A Hierarchical Framework for Modeling and Forecasting Web Server Workload

Hierarchical Forecasting of Web Server Workload Using Sequential Monte Carlo Training

“…1 shows the server workload obtained by aggregating the hypertext transfer protocol (HTTP) service requests for a commercial website over 5-min intervals over a 13-day period, giving a total of 288 intervals in a day. This is the same data as used in [12], and we will employ this time series throughout this paper as a working example to demonstrate the performance of the proposed algorithm. The time series is first converted into logarithmic domain to reduce the dependence of the local variability on the local mean of the untransformed data.…”

“…Note that for a given , the time index for the observation in the th time interval in the th period is given by , and . Although, several methods exist for modeling such a time series, we follow the hierarchical approach developed in [12], which not only provides point predictions, but also simultaneous confidence bands, that can be used to support flexible (probability-based) service-level agreements. Furthermore, by allowing the model parameters to change with time, we can handle nonstationarity in both the long-term patterns as well as short-term fluctuations.…”

“…Then, from (2), using the full basis, we can write (11) where are the coefficients associated with the complete basis decomposition, is the matrix of all the basis functions, and its inverse has an analysis filter bank interpretation. In other words, denoting the th row of by , (11) can be written as (12) which is nothing but the output obtained on passing through a filter bank consisting of finite-impulse-response (FIR) filters, with being the impulse response of the th filter. After having obtained the analysis filter bank output defined in (12), the data can be reconstructed according to (13) which can be considered as the decomposition of into component waveforms, whose shapes are determined by the basis functions .…”

“…In [12], two measures on the component waveforms are suggested to quantify the behavior of to aid in the selection of , namely, the coherence measure and the energy measure. The coherence measure is defined as (14) where is the sample mean and is the sample variance of , obtained as and (15) The th waveform is said to be completely coherent if , and incoherent if…”

“…In this paper, we follow the hierarchical approach proposed in [11] and [12] in which the time-series prediction is decomposed into two steps: first a prediction of the long-term component, which primarily captures the nonstationarity of the data, is performed, and then the residual short-term process, which captures both the long-term prediction error and the short-term component of the time series, is processed. As demonstrated by the results, the two-scale decomposition captures the underlying statistics of the data fairly well.…”