  We analyze asymptotically a differential-difference equation, that arises in
a Markov-modulated fluid model. Here there are N identical sources that turn
"on" and "off", and when "on" they generate fluid at unit rate into a buffer,
which process the fluid at a rate c < N. In the steady state limit, the joint
probability distribution of the buffer content and the number of active sources
satisfies a system of N + 1 ODEs, that can also be viewed as a
differential-difference equation analogous to a backward/forward parabolic PDE.
We use singular perturbation methods to analyze the problem for N large with
appropriate scalings of the two state variables. In particular, the ray method
and asymptotic matching are used.
