This is a differential equation. To get a closed-form answer we need to use continuous time and get rid of the notion of "cycles".

w = volume of particles on the wafer
c = density of particles in the chamber

The rate of accumulation on the wafer is proportional to the density of chamber particles.

dw/dt = kc
dc/dt = -akc-lc ; let l' = ak+l where a measures the ratio of units
dc/dt = -l'c
dc/c = -l' dt
ln c = -l't + C1
c = exp(-l't+C1)
and we want to choose C1 so that c(0) = the desired concentration

dw/dt = k(exp(-l't)+C1)
dw = k(exp(-l't)+C1)dt
w = -k/l' exp(-l't) + C1t + C2
and we want to choose C2 so that w(0) = 0 (it'll be k/l' )

so it is always exponential. However, if the ratio -k/l' is small compared to C1, or if l't is large making exp(-l't) small compared to C1, the function will behave very much like C1t + C2.