This paper is concerned with a class of stochastic recursive sequences that arise in various branches of queueing theory. First, we make use of Kingman's subadditive ergodic theorem to determine the stability region of this type of sequence, or equivalently, the condition under which they converge weakly to a finite limit. Under this stability condition, we also show that these sequences admit a unique finite stationary regime and that regardless of the initial condition, the transient sequence couples in finite time with this uniquely defined stationary regime. When this stability condition is not satisfied, we show that the sequence converges a.s. to $\infty$ and that certain increments of the process form another type of stochastic recursive sequence that always admit at least one stationary regime. Finally, we give sufficient conditions for this increment sequence to couple with this stationary regime.
"On a Class of Stochastic Recursive Sequences Arising in Queueing Theory." Ann. Probab. 20 (1) 350 - 374, January, 1992. https://doi.org/10.1214/aop/1176989931