.. include:: replace.txt
.. highlight:: cpp

Random Variables
----------------

|ns3| contains a built-in pseudo-random number generator (PRNG). It is important
for serious users of the simulator to understand the functionality,
configuration, and usage of this PRNG, and to decide whether it is sufficient
for his or her research use.  

Quick Overview
**************

|ns3| random numbers are provided via instances of
:cpp:class:`ns3::RandomVariableStream`.

* by default, |ns3| simulations use a fixed seed; if there is any randomness in
  the simulation, each run of the program will yield identical results unless
  the seed and/or run number is changed.  

* in *ns-3.3* and earlier, |ns3| simulations used a random seed by default; this
  marks a change in policy starting with *ns-3.4*.

* in *ns-3.14* and earlier, |ns3| simulations used a different wrapper class
  called :cpp:class:`ns3::RandomVariable`.  As of *ns-3.15*, this class has been
  replaced by :cpp:class:`ns3::RandomVariableStream`; the underlying pseudo-random
  number generator has not changed.

* to obtain randomness across multiple simulation runs, you must either set the
  seed differently or set the run number differently.  To set a seed, call
  :cpp:func:`ns3::RngSeedManager::SetSeed` at the beginning of the program; to set
  a run number with the same seed, call :cpp:func:`ns3::RngSeedManager::SetRun` at
  the beginning of the program; see :ref:`seeding-and-independent-replications`.

* each RandomVariableStream used in |ns3| has a virtual random number generator
  associated with it; all random variables use either a fixed or random seed
  based on the use of the global seed (previous bullet);

* if you intend to perform multiple runs of the same scenario, with different
  random numbers, please be sure to read the section on how to perform
  independent replications: :ref:`seeding-and-independent-replications`.

Read further for more explanation about the random number facility for |ns3|.

Background
**********

Simulations use a lot of random numbers; one study 
found that most network simulations spend as much as 50% 
of the CPU generating random numbers.  Simulation users need
to be concerned with the quality of the (pseudo) random numbers and
the independence between different streams of random numbers.  

Users need to be concerned with a few issues, such as:

* the seeding of the random number generator and whether a 
  simulation outcome is deterministic or not,
* how to acquire different streams of random numbers that are 
  independent from one another, and 
* how long it takes for streams to cycle

We will introduce a few terms here:  a RNG provides a long sequence
of (pseudo) random numbers.
The length of this sequence is called the *cycle length*
or *period*, after which the RNG will repeat itself.  
This sequence can
be  partitioned into disjoint *streams*.  A stream of a
RNG is a contiguous subset or block of the RNG sequence.
For instance, if the
RNG period is of length N, and two streams are provided from this
RNG, then
the first stream might use the first N/2 values and the second
stream might produce the second N/2 values.  An important property
here is that the two streams are uncorrelated.  Likewise, each
stream can be partitioned disjointedly to a number of 
uncorrelated *substreams*.  The underlying RNG hopefully
produces a pseudo-random sequence of numbers with a very long
cycle length, and partitions this into streams and substreams in an 
efficient manner.

|ns3| uses the same underlying random number generator as does |ns2|:  the
MRG32k3a generator from Pierre L'Ecuyer.  A detailed description can be found in
http://www.iro.umontreal.ca/~lecuyer/myftp/papers/streams00.pdf.  The MRG32k3a
generator provides :math:`1.8x10^{19}` independent streams of random numbers,
each of which consists of :math:`2.3x10^{15}` substreams. Each substream has a
period (*i.e.*, the number of random numbers before overlap) of
:math:`7.6x10^{22}`. The period of the entire generator is :math:`3.1x10^{57}`. 


Class :cpp:class:`ns3::RandomVariableStream` is the public interface to this
underlying random number generator.  When users create new random variables
(such as :cpp:class:`ns3::UniformRandomVariable`,
:cpp:class:`ns3::ExponentialRandomVariable`, etc.), they create an object that uses
one of the distinct, independent streams of the random number generator.
Therefore, each object of type :cpp:class:`ns3::RandomVariableStream` has,
conceptually, its own "virtual" RNG.  Furthermore, each
:cpp:class:`ns3::RandomVariableStream` can be configured to use one of the set of
substreams drawn from the main stream.

