A functional law of the iterated logarithm for multi-class queues with batch arrivals

• Published in: Annals of operations research Vol. 300; no. 1; pp. 51 - 77
• Main Authors: Guo, Yongjiang, Hou, Xiyang, Liu, Yunan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2021; Springer; Springer Nature B.V
• Subjects: Business and Management; Combinatorics; Logarithms; Operations research; Operations Research/Decision Theory; Original Research; Queues; Queuing theory; Stochastic processes; Theorems; Theory of Computation; Workload; Workloads; China; Strong approximation; 90B22; Functional law of the iterated logarithm; First come first served service discipline; Law of the iterated logarithm; Multi-class queue; 60B10; 60K25; 60F17
• ISSN: 0254-5330; 1572-9338
• DOI: 10.1007/s10479-020-03864-6

A functional law of the iterated logarithm (LIL) and its corresponding LIL are established for a multiclass single-server queue with first come first served (FCFS) service discipline. The functional LIL and its LIL quantify the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. The functional LIL and LIL are established in three cases: underloaded, critically loaded and overloaded, for performance measures including the total workload, idle time, queue length, workload, busy time, departure and sojourn time processes. The proofs of the functional LIL and LIL are based on a strong approximation approach, which approximates discrete performance processes with reflected Brownian motions. Numerical examples are considered to provide insights on these limit results.