Mesquite Software : Documentation : C++ Guide : Random Numbers

Most simulations are random number driven. In such simulations, random numbers are used for interarrival times, service times, allocation amounts, and routing probabilities. For each application of random numbers in a simulation, a distribution must be chosen. The distribution determines the likelihood of different values occurring. A distribution is uniquely specified by the name of its family (such as uniform, exponential, or normal) and its parameter values (such as the mean and standard deviation). Discussions of distributions and their uses in models can be found in texts [Lake 96].

Random numbers generated by computers are actually pseudo-random. A sequence of values is generated using a recurrence relation that calculates the next value in the sequence from the previous value. The sequence is begun by specifying a starting value called a seed. A good random number generator has the property that the numbers it produces have no discernible patterns that distinguish them from truly random numbers.

Most CSIM users need only read the following two sections, which describe single stream random number generation. Those interested in building multiple-stream simulations should read the remaining sections as well.

CSIM includes a library of functions for generating random numbers from more than 20 different distributions. Continuous distributions have values that are floating-point numbers; values from these distributions are most often used for amounts of time. Discrete distributions have values that are integers; values from these distributions are often used for quantities of resources.

The following prototypes are for the functions that generate values from continuous distributions. The parameters min and max specify the minimum and maximum values that will be generated. The parameters mean, var, stddev, and mode specify respectively the mean, variance, standard deviation, and mode of the distribution. The parameters shape1, shape2, shape, alpha, and beta are all shape parameters whose meaning can be found in any text that describes these distributions.

Prototype: double hyperx(double mean, double var)

Prototype: double weibull(double shape, double scale)

The following prototypes are for the functions that generate values from discrete distributions. The parameters min and max specify the minimum and maximum values that will be generated. The parameter mean specifies the mean of the distribution. The parameters prob_success, num_trials, and success_num are respectively the probability of success, the number of trials, and the success number. A text that describes theses distributions should be consulted for the detailed meaning of these parameters.

Prototype: long uniform_int(long min, long max)

Prototype: long bernoulli(double prob_success)

Prototype: long binomial(double prob_success, long
num_trials)

Prototype: long geometric(double prob_success)

Prototype: long negative_binomial(long success_num,
double prob_success)

Prototype: long poisson(double mean)

Two functions must be used to efficiently generate values from an empirical distribution.

Their prototypes are shown below.

Prototype: void setup_empirical(long n, double prob[],
double cutoff[], long alias[])

Prototype: double empirical(long n, double cutoff[],
long alias[], double value[])

The setup_empirical function must be called once, prior to any calls to function empirical. It takes as input the number of values, n, in the distribution and an array, prob, that specifies the probability of generating each value. It calculates two sets of values and stores them in the arrays cutoff and alias. The contents of these arrays need not be understood to use this distribution. All arrays must be of size at least n+1. Function empirical is called to generate a value from an empirical distribution that has already been set-up. The function takes as input the same parameters n, cut-off, and alias as the setup_empirical function. It also takes an array, value, that contains the values to be generated with the probabilities that were specified in array prob. Each call returns one of the values in the array value.

18.2 Changing the Seed of the Single Stream

By default, the single stream from which all random numbers are generated is seeded with the value of 1. Unless the seed is changed, every execution of every CSIM program will use the same sequence of random numbers. The seed can be changed by calling the reseed function.

Prototype: void reseed(stream *s, long n)
Example: reseed(NIL, 13579);

In simulations that use a single random number stream, the value of the first parameter in the function call should always be NIL. The second parameter is the positive integer that is to be used as the seed. The choice of the seed value will not affect the randomness of the numbers that are produced. Although it is most common to call reseed once at the beginning of a CSIM program, the reseed function can be called any number of times and from any place within a program.

The current state of the stream can be retrieved by calling the stream_state function.

Prototype: long stream::state()
Example: i = stream_state(NIL);

If stream_state is called immediately after reseeding the stream, the seed value will be returned. Otherwise, the positive integer used to produce the most recently generated random number will be returned.

In a single stream simulation, all random numbers are produced from a single stream of pseudo-random integers. The random numbers used for a particular purpose (for example, interarrival times) are generated from a subsequence of these random integers. It is of concern to some people that the subsequence of integers may not be "as random" as the stream from which they were extracted. This concern can be alleviated by using a separate stream of pseudo-random integers for each application of random numbers in the model. So, separate streams would be used for the service times at each facility, for the allocation amounts of each storage, and so forth.

Multiple streams are also used to guarantee that exactly the same sequence of random numbers is used for the interarrival times (for example) in two different models. This technique is called common random numbers and is described in simulation texts.

There is virtually no difference in the time required to generate random number from a single stream or from multiple streams. Multiple stream simulations require slightly more programming: the multiple streams must be declared, initialized, and (perhaps) seeded, and each call to a function that generates random numbers must specify the stream to be used.

18.4 Managing Multiple Streams

A stream is declared in a CSIM program using the class stream.

Dynamic Example: stream *s;

Before a stream can be used, it must be initialized by calling the create_stream constructor.

Prototype: stream::stream(void)
Example: s = new stream();

By default, streams are created with seeds that are spaced 100,000 values apart. CSIM contains a table of 100 such seed values; if more than 100 streams are created, the seed values are reused.

The seed value for any stream can be changed by calling the reseed function.

Prototype: void stream::reseed(long n)
Dynamic Example: s->reseed(24680);

The second parameter is a positive integer that is to be used as the new seed. Although it is most common to call reseed once for each stream at the beginning of a CSIM program, streams can be reseeded any number of times and at any place in the program.

The current state of a stream can be retrieved by calling the stream_state function.

Prototype: long stream::state()
Dynamic Example: i = s->state();

If state is called immediately after reseeding a stream, the seed value will be returned. Otherwise, the positive integer used to produce the random number most recently generated from the stream will be returned.

If a dynamic stream is no longer needed, its storage can be reclaimed as follows:

Dynamic Example: delete s;

Once a stream has been deleted, it must not be further referenced.

The same 18 distributions are available for generating random numbers from multiple streams as are available for generating random numbers from a single stream. The following are two examples:

Multiple Stream Prototype: double stream::uniform
(double min, double max)
Multiple Stream Prototype: double stream::triangular
(double min, double max,
double mode)

In all other ways, the functions and their parameters are exactly the same. It is the programmer's responsibility to ensure that a stream is used for only one purpose and that a separate stream is used for each application of random numbers in the model.