Question: The purpose of this problem is to illustrate how the relative throughputs of competing sessions can be affected by the priority rule used to serve them. Consider two sessions A and B sharing the first link L of their paths as shown in Fig. 6.28. Each session has an end to-end window of two packets. Permits for packets of A and B arrive dA and dB seconds, respectively, after the end of transmission on link L. We assume that dA is exponentially distributed with mean of unity, while (somewhat unrealistically) we assume that dB = 0. Packets require a transmission time on L which is exponentially distributed with mean of unity. Packet transmission times and permit delays are all independent. We assume that a new packet for A (B) enters the transmission queue of L immediately upon receipt of a permit for A (B).
(a) Suppose that packets are transmitted on L on a first-come first-serve basis. Argue that the queue of L can be represented by a Markov chain with the 10 queue states BB, BBA, BAB, ABB, BBAA, BABA, BAAB, ABBA, ABAB, and AABB (each letter stands for a packet of the corresponding session). Show that all states have equal steady-state probability and that the steady-state throughputs of sessions A and B in packets/sec are 0.4 and 0.6, respectively.
(b) Now suppose that transmissions on link L are scheduled on a round-robin basis. Between successive packet transmissions for session B, session A transmits one packet if it has one waiting. Draw the state transition diagram of a five-state Markov chain which models the queue of L. Solve for the equilibrium state probabilities. What are the steady-state throughputs of sessions A and B?
(c) Finally, suppose that session A has nonpreemptive priority over session B at link L. Between successive packet transmissions for session B, session A transmits as many packets as it can. Session B regains control of the link only when A has nothing to send. Draw the state transition diagram of a five-state Markov chain which models L and its queue. Solve for the equilibrium state probabilities. What are the steady-state throughputs of sessions A and B?