The problem of achieving fairness in the allocation of the bandwidth resource on a link shared by multiple flows of traffic has been extensively researched over the last decade. However, with the increasing pervasiveness of optical networking and the occasional trend toward using over-provisioning as the solution to congestion, a router's processor also becomes a critical resource to which, ideally speaking, all competing flows should have fair access. For example, if the network is not fair in allocating processing resources, denial of service attacks based on an excessive use of the router processor (such as by using unnecessary optional headers) become possible. In this paper, we investigate the issue of achieving fairness in the joint allocation of the processing and bandwidth resources. We first present a simple but powerful general principle for defining fairness in such systems based on any of the classic notions of fairness such as max-min fairness, proportional fairness and utility max-min fairness defined for a single resource. We apply our principle to a system with a shared processor and a shared link with max-min fairness as the desired goal. We then propose a practical and provably fair packet-by-packet algorithm for the joint allocation of processing and bandwidth resources. We demonstrate the fairness achieved by our algorithm through simulation results using real gateway traffic traces. The principles and the algorithm detailed in this paper may also be applied in the allocation of other kinds of resources such as power, a critical resource in mobile systems.