An alternate implementation would be to allow each RandomVariable to have its
own (differently seeded) RNG.  However, we cannot guarantee as strongly that the
different sequences would be uncorrelated in such a case; hence, we prefer to
use a single RNG and streams and substreams from it.

.. _seeding-and-independent-replications:

Seeding and independent replications
************************************

|ns3| simulations can be configured to produce deterministic or random results.
If the |ns3| simulation is configured to use a fixed, deterministic seed with
the same run number, it should give the same output each time it is run.

By default, |ns3| simulations use a fixed seed and run number.  These values
are stored in two :cpp:class:`ns3::GlobalValue` instances: ``g_rngSeed`` and
``g_rngRun``.

A typical use case is to run a simulation as a sequence of independent trials,
so as to compute statistics on a large number of independent runs.  The user can
either change the global seed and rerun the simulation, or can advance the
substream state of the RNG, which is referred to as incrementing the run number.

A class :cpp:class:`ns3::RngSeedManager` provides an API to control the seeding and
run number behavior.  This seeding and substream state setting must be called
before any random variables are created; e.g::

  RngSeedManager::SetSeed (3);  // Changes seed from default of 1 to 3
  RngSeedManager::SetRun (7);   // Changes run number from default of 1 to 7
  // Now, create random variables
  Ptr<UniformRandomVariable> x = CreateObject<UniformRandomVariable> ();
  Ptr<ExponentialRandomVariable> y = CreateObject<ExponentialRandomVarlable> ();
  ...

Which is better, setting a new seed or advancing the substream state?  There is
no guarantee that the streams produced by two random seeds will not overlap.
The only way to guarantee that two streams do not overlap is to use the
substream capability provided by the RNG implementation.  *Therefore, use the
substream capability to produce multiple independent runs of the same
simulation.* In other words, the more statistically rigorous way to configure
multiple independent replications is to use a fixed seed and to advance the run
number.  This implementation allows for a maximum of :math:`2.3x10^{15}`
independent replications using the substreams. 

For ease of use, it is not necessary to control the seed and run number from
within the program; the user can set the ``NS_GLOBAL_VALUE`` environment
variable as follows:

.. sourcecode:: bash

  $ NS_GLOBAL_VALUE="RngRun=3" ./waf --run program-name

Another way to control this is by passing a command-line argument; since this is
an |ns3| GlobalValue instance, it is equivalently done such as follows:

.. sourcecode:: bash

  $ ./waf --command-template="%s --RngRun=3" --run program-name

or, if you are running programs directly outside of waf:

.. sourcecode:: bash

  $ ./build/optimized/scratch/program-name --RngRun=3

The above command-line variants make it easy to run lots of different
runs from a shell script by just passing a different RngRun index.

Class RandomVariableStream
**************************

All random variables should derive from class :cpp:class:`RandomVariable`. This
base class provides a few methods for globally configuring the behavior
of the random number generator. Derived classes provide API for drawing random
variates from the particular distribution being supported.

Each RandomVariableStream created in the simulation is given a generator that is a 
new RNGStream from the underlying PRNG. Used in this manner, the L'Ecuyer
implementation allows for a maximum of :math:`1.8x10^19` random variables.  Each
random variable in a single replication can produce up to :math:`7.6x10^22`
random numbers before overlapping.

Base class public API
*********************

Below are excerpted a few public methods of class :cpp:class:`RandomVariableStream`
that access the next value in the substream.

::

  /**
   * \brief Returns a random double from the underlying distribution
   * \return A floating point random value
   */
  double GetValue (void) const;
  
  /**
   * \brief Returns a random integer from the underlying distribution
   * \return  Integer cast of ::GetValue()
   */
  uint32_t GetInteger (void) const;

We have already described the seeding configuration above. Different
RandomVariable subclasses may have additional API.

Types of RandomVariables
************************

The following types of random variables are provided, and are documented in the
|ns3| Doxygen or by reading ``src/core/model/random-variable-stream.h``.  Users 
can also create their own custom random variables by deriving from class
:cpp:class:`RandomVariableStream`.

