EE-547 Midterm 1 October 18, 2007 1. Enter your name, student ID number, and e-mail address in the space provided on this page. You have 1 hour 15...

(3) (40 points) Consider the following single-server queueing system with finite buffersize N. Packets arrive according to a Poisson process with rate . The size of eachpacket is i.i.d exponentially distributed with mean 1/μ. The server has a capacity of1. When there is only one packet in the system, the server will serve this packet atthe rate of 1. Hence, a single packet with length l will take time l to complete service.However, whenever there are two or more packets in the system, the server will splitits capacity into two parts, and serve the first two packets in the queue simultaneously,each at the rate of 1/2. For example, if two packets arrive at an empty system at thesame time, and their sizes are l and 2l, respectively, then the server will first serve twopackets simultaneously. Hence, the short packet will complete service first, after a timeperiod of 2l. Then, if there are no other packets in the system, the server will servethe remaining size l of the second packet at the rate of 1. Therefore, the remainingpacket will complete service after another time period of l. On the other hand, if thereis a third packet that arrives before the short packet completes service, then again theremaining long packet will be served simultaneously with the third packet.Let Pm denote the steady-state probability of having m packets in the queue.(a) (15 points) Assume that at a particular time-instant t0, there are m packets inthe queue, and m 2. Let t0 + T denote the first time-instant after t0 when apacket completes service. Carefully derive the probability P[T t] for all t.(b) (10 points) Draw the state transition diagram of the continuous time Markovchain that can be used to determine Pm, the probability that there are m packetsin the system. (Hint: You can use Part (a) to draw a conclusion on the rate withwhich customers depart from the system.)(c) (5 points) Write down the balance equations for solving Pm.(d) (10 points) Assuming that the buffer size is N, compute PL, the probability thatan incoming packet is dropped.