$3)$Consider a make$-$to$-$order $($MTO$)$ production system, where demand orders arrive according to a Poisson process with rate $\backslash$lambda $>0.$ Upon arrival, each order waits in a first$-$in$-$first$-$out $($FIFO$)$ queue until the system starts processing the order. The processing time of an order is distributed as an exponential distribution with processing rate $,$ and assume that there is a single processor $($so orders are fulfilled one at a time$).$ The demand orders are impatient, and while waiting to be processed, may get cancelled. $($Note that a job currently undergoing processing will not be cancelled.$)$ Assume that each waiting order gets cancelled after waiting a period of time that is distributed independently and identically as an exponential distribution with rate $\backslash$gamma $.$ Note that if there are k orders waiting to be fulfilled, the rate at which cancellations happen is k$\backslash$gamma $.$

$($a$)(3$ points$)$ Model the production system as an birth$-$death chain. Specify the states and the state space, and draw the state transition diagram. Make sure to correctly label the transition rates for each transition.

$($b$)(2$ points$)$ Write the detailed balance equations that the steady state distribution satisfies.

$($c$)(3$ points$)$ Assume $=\backslash$gamma $.$ Solve the detailed balance equations to find the steady state distribution. $($See useful formulae on the first page for series summations.$)$

$($d$)(2$ points$)$ Assume $=\backslash$gamma $.$ What is the throughput of this system? $($Note: throughput of a system is the rate at which jobs are successfully completed.$)$

$($e$)(2$ points$)$ Assume $=\backslash$gamma $.$ What is the steady state probability that an arriving order is cancelled rather than fulfilled?