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Statistical Effects in Independent Demand Aggregation

Suppose that we are a service provider with m customers, which conveniently are named 1,2,3, . . . m.4 Let’s also suppose that we can characterize each of their demands over time as D1(t), D2(t), D3(t), . . . Dm(t). To illustrate the smoothing effect of statistical multiplexing, suppose that these demands are independent and uncorrelated; in other words, knowing that one customer has a given demand at a given time is of no help in determining what the demands of any of the other customers are at that time. Moreover, to keep things simple, let’s assume that the demands have identical variances (and thus standard deviations) and identical means. Note that the underlying distributions don’t need to be the same—one can be a triangular, one can be normal, one can be uniform, one can be exponential; all we are looking for is the same mean and standard deviation.

Under those circumstances, since the standard deviations and means of each customers demand function are identical, the coefficient of variation of each function is identical.

What is the aggregate variation if we multiplex the m demands into a shared resource pool offered by a service provider?

To solve this, we note that the standard deviation of each customer’s demand function is identical, say, σ, where σ = σ(Di (t)). Similarly, the mean of each customer’s demand function is identical, say, μ, where μ = μ(Di (t)). Therefore, the coefficient of variation of any demand function ...