Describe the problem you, or the individual, group or organization you represent, is facing that can be potentially solved via the public services in question. For example, if you are writing on behalf of a recovering heroin addict appealing for Medicare to cover a prescription for a detoxification drug, you could describe the economic hardships the individual is struggling with that prevents her from purchasing the drug out of pocket.
Outline how the public services for which you are appealing can either solve or help to solve the specific problem you highlighted. For example, if you are writing to appeal a municipality to alter its trash collection route to include your home, you could explain that the route change allows you dispose of your trash in a environmentally friendly and economical fashion.
Explain why the public services for which you are appealing are the best possible solution for the problem you described. For example, if appealing for a more frequent police patrol of your retirement community, you could explain that the only alternative would be for the citizens of the community to form a neighborhood watch, which is unreasonable, as most of the citizens have eye problems.
Articulate the unique need you, or the individual, group or organization you represent, has for the service in question. Describe how long you have survived without the service and how it has been a detriment to your well-being. Additionally, articulate how the addition of the service will improve your quality of life, productivity and personal well-being.