Simulations

ComplexModels.com is Joe Weinman's simulation web site addressing the structure, dynamics, and financial analysis of utility and cloud computing, random graphs, power law preferential attachment graphs, and other simple models that illustrate complex, emergent characteristics or behavior.

There is a theory that networks should be "stupid" and provide "dumb pipes," i.e., all intelligence should be at the endpoints. This simulation shows that route optimization based on consideration of both server response time and path latency can lead to optimized performance. The degree of improvement is coupled to the number of optional path and server combinations: the more options, the greater the speedup, but with diminishing returns to scale. The simulation model supports theoretical analysis based on expected values of minimum values given order statistics calculated using the Beta distribution. Go directly to the simulation model or first review how it works.

There are a variety of tradeoffs in the design of networks, whether they be highways, airport location and flight routing, or communications networks. One strategy is the point-to-point network, which offers the shortest possible distance and therefore latency between any two points, but at the expense of a quadratically increasing number of connections. Another option is the hub-and-spoke network, which requires fewer total links growing linearly in the number of nodes, but can increase latency, as any traveler who has had to change planes and fly indirect routes knows. This model provides Monte Carlo simulation support for the theoretical expected values. Go directly to the simulation model or first review how it works.

A simulation of a duopoly with one flat-rate, one pay-per-use provider, and customers with dispersed heterogeneous consumption, based on Joe Weinman's "The Market for Melons" analysis. The flat-rate price always reflects the average consumption of the patrons of the flat-rate provider, whereas the pay-per-use provider price is based on each customers exact usage. Initially, heavy users migrate to the flat-rate plan and light users migrate to the pay-per-use one. This causes price rises with more repercussions, until eventually virtually all customers are patrons of pay-per-use. Go directly to the simulation model or first review the paper (PDF) to see how it works.

This model illustrates the relationship between various demand curves, capacity strategies addressing those types of demand singly or in combination, and the economics of tradeoffs. Is it better to use solely static capacity engineered to peak demand? Or to use less static capacity and accept penalties associated with inability to meet peak requirements (e.g., loss of revenue associated with unserved customers)? Or use purely utility resources to meet all demand? Or to use a hybrid approach? The answer is: it depends, and this model shows the relationships between the different factors that would lead to an optimal selection. Go directly to the simulation model or first review how it works.

This model illustrates the benefits of pooling resources, which may be geographically dispersed. Aligning each demand source individually with a partitioned set of resources means that each set of resources must be engineered to meet the peak demand associated with that resource. Pooling resources and meeting demand via any available resource has the benefit that the capacity required is not the sum of the peak demand, but the peak of the sum of the demands, which is provably no larger, and is typically smaller. Consequently, pooling resources and distributing load across them is usually a good idea (subject to ancillary costs not addressed in this model. A special case of this is geographic dispersion across time zones, where some demand types, e.g., 9 to 5 weekday work, can lead to dramatic savings through a "follow the sun" approach. Go directly to the simulation model or first review how it works.

This model illustrates the benefits of dispersion for latency reduction, as well as the diminishing returns of additional node builds, leading to a "sweet spot" number of nodes. This model is relevant in scenarios as diverse as locating coffee shops, fast food restaurants, distributed interactive computing environments, and content delivery architectures. Dispersion can have additional benefits, e.g., enhanced business continuity, as well as costs, e.g., operations and real estate, but these are not modeled here. Go directly to the simulation model or first review how it works.

This model illustrates the two main benefits of aggregating demand: reduction in peak capacity requirements as well as increase in utilization. The increase in utilization comes about in two ways. One is that the same demand is served by a smaller-sized pro forma capacity, but the second is that aggregation of random variables preserves absolute variance, but reduces relative variance. The so-called "coefficient of variation" is a measure of variability relative to the mean, and this is reduced as more and more random variables are summed. In a cloud or utility environment, this is one of the drivers of service provider economics. Go directly to the simulation model or first review how it works.

This model also illustrates the benefits of aggregating demand: reduction in peak capacity requirements as well as increase in utilization. Strictly speaking, the peak does not change across all possible combinations, but the mass near the peak does, therefore reducing the cost of unserved demand. The distribution of the sum increasingly is normal, in accordance with the Central Limit Theorem. This model shows the effect by determining the distribution of sums of dice that are randomly rolled, and seeing how summing more and more dice results in a tighter (reduced coefficient of variation) distribution. Go directly to the simulation model or first review how it works.

Mathematical modeling and empirical analysis of Internet connectivity by Professor and author Albert-Laszlo Barabasi has demonstrated that network evolution by "preferential attachment" leads to so-called power-law, aristocratic, or scale-free node degree distributions. In other words, the "rich" nodes get "richer" as new entrants to the network preferentially decide to attach to them. This simulation shows the difference between preferential attachment and random attachment, up to just over 1000 nodes, and enables viewing of the resulting network from both a node degree distribution perspective as well as an actual graph. Go directly to the simulation model or first review how it works.

In "Group Theory in the Bedroom", Brian Hayes reviews so-called "Yard-Sale Simulations," or asset exchange models, where pairwise traders operate in a market on a zero-sum basis to randomly win or lose trades. For very small "bets" and short time periods, an initially equal distribution of wealth will evolve to a normal distribution, as wins and losses tend to cancel each other out. However, for larger bets and/or longer periods of time, the market evolves to concentrate wealth increasingly, eventually resulting in a "winner-take-all" results. This is because once a trader goes broke, no matter how unlikely that is, they cannot reenter the market. Interestingly, even when wealth is initially equal across all traders, it can rapidly evolve to a power-law or exponential distribution. Go directly to the simulation model or first review how it works.

This is a simplified version of the Yard-Sale simulation. Investors either win or lose on each trial, with the amount of the gain or loss dependent on the investment rules. Due to Brownian effects, eventually investors can go broke, at which point they exit the system. This results in increasing concentration of wealth. Different rules lead to different distributions: an equal distribution of wealth evolves to a normal distribution shifts to what appears to be a power-law or exponential distribution. Go directly to the simulation model or first review how it works.

This illustrates the random processes at work behind such phenomena as Brownian motion. Ten particles begin the simulation in the same location, but then move, each trial, based on a rule. Some variations are one-dimensional, others are two-dimensional. The one-dimensional rule is similar to a simplified version of investor risk model, which is a simplified version of the asset exchange model (yard-sale simulation). The only difference is that any particle (i.e., investor), can go deeply into debt, so remains in the simulation. Consequently, rags to riches is not unlikely. Go directly to the simulation model or first review how it works.