The Appendix at the end of the paper completes the characterization of the equilibrium by providing solutions for the aggregate variables of the model (e.g., the

stationary distribution of firms over productivity and demand parameters).

Description of the Model: [GI.sup.[X]]/C-MSP/1/[infinity] Queue and the Analysis of the

Stationary Distribution of Waiting Time

In Section 2, we present some lemmas concerning the existence of a global positive solution and ergodic

stationary distribution. In Section 3, we prove that the disease is persistent under one condition.

(P2) Under what conditions, will system (5) have a unique ergodic

stationary distribution?

Assume that [R.sup.s.sub.0] > 1; there exists a

stationary distribution [pi](x) and the ergodicity holds for any initial value (S(0), I(0), Q(0), R(0)) [member of] [R.sup.4.sub.+] in system (3).

For an MTD game of path hopping with transition matrix M and

stationary distribution [pi] = (n(0), [pi](1), ..., n(K)), where K is the winning state, LT, which is the expected number of times the adversary wins in the first T time steps, is less than or equal to T x [pi](K), assuming that the game starts with the 0 = (1, 0, ..., 0) distribution.

In section 2, a two dimensional queue length process is represented by QBD process such that it is amenable for matrix analytical treatments and thus the

stationary distribution of the queue length process and expected queue lengths are obtained.

In this section, we will derive the

stationary distribution of {([L.sub.l], [J.sub.l]), l [greater than or equal to] 1} at the arrival epochs by using the matrix geometric approach.

Under this assumption, the Markov chain has a unique

stationary distribution [pi] = ([[pi].sub.1], [[pi].sub.2], ..., [[pi].sub.m]) which is the solution of the system of linear equations [pi][TAU] = 0 subject to [[summation].sup.m.sup.j=1] [[pi].sub.j] = 1 and [[pi].sub.j] > 0 for all j [member of] S.

Let [p.sup.*.sub.f] be the

stationary distribution of state f.

Given the invariant density or ergodic distribution, we can find some dynamical map f(.) such that: [x.sub.n+1] = f([x.sub.n]) for which the collection {[x.sub.n]} of prime gaps follow the

stationary distribution referred to as the Inverse Frobenius-Perron map.

When [r.sub.i] = 1 and [r.sub.i] = 0 for all j [not equal to]i, the

stationary distribution of the RWR represents the affinity of each node to the specific node i.