5.2 Infinite array of linear concatenated sectors One infinite server spatially distributed queueing system that has provided certain physical insights into alternative dispatching procedures is a linear concatenated sector system. On the x-axis, assume that sector i covers the interval from x = i/2 to x = (i/2) + 1 for i even and from x = -(i + 1)/2 to x = -(i - 1)/2 for i odd. Response unit i is assigned to patrol uniformly sector i when it is available for dispatch assignment. Each unit is assumed to be available with probability (1 - ), independently of the status of all other units. (It should be clear that the independence assumption is an approximation.) The position of each available unit is selected from a uniform distribution over the length of the unit's sector. The random variable indicating the position of unit i is Xi; a particular experimental value of the random variable is xi.
Assume that an incident is reported from some point x in sector 0 (0 x 1) and that the dispatcher must select an available unit to assign to the incident. The incident position x is drawn from a uniform probability density over [0, 1]. The dispatcher may use any one of the following three selection criteria:
Let i() = expected travel distance for strategy i, given a utilization factor of (i = 1, 2, 3).
Do these results make intuitive sense for limiting values of ? What practical significance do they have?