Stochastic-Process Limits

Time-dependent properties of symmetric queues

“…The continuity result in Whitt [16] implies that the sequence of workload processes converge in D to the workload process corresponding to the given service-time distribution. Since the first passage time is a continuous functional in D (due to the fact that when the workload first hits zero, it crosses its w.p.1; see Whitt [17]), we see that we have proved the following theorem.…”

Pathwise comparison results for stochastic fluid networks

“…Proof Before proceeding we require the following fact (Corollary 14.3.5 of [10] and Theorem 3.7 of [9]): ∀j ∈ P i states the processes W (1) and W (2) are continuous at all continuity points of the input processes J (1) and J (2) respectively. The proof of the main result will follow by contradiction.…”

“…The particular case of SFNs with constant routing matrix has been extensively studied in a series of papers by Kella [3,4], Kella and Whitt [5][6][7] and in the book by Whitt [10]. In particular, the papers of Kella [4] and Kella and Whitt [7] provide stability conditions for SFNs with Lévy and stationary increment inputs respectively, through the use of comparison theorems.…”

On the asymptotic optimality of the cμ/θ rule under ergodic cost

“…9.3] a priority queue is considered as one of a large class of processes for which convergence to a fluid model holds. Further properties of priority queues in heavy traffic are analyzed in [4,Sect. 5.10].…”