P$4.($a$)(10$ points$)$ A multiprocessor system consists of $k$ processors. The system can provide

service if at least one processor remains operational. All supporting components are reliable.

Each processor has a failure rate of $($lambda$),$ and the time to failure is exponentially

distributed. Draw a fault tree and a Markov chain to model the failure of this system for $k=7.$

$($b$)(20$ points$)$ The same system is deployed to achieve task$-$level fault tolerance. All processors

have a failure rate of $.$ The system also includes a repair service that can only repair at most

one failed processor with the rate $.$ A repaired processor is returned to a free processor pool.

The tasks arrive at the rate $$ and depart $($completed$)$ at a rate $.$ Three processors are assigned

a task to work in a TMR configuration if three processors are available. Otherwise, the task is

started on two processors to work in duplex mode. A task is not accepted if at least two

processors are not available. Once a task has started, a processor assigned to it may fail. In that

case, the task continues in a degraded mode $($TMR operates as duplex, and duplex operates as

simplex$).$ If all processors assigned to a task fail, the task is terminated. When a task departs

$($completes$),$ the processors still executing it are released to the free processor pool.

We want to model this system with node failures, node repairs, and task execution. We want to

keep track of the number of operational and failed nodes and the number of tasks operating in

triplex, duplex, and simplex modes. Draw a Markov chain to model the processor failures,

repairs, task arrival, and task departure processes for $k=7.$