Internet Traffic Tends To Poisson and Independent as the Load Increases

The burstiness of Internet traffic was established in pioneering work in the early 1990s, which demonstrated that packet arrival times are not Poisson, and packet and byte counts in fixed-length intervals are long-range dependent [17, 20]. Here we demonstrate that these results are one end of a continuum of traffic characteristics. At the other end are Poisson behavior and independence. Our study focuses on packets, what devices actually see; we study the statistical properties of packet inter-arrival times and packet sizes. As the traffic load increases — that is, as the number of simultaneous transport connections increases — arrivals tend to Poisson and sizes tend to independence. More specifically, long-range dependence of inter-arrivals and sizes decreases to independence, and the marginal distribution of inter-arrivals tends toward exponential; this happens (1) through time on a single link as the load increases due to daily variation, or (2) at a single point in time as the load increases going from lightly loaded links at the edges of the Internet to heavily loaded links at the core. Convergence is rapid; the packet traffic gets quite close to Poisson and independent loads far less than the maximum we observe. 1. FOCUS This article focuses on packet traffic on Internet links. We study two traffic variables, packet size and packet inter-arrival time. Let be measurements of the packet sizes on a link in a single direction where corresponds to the first arriving packet, to the second, and so forth. Let be measurements of the interarrival times where is for the time between packets 1 and 2, is for the time between packets 2 and 3, and so forth. We will suppose that both variables are stationary time series in over short blocks no larger than 5 minutes. We focus on packet traffic variables because it is packets that network devices must send and receive, and the burstiness of traffic as seen by the devices is determined by the statistical characteristics of and . So our goal is to characterize as thoroughly as possible the statistical properties of these variables. It is for this reason that we do not study packet or byte counts in fixed-length time intervals. As we will demonstrate, it is not possible to readily determine the statistical properties of the packet variables from such counts. We focus on empirical study to derive our results about changes in traffic with load, analyzing packet header traces for 6 links. We rely heavily on mathematical statistical models, but they are validated by the data. This empirical study is necessary to establish the results. Theory is important for understanding packet processes, and we will invoke theory, but it is not enough to determine the statistical characteristics of packet processes. Depending on the assumptions made about the source traffic, theoretical arguments lead either to burstiness or smoothness as the load increases; this was recognized in the very beginning in the initial work on long-range dependence of Internet traffic [17]. For each of the two packet variables and , we study the marginal distribution and the time dependence. We study marginal distributions using quantiles (percentiles). We study time dependence using the power spectrum, the Fourier transform of the autocovariance function; for the variable , we first transform so the that the marginal distribution is , that is, normal with mean 0 and variance 1, then we estimate the power spectrum using the transformed values . We study marginal distributions and dependence for many short blocks separately, and then determine how the statistical properties change with the traffic load. If arrivals are Poisson, then the have an exponential marginal distribution, and the are white noise, so the power spectrum is a constant. If the are independent, then its power spectrum is constant as well. Closeness to Poisson arrivals and independent sizes will be of great importance in our study.