Network clouds are typically private clouds owned by the network provider, consisting of a large number of geo-distributed sites with heterogeneous capabilities and small capacities. Each of these small clouds often run specialized service chains of Virtual Network Functions (VNFs), which need to meet strict Service Level Objectives (SLOs), especially along the lines of availability (e.g., First responder services). Hence, VNFs in such thinly provisioned clouds may need to be moved (rehomed), both within and across sites, much more frequently than in traditional public clouds (like Amazon's EC2 cloud), in order to meet the performance SLOs, when reacting to various cloud events like hotspots, interference from co-located VMs, failures and upgrades. Rehoming is also required by the infrastructure (platform) providers for various other reasons such as consolidation of resources for saving energy and improving the platform utilization. In this paper, we propose a model-based approach to show that naive strategies for rehoming, applied uniformly across all VNFs of the service chain, are often sub-optimal when considering different metrics like user-perceived service disruption time and the time taken to complete the rehoming action. Our model leverages the transparency between the services and platforms on private clouds (grayness), and provides appropriate rehoming recommendations based on various factors including service characteristics and runtime platform dynamics. We validate our models using a simple, yet ubiquitously deployed service chain, and using out-of-the-box rehoming options provided by Openstack, the most commonly used open-source cloud. Our results show that our graybox approach is able to achieve significant reductions in service disruption times and time taken for the rehoming action.