$($queuing$-$related$)$: Consider the following variant of the M$/$M$/1$ queue. Instead of assuming there are no limits on how many jobs can be in the system, assume now that a job is rejected $($not allowed to join the queue$)$ if it finds k jobs already in the system when it arrives. A job that is rejected is considered lost. a$)$ Characterize the distribution of the number of jobs in the system $($i$.$e$.,$ derive an expression for the probability of n jobs being in the system$).$ b$)$ Derive an expression for expected time in system, expected number of jobs in the system, utilization, and throughput rate. c$)$ What is the relationship between throughput and the parameter k$?$ What is the relationship between expected number of jobs in the system and k$?$ Generate graphs for an example system $($you can decide on the parameters$)$ showing the relationship between throughput and k and number of jobs and k$.$ What can you say about the tradeoff between throughout and WIP based on what you observe?