* class :cpp:class:`UniformRandomVariable`
* class :cpp:class:`ConstantRandomVariable`
* class :cpp:class:`SequentialRandomVariable`
* class :cpp:class:`ExponentialRandomVariable`
* class :cpp:class:`ParetoRandomVariable`
* class :cpp:class:`WeibullRandomVariable`
* class :cpp:class:`NormalRandomVariable`
* class :cpp:class:`LogNormalRandomVariable`
* class :cpp:class:`GammaRandomVariable`
* class :cpp:class:`ErlangRandomVariable`
* class :cpp:class:`TriangularRandomVariable`
* class :cpp:class:`ZipfRandomVariable`
* class :cpp:class:`ZetaRandomVariable`
* class :cpp:class:`DeterministicRandomVariable`
* class :cpp:class:`EmpiricalRandomVariable`

Semantics of RandomVariableStream objects
*****************************************

RandomVariableStream objects derive from :cpp:class:`ns3::Object` and are
handled by smart pointers.

RandomVariableStream instances can also be used in |ns3| attributes, which means
that values can be set for them through the |ns3| attribute system.
An example is in the propagation models for WifiNetDevice::

    TypeId
    RandomPropagationDelayModel::GetTypeId (void)
    { 
      static TypeId tid = TypeId ("ns3::RandomPropagationDelayModel")
        .SetParent<PropagationDelayModel> ()
        .AddConstructor<RandomPropagationDelayModel> ()
        .AddAttribute ("Variable",
                       "The random variable which generates random delays (s).",
                       StringValue ("ns3::UniformRandomVariable"),
                       MakePointerAccessor (&RandomPropagationDelayModel::m_variable),
                       MakePointerChecker<RandomVariableStream> ())
        ;
      return tid;
    }

Here, the |ns3| user can change the default random variable for this
delay model (which is a UniformRandomVariable ranging from 0 to 1) through
the attribute system.

Using other PRNG
****************

There is presently no support for substituting a different underlying
random number generator (e.g., the GNU Scientific Library or the Akaroa
package).  Patches are welcome.

Setting the stream number
*************************

The underlying MRG32k3a generator provides 2^64 independent streams.
In ns-3, these are assigned sequentially starting from the first stream as 
new RandomVariableStream instances make their first call to GetValue().

As a result of how these RandomVariableStream objects are assigned to
underlying streams, the assignment is sensitive to perturbations of
the simulation configuration.  The consequence is that if any aspect of the
simulation configuration is changed, the mapping of RandomVariables to
streams may (or may not) change.

As a concrete example, a user running a comparative study between routing
protocols may find that the act of changing one routing protocol for another
will notice that the underlying mobility pattern also changed.  

Starting with ns-3.15, some control has been provided to users to allow
users to optionally fix the assignment of selected RandomVariableStream 
objects to underlying streams.  This is the ``Stream`` attribute, part
of the base class RandomVariableStream.  

By partitioning the existing sequence of streams from before:

.. sourcecode:: text

   <-------------------------------------------------------------------------->
   stream 0                                                   stream (2^64 - 1)

into two equal-sized sets:

.. sourcecode:: text

   <--------------------------------------------------------------------------> 
   ^                                    ^^                                    ^
   |                                    ||                                    |
   stream 0            stream (2^63 - 1)  stream 2^63         stream (2^64 - 1)
   <- automatically assigned -----------><- assigned by user ----------------->

The first 2^63 streams continue to be automatically assigned, while
the last 2^63 are given stream indices starting with zero up to
2^63-1.

The assignment of streams to a fixed stream number is optional; instances
of RandomVariableStream that do not have a stream value assigned will
be assigned the next one from the pool of automatic streams.

To fix a RandomVariableStream to a particular underlying stream, assign
its ``Stream`` attribute to a non-negative integer (the default value
of -1 means that a value will be automatically allocated).

Publishing your results
***********************

When you publish simulation results, a key piece of configuration 
information that you should always state is how you used the
the random number generator.

* what seeds you used,
* what RNG you used if not the default,
* how were independent runs performed,
* for large simulations, how did you check that you did not cycle.

It is incumbent on the researcher publishing results to include enough
information to allow others to reproduce his or her results. It is also
incumbent on the researcher to convince oneself that the random numbers used
were statistically valid, and to state in the paper why such confidence is
assumed.

Summary
*******

Let's review what things you should do when creating a simulation.

* Decide whether you are running with a fixed seed or random seed; a fixed seed
  is the default, 
* Decide how you are going to manage independent replications, if applicable, 
* Convince yourself that you are not drawing more random values than the cycle
  length, if you are running a very long simulation, and
* When you publish, follow the guidelines above about documenting your use of
  the random number generator.
