1. A computer-implemented method, comprising: establishing, by one or more computer systems, a network connection with at least one customer device associated with a customer that is located in a separate facility from the one or more computer systems, the at least one customer device having connectivity with at least one host via the network connection; assigning one or more virtual machines to a virtual network for the at least one customer device, the virtual network overlaid on a substrate network operated by a service provider; providing, to a first customer device and over the network connection, at least one Internet Protocol (IP) address representing the first customer device to serve as an identifier for use over a public network by the first customer device; receiving, through an application programming interface (API), a remapping request from a second customer device, the remapping request identifying at least one of the one or more virtual machines; assigning, to the first customer device, the specified virtual machine of the one or more virtual machines identified in the received remapping request; reassigning the identifier of the first customer device to the specified virtual machine within the virtual network, the reassigning used to cause network traffic addressed to the IP address representing the first customer device to be sent to the specified virtual machine; processing, at the specified virtual machine, the network traffic using one or more network-related services comprising at least one network-related service configured to manage network traffic for the at least one host represented by the identifier; and forwarding at least a portion of the processed network traffic from the specified virtual machine to the first customer device via the network connection.