An Infinite Closed Jackson Network (ICJN; also known as Zero-Range process) on the infinite set of queues J is defined by its generator

A f(h)= å i Î J  å j Î J  [ 1{hi > 0}gipij( f(¼hi-1¼hj+1¼)-f(h)) ],

where h_i is the number of customers in queue i, g_i is the service rate at queue i, p_ij is the probability that customer leaving the queue i would join queue j and f:J®R is a real-valued function measurable w.r.t. cylindric topology. Under conditions sup_ig_i < ¥, sup_iå_{j Î J}g_jp_ij < ¥ there exists a unique Feller process h(t;h⁽⁰⁾) with generator A such that h(0;h⁽⁰⁾)=h⁽⁰⁾ a.s. [1].

ICJN with infinite number of customers (å_ih⁽⁰⁾_i=¥) is not ergodic because of continuum family of invariant measures, and there exists a number of sufficient conditions for devastating (overloading) of the queues with customers leaving to (coming from) infinite part of J [2,1]. The only example (dual to simple exclusion process) of convergence of h(t;h⁽⁰⁾) to equilibrium measure was constructed in [3] for the case J=Z={0,±1,±2,¼}, g_i=1, p_i,i+1=p_i,i-1=1/2 for all i Î Z, with periodic initial deterministic configuration h⁽⁰⁾_i=h⁽⁰⁾_i+p. Then h(t;h⁽⁰⁾)[ w. || (® )]G(r), where G(r)=Õ_{i Î J}G_i(r), and G_i(r) are distributions of geometric r.v. with expectation EG_i(r)=r(h⁽⁰⁾)=(h⁽⁰⁾₁+¼+h⁽⁰⁾_p)/p.

Here the new examples are constructed for IJCN on J=N={1,2,¼}.

Theorem. For any M Î N there exist (small) numbers d_i > 0, i Î N such that IJCN h(t;h⁽⁰⁾) with g_ip_i,i+1=g_i+1p_i+1,i=d_i, p_i,i=1-p_i,i+1-p_i,i-1 for all i Î J (p_1,0 º 0), converges weakly to G(r) for any initial configurations h⁽⁰⁾ bounded by M (h_i⁽⁰⁾ £ M for all i) and having density r(h⁽⁰⁾)=lim_N®¥(h⁽⁰⁾₁+¼+h⁽⁰⁾_N)/N: h(t;h⁽⁰⁾)[ w. || (® )]G(r). If h_i⁽⁰⁾ £ M for all i and limiting density r(h⁽⁰⁾) is not defined, then h_i(t;h⁽⁰⁾) is stochastically bounded for each i Î N.

The proof uses variational metric and stochastical comparision of random processes and can be extended to non-neighbour transitions p_ij ¹ 0, |i-j| ³ 2.

[1] D.V. Khmelev and E. Spodarev, J. Math. Sci. (New York) 106 (2001), no. 2, 2820-2829. [2] M.Ya.Kelbert, M.L.Konzevich, and A.N. Rybko. Teor. ver. i yeyo prim., XXXII(2):379-382, 1988. [3] A. Galves and H. Guiol. Markov Process. Related Fields, 3(3):323-332, 1997.

Footnotes:

¹ Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5S 3G3

E-mail: dkhmelev(at)math dot toronto dot edu