An efficient method for the exact numerical simulation of semi-Markov proce
sses is used to study minimal models of the control of eye movements in rea
ding. When we read a text, typical sequences of fixations form a rather com
plicated trajectory - almost like a random walk. Mathematical models of eye
movement control can account for this behavior using stochastic transition
rules between few discrete internal states, which represent combinations o
f certain stages of lexical access and saccade programs. We show that exper
imentally observed fixation durations can be explained by residence-time-de
pendent transition probabilities. Stochastic processes with this property a
re known as semi-Markov processes. For our numerical simulations we use the
minimal process method (Gillespie algorithm), which is an exact and effici
ent simulation algorithm for this class of stochastic processes. Within thi
s mathematical framework, we study different forms of coupling between eye
movements and shifts of covert attention in reading. Our model lends suppor
t to the existence of autonomous saccades, i.e., the hypothesis that initia
tions of saccades are not completely determined by lexical access processes
.