What are the eigen values of the matrix $$P$$ given below? $$$P = \left( {\matrix{ a & 1 & 0 \cr 1 & a & 1 \cr 0 & 1 & a \cr } } \right)$$$

Consider three $$CPU$$-intensive process, which require $$10,20$$ and $$30$$ time units and arrive at times $$0,2$$ and $$6$$ respectively. How many context switches are needed if the operating system implements a shortest remaining time first scheduling algorithm? Do not count the context switches at time zero and at the end.

Consider three processes (process id $$0,1,2,$$ respectively) with compute time bursts $$2, 4,$$ and $$8$$ time units. All processes arrive at time zero. Consider the longest remaining time first $$(LRTF)$$ scheduling algorithm. In $$LRTF$$ ties are broken by giving priority to the process with the lowest process id. The average turn around time is

Consider three processes, all arriving at time zero, with total execution time of $$10,20,$$ and $$30$$ units, respectively. Each process spends the first $$20$$% of execution time doing $${\rm I}/O$$, the next $$70$$% of time doing computation, and the last $$10$$% of time doing $${\rm I}/O$$ again. The operating system uses a shortest remaining compute time first scheduling algorithm and scheduling a new process either when the running processes gets blocked on $${\rm I}/O$$ or when the running process finishes its compute burst. Assume that all $${\rm I}/O$$ operations can be overlapped as much as possible. For what percentage of time does the $$CPU$$ remain idle?