Future mobile multimedia systems will have wearable computing devices as their front ends, supported by database servers, I/O servers, and compute servers over a backbone network. Multimedia applications on such systems are demanding in terms of network and compute resources, and have stringent Quality of Service (QoS) requirements. Providing QoS has two aspects. On the one hand, the QoS requirements for the relevant resources have to be defined and suitable policies for meeting these requirements have to be devised and analyzed. On the other hand, the architecture of the system components and the mechanisms enabling the implementation of these policies have to be designed. In this paper we propose an architecture called HARMONY for providing QoS in mobile computing environments. The HARMONY architecture is a layered architecture that provides mechanisms for the management of network and compute resources, in particular for call admission control taking into account the simultaneous requests for both types of resources by the mobile units. It also provides a mechanism for mobility management of mobile units as they move from one cell to another in a mobile computing environment. The network resources are reserved based on the Entropy model. In order to provide compute guarantees, we provide a novel scheme for off-loading tasks from the mobile units to the compute servers in the backbone network. We propose a load-balancing scheme to minimize the call blocking probability due to lack of compute resources, which redistributes the total load in the system across all compute servers so that these are equally loaded. Through a quantitative analysis of the HARMONY architecture we establish its effectiveness in providing quality of service in mobile computing environments.