Develop the state transition diagram of the joint Markov chain for the output
of an M/M/1/∞queue feeding fluid bits into a constant rate leaky bucket. The
state is an ordered pair of the number of packets in the M/M/1/∞queue and the
random number of “integer fluid units.†The latter quantity, the integer random
number of fluid bits increases by one when the buffer content is increasing
and the environment completes pumping a full fluid packet. It attempts to
decrease by one when the environment is not pumping in and the leaky bucket
successfully completes pumping out the equivalent of one full fluid packet.
Solve for the equilibrium state probabilities of this Markov chain and obtain
an expression for the cdf of the fluid buffer content. Compare this answer with
that developed in Section 10.6.
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