Mobile edge computing (MEC) has emerged as a promising technology that offers resource-intensive yet delay-sensitive applications from the edge of mobile networks. With the emergence of complicated and resource-hungry mobile applications, offloading user tasks to cloudlets of nearby mobile edge-cloud networks is becoming an important approach to leverage the processing capability of mobile devices, reduce mobile device energy consumptions, and improve experiences of mobile users. In this article we first study the provisioning of virtualized network function (VNF) services for user requests in an MEC network, where each user request has a demanded data packet rate with a specified network function service requirement, and different user requests need different services that are represented by virtualized network functions instantiated in cloudlets. We aim to maximize the number of user request admissions while minimizing their admission cost, where the request admission cost consists of the computing cost on instantiations of requested VNF instances and the data packet traffic processing of requests in their VNF instances, and the communication cost of routing data packet traffic of requests between users and the cloudlets hosting their requested VNF instances. We study the joint VNF instance deployment and user requests assignment in MEC, by explicitly exploring a non-trivial usage tradeoff between different types of resources. To this end, we first formulate the cost minimization problem that admits all requests by assuming that there is sufficient computing resource in MEC to accommodate the requested VNF instances of all requests, for which we formulate an Integer Linear Programming solution and two efficient heuristic algorithms. We then deal with the problem under the computing resource constraint. We term this problem as the throughput maximization problem by admitting as many as requests, subject to computing resource capacity on each cloudlet, for which we ...