Economic implications of allowing competition with essential services i.e police, fire department, etc I'm wondering what are the implications of allowing competition between firms that deal with essential services in public welfare. I.e assuming there is no government regulation of such services, what would happen if there was competition between firms in the policing industry or if there was competition allowed among different fire departments? 

What would the industry look like and what kind of methods would we use to analyse such a situation?
As already suggested in the comments of the OP, we find cases where the provision of essential services behave much like any other market when privatized.
As suggested by Henry it was found that in the case of fire fighters, they simply let the homes burn down. On the London Fire Brigade's Website they discuss the issue in question with great detail.

People paid a fee to an insurance company to insure their property against damage. The most common risk at this time was fire. The first insurance company was named Phoenix, after the Greek mythological bird that rose from the ashes, and was established by Nicolas Barbon.
Building insurance was very profitable and many more insurance companies were set up with the insurance companies establishing their own fire brigades. These brigades were sent to insured properties if a fire occurred to minimise damage and cost.
Firemarks were used to identify - and advertise - different insurance companies. They were placed on the outside of an insured building and brigades would use these firemarks to determine whether a building was insured by them. If a building was on fire, several brigades would attend; if they did not see their specific firemark attached to the building, they would leave the property to burn. You can still see some old firemarks on buildings today.

Similarly we in 2010 there was a case of Tennessee firefighters letting a home burn when the owner didn't pay fire-protection fee. Its clear that if an essential service is allowed to be privatized, they will function like any other business.