We construct and analyze a stochastic model of eukaryotic flagellar length control. Flagella are microtubule-based structures that extend to about 10 μ
m from the cell and are surrounded by an extension of the plasma membrane. Flagellar length control is a particularly convenient system for studying organelle size regulation since a flagellum can be treated as a one-dimensional structure whose size is characterized by a single length variable. The length of a eukaryotic flagellum is important for proper cell motility, and a number of human diseases appear to be correlated with abnormal flagellar lengths. Flagellar length control is mediated by intraflagellar transport (IFT) particles, which are large motor protein complexes within a flagellum that transport tubulin (the basic building block of microtubules) to the tip of the flagellum. The critical length of the flagellum is thus thought to be determined by the dynamical balance between length-dependent transport and assembly of microtubules and length-independent disassembly at the tip. In our model we assume that IFT particles are injected into a flagellum according to a Poisson process, with a rate that depends on a second stochastic process associated with the binding and unbinding of IFTs to sites at the base of the flagellum. The model is thus an example of a doubly stochastic Poisson process (DSPP), also known as a Cox process. We use the theory of DSPPs to analyze the effects of fluctuations on IFT and show how our model captures some of the features of experimental time series data on the import of IFT particles into flagella. We also indicate how DSPPs provide a framework for developing more complex models of IFT.