Macbook Air WiFi Connection Timeout problem
Troubleshooting can be a process of elimination and often requires patience. Below are a few things you can try. 

Try another 'Wi-Fi' network

It'd be interesting to see whether your MBA can maintain an internet connection from another Wi-Fi network. If you have an iPhone you can share an internet connection from its cellular connection and connect to that via your Mac's Wi-Fi port? Does this allow your MBA to browse the web, send/receive emails, etc and maintain the connection for longer than the five or so minutes you're having problems with?

Another way to test this is via a public Wi-Fi services (McDonalds, Starbucks, etc).

Test your Wi-Fi network

If your MBA connects fine to other Wi-Fi networks, then it's time to test your home Wi-Fi connection with other devices (such as an iPhone, iPad, etc) to see if they also drop a connection. If they do then you know it's your Wi-Fi network that needs investigating. 

On the other hand, if they connect fine, it may be time to reboot your Wi-Fi router. This is because your router may have a limit on how many devices can be connected at any one time, or may struggle when too many devices are connected.

To do this, switch the router off for about a minute and then switch it back on. Wait a couple of minutes and then try connecting your MBA and testing to see what happens. 

Remove and then add back your Wi-Fi service

I would totally remove your Wi-Fi service, restart, and add your Wi-Fi service back.

To do this:


Go to Apple > System Preferences > Network
Select the Wi-Fi service on the left-hand side
Click on the cog icon at bottom-left and select Make Service Inactive
Now delete the service by clicking on the minus sign (i.e. the - button) at left of the cog
Click on the Apply button
Exit Network preferences
Restart your MBA
Go to Apple > System Preferences > Network
Click on the plus sign (i.e. the + button) at left of the cog
In the pop-up window, ensure that Wi-Fi is selected from the drop-down menu
Click on the Create button
Make sure Wi-Fi is on and connected (it should remember your previous settings, but if not just re-add your Wi-Fi network again)


Run Wireless Diagnostics

You can use Wireless Diagnostics to analyse your Wi-Fi network. Follow these steps:


Close any apps you have running
Connect to the Wi-Fi network
Hold down the option key and go to the Wi-Fi status menu and select Open Wireless Diagnostics
Enter your administrator name and password


Now follow the steps to see whether any problems are found.