execute on that processor until completion. (a) What is the average number of jobs in the system at any time instant? (b) What is the probability that the fast processor is busy? (c) What is the probability that the slow processor is busy? (d) What is the average time for completing a job? (Hint: Draw system’s state transition diagram. In a statistically equilibrium, from the perspective of a state, the total outgoing rate is equal to the total incoming rate.)
As the hint inidicates, I tried to draw a state transition diagram but I'm confused on how I will choose the random variable X:
- if I consider it as the number of jobs in the system, then I would have the following: diagram 1
- if I consider it as a binary number where the most significant bit represents whether the fast server is on or off and the other whether the slow server is on or off (00 = both are off, 01 = only slow server is on, etc...), then I have the following: digram 2.
In the first case, I would have to calculate the value of mu1 and mu2 which I don't know how... and the second case, I don't think it's a Marvokian Process anymore because there is a loop make to the initial state. How do I solve this problem?
It would be really helpful if someone can help me out. Thanks!