Heavy traffic limits for queues with non-stationary path-dependent arrival processes.

Area of Science:

• Queueing Theory
• Stochastic Processes
• Applied Probability

Background:

• Standard queueing models often assume independent arrival processes.
• Non-stationary arrival processes with self-reinforcing properties are relevant in various applications.
• Analyzing transient workload distributions in queues is crucial for performance evaluation.

Purpose of the Study:

• To develop a diffusion approximation for the transient workload distribution in a single-server queue.
• To model queues with a non-stationary Polya arrival process, characterized by path-dependent arrival rates.
• To analyze the impact of self-reinforcing arrival patterns on queueing system performance.

Main Methods:

• Utilizing heavy-traffic limits for sequences of Polya processes.
• Employing diffusion approximation techniques for transient queueing analysis.
• Analyzing the limiting behavior of P/GI/1 queues with approaching constant service rates.

Main Results:

• The transient workload distribution is approximated by a Gaussian-Markov process.
• The path-dependent nature of the Polya arrival process is effectively captured.
• The heavy-traffic analysis provides insights into the queue's behavior under specific limiting conditions.

Conclusions:

• The developed diffusion approximation offers a tractable method for analyzing complex queueing systems.
• The model is applicable to scenarios where arrival rates exhibit history-dependent, self-reinforcing behavior.
• This research contributes to a deeper understanding of non-stationary queueing dynamics and workload analysis.