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.

For each of the questions in part I and part II$,$ give a Continuous$-$Time Markov Chain $($CTMC$)$ formulation with a clear state description and the corresponding state space.

Part I. Assume that there are always $($infinitely$)$ many jobs waiting to be processed. The jobs are first processed on server $1$ with an exponential rate of ${}_1$ and then on server $2$ with an exponential rate of ${}_2.$

For the CTMC formulation you give, draw the state$-$transition diagram assuming that $K=3.$