1 Stochastic Performance Analysis of Non-Feedforward Networks Chengzhi Li and Wei Zhao Abstract— Many Internet applications are both delay and loss sensi- tive, and need network performance guarantees that include bandwidth, delay/delay jitter, and packet loss rate. It is very important to quan- tify and exploit the capabilities of guaranteed service provisioning of com- munication networks. In this paper, we study the queueing behaviors of non-feedforward networks 1 with FIFO scheduling discipline and Regu- lated, Markov On-Off, and Fractional Brownian traffic sources. We de- velop 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 deriva- tion of a probabilistic bound on the delays experienced by traffic in non- feedforward, based on an assumption, i.e., the tail probability of the dif- ference 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 algo- rithms for non-feedforward networks to provide performance guarantees to various applications with diverse performance requirements. Index Terms: Non-feedforward Network, Probabilistic Delay Bound, Large Deviations, Fixed Point. I. I NTRODUCTION W ITH the rapid evolution of communication technology and the explosive growth in network services, the Inter- net is becoming a ubiquitous means for delivering various infor- mation with heterogeneous network traffic characteristics and diverse performance requirements. Many Internet applications are both delay and loss sensitive, and need network performance guarantees that include bandwidth, delay/delay jitter, and packet loss rate, e.g., IP telephony and interactive multimedia over the Internet. Thus, it is very important to quantify and exploit the capabilities of guaranteed service provisioning of communica- tion networks. Since the emergence of the Internet, extensive efforts have been made by the communications and networking research community to establish comprehensive frameworks for network quality of service (QoS) offering. Many elegant re- sults including network calculus and effective bandwidth have been obtained for the performance guarantees of feedforward networks, as illustrated in books [9], [5], survey papers [28], [16], [17], [14], recent papers [12], [15], and references therein. There exist non-feedforward networks, e.g., the networks with the IETF Differentiated Services (DiffServ) architecture [2] and non-feedforward routing protocols [25]. Since the vir- tual feedback phenomenon may be caused by the union of mul- Dr. Chengzhi Li is with the University of Houston, Texas, USA. Dr. Wei Zhao currently serves as the Rector of the University of Macau, Macau, China. 1 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. tiple acyclic traffic routes [13], [23], [8], the queueing behav- iors of non-feedforward networks are much more complicated than those of feedforward networks. A non-feedforward net- work even with low link utilization (traffic load) may be unsta- ble, i.e., the queue length at some nodes may be un-bounded [1]. Thus, the obtained results for feedforward networks, which are stable if and only if the utilization of each link is smaller than 100% [13], [23], [8], cannot be applied to non-feedforward networks. Several results have been provided for the deterministic sta- bility criterion and delay bounds for non-feedforward networks. One of the earliest such results was given in [13] for a four node network with a special ring topology and first come first out (FIFO) scheduling discipline. A conjecture was raised in [13], [23] that networks with ring topology and work conserving scheduling disciplines are stable if the utilization of each link is less than 100%. This conjecture was proven in [26]. Based on the fixed-point theory, a general framework has been provided in [21] for computing deterministic end-to-end delay bounds for non-feedforward networks with FIFO or static priority (SP) scheduling discipline. Moreover, a stability criterion was devel- oped in [7] for deterministic non-feedforward networks with a routing scheme analogous to one used in the Jackson network and work conserving scheduling disciplines. For networks with the DiffServ architecture and aggregated scheduling disciplines, a deterministic bound on the end-to-end delays experienced by any expedited forwarding (EF) packet was provided in [10] for rather low link utilization. A similar result was also obtained independently and contemporaneously in [11]. The multiplexing, which is at the core of the transmission technologies of high speed communication networks, allows multiple traffic flows to share a common link with a finite buffer to absorb traffic fluctuations. Unfortunately, the aforementioned deterministic results cannot be used to exploit the statistical mul- tiplexing gains of non-feedforward networks. To the best of the authors‘ knowledge, there only exist a few papers [18], [8], [3], [27] that address the issues of stochastic performance analysis for non-feedforward networks. In [18], a simple network with two nodes was studied. In [8], the results of [7] have been extended to stochastic non- feedforward networks. The obtained stochastic stability crite- rion relies on an assumption that for any given node, the traffic arrivals from any other nodes form independent sequences of in- dependent and identically distributed (i.i.d.) Bernoulli random variables. This assumption may not be suitable for networks with the virtual feedback phenomenon. In [3], an interesting, but radical conjecture was proposed for networks with DiffServ architecture and non-preemptive aggregated priority scheduling discipline. That is, the delay jitter remains negligible through- out the network if two conditions are satisfied. The first con-