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Nowadays, most telecommunication services adhere to the Service Function Chain (SFC) paradigm, where network functions are implemented via software. In particular, container virtualization is becoming a popular approach to deploy network functions and to enable resource slicing among several tenants. The resulting infrastructure is a complex system composed by a huge amount of containers implementing different SFC functionalities, along with different tenants sharing the same chain. The complexity of such a scenario lead us to evaluate two critical metrics: the steady-state availability (the probability that a system is functioning in long runs) and the latency (the time between a service request and the pertinent response). Consequently, we propose a latency-driven availability assessment for multi-tenant service chains implemented via Containerized Network Functions (CNFs). We adopt a multi-state system to model single CNFs and the queueing formalism to characterize the service latency. To efficiently compute the availability, we develop a modified version of the Multidimensional Universal Generating Function (MUGF) technique. Finally, we solve an optimization problem to minimize the SFC cost under an availability constraint. As a relevant example of SFC, we consider a containerized version of IP Multimedia Subsystem, whose parameters have been estimated through fault injection techniques and load tests.