Geometric probability models to analyze strategies for finding a buried cable

Different strategies can be used to find a straight cable buried underground. The original problem considered by Faber et al. focused on a telephone company that wishes to dig a trench to locate a straight cable. The cable is known to pass within a given distance, a, from a marker erected above the putative location of the cable. Faber et al. showed that the shortest simply connected curve guaranteed to find the cable is a U-shaped curve whose length is about 18% less than that of a circular trench of radius a. This problem can be regarded as minimizing the maximum length that a trench digger must dig. In reality, once the cable is found, digging can stop. So far, however, no attempt has been made to evaluate the trench shape on characteristics other than the maximum trench length. In this paper, we present geometric probability models to analytically derive the distribution of trench length and calculate the expected value and variance for both the short-length (U-shaped) trench and a circular trench. Our main result is that the expected digging length is about 5% less for the circular trench than for the U-shaped trench.