Modelos de tráfico

Time is a tricky thing. We take it for granted in everyday life, but the more closely you think about it, the more intricate and complicated it becomes. That this is true is evident from the many different schools of thought on how to model 'time' in a mathematical or philosophical way (e.g. Cleugh, 1937). The first point of divergence is the distinction made between absolute time, where time is measured from a fixed starting point (as in a calendar), and relative time, where any one point in time is measured relative to any other. Relative time allows imprecision in that it is not necessary to know the exact time at which an event occurred, just whether it occurred before or after another event. Absolute time, by contrast, is necessarily much more precise: it describes exactly when an event occurred.

This is not to say that relative time cannot have precision – it is possible to say, for example, that one event occurred 3.142 seconds after another, in which case it is quantitative and has a metric for time (i.e. time is measurable). Absolute time always has a metric, since it measures time since a common starting point, whereas relative time just means it is not necessary to have a common frame of reference to relate all events to each other.

Figure 10 – Absolute versus relative time

Figure 10 shows the difference between absolute and relative time. Absolute time has a starting point and all events are measured from that point, so X may occur at time 15, Y at time 33, and

Z at time 57, for example. In relative time, it is only possible to know that Y occurred after X (or X occurred before Y) and similarly Z occurred after Y. If a metric exists, we can also say that Y occurred 18 time units after X, for example.

A second division between different models of time is whether or not time is represented as being continuous (also known as dense time) or discrete. In a continuous model of time, there are no gaps between moments – there is infinite precision. In a discrete model of time, time is represented by a series of moments or states that are clearly delineated. The common analogy used in this case is the difference between real numbers and natural numbers; in the former case, there is always another number between any two numbers, in the latter case, there is not.

Figure 11 – Continuous versus Discrete time

Figure 11 shows the difference between continuous and discrete time. In the former case, it is possible for events to occur at any point along the scale, e.g. at 1.5, 2.222, 7.123 etc. In discrete time, events can only occur at discrete moments, not in between, e.g. only at 1, 2, or 7. Continuous time is more frequently used in absolute time models whereas discrete time is often used in relative models of time (e.g. in which each discrete time value is a separate state).

Yet another variation, more philosophical this time, is between linear time and branching (or non-linear) time. With linear time, time is represented by a single time line, but with branching time, there can be multiple paths in the future (and optionally in the past), and events cause different timelines to split from the original line (e.g. in one path the event was true, in the other the event was false). These two options are also known as deterministic and non-deterministic time, because branching time allows the possibility of events occurring in some future time lines but not others. One further point to note here is the notion of bounds. A linear time model can be bound in the past (i.e. it has a fixed starting point) and in the future (it has an end point), one or the other, or neither. Similarly, branching time can branch only in the future, only in the past, or both.

Figure 12 – Linear vs. Branching time

There is one more common distinction in time models – the choice of the unit of time itself. The first option is the notion of a point-based time, where essentially time is represented by a series of moments in time (either continuous or discrete). Intervals are represented as a collection of time points. For example, looking at the linear time line in Figure 12, we could define an interval as {1, 2, 3} containing three moments and therefore lasting three units of time, or we could simply define it as [1,3] by giving a start and end point. The other option is to use interval-based time, in which the interval is assumed to be the atomic unit of time. Again using Figure 12, the base interval unit would be the time between 0 and 1. Whichever interval is chosen is assumed to be atomic, i.e. indivisible, so this obviously has implications for the granularity of time the model can represent.

A good analogy to these two ideas is the notion of a measurement of length. We can define a distance using a standard unit of measurement, e.g. the metre. This is the interval-based method. The problem with this is that we must make sure our unit is sufficiently small to be able to achieve the level of precision we require, otherwise something may need to take a fraction of an interval, which the model does not allow. The alternative is the point-based method, which is analogous to measuring distance using a series of markers, like milestones. Every time you pass a milestone, you have travelled further. The obvious flaw with this approach is that there are mile-long "gaps" between milestones, i.e. in other words, potentially minute periods of time between the points. This can pose problems: for example, if one period of time starts when another ends, do they overlap at the same point in time (so that both are true at that point) or is there a minute amount of time between the first ending and the second starting? This problem is illustrated in Figure 13:

Figure 13 – The Point-Based Time Problem

In the first possibility, we define the duration of event X as [0,10] and Y as [10,15] – but in that case, is point 10 included in both events? The second possibility is to make sure the events do not overlap, e.g. X = [0,9] and Y = [10,15]. But in that case there appears to be a "gap" between points 9 and 10 that the events do not cover.

In Allen (1983), the author categorises the different approaches to representing time into four general groups:

• State space approaches

State space approaches work by modelling the system as a database of information. Each state is a database representing the system at one time; when an event occurs, it causes a transition from one state to another, thereby representing a progression through time. The items of information in the database may be true for one state but false for another and so simulate the changes to the system over time. Events are persistent in that once an event causes a change in one facet of the system, the changed facet of that state persists unless another event changes it again. State-space approaches are necessarily discrete, point-based, and relative, but can be either branching or linear.

• Date-base/Date-line systems

In these systems, information is stored in a database and indexed with a date. By comparing the dates of two items of information, it is possible to discern the temporal order in which they occurred. Unfortunately, it is often difficult to assign the necessary precision or imprecision to dates, making it difficult say that two events did not occur or did occur at the same time, for example. This scheme relies on the use of linear, absolute times for all items of information, but the model of time can be continuous or discrete and either point-based or interval-based.

• Before/After chaining

Before and after chaining is a way of explicitly modelling relative times of events by linking them according to the order in which they occur. However, for large numbers of events there are problems of storage and overly-expensive searching, since searching is typically linear. It is also difficult to represent durations (since each 'event' is a link in the chain). Before/after chaining is usually independent of whether time is point-based or interval-based, or continuous or discrete.

• Formal Models

Formal models apply to a range of disciplines, from philosophy to artificial intelligence. One such formal model is situation calculus, which is the principle behind state-space approaches; in situation calculus, time is represented as a series of states, each of which represents a system at a point in time. Transitions between the states are actions or events. The main drawback with situation calculus is that only one state is true at a time – there is no notion of overlap.

Allen also suggests that the formal concept of point-based time is unhelpful, because even a seemingly instantaneous event can usually be further decomposed (Allen, 1983), and if the original event occurs at one point, then it is not clear when the sub-events occur. Instead, he suggests the use of an 'interval' unit instead – the smallest relevant period of time. We use this concept all the time – rather than measuring time in seconds all the time, we measure it in minutes, hours, or days etc according to our needs. In effect, we treat a period of time (a time interval) as if it were indivisible. Allen's time intervals, unlike points, are consecutive – there are no gaps between them, however small. Furthermore, to allow conclusions to be drawn about the relative time of these time-intervals, Allen provides 13 temporal relations to cover all possible relations between two intervals in time:

Figure 14 – Allen's 13 Temporal Relations

The choice of time model (or lack thereof) can have important repercussions when looking at different approaches to modelling time, whether in fault trees or elsewhere. In particular, the choices of relative vs. absolute and linear vs. branching time have significant impacts on how any logic underlying any such model would work. However, it is often the case that certain models of time suit certain type of applications better than others, and in such cases it is possible to see that some choices are better than others.