DONT COPY ANSWER ALREADY POSTED

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) 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. Hint: This system has an infinite capacity for waiting customers, so use to indicate that there are more states beyond 3. Also, a death or departure in this system may be caused by a service completion (for i 1) or an impatience timer expiring (for i 2). In either case, the state of the system would decrease from i to i 1. 1

(b) Determine the balance equations for this queueing system.

(c) Solve for the steady-state probability of the queuing system being empty (i.e., 0)