Many Internet applications are both delay and loss sensitive, and need network performance guarantees that include bandwidth, delay/delay jitter, and packet loss rate. It is very important to quantify and exploit the capabilities of guaranteed service provisioning of communication networks. In this paper, we study the queueing behaviors of non-feedforward networks (a non-feedforward network is a network in which at least one set of acyclic traffic routes forms a cycle; a feedforward network is a network in which any set of acyclic traffic routes does not form a cycle) with FIFO scheduling discipline and Regulated, Markov On-Off, and Fractional Brownian traffic sources. We develop a new methodology to analyze the probabilistic bounds on the delays experienced by traffic. By leveraging the large deviations and fixed-point techniques, we turn probability problems into deterministic optimization problems and translate a probabilistic delay bound into a fixed point of a non-linear real function. Our contribution in this paper is the derivation of a probabilistic bound on the delays experienced by traffic in non-feedforward networks, based on an assumption, i.e., the tail probability of the difference between the beginning time of a busy interval of a server and the earliest arriving time at the corresponding network ingress of the traffic arrivals that arrive at this server during this busy interval can be bounded by the maximum of the violation probabilities of the accumulative upper stream delay bound suffered by this server's traffic arrivals. Consequently, our new results not only consummate the theory of stochastic analysis of network performance, but also facilitate the design of protocols and algorithms for non-feedforward networks to provide performance guarantees to various applications with diverse performance requirements. © Springer Science + Business Media, LLC 2009.