5.12 Fire problem Consider three fire stations, A, B, and C, located on the X-axis at -1, 0, and +1, respectively. The fire trucks at these stations service fires that are reported in the region X = -2 to X = +2. Fires within this region are independently uniformly distributed; they are generated at the rate per hour. Travel time to and from the fires is instantaneous. Service time at the scene is negatively exponentially distributed with mean 1/. There is one truck at each station and the dispatcher will assign the closest available truck. If no truck is available, reports of fires enter a queue that is depleted in a first-come, first-served manner.

1. Write down the equations whose solution would provide the utilization factor (fraction of time busy) of each of the trucks. You need not solve the equations.

2. Write down the equations whose solution would provide the fraction of calls generated in the interval X = -½to X = +½ that are serviced by unit C. You need not solve the equations.

3. Find approximate solutions to parts (a) and (b) by employing the hypercube approximation procedure of Section 5.5 and Problem 5.11.