Page 227 - DCAP601_SIMULATION_AND_MODELING

Unit 12: Design and Evaluation of Simulation Experiments (II)

Poisson’s Equation Notes

For that purpose, it is useful to introduce Poisson’s equation. The stationary probability vector
 and the fundamental matrix Z can be viewed as solutions x to Poisson’s equation
xQ = y ;

for appropriate (row) vectors y. It can be shown that Poisson’s equation has a solution x if and
T
only if ye = 0, where e is the row vector of 1’s and e is its transpose. Then all solutions are of the
T
form
x = –yZ + (xe ) .
T
For example,  is obtained by solving Poisson’s equation when y is the zero vector (and
normalizing by requiring that xe = 1). Then elements of Z can be obtained by choosing other
T
vectors y, requiring that xe = 0.
T
In passing, we remark that there also is an alternate column-vector form of Poisson’s equation,
namely,
T
Qx = y ,
T
T
which has a solution if and only if y = 0. Then all solutions are of the form
T
T
x = —Zy + (x )e .
T
T
It is significant that, for a CTMC, the asymptotic bias  defined in (2) and the asymptotic
2
variance  defined in (3) can be expressed directly in terms of , Z, the function f and (for ) the
initial probability vector, say  , i.e.,
m m

 ( )   Zf    i Z f j
, i j
i 0 j 0
and
m m
2
  2( f  )Zf  2  f  i Z , i j j . f
i
i 0 j 0
Moreover, the asymptotic parameters ( )  and  are themselves directly solutions to Poisson’s
2
equation. In particular,
 ( )  xf T ,

T
where x is the unique solution to Poisson’s equation for y = —  +  with xe = 0. Similarly,
2
  2xf T ,
where x is the unique solution to Poisson’s equation for y = –¡(f — ) with xe = 0.
T
i i i
Birth-and-Death Processes

Birth-and-death (BD) processes are special cases of CTMC’s in which Q = 0 when |i—j| > 1; then
i,j
we often use the notation Q   and Q —1  , and refer to  as the birth rates and  as the
i,i+1 i i,i i i i
death rates. For BD processes and skip-free CTMC’s (which in one direction can go at most one
step), Poisson’s equation can be efficiently solved recursively.

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