A theoretical method of evaluating degradations of variable rate coders in a multichannel digital speech interpolation (DSI) system is developed. Each of the coder outputs has a variable rate based on the algorithm. The DSI system multiplexes the outputs of these variable rate coders into a fixed bit rate channel. During periods of high activity all active users are served, but at a reduced rate depending on the demand. The degradation due to high activity is shared by all active users. This system avoids speech clipping and "freeze-out" distortion. Theoretical expressions of the system overload probability and the probability of degradation to a particular user in the DSI system are derived. Two types of variable rate coders, namely, a constant quality subband coder and a constant noise subband coder, are chosen and used as examples. Comparisons of the degradations are made between the theoretical results and computer simulated results for the two types of variable rate coders, and close agreement is observed. The theory is applicable to other variable rate coding algorithms as well. In this study, all of the simulations are made at 40 percent speech activity and the average rate of the variable rate coders is close to 16 kbits/s. Objective quality measures indicate that in a system with a trunk size larger than 40, the variable rate coder DSI system can achieve a 2:1 compression with a degradation of less than 1 dB compared to non-DSI variable rate coders. This corresponds to a total gain of 8:1 when compared to 64 kbit/s PCM.