Products And Performance Throughout Las Vegas

Scientists dedicated to the discipline known as queuing theory have developed and continue to develop accurate solutions for particular cases of queuing systems with input and service rates represented by specific functions. Also, another research current queueing system develops sufficient approximations that can be used without knowing the specific statistical function that would represent the queue. In practice, it can be considered that three elements are necessary: a) to know all the distributions to have a clear knowledge of the possibilities of characterization a certain distribution. B) to know a procedure to establish, via statistical inference, what is the distribution of a given sample. C) to be able to calculate the mean and the standard deviation (and with it the coefficient of variation) of a process in function of a sample. The statistical distributions of discrete type take values of a finite set of possibilities.

In queuing theory they are relevant because they represent the number of clients in a time interval. If the possible occurrences are a finite and uniform set of values (eg the launching of a perfect die) it is known as the uniform discrete variable. If the variable is given between a and b values, the mean of the distribution is (a+ b) and the varianceif the probability of each occurrence is different, simplest of all bernoulli distributions where the variable can only take two values (eg boy or girl, a or b) with a certain probability p for the first member of the pair, which is often queueing system software called success. The mean is p and the variance is p (-p) the binomial distribution represents the probability of obtaining k events a with probability p, from n trials. It is therefore the sum of bernoulli n probability p. The mean is np and the variance is np.

The geometric distribution represents the probability of obtaining the first occurrence a at launch n. This variable has an infinite range but remains discrete. The mean is p and the variance is (-p) p^ it also has an infinite range known as poisson in which occurrences are represented for a large and independent set of events distributed over space or time the distribution has interesting mathematical properties that make it widely used. The mean is λ and the variance is also λ.when the occurrences can take values within a continuous range the distributions are of the continuous type. In queuing theory they are especially suitable for representing time intervals between consecutive events. The uniform continuous takes values equivalent to a given range [a, b].

The mean of that function is (a+ b) and the variance is (b-a)^ the exponential (or negative exponential) is the complement of the poisson distribution. Its mean is λ and the variance is λ^. It is used in queuing theory to express the time elapsing between two consecutive occurrences of independent events. The erlang [k, β] is a distribution that is the sum of k exponents of mean β k. The mean of such distribution is β and the variance is β^ k. In fact the erlang distribution is a part of a broader class which are gamma distributions.

Each gamma function is defined by two parameters α and β. The mean queueing system software is βα and the variance is αβ^ the wei bull distribution is the one that is usually used to describe the time between two consecutive failures of the same machine, while the lognormal distribution is used to describe the time that is used for the repair of the machines. The selection of the statistical distribution that best fits the observed reality should be performed using the standard statistical procedures of data capture and hypothesis validation. If approximate formulas are used, as queueing system simulation explained later in this book, the average and the coefficient of variation must be known from the sample taken, which requires an approximation that also uses basic statistical procedures.