Estimation for the Simple Linear Boolean Model

We consider the simple linear Boolean model, a fundamental coverage process also known as the Markov/General/∞ queue. In the model, line segments of independent and identically distributed length are located at the points of a Poisson process. The segments may overlap, resulting in a pattern of “clumps”–regions of the line that are covered by one or more segments–alternating with uncovered regions or “spacings”. Study and application of the model have been impeded by the difficulty of obtaining the distribution of clump length. We present explicit expressions for the clump length distribution and density functions. The expressions take the form of integral equations, and we develop a method of successive approximation to solve them numerically. Use of the fast Fourier transform greatly enhances the computational efficiency of the method. We further present inference procedures for the model using maximum likelihood techniques. Applications in engineering and biomedicine illustrate the methods.