The basic logic used for extracting random values from probability distribution is based on a Cumulative Distribution Function (CDF) and a Random Number Generator (RNG). The CDF hasYvalues that range from 0 to 1. RNGs produce a set of numbers which are uniformly distributed across this interval. For everyYvalue there exists a unique random variate value, X, that can be calculated.
All commercial simulation packages do not require the simulationist to write a program to generate random variates or observations. The coding is already contained in the package using special statements. In such a case, a model builder simply: (a) selects a probability distribution from which he desires random variates; (b) specifies the input parameters for the distribution; and (c) designates a random number stream to be used with the distribu- tion.
Standard probability distributions are usually perceived in terms of the forms produced by their Probability Density Functions (pdf). Many probability density functions have para- meters that control their shape and scale characteristics. There are several standard contin- uous and discrete probability distributions that are frequently used with simulation. Examples of these are: the exponential, gamma, normal, uniform continuous and discrete, triangular, Erlang, Poisson, binomial, Weibull, etc. Standard probability distributions are used to repre- sent empirical data distributions. The use of one standard distribution over the other is dependent on the empirical data that it is representing, or the type of stochastic process that is being modeled. It is essential to understand the key characteristics and typical applica- tions of the standard probability distributions as this helps analysts to find a representative distribution for empirical data and for processes where no historical data are available. Next is a brief review of the main characteristics of the most often used probability distributions for simulation [1–4].
† Bernoulli distribution. This is considered the simplest discrete distribution. A Bernoulli variate can take only two values, which are denoted as failure and success, orxẳ0 and
xẳ1, respectively. Ifprepresents the probability of success, thenqẳ12pis the prob- ability of failure. The experiments to generate a Bernoulli variate are called Bernoulli trials. This distribution is used to model the probability of an outcome having a desired class or characteristic; for example, a packet in a computer network reaches or does not reach the destination, and a bit in a packet is affected by noise and arrives in error. The Bernoulli distribution and its derivative can be used only if the trials are independent and identical.
† Discrete uniform. This distribution can be used to represent random occurrence with several possible outcomes. A Bernoulli (1/2) and Discrete Uniform (DU)ð0;1ị are the same.
† Uniform distribution (continuous). This distribution is also called the rectangular distribu- tion. It is considered one of the simplest distributions to use. It is commonly used if a random variable is bounded and no further information is available. Examples include:
distance between source and destination of message on a network, and seek time on a disk.
In order to generate a continuous uniform distribution,Uða;bị, you need to: generateu, Uð0;1ịand returnaẳ ðb2aịu. The key parameters are:aẳlower limit andbẳupper limit, whereb.a. The continuous uniform distribution is used as a ‘first’ model for a quantity that is felt to be randomly varying between two bondsaandb, but about which little else is known.
† Exponential distribution. This is considered the only continuous distribution with memoryless property. It is very popular among performance evaluation analysts who work in simulation of computer systems and networks as well as telecommunications. It is often used to model the time interval between events that occur according to the Poisson process.
† Geometric distribution. This is the discrete analog of the exponential distribution and is usually used to represent the number of failures before the first success in a sequence of Bernoulli trials such as the number of items inspected before finding the first defective item.
† Poisson distribution. This is a very popular distribution in queuing, including telephone systems. It can be used to model the number of arrivals over a given interval such as the number of queries to a database system over a duration,t, or the number of requests to a server in a given duration of time, t. This distribution has a special relation with the exponential distribution.
† Binomial distribution. This distribution can be used to represent the number of successes int independent Bernoulli trials with probability p of success on each trial. Examples include the number of nodes in a multiprocessor computer system that are active (up), the number of bits in a packet or cell that are not affected by noise or distortion, and the number of packets that reach the destination node with no loss.
† Negative binomial. It is used to model the number of failures in a system before reaching the kth success such as the number of retransmissions of a message that consists ofk packets or cells and the number of error-free bytes received on a noisy channel before thek in-error bytes.
† Gamma distribution. Similar to the exponential distribution, this is used in queuing model- ing of all kinds, such as modeling service times of devices in a network.
† Weibull Distribution. In general, this distribution is used to model lifetimes of components such as memory or microprocessor chips used in computer and telecommunications
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systems. It can also be used to model fatigue failure and ball bearing failure. It is consid- ered the most widely used distribution to represent failure of all types. It is interesting to point out that the exponential distribution is a special case of the Weibull distribution when the shape parameterais equal to 1.
† Normal or Gaussian distribution. This is also called the bell distribution. It is used to model errors of any type including modeling errors and instrumentation errors. Also, it has been found that during the wearout phase, component lifetime follows a normal distribu- tion. A normal distribution with zero mean and a standard deviation of 1 is called standard normal distribution or a unit normal distribution. It is interesting to note that the sum of large uniform variates has a normal distribution. This latter characteristic is used to generate the normal variate, among other techniques such as the rejection and Polar techniques. This distribution is very important in statistical applications due to the central limit theorem, which states that under general assumptions, the mean of a sample ofn mutually independent random variables, that have distribution with finite mean and variance, is normally distributed in the limitn!1.
† Lognormal distribution: The log of a normal variate has a distribution called lognormal distribution. This distribution is used to model errors that are a product of effects of a large number of factors. The product of a large number of positive random variates tends to have a distribution that can be approximated by lognormal.
† Triangle distribution. As the name indicates, the pdf of this distribution is specified by three parameters (a,b,c) that define the coordinates of the vertices of a triangle. It can be used as a rough model in the absence of data.
† Erlang distribution. This distribution is usually used in queuing models. It is used to model service times in a queuing network system as well as to model the time to repair and time between failures.
† Beta distribution. This distribution is used when there is no data about the system under study. Examples include the fraction of packets or cells that need to be retransmitted.
† Chi-square distribution. This was discovered by Karl Pearson in 1900 who used the symbol x2 for the sum. Since then statisticians have referred to it as the chi-square distribution. In general, it is used whenever a sum of squares of normal variables is involved. Examples include modeling the sample variances.
† Student’sdistribution. This was derived by Gosset who was working for a winery whose owner did not appreciate his research. In order not to let his supervisor know about his discovery, he published his findings in a paper under the pseudonym student. He used the symboltto represent the variable and hence the distribution was called the ‘student’s t distribution’. It can be used whenever a ratio of normal variate and the square root of chi- square variable is involved and is commonly used in setting confidence intervals and int- tests in statistics.
† F-Distribution. This distribution is used in hypothesis testing. It can be generated from the ratio of two chi-square variates. Among its applications is to model the ratio of sample variances as in theF-test for regression and analysis of variances.
† Pareto distribution. This is also called the double-exponential distribution, the hyperbolic distribution, and the power-law distribution. It can be used to model the amount of CPU time consumed by an arbitrary process, the web file size on an Internet server, and the number of data bytes in File Transfer Protocol (FTP) bursts [8].