This paper considers random cellular systems in one, two, or three dimensions. We study the carrierto-interference ratio (CIR) and the carrier-to-interference-plus-noise ratio (CINR) at the mobile-station (MS), by using the so-called shotgun cellular system (SCS) model for the cellular system. In a SCS, the base-stations (BS) are placed randomly according to a non-homogeneous Poisson point process. For the SCS, analytical expressions for the tail probabilities of CIR and CINR are obtained. These results are generalized for an SCS affected by random shadow fading with any general distribution, for any arbitrary path-loss models, and for the MS located anywhere in the SCS. Further, for a special case of the SCS, namely, the homogeneous SCS, it is shown that the shadow fading has no effect on the CIR, and the effect of random transmission gains, power control, multiple channel reuse groups are analyzed. The techniques developed for the analysis of the CIR and the CINR holds applications beyond cellular networks and can be used to study several important scenarios in both uplink and downlink for cognitive radios, femtocell networks and other heterogeneous and multi-tier networks.