Before revision:  For proper functioning, particularly as sensors, the length of a specific protrusion must not cross a narrow band bounded by two thresholds. However, fluctuations may drive the length beyond these thresholds. We study generic stochastic models of length control of long cell protrusions; the special cases of this model correspond to specific examples of different types of cell appendages . Exploiting the techniques of level crossings developed for random excursions of stochastic process, we have derived analytical expressions of passage times for hitting various thresholds, sojourn times of random excursions beyond the threshold and the extreme lengths attained during the lifetime of these protrusions.
After revision:  We develop here a stochastic model motivated by length fluctuations of a type of appendage of an eukaryotic cell called flagellum (also called cilium) . Exploiting the techniques of level crossings developed for random excursions of stochastic process, we have derived analytical expressions of passage times for hitting various thresholds, sojourn times of random excursions beyond the threshold and the extreme lengths attained during the lifetime of these protrusions.