Consider a manufacturing system with $K$ fixtures $($pallets$,$ kanbans$).$ A job to be processed in this system is

assembled with a fixture. When there is not any available fixture, jobs arriving wait in queue to be taken

into the system one by one as the fixtures become available. An arriving job that cannot be immediately

assembled with a fixture is said to be backordered and waits in queue for an available fixture.

The jobs arrive according to a Poisson process with rate $.$ There is only one manufacturing stage $($server$)$

and its exponential service rate is $.$

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Hint. Consider an $($artificial$)$ exponential server with rate $$ to process the available fixtures. Let $n_f$ be the

state variable defined for this $($artificial$)$ server. When $n_f$ takes a positive value, it denotes the number of

available fixtures to be processed by the artificial server. When $n_f$ takes a negative value, $|n_f|$ denotes the

number of bacordered jobs waiting for an available fixture.

a$)$ Give a Continuous$-$Time Markov Chain $($CTMC$)$ formulation with a clear state

description and the corresponding state space.

b$)$ Draw a part of the transition diagram to show the transitions emanating from a typical

state.