Piecewise Linear Representation (PLR) has been a widely used method for approximating data streams in the form of compact line segments. The buffer-based approach to PLR enables a semi-global approximation which relies on the aggregated processing of batches of streamed data so that to adjust and improve the approximation results. However, one challenge towards applying the buffer-based approach is allocating the necessary memory resources for stream buffering. This challenge is further complicated in a multi-stream environment where multiple data streams are competing for the available memory resources, especially in resource-constrained systems such as sensors and mobile devices. In this paper, we address precisely those challenges mentioned above and propose efficient buffer management techniques for the PLR of multiple data streams. In particular, we propose a new dynamic approach called Dynamic Buffer Management with Error Monitoring (DBMEM), which leverages the relationship between the buffer demands of each data stream and its exhibited pattern of data values towards estimating its sufficient buffer size. This enables DBMEM to provide a global buffer allocation strategy that maximizes the overall PLR approximation quality for multiple data streams as shown by our experimental results.