First of all, as you will probably have more than just a few addresses to ping, checkig them one after the other would consume much time. Paralleism is just for you. See this article:
http://www.aspsnippets.com/Articles/Parallelism---Parallel-Task-Execution-in-.Net-4.0.aspx[
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On the other hand, asp.net and the web in general is stateless. Yor web applicatin will be asleep as long as nobody is requesting anything from it. Thus doing scheduled tasks in a web application is not really a good a good approach - if hosted in a regular web server, like IIS. But you have also WebAPI at hands and you can host your web service inside your own windows service for example (see:
ASP.NET WebAPI Hosting Techniques[
^]).
So you either
A) make a simple windows service to ping thos host at specified intervals, ans use IIS as web server
B) use windows task scheduler to call your console application which does the same, and use IIS as web server
C) make a windows service that does the ping, and in the same time it provides the web service.
This latest one is the most complicated to make, but it is the easiest to manage as a product.