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 $.$

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

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

number of available fixtures to be processed by the artificial server. When $nf$ takes a negative value,

$|nf|$ 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.

c$)$ Draw a part of the transition diagram to show the transitions ending in a typical state.

d$)$ Give the steady$-$state balance equation for a typical state. e$)(4$ points$)$ Give formulations of the

following steady$-$state performance measures in terms of the steady$-$state probabilities.

expected number of customers at server $($including the customers waiting in line to be processed at

the server$)$

expected number of backorders

expected number of idle fixtures

expected percent utilization of the fixtures In order to answer the numerical questions in parts $($f$),$

$($g$)$ and $($h$),$ truncate the state space assuming that $nfB.$ In this case, the maximum number of

backorders allowed in the system is equal to $|B|.$ A job that arrives when there are $|B|$ backorders in

the system would be lost.