This is my third time posting it the same question Please dont copy the answer from ChatGpt. Consider a manufacturing system with three machines for their operations. Each machine

can handle any job such that any job is ready to leave the system after being processed on

any of the three machines.

Recently, management has invested resources to replace one of the machines with a new

machine that is identical to the old one. As a result, the service rate of the new machine is

greater than the service rates of the remaining two old machines. However, the service rates

of the two old machines are the same.

Jobs arrive the system following an exponential distribution with a rate of $$ jobs per hour.

Similarly, the service operations of the three machines follow an exponential distribution.

The service rate of the new machine is ${}_1$ jobs per hour, while the service rates of the old

machines are each ${}_2$ and ${}_3$ jobs per hour respectively, such that ${}_2={}_3.$

If the total number of jobs that can be in the system at any time is $5,$ use this information

to answer the following questions.

a$)$ Draw and label a state space diagram, showing all states and the rates of transition

between states.

b$)$ How many equations would your set of balance equations have for this problem?

Hint:

You can model this system as having two stations where one station has just the new

machine $($service rate of ${}_1),$ while the second station contains the two old machines

$($with service rates ${}_2={}_3).$ Incoming jobs are processed by a single machine in any

of the two stations.