Abstract--We consider the stability and performance of a model for networks supporting services that adapt their transmission to the available bandwidth. Not unlike real networks, in our model, connection arrivals are stochastic, each has a random amount of data to send, and the number of ongoing connections in the system changes over time. Consequently, the bandwidth allocated to, or throughput achieved by, a given connection may change during its lifetime as feedback control mechanisms react to network loads. Ideally, if there were a fixed number of ongoing connections, such feedback mechanisms would reach an equilibrium bandwidth allocation typically characterized in terms of its "fairness" to users, e.g., max-min or proportionally fair. In this paper we prove the stability of such networks when the offered load on each link does not exceed its capacity. We use simulation to investigate performance, in terms of average connection delays, for various fairness criteria. Finally, we pose an architectural problem in TCP/IPs decoupling of the transport and network layer from the point of view of guaranteeing connection-level stability, which we claim may explain congestion phenomena on the Internet.Index Terms--ABR service, bandwidth allocation, Lyapunov functions, performance analysis, proportional fairness, rate control, stability, TCP/IP, weighted max-min fairness.
We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to −∞ and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the process S in continuous time reduces to a Lévy process with heavy-tailed Lévy measure. A central point of the paper is that we make full use of the so-called "principle of a single big jump" in order to obtain both upper and lower bounds. Thus, the proofs are entirely probabilistic. The paper is motivated by queueing and Lévy stochastic networks.
We consider a stochastic directed graph on the integers whereby a directed edge between i and a larger integer j exists with probability p j−i depending solely on the distance between the two integers. Under broad conditions, we identify a regenerative structure that enables us to prove limit theorems for the maximal path length in a long chunk of the graph. The model is an extension of a special case of graphs studied in [18]. We then consider a similar type of graph but on the 'slab' Z × I, where I is a finite partially ordered set. We extend the techniques introduced in the in the first part of the paper to obtain a central limit theorem for the longest path. When I is linearly ordered, the limiting distribution can be seen to be that of the largest eigenvalue of a |I| × |I| random matrix in the Gaussian unitary ensemble (GUE).
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