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Data in the form of fixed-length packets arrive in slots on the FOUR input lines of a multiplexer. A slot contains a packet with probability p, independent of the arrivals during other slots or on the other line. The multiplexer transmits one packet per time slot and has the capacity to store THREE packets only. If no room for a packet is found, the packet is dropped.

a) COMPUTE the probability of j (for all possible j values) packets arriving on the four input lines during any given time slot.

b) DRAW the state transition diagram and SPECIFY the transition matrix P – The state is taken to be the number of packets in the multiplexer.

c) If p is equal to 0.4, what is the probability that the MUX will contain 3 packets after 10 time slot (i.e. at the start of the 11th time slot)? Assume that we start with an empty MUX.

d) PLOT the PMF for the state distribution for the first 10 time slots (i.e. n =0,1,2,…,10). (Hint: Write a C++, Java (or any other programming language) a program to simulate the process).

e) DEFINE and WRITE an expression for each of the following quantities:

• Multiplexer throughput in terms of packets per time slot.

• Multiplexer mean number of dropped packets per slot.