- Name IMGcontextpriority Name Strings EGLIMGcontextpriority Contributors Ben Bowman, Imagination Techonologies Graham Connor, Imagination Techonologies Contacts Ben Bowman, Imagination Technologies (benji 'dot' bowman 'at' imgtec 'dot' com) Status Complete Version Version 1.1, 8 September 2009 Number EGL Extension #10 Dependencies Requires EGL 1.0.
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3.7.1 Simple ModelExample 12: Design of a Response Distfict Suppose that we have once more the situation described inExercise 3.1, where requests for assistance are medical emergencies and the urban response unitis an ambulette. Under the assumptions that (1) locations of a medical emergency(X1, Y1) and of the ambulette(X2, Y2) are independent anduniformly distributed over the response district, and (2) travel is parallel to the sidesof the rectangular response area, the travel distance [from (3.11)] is given by D =|X1 -X2| + |Y1 -Y2| From Exercise 3.1, we then have that E[D) = 1/3[Xo + YJ (3.12a) where X0 andY0 are the sides of the rectangle (see Figure 3.3). In this example wewish to formulate and solve the problem of optimal district design and to investigate thesensitivity of our results to suboptimal designs. Solution To find the district dimensions which lead to theminimum expected travel distance, we must keep the area of the response district
Results such as those of (3.79) and (3.80) canbe derived for various district shapes. The first three columns of Table 3-1summarize the equivalents of (3.79) for a square district, a square district rotated by 45' withrespect to the right-angle directions of travel, and a circular district. Thefollowing four cases are included:8
In all cases it is assumed that the locations ofrequests for service are uniformly distributed in the district and independent of thelocation of the service unit. When the constants in Table 3-1 are multiplied by , thesquare root of the area of the district in question, E[D] is obtained. In some instances(e.g., a square district with a randomly positioned response unit and Euclidean travel) theconstant of interest is not known exactly and the best known approximation, totwo-decimal-place accuracy, is shown. Some of these constants have already been derived in thischapter or will be derived in the Problems.
The last column of Table 3-1 lists values thatcan be used for c in (3.84) for the four combinations of response unit locations andmetrics that we have examined here. In all cases, we have selected the largest value of clisted in each row of the three leftmost columns of Table 3-1. When the effective travel speed is independent ofthe distance covered, one can use the constants in the fourth column of Table 3-1 toapproximate the expected travel time, E[T), as well. In that case we have
E[Teast-west] E[Tnorth-south] That is, it takes on the average about as much timeto traverse the district from east to west as from north to south.
8 few results for metrics other than Euclideanor right-angle are derived in the Problems. |