In recent years, Edge AI has become more prevalent with applications across various industries, from environmental monitoring to smart city management. Edge AI facilitates the processing of Internet of Things (IoT) data and provides privacy-enabled and latency-sensitive services to application users using Machine Learning (ML) algorithms, e.g., Time Series Classification (TSC). However, existing TSC algorithms require access to full raw data and demand substantial computing resources to train and use them effectively in runtime. This makes them impractical for deployment in resource-constrained Edge environments. To address this, in this paper, we propose an Adaptive Brownian Bridge-based Symbolic Aggregation Vector Space Model (ABBA-VSM). It is a new TSC model designed for classification services on Edge. Here, we first adaptively compress the raw time series into symbolic representations, thus capturing the changing trends of data. Subsequently, we train the classification model directly on these symbols. ABBA-VSM reduces communication data between IoT and Edge devices, as well as computation cycles, in the development of resource-efficient TSC services on Edge. We evaluate our solution with extensive experiments using datasets from the UCR time series classification archive. The results demonstrate that the ABBA-VSM achieves up to 80% compression ratio and 90-100% accuracy for binary classification. Whereas, for non-binary classification, it achieves an average compression ratio of 60% and accuracy ranging from 60-80%.