4. Consider an M/M/1-type queue where customers arrive according to a Poisson process with rate =4 and are served with rate =2. Assume that customers are impatient, such that if they do not immediately enter service upon arrival to the system, then they begin an exponentially distributed impatience timer with rate =1, and if this timer expires prior to reaching the server then they renege and leave the system before receiving service. If a customer reaches the server prior to the conclusion of their impatience timer, then the timer is "turned off" and they are no longer at risk of leaving the system prior to service completion. (a) [2 points] Draw a state-transition diagram that illustrates all possible transitions between at least the first four states (0,1,2,3) of this birth and death process. (b) [1 points] Determine the balance equations for this queueing system. (c) [4 points] Solve for the steady-state probability of the queuing system being empty (i.e., 0 ). Hint: The following identity will be needed to simplify your final answer. For some constant c, n=0n!cn=ec