The following files are provided in this package:

README.txt (this file)
RUGmatrices.m
pivectors.m
Gmatrix.m

--

RUGmatrices.m
pivectors.m

These two files are used to compute pi vectors of a QBD process.

Here, the generator matrix of the QBD process is:

[ A1{1} A0{1}                                    ]
[ A2{2} A1{2} A0{2}                              ]
[       A2{3} A1{3} A0{3}                        ]
[              :     :                           ]
[                        A2{n} A1{n} A0{n}       ]
[                              A2{n} A1{n} A0{n} ]
[                                     :     :    ]
 
Note that the QBD process repeats after level n.
 
We define cells, A0, A1, and A2:
 A0 = {A0{1} A0{2} ... A0{n}}
 A1 = {A1{1} A1{2} ... A1{n}}
 A2 = {A2{1} A2{2} ... A2{n}},


Example: 

Consider the QBD process below:

   ----1--->   ----1--->
[1]         [2]
   <---2----   <---2----
^ |         ^ |
| |         | |
3 3         3 3
| |         | |
| v         | v
   ----1--->   ----1--->
[1]         [2]
   <---2----   <---2----

A0, A1, A2 are given by:

A0{1} = [1 0;
         0 1];
A0{2} = A0{1};
A2{2} = [2 0;
         0 2];
A1{1} = [-4  3;
          3 -4];
A1{2} = [-6  3;
          3 -6];

Thus,

n = 2;

Let

epsilon = 1e-6;

Now, we can compute pi vectors, pi:

[R,U,G] = RUGmatrices(A0,A1,A2,n,epsilon);
[pi,meannum] = pivectors(A0,A1,A2,R,U{1},2);


The output is

pi{1} = [0.25 0.25];
pi{2} = [0.125 0.125];

If you need pi{i} for i>2,

pi{3} = pi{2} * R{2}
      = [0.0625 0.0625];

pi{4} = pi{3} * R{3}
      = [0.03125 0.03125];

      :
      :




Gmatrix.m is used to compute G matrix of the repeating part in RUGmatrices.m