Abstract
We propose a representation of wireless workload patterns as large, sparse matrices and provide a method for stochastically generating experimental workloads from a given matrix. The essential property of the algebraic representation is that the summation of vectors naturally yields a faithful description of the aggregate behavior of the corresponding flows. This deceptively simple property allows us to express many common concepts from traffic modeling succinctly in terms of a few linear transformations. The algebraic representation has many benefits: (1) it makes the meaning of generally understood but vague concepts, such as “uniform behavior,” mathematically precise and unambiguous; (2) it allows us to see clearly, through the lens of linear algebra, the implications of common modeling assumptions; (3) the implementation of traffic models becomes unprecedentedly simple and orthogonal, requiring only a handful of high-level matrix operations, which can be freely composed; (4) the vast body of algebraic theory and highly optimized numerical software may immediately be applied to traffic modeling. We use the paired differential simulation methodology introduced by the authors in previous work to experimentally demonstrate that the general matrix model accurately reproduces realistic network performance (Karpinski et al. 2007a, b). We use the same experimental methodology to explore the implications of various assumptions and simplifications that are commonly made in traffic modeling.
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Notes
We use the common definition of a flow as a sequence of packets sharing the same “5-tuple”: IP protocol type, source and destination nodes, and TCP/UDP port numbers.
The statistical term “marginal” refers to the margins of actuarial tables formerly used for statistical computations. The rows and columns of the table represent possible values of two properties. Each entry in a table contains a count of the number of events falling into the joint category for that row and column. The margins contain sums of the rows and columns, thus giving the unconditional distributions of each property.
This is violated by some quality of service (QoS) schemes. However, we can simply add QoS metadata—such as traffic classes or urgency flags—to our models of user behavior and the rest of our arguments remain valid. The network is still disinterested in the exact content of the data being transported; only the QoS metadata is relevant.
Even this is unclear: generated traffic is only compared with trace traffic using the very same metrics that are explicitly specified as part of the model. Unsurprisingly, the distributions of sampled metrics closely match the distributions those metrics were sampled from. Realism is not double-checked using other statistical metrics or actual performance metrics.
There are many schemes to correct the discrepancy in bytes when p does not divide b evenly. The simplest is to ignore it; we leave more complex schemes to the reader.
Through out this paper, 1 denotes a matrix of all ones; dimensions of the matrix are inferred by context. Where necessary, we disambiguate the dimensions by explicitly specifying the dimensions of the final matrix product, as we do here.
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Karpinski, S., Belding, E.M., Almeroth, K.C. et al. Linear Representation of Network Traffic. Mobile Netw Appl 14, 368–386 (2009). https://doi.org/10.1007/s11036-008-0110-0
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DOI: https://doi.org/10.1007/s11036-008-0110-0