Modeling a Physical Process
Most modern industrial and nonindustrial operations are composed of sequences of physical processes, such as making a product, scheduling trains, electronic transactions, and so forth. In an effort to optimize these physical processes, “what-if” analyses or goal-seeking procedures need to be applied to each physical process. This approach has considerable advantages, because experimenting on a model is cheaper, less dangerous, and often can do operations impossible or impractical in the real world [Arsham, 2000]. However, in order to ensure that results from the virtual models will represent the behavior of the physical models, the virtual process should replicate all behavioral aspects of the physical process.
For example, training pilots in a flight simulator is significantly cheaper and safer than flying a real plane, since the simulator will not endanger the pilot and others. Even when considering the expense of developing such simulators, building the virtual models is still cheaper than the crashes that would likely take place if training in real commercial or military jets. A flight simulator is a model that normally has two components: the physical component and the digital component. The physical component is the mock-up of a real cockpit, and the digital component is the computer that controls the simulation.
Simulations can be done with a completely physical model, as in an architect’s scaled model. Electronic component manufacturers use digital models to test and debug circuits prior to manufacture [Arsham, 2000]. Most models in the mining industry are digital computer models. Such models can function either as a simulation system or an expert system.
There are many different simulation types, such as discrete event, continuous, hybrid, and so forth. A simulation may incorporate Monte Carlo techniques, Markov chains, and/or apply to queue systems.
- Discrete event simulations process discrete events that occur at random times through a central processing unit. In discrete event systems many events can occur simultaneously [Schriber, 1997]. An example of a discrete event process is that of cars approaching an intersection. Cars arrival times are random events and each arrival time is a discrete event.
- Continuous simulations comprise variables that act as a function of time. An example of an application of a continuous simulation is a continuously changing system like a car’s suspension [Gordon, 1969].
- There are also combinations of discrete and continuous simulations called combined simulations. Hybrid simulations are analytical subsystems in discrete models.
- Monte Carlo simulations use stochastic processes to describe nonprobabilistic problems. In other words, a Monte Carlo system uses a random event generator, such as an electronic die, to create the random variables or events needed for the simulation
- Queuing systems refer to specific groups of physical processes where serving and requesting mechanisms are in place. If the serving units cannot serve the requesting items, then the requests are placed in a queue until the servicing mechanisms are available.
- A Markov process looks at a sequence of events and calculates the probability of one event following another. A string of such events is then called a Markov chain.
In almost every application that requires a model of a physical process, there is a statistical component that must be considered. For any single process, there is a statistical uncertainty associated with it. The type of uncertainty that needs to be described in the model depends on the desired output from the model. For instance, a robot arm loading boxes onto a conveyor belt will take an average amount of time to place a new box onto the belt. If the model’s purpose is to simulate the operation and loading of the belt, then the amount of time for the arm to load a box should be studied to find the time distribution. However, if the model is of the robot arm itself, then the rates of each movement should be studied. The results from these studies can be shown to be consistent with existing probability functions [Tocher, 1963]. A process that has no uncertainty can be handled deterministically; in other words, there is a 100% probability that the next value will be the same as the previous. A probability function is a function of a random variable that yields the probability of a continuous random variable in a system
Random variables take on values based solely on chance. The development of probability distributions based on random variables is the central mechanism of stochastic analysis and statistics. Figure 2.1, shows the probability of rolling a specific side of a die. Because there are only six possibilities, this is a discrete distribution. Figure 2.2 shows a continuous distribution based on the normal distribution, discussed later. Continuous distributions are described by equations that are parameterized to certain values that describe the statistical make up of the process described [Hamburg, 1987].
For simulations to calculate a value for a random variable, the simulator must create a random number. Computers generate random numbers in a uniform fashion. This means that the probability of a random number being any number in the given range is the same. This distribution is discussed below. Because computers calculate random number in this manner, the probability must be translated into the distribution that is being used to model the process. This is done using the cumulative distribution function that is the sum of the probabilities to that value. Figure 2.3 shows the cumulative distribution function for the probability function in Figure 2.1. Cumulative distribution functions always range from zero to one. The random number is taken as the cumulative probability, and the value is read from the range. Figure 2.4 shows the cumulative distribution for the example shown in Figure 2.2. For the purposes of simulating physical processes, the uniform, normal, exponential, and Poisson distributions are commonly used [Hamburg, 1987].
The uniform distribution can also be referred to as the rectangle distribution. The distribution is used to model really random events. This is because the probability of any value is exactly the same as any other value. It is the simplest of all the distributions used in simulations [Hamburg, 1987]. Equation (1) shows the function for calculating the probability density function for the uniform distribution (see Figure 2.5)
The normal distribution is central to statistical theory and practice. Graphically, the normal curve appears as a bell and can be defined based on two parameters: the mean value and the standard deviation, as shown in equation (2), where σ is the standard deviation, µ is the mean, x is the value of interest. Figure 2.2 shows the complete normal distribution function. The normal distribution is applicable to most types of random variables that describe the majority of physical processes. A characteristic of this distribution is that two-thirds of the area under the normal distribution curve lies within one standard deviation from the mean. Additionally, 99% of the area is within three standard deviations. Hence, the normal distribution describes well processes that do not vary widely. Many statistical calculations are based on the normal distribution [Hamburg, 1987]. In mining applications, the normal distribution is used for travel times. It is particularly good for travel times of vehicles that have a governor and their top speed is limited [Sturgul, 2000]
1.1 ENERGY ISSUES.
1.2 COAL AND ENERGY
1.3 NEED FOR DEVELOPMENT OF A ROOM-AND-PILLAR CONTINUOUS MINING SIMULATOR.
2. MODELING A PHYSICAL PROCESS
2.1 PROBABILITY DISTRIBUTIONS .
2.2 EXPERT SYSTEMS
2.3 SUMMARY AND CONCLUSIONS.
3. EXISTING SIMULATION PACKAGES
3.1 SIMULATORS IN TRADITIONAL SIMULATION LANGUAGES
3.2 SIMULATORS IN DEDICATED PROGRAMMING LANGUAGES
3.3 CONCLUSIONS AND SUMMARY
4. SIMULATOR STRUCTURE
4.1 A FRAME-BASED EXPERT SYSTEM
4.2 SUPPORT OBJECTS
4.3 EQUIPMENT OBJECTS
4.4 REPORT GENERATION
4.5 SUMMARY AND CONCLUSIONS
5. SUPPORT SYSTEMS
5.1 DATABASE SYSTEM
5.2 FRONT END
6 CASE STUDIES
6.1 A SIMPLE CASE
6.2 CONSIM EXAMPLE REVISITED
6.3 LONGWALL DEVELOPMENT SECTION
6.4 CONCLUSIONS AND SUMMARY
7. CONCLUSIONS AND FURTHER WORK
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