Fire Station Location 101

[XLFD]

I apologize beforehand for this math-based article, and swear I will keep it at a level that requires no algebra and only the basics of geometry.  In this exercise, we will look at ideal situations, not real world situations.  At the end, feel free to ask any questions if you do not understand any concepts.  I may gloss over an explanation because I am also presuming the reader has a high school level of math comprehension.

Let's consider a boom town called Squareville.  Squareville is 12 blocks long and 12 blocks wide and they need a new fire department to service their area.  Squareville cannot grow bigger due to geographical/political features.  A total of 16 firefighters are ready to go, and they are evenly distributed throughout the city (to keep it simple).  Here is a representation of Squareville and their firefighters by location, the orange spots with letters denoting each firefighter A-P.

The town council has three possible locations determined for the fire station:  1) right in the very center of Squareville.  2) On Firefighter D's property in the upper right corner or 3) On the corner of Firefighter H's location. 

Now the firefighters are all volunteer and are around their home most of the time, so again for simplicity sake, consider their starting point at their residence.  The council wants their department to have the best response times for any fire that may happen in Squareville.   Response times are a factor of how long on average it takes a firefighter to get to the station and how long on average it takes to get to the fire.  Here is a case by case analysis, followed by an interesting fourth case. 

1)  Fire station is located in the center of Squareville

Most people seemed to think that having the station in the center of town would be prudent, though they couldn't exactly say why.  Let's examine that scenario as far as blocks traveled and then the related response times:

Firefighter (FF) J is four blocks from the station(2+2), as is FF F,G, and K.

FF L is eight blocks from the station (2+6), as are seven others on the edge of town but not in the corners.

FF A and the other three corner ones are 12 blocks away (6+6). 

The total number of blocks traveled is thus (4X4) + (8X8) + (12X4) = 128.  The average blocks traveled per firefighter is 128/16 equals eight.  If we presume three-fourths of the FF are available to go to the station per fire call, and protocol calls for at least three men per truck, a truck can go out in ideal situations at the end of four time units (the time it takes to travel one block), two more can go out at eight time units and one more at twelve time units. 

Is this going to be the same for wherever the station winds up?

2)  The Station is located in the upper right corner of Squareville.

Obviously, Firefighter D would be at the station most of the time, so he would like it, but what about FF M on the opposite corner?  Here is a schematic showing the distances of each firefighter from the corner station:

The diagonal red lines show FFs who are equidistant from the new proposed station, the green numbers representing the number of blocks away.  Here the blocks traveled are (0X1) + (4X2) + (8X3) + (12X4) + (16X3) + (20X2) + (24X1) = 192.  The average responder must travel 12 blocks (192/16), and take twelve time units to get there.

Furthermore, chances are likely they will not be able to roll in the first four time units unless all three nearby show up (remember we presume only 3/4 of the FF's show up).  Likewise, at eight time units they should be getting the first truck out but with only four or five FF, they cannot get a second one until twelve time units, and the third won't leave until after sixteen time units.  In the centralized location, four fully loaded trucks went out after twelve minutes. 

Having the fire station in any corner will have the same result, where the average firefighter will have to travel four more blocks and take four more time units to get to the station than if the station was in the center.  The central location can roll three trucks in the first eight time units, whereas the first one just gets underway in that same time in the corner.

3)  The station is located near Firefighter H on the edge of town

One would think this location would be in between the first two results, and it is directly between them.  Here is the schematic where we again connect the firefighters that are equidistant from the proposed station.  As you can see one is at the station, one is 20 blocks away, three are 4 and 16 blocks away, and four are 8 and twelve blocks away, in a bit of symmetry.

Thus the total blocks traveled to the station is (0X1) + (4X3) + (8X4) + (12X4) + (16X3) + (20X1) = 160.  Each much travel an average of 10 blocks (160/16) taking ten time units.  Enough people are available for a truck at four time units, one more truck can leave at eight and twelve time units.

4)  An interesting observation about other centralized points:

One may wonder what would happen if the station was positioned at one of the four firefighters homes located towards the center of Squareville.  One might assume that it would be a bit worse than the exact center, but the surprising result is that it isn't! 

Without loss of generality, presume the fire station is located at 'G', as shown we can figure out how far each firefighter is away:

We have one at four and 16 blocks away, four at 4 and 12 blocks away, and six at eight blocks away.  This figures to (0X1) + (4X4) + (8X6) + (12X4) + (16X1) = 128 blocks traveled or an average of 8 blocks traveled, the same as if the station was at the center.  This would seem to indicate that any centralized location works best for Squareville, there's even a good chance to get three trucks out within eight time units. 

A Convenient Result about average response times en route to emergencies from the station

If we presume that each fire call for Squareville is equally distributed throughout town, then the average time it takes to get to the scene from the station is exactly the same average it takes for the evenly distributed firefighters to get to the station. 

This means that the difference in total response time between a corner station and a central station is even more dramatic.  It's the difference between 24 time units and 16 time units for travel to and from the station.  That is significant when you consider how quickly a fire can grow, and how dangerous it can be to have emergency vehicles running code for longer than they have to.

The Real World

Real world situations may offer a bit of complexity thrown in, such as having thru-streets available, having areas of high building density requiring a quicker response, areas like school zones and low speed limit zones, etc. that might hinder quick and safe responses.  This will be covered in Fire Station Location 201, please read that after you finish this.

The relevance for Ludington should be clear.  We currently have a centralized location for the fire station just off the downtown, it would be a little above Firefighter J's location if Squareville could be placed over Ludington.  The city brought a piece of property for a future fire station close to where firefighter K would be, then later decided against it.

They have now spent over $100,000 to pick up a property for the future fire station at a position between D and H, towards the northeast corner of the district, and well away from many of the city's potential hazardous situations.   Pere Marquette Fire Department is now closer to most Fourth Ward locations and beyond. 

As we have seen, without any other variables besides location, the total response time could be as much as 50% more time, 50% more blocks where emergency vehicles are traveling through the streets.  Fire station location matters.

[Willy]

Very interesting X. I never thought about looking at response time in this manner. Is there a logical reason why the fire station is going to be relocated? I would think that for such an expensive proposition that a vote by the citizens regarding the station relocation and costs makes more sense. I found the video below and thought you might enjoy it.

[XLFD]

An otherwise nice video, but although it's labeled Ludington Fire Department there are interludes to other departments and a half minute dedicated to the Officer Maltby shooting, in which LFD was not involved with beyond standby.  There was a couple fires I remembered referred to in the paper, especially the fatal Robert Street fire back in 2001, the first fully involved fire I was involved in. 

The reason why they want to move the LFD station is that they want to free up the premium downtown space that it's taking up, particularly for John Henderson's pet project about establishing a convention center and/or another hotel in the downtown on the full bowling alley block.  Our other hotels on Ludington Avenue will be screwed over by all the back room deals between government entities (led by John Henderson), the CVB (led by Brandy Henderson) and mostly anonymous LLC's made to make such a project hit the ground running.

[XLFD]

I tried to find other "fire station location" examples on the web, since locating a fire station in a city with volunteer firefighters should be a classic math problem.  But I couldn't find any similar analysis on such a subject.  I took it on myself using a simple case of a small square city.

The best way to look at it is to see what happened with the Ludington DPW.  It used to be located across the fire station, and their main duties revolved around the downtown and lakefront area.  In 2005 they moved out to the edge of the city limits on First Street where it's at least a two mile journey to make it to the downtown and lakefront.  Over the last ten years, the cost in gas, the amount of vehicle wear and tear, and the amount of employee time used to cover these extra travels is significant.