A functional law of the iterated logarithm for multi-class queues with batch arrivals
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A functional law of the iterated logarithm for multi-class queues with batch arrivals Yongjiang Guo1 · Xiyang Hou2 · Yunan Liu3 Accepted: 4 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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. Keywords Functional law of the iterated logarithm · Law of the iterated logarithm · Multi-class queue · First come first served service discipline · Strong approximation Mathematics Subject Classification 60K25 · 60F17 · 60B10 · 90B22
1 Introduction In this paper, we develop a functional law of the iterated logarithm (LIL) and its corresponding LIL for a multi-class batch-arrival (G I B /G I ) K /1/FCFS queue, which has one server, K
B
Yunan Liu [email protected] Yongjiang Guo [email protected] Xiyang Hou [email protected]
1
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People’s Republic of China
2
School of Management, Huazhong University of Science and Technology, Wuhan 430074, Hubei, People’s Republic of China
3
Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, USA
123
Annals of Operations Research
customer classes and first come first served (FCFS) service discipline, a renewal batch-arrival process (the first G I ) and independent and identically distributed service times (the second G I ). In the literature of queueing theory, the multi-class batch-arrival queueing system under FCFS service discipline largely captured researchers’ attention in recent 30 years because these queueing models are closely related to many actual applications in life, including data packetization in communication system, central processing unit (CPU) scheduling in computer system and packet-switched communication networks, see Kulkarni and Glazebrook (2002) and Whitt (1983a). Many stochastic approximation theories have been established for this type of queueing networks with arrival in batches and FCFS service discipline, including stationary probability distribution for single-server queue (Van Ommeren 1990) and infinite-server queue (Falin 1994; Whitt 1983b), and for multi-class discrete-time queueing system (Clercq et al. 2013), fluid approximation for multiclass queueing networks
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