I've been trying to figure out a way to find the transient state solutions for n state Birth-Death Process by solving a system of matrix equations instead of looking at each differential equaions one by one.  Let P(t) be the transient state distribution matrix at time t, I be the identity matrix, Q be the rate of change matrix, and A_i's be the coefficient matrices for each e^r_i where r_i is the nonzero eigenvalues of Q.  Then first P(0) = I, P'(0) = Q, and Q has 3 properties: first the main diagonal of Q has negative entries; second, everything off diagonal is positive; third, the sum of entries from each row is zero.

 

1. Q is a matrix of real entries, thus the characteristic equation of det(Q - rI) always has solution in the complex field.

2. That Q has an eigenvalue of zero depends on if such Brith-Death Process has a steady state(just proposition).

3. Q will not have positive eigenvalues (not yet proven)

 

Hence consider the case when Q has an eigenvalues of zero, and let it be r_0, that is, r_0=0

Category one:  Q has n distinct real eigenvalues. Let R = {r_0, r_1, ..., r_(n-1)}, then r_i != (not equal to) r_j if i != j.

 

So P(t) = A_0 + A_1 * e^(r_1*t) + A_2 * e^(r_2*t) + ... + A_(n-1) * e^((n-1)*t)

 

case one: For a 2 state Birth-Death Process, we will have the following equations:

I   =  A_0 + A_1

Q  =            A_1 * r_1

 

where the equations can be written as the vector equation:

[I, Q]^T (the transpose of the row vector) =   | 1   1   | * [A_0, A_1]^T

                                                                   | 0   r_1|

 

which is a linear transformation from [A_0, A_1] to [I, Q], (since all of them are 2*2 matrices) using the transform matrix:

|1  1   |

|0  r_1|

 

is it possible? yes, similar to the partition of a matrix so that one can do the matrix multiplication easier and faster.  the multiplication between each entry in the transform matrix and the new vector [A_0, A_1]^T is defined, because it's just the scalar multiplication of a matrix.

 

then A_0 = I - [1/r_1] * Q; A_1 = [1/r_1] * Q

 

case two: for a 3 state Birth-Death Process, they satisfy the following equations:

I        = A_0 + A_1                       +  A_2

Q       =           A_1 * r_1              +  A_2 * r_2

Q^2   =           A_1* (r_1)^2         + A_2 * (r_2)^2

 

So the transform matrix is 

| 1   1              1         |

| 0   r_1           r_2      |

| 0   (r_1)^2    (r_2)^2|

 

by getting an inverse of this matrix, we can get

A_0 = I - [1/(r_1*r_2)] * Q^2 + [(r_1 + r_2)/(r_1*r_2)] * Q

A_1 = [1/(r_1*(r_1-r_2))] * (Q^2 - r_2 * Q)

A_2 = [1/(r_2*(r_2-r_1))] * (Q^2 - r_1 * Q)

 

case three: for a 4 state Birth-Death Process, the matrices satisfy the following equations:

Q^3   =           A_1 * (r_1)^3           + A_2 * (r_2)^3     + A_3 * (r_3)^3

 

So then transform matrix is :

| 1   1            1            1          |

| 0   r_1         r_2         r_3       |

| 0   (r_1)^2  (r_2)^2  (r_3)^2  |

| 0   (r_1)^3  (r_2)^3  (r_3)^3  |

 

Similar process of solving for inverse matrix, we can get:

A_0  = I - [1/(r_1*r_2*r_3)] * Q^3 + [(r_1+r_2+r_3)/(r_1*r_2*r_3)] * Q^2 - [(r_1*r_2 + r_2*r_3 + r_3*r_1)/(r_1*r_2*r_3)] * Q

A_1 = [1/(r_1*(r_1-r_2)*(r_1-r_3))] * (Q^3 - (r_2+r_3) * Q^2 + r_2*r_3 * Q)

A_2 = [1/(r_2*(r_2-r_1)*(r_2-r_3))] * (Q^3 - (r_1+r_3) * Q^2 + r_1*r_3 * Q)

A_3 = [1/(r_3*(r_3-r_1)*(r_3-r_2))] * (Q^3 - (r_1+r_2) * Q^2 + r_1*r_2 * Q)

 

Remark: as one can see, there's a pattern in the solutions for the A_i's.

 

case four: for a 5 state Birth-Death Process, the matrices satisfy the following equations: