In this paper, we introduce the concept of Random Access Channel (RACH) resource slicing and investigate the performance of a sliced Radio Access Network (RAN). We consider three generic service types that the fifth-generation (5G) wireless technology is claimed to support, and refer to them as enhanced mobile broadband (eMBB), massive machine-type communications (mMTC), and ultra-reliable low latency communications (URLLC) slices. Each slice can be configured with dedicated and shared subsets of RAN and RACH resources, also known as RACH preambles. We propose an analytic framework that allows us to obtain the blocking probability of each slice in a closed-form. We compare different resource allocation policies in sliced radio access system and investigate their impact on a slice performance.