Overview
This study examines growing open Jackson networks, focusing on their behavior under a fluid scaling of the queue-length process. It proposes an approximation of the queueing system by an infinite-dimensional fluid limit, which is then characterized using an infinite-dimensional Skorokhod problem. A new theoretical framework for this problem is developed, establishing conditions for existence, uniqueness, and Lipschitz continuity of its solutions. The research also addresses the convergence of processes within these networks to the derived fluid limit.
Research Context
The investigation centers on open Jackson networks. In these networks, each station operates as a single-server queue adhering to a first-come first-served discipline. The arrival processes are Poisson, and service times are exponentially distributed, with rates specific to each node. The research specifically considers these networks in a 'growing' state, implying a dynamic rather than static configuration. The analytical approach involves applying a fluid scaling to the queue-length process, a common technique for simplifying the analysis of complex queueing systems under high load conditions.
Approach
The primary methodological steps included:
- Fluid Scaling Application: A fluid scaling was applied to the queue-length process of the growing open Jackson networks. This aims to approximate the discrete queueing system with a continuous fluid model.
- Infinite-Dimensional Fluid Limit: The research posited that, under specific conditions, the scaled queueing system can be approximated by an infinite-dimensional fluid limit. This limit integrates a kernel function in place of the conventional transition matrix.
- Infinite-Dimensional Skorokhod Problem Formulation: The derived fluid limit was characterized by an infinite-dimensional Skorokhod problem.
- New Theory Development for Skorokhod Problem: The study developed a new theoretical framework for this infinite-dimensional Skorokhod problem. This involved considering a broader class of reflection operators and general infinite-dimensional processes.
- Existence and Uniqueness Establishment: The theory established existence and uniqueness of a solution for the infinite-dimensional Skorokhod problem. This was contingent on the reflecting operator having a spectral radius of less than 1.
- Lipschitz Continuity Proof: Lipschitz continuity of the solution was also established under the same condition for the reflecting operator.
- Intermediate Process Introduction: An intermediate process was introduced. This process was created by removing compensated Poisson components. It was then lifted to an infinite-dimensional process for analysis.
- Convergence Proof for Intermediate Process: The established Lipschitz property of the infinite-dimensional Skorokhod mapping was leveraged to prove the convergence of this intermediate process.
- Estimation of Process Difference: Necessary estimates were proven for the difference between the original queueing process and the intermediate process. This was achieved by utilizing martingale properties.
- Empirical Measure Convergence: The study considered the empirical measure of the queueing processes, demonstrating its convergence to the measure associated with the path of the infinite-dimensional fluid limit. This extended to the convergence of specific performance-related functionals.
Findings
- Under certain conditions, a fluid scaling of the queue-length process in growing open Jackson networks approximates the queueing system by an infinite-dimensional fluid limit.
- This infinite-dimensional fluid limit can be characterized by an infinite-dimensional Skorokhod problem, which utilizes a kernel function instead of a transition matrix.
- A new theory for the infinite-dimensional Skorokhod problem was developed, encompassing a broader class of reflection operators and general infinite-dimensional processes.
- Existence and uniqueness of a solution to this Skorokhod problem were established, provided the reflecting operator has a spectral radius less than 1.
- Lipschitz continuity of the solution was demonstrated when the reflecting operator's spectral radius is less than 1.
- Convergence of an intermediate process (derived by removing compensated Poisson components) was proven by utilizing the Lipschitz property of the infinite-dimensional Skorokhod mapping.
- Estimates for the difference between the original and intermediate processes were established through martingale properties.
- The empirical measure of the queueing processes converges to the measure associated with the path of the infinite-dimensional fluid limit.
- Convergence of specific performance-related functionals was extended through the convergence of the empirical measure.