Problem exposition

 

We consider a large population of individual ‘units’.  At any time, t, each unit is in one and only one of n distinct ‘states’.

Over time a unit may remain where it is or switch to one of the other states.  The likelihood of switching out of a state is a function of its residence time, s, the time it has been in the state, and the state it is in.  Once a unit switches it starts in its new state with residence time s=0.  The state a unit switches into is also state and residence time dependent.

 

We describe the switching time distribution by suitable residence dependent function for each state.  Similarly the residence dependent destinations may be described by a set of transition probabilities.

 

What if all units are not quite the same – in terms of their residence dependence – so there is extra variability/dispersion?

 

First Problem

 

Given a set of n switching time density functions,

ri(s) : R+ ®  R +,       ,       i=1,…,n

then we write     

 - the primitive,     i=1,…,n

and

r(s) = diag{ri(s),…,rn(s)}

      R(s) = diag{Ri(s),…,Rn(s)}.

 

 

Given:  A(s) : Ñ+ ® {n´n non negative matrices, identically zero on the diagonal such that A(s).u = u }, where u = (1,1,…,1)T, for all s ³ 0

 

Aij(s) = P[switch from state i to state j | switch happens after residence time s].

 

Suppose a unit is in state i.  What is the probability it switches out between (s, s+dt) given that it did not switch prior to s?

 

Let Ci(s,t) = # of units in state i at time t with current residence time of s.

 

For s>0,

 + HOT

                                                + HOT

 

 

     t>0,   s>0.

 

 

 

The new arrivals switching into states determine the boundary condition:

.

 

In vector notation, C = (Ci,…,Cn)T, we have

     t>0.        (1)

And

     s>0,   t>0.        (2)

 

 

[Check conservation of mass holds:

                            º 0  as required]

 

Solution to (2) is

        (3)

 

since directly:

.

 

Let .  Then (1) and (3) imply

        (S)

 

A steady state solution: 

   and  

(Note :  is left eigenvector for unit eigenvalue)

 

Hence  exists.

 

What can be said more generally about the general solutions of (S) and hence of (1) and (2)?

 

What sort of perturbations should we consider in order to account for individual variability?  (For example, suppose the ris and As depend very slightly on the unit’s identity.  Can we justify (2) becoming

for some s > 0? 

 

What can be said about the solution to this coupled system?

 

What further extensions could/should we consider?