A control circuit (such as a smartphone processor) having access to a wireless two-way transceiver (such as a short-range transceiver) determines a need to offload execution of a task to a server and then determines present unavailability of that server. This control circuit then transmits a first message to the server to prompt the server to awaken. The control circuit then transmits a second message to the server to establish a wireless connection that the control circuit employs to offload execution of the task to the server. By one approach, the aforementioned first message can include a unique identifier for the server. If desired, this unique identifier can have been provided earlier by the server via a general broadcast.