This book provides a mathematical analysis of bandwidth allocation on end-to-end communication networks with multiple traffic classes, where bandwidth is determined under the budget constraint. Due to the limited budget, there exists a risk that network service providers cannot assert a 100% guaranteed availability for the stochastic traffic demand at all times. The installed bandwidth allocation would be used to maintain the guaranteed connection availability while the blocking probability is kept below a certain negotiated level. We derive the blocking probability of connections as a function of bandwidth, traffic demand and the available number of virtual end-to-end paths for all service classes. Monotone and convex properties of the blocking probability are shown in both theoretical construction and numerical examples. The sensitivity analysis and economic elasticity notions are also proposed to investigate the long-run average revenue of network service providers. The results derived in this book can be applied in the design of broadband communication networks under the budget control for sharing bandwidth in terms of blocking/congestion costs.