Function File: t = ctmcmtta (Q, p)

Compute the Mean-Time to Absorption (MTTA) of the CTMC described by the infinitesimal generator matrix Q, starting from initial occupancy probabilities p. If there are no absorbing states, this function fails with an error.

INPUTS

Q(i,j)

N \times N infinitesimal generator matrix. Q(i,j) is the transition rate from state i to state j, i \neq j. The matrix Q must satisfy the condition \sum_{j=1}^N Q_{i,j} = 0

p(i)

probability that the system is in state i at time 0, for each i=1, …, N

OUTPUTS

t

Mean time to absorption of the process represented by matrix Q. If there are no absorbing states, this function fails.

REFERENCES

• G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998.

See also: ctmcexps.

Demonstration 1

 mu = 0.01;
 death = [ 3 4 5 ] * mu;
 birth = 0*death;
 Q = ctmcbd(birth,death);
 t = ctmcmtta(Q,[0 0 0 1])

t =  78.333

Demonstration 2

The following code

 N = 100;
 birth = death = ones(1,N-1); birth(1) = death(N-1) = 0;
 Q = diag(birth,1)+diag(death,-1); 
 Q -= diag(sum(Q,2));
 t = zeros(1,N/2);
 initial_state = 1:(N/2);
 for i=initial_state
   p = zeros(1,N); p(i) = 1;
   t(i) = ctmcmtta(Q,p);
 endfor
 plot(initial_state,t,"+");
 xlabel("Initial state");
 ylabel("MTTA");

Produces the following figure

t =  78.333

Figure 1

Package: queueing