Asymptotic results on infinite tandem queueing networks

We consider an infinite tandem queueing network consisting of (.)/GI/1/infi nity stations with i.i.d. service times. We investigate the asymptotic beha vior of t (n, k), the inter arrival times between customers n and n + 1 at station k, and that of w(n, k), the waiting time of customer n at station k . We establish a duality property by which w(n, k) and the "idle times" y(n , k) play symmetrical roles. This duality structure, interesting by itself, is also instrumental in proving some of the ergodic results. We consider t wo versions of the model: the quadrant and the half-plane. In the quadrant version, the sequences of boundary conditions {w(0, k), k is an element of N} and {t(n, 0), n is an element of N}, are given. In the half-plane versio n, the sequence {t(n, 0), n is an element of Z} is given. Under appropriate assumptions on the boundary conditions and on the services, we obtain ergo dic results for both versions of the model. For the quadrant version, we pr ove the existence of temporally ergodic evolutions and of spatially ergodic ones. Furthermore, the process {t(n, k), n is an element of N} converges w eakly with k to a limiting distribution, which is invariant for the queuein g operator. In the more difficult half plane problem, the aim is to obtain evolutions which are both temporally and spatially ergodic. We prove that 1 /n Sigma (n)(k=1) w(0, k) converges almost surely and in LI to a finite con stant. This constitutes a first step in trying to prove that {t(n, k), n is an element of Z} converges weakly with k to an invariant limiting distribu tion.