Exercise 2: A hardware system has two processors P1 and P2 that are identical. Time in the system

is divided into slots. The system is governed by the following set of rules:

With a probability p a , a maximum of one job arrives each time slot in an independent way.

The job starts immediately on a free processor (idle) if any. If both are free the job starts at

P1.

Jobs are not buffered. If P1 and P2 are busy, arriving jobs are lost.

A job is ended by a processor in a slot with probability p e . Since job completion is

independent, the possibility that both processors end their jobs at the same time slot is p e 2 .

A processor can end a job in a time slot and starts another.

a) Represent the problem as a Markov chain.

Hint: Consider the following states: State 0: No jobs, State 1: one job in the system (in P1)

and State 2: 2 jobs in the system.

Hint: you need to think about all possibilities when looking for the transition probability.