Q1. In a small bank, there is just one teller on duty. The average rate of arrival of clients is 25 per hour. On an average the teller can serve at the rate of one client every 5 minutes. Assuming the arrival of clients follows a Poisson distribution and service times are exponential a) What is the average number of clients in the waiting line and in the bank? b) What is the average waiting time in the line and the average waiting time in the bank?

Q2. A specific router takes 3 milliseconds on average to transmit a packet. The mean rate of arrival of packets to this router is 150 packets per second. Assuming an M/M/1 queue model a) What is the probability of buffer overflow if this router has 12 buffers? b) What is the buffer size needed to keep packet loss below one packet per million?

Q3. Consider a generalized processor sharing (GPS) system with a processing capacity of 2 packets/sec serving 4 buffers and respective weights of w1=0.3, w2=0.2, w3=0.4 and w4=0.1. Assume a scenario where packets arrive simultaneously as follows: 4 packets at buffer 1, 3 packets at buffer 2, 2 packets at buffer 3 and 2 packets at buffer 4. Find the following: a) Serving time for all packets in the system b) First buffer to become empty and the time that it occurs c) Second buffer to become empty and the time that it occurs. Q4. Consider Host A and Host B connected via a packet-switched network and transmitting realtime voice data (VoIP). Initially, Host A converts the analog data to a digital 64 kbps bit stream. It then encapsulates the bits into 56-byte packets (do not consider overhead) and transmits the packets. Host A and Host B are connected by a 2 Mbps link with 10 ms propagation delay. Once Host B receives an entire packet, the digital-to-analog conversion is done. How much time elapses from the time a bit is created in Host A to the final decoding in Host B?

Q5. Consider a crossbar switch with n inputs and k outputs a) Why is the switch is called a concentrator when n > k? Under what traffic conditions is this switch appropriate? b) Why is switch is called an expander when n < k? Under what traffic conditions is this switch appropriate? c) Assume an N x N switch consists of three stages: an N x k concentration stage; a k x k crossbar stage; and a k x N expansion stage. What is the number of crosspoints needed in this 3-stage switch? d) When does the three-stage switch in part (c) fail to provide a connection between an idle input and an idle output line?