Classification of Models

Any model can be classified along several broad axes, each splitting the space into camps and giving the modeler a vocabulary for what kind of model is this?. No single axis is the right one; in practice a model is described by where it sits along several of them.

Discrete vs. continuous

The first axis splits models by what kinds of values their quantities are allowed to take.

A discrete model describes the system using countable quantities — binary flags, integer counts, queue lengths. Often the system itself is captured as a graph or automaton: it sits in one of finitely many states and jumps between them as events fire.

A continuous model describes the system using real-valued, smoothly varying quantities — mass, position, velocity, temperature, density. These are tied together by algebraic equations (static relations) or differential equations like ODEs and PDEs (rates of change).

Phenomenon character vs. model character

A natural-but-incorrect intuition is that a discrete phenomenon must be modeled discretely, and a continuous phenomenon must be modeled continuously. The character of the underlying process and the character of the model used to describe it are two separate calls — the modeler is free to mismatch them whenever the mismatched view answers the question better.

Traffic flow through a city. Real traffic is unambiguously discrete — there is no “1.7 cars” arriving at an intersection. Yet two readings of the same scene are both legitimate.

A discrete model tracks the integer-valued state directly — how many cars are at this light, in this lane, in this queue right now?. Events fire (a car joins a queue, a light turns green) and the counts step up or down by one.

A continuous model treats the same traffic as a fluid — picture honey poured through a network of channels and gates, individual vehicles blurring together the way drops of honey do. Streets become the channels, intersections the gates, and “traffic” is now a continuous density moving along each segment at a continuous flow rate. The mathematics of fluids captures city-scale behavior cleanly even though no real vehicle smears the way honey does.

Both views are valid. The choice is governed by the question — which of these light cycles produces the longest queue? points to the discrete model, where does congestion form when this avenue closes? points to the continuous model.

Deterministic vs. stochastic

The second axis splits models by whether their predictions are exact or probabilistic.

A deterministic model produces a single, fully determined output for any given input. Run it with the same starting state and it returns the same answer every time; nothing is left to chance.

A stochastic model produces outputs that carry randomness. The output for any given input is not a fixed value but a random variable — a quantity that takes different values with specified probabilities.

Phenomenon character vs. model character

The same caveat applies as on the previous axis: a deterministic phenomenon does not have to be modeled deterministically, and a random phenomenon does not have to be modeled stochastically. Four short cases sketch the spectrum.

  • Rolling dice — random reality, stochastic model. A single throw of a die is the textbook random experiment: the outcome cannot be predicted in advance. The model follows the reality — the outcome is a random variable taking values 11 through 66, each with probability 1/61/6.
  • Crash test — deterministic reality, deterministic model. A car decelerating into a barrier is governed by Newtonian mechanics — given the initial speed, mass, and structural properties, the resulting deformation is fully determined. The standard simulations match this character: deterministic equations of motion, deterministic stress propagation, a single predicted outcome per scenario.
  • Weather forecast — deterministic or chaotic? Atmospheric dynamics is governed by the deterministic equations of fluid mechanics; in principle the forecast follows from initial conditions. But the system is famously chaotic — tiny errors in initial measurements amplify into wildly different outcomes — and after a few days the deterministic forecast becomes meaningless. Whether the right model is deterministic or probabilistic is therefore genuinely an open question, depending on the forecast horizon.
  • Internet packets — deterministic reality, stochastic model. Packet arrivals at a service unit on the Internet are, in principle, deterministic — every packet was sent by a specific machine for a specific reason at a specific time. But to an observer at the receiving end, with no visibility into the senders, the arrivals are indistinguishable from random. And what that observer typically wants to know is not when does this specific packet arrive? but what is the average load on this server? — a stochastic model captures the average-load question cleanly, even though the underlying process has nothing random about it.