$6.$ A manufacturing cell has $4$ operators $($working independently in parallel$)$ and a shared buffer space that can fit at most $6$ incoming jobs. During a production period, jobs arrive to the cell with rate of $10$ jobs per hour $($Poisson arrival, i$.$e$.,$ exponential inter$-$arrival time$).$ The average processing time of a job is $30$ minutes $($exponentially distributed service time$).$ Answer the following questions using the analytical formulas learned in class. $($a$)$ Explain why the M$/$M$/4/10$ queueing model should be used to model this problem. $($b$)$ Show that the long$-$term average waiting time of a job in the buffer is $0\mathrm{.}4554$ hr or $27\mathrm{.}3$ min. $($c$)$ Show that the probability that an arriving job cannot be admitted into the cell due to full occupancy is $0\mathrm{.}2301.($d$)$ If the desired the rejection probability is no more than $10\%$ in this case, what would you suggest? Use either calculations or simulation to justify your answer.