3.15 Rectangular grid of two-way streets Consider an n x m rectangular grid of two-way streets running north-south and east- west as shown in Figure P3.15. Assume that incident positions are distributed uniformly over the grid. A response unit patrols the grid in a uniform manner. The incident location and the response unit location are independent. Let

D = travel distance between the response unit and the incident, assuming the unit follows a shortest path that remains on the streets of the grid

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a. (Optional) By carefully and patiently conditioning on the various possible locations for the incident and response unit, show that

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where the left-hand inequality becomes an equality when n or m is zero and the right-hand inequality becomes an equality when n = m. Thus, the continuous approximation, (3.12a), is never in error by more than I block length.