Heavy traffic limit theorems for a sequence of shortest queueing systems

1995 ◽  
Vol 21 (1-2) ◽  
pp. 217-238 ◽  
Author(s):  
Hanqin Zhang ◽  
Guang -Hui Hsu ◽  
Rongxin Wang
Keyword(s):  
Limit Theorems ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Heavy Traffic Limit

scholarly journals Heavy traffic analysis for single-server SRPT and LRPT queues via EDF diffusion limits

2021 ◽  
Author(s):  
Łukasz Kruk
Keyword(s):  
Limit Theorems ◽  
Processing Time ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Single Server ◽  
Network Applications ◽  
Heavy Traffic Analysis ◽  
Heavy Traffic Limit ◽  
Heavy Tailed ◽  
Earliest Deadline

AbstractExtending the results of Kruk (Queueing theory and network applications. QTNA 2019. Lecture notes in computer science, vol 11688. Springer, Cham, pp 263–275, 2019), we derive heavy traffic limit theorems for a single server, single customer class queue in which the server uses the Shortest Remaining Processing Time (SRPT) policy from heavy traffic limits for the corresponding Earliest Deadline First queueing systems. Our analysis allows for correlated customer inter-arrival and service times and heavy-tailed inter-arrival and service time distributions, as long as the corresponding stochastic primitive processes converge weakly to continuous limits under heavy traffic scaling. Our approach yields simple, concise justifications and new insights for SRPT heavy traffic limit theorems of Gromoll et al. (Stoch Syst 1(1):1–16, 2011). Corresponding results for the longest remaining processing time policy are also provided.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

Heavy traffic limit theorems for a queueing system in which customers join the shortest line

1989 ◽  
Vol 21 (02) ◽  
pp. 451-469 ◽  
Author(s):  
Zhang Hanqin ◽  
Wang Rongxin
Keyword(s):  
Stochastic Processes ◽  
Central Limit ◽  
Limit Theorems ◽  
Queueing System ◽  
Heavy Traffic ◽  
Traffic Intensity ◽  
Heavy Traffic Limit ◽  
Functional Central Limit Theorems ◽  
Service Channels ◽  
Functional Central Limit

The queueing system considered in this paper consists of r independent arrival channels and s independent service channels, where, as usual, the arrival and service channels are independent. In the queueing system, each server of the system has his own queue and arriving customers join the shortest line in the system. We give functional central limit theorems for the stochastic processes characterizing this system after appropriately scaling and translating the processes in traffic intensity ρ > 1.

Complements to heavy traffic limit theorems for the GI/G/1 queue

1972 ◽  
Vol 9 (1) ◽  
pp. 185-191 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Rate Of Convergence ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Sufficient Conditions ◽  
Critical Value ◽  
Traffic Intensity ◽  
Heavy Traffic Limit ◽  
Convergence Of Moments

A bound on the rate of convergence and sufficient conditions for the convergence of moments are obtained for the sequence of waiting times in the GI/G/1 queue when the traffic intensity is at the critical value ρ = 1.

Multiple channel queues in heavy traffic. II: sequences, networks, and batches

1970 ◽  
Vol 2 (02) ◽  
pp. 355-369 ◽  
Author(s):  
Donald L. Iglehart ◽  
Ward Whitt
Keyword(s):  
Stochastic Processes ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
General Setting ◽  
Critical Value ◽  
Multiple Channel ◽  
Heavy Traffic Limit ◽  
Service Channels ◽  
Traffic Behavior

This paper is a sequel to [7], in which heavy traffic limit theorems were proved for various stochastic processes arising in a single queueing facility with r arrival channels and s service channels. Here we prove similar theorems for sequences of such queueing facilities. The same heavy traffic behavior prevails in many cases in this more general setting, but new heavy traffic behavior is observed when the sequence of traffic intensities associated with the sequence of queueing facilities approaches the critical value (ρ = 1) at appropriate rates.

Complements to heavy traffic limit theorems for the GI/G/1 queue

1972 ◽  
Vol 9 (01) ◽  
pp. 185-191 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Rate Of Convergence ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Sufficient Conditions ◽  
Critical Value ◽  
Traffic Intensity ◽  
Heavy Traffic Limit ◽  
Convergence Of Moments

A bound on the rate of convergence and sufficient conditions for the convergence of moments are obtained for the sequence of waiting times in the GI/G/1 queue when the traffic intensity is at the critical value ρ = 1.

Multiple channel queues in heavy traffic. II: sequences, networks, and batches

1970 ◽  
Vol 2 (2) ◽  
pp. 355-369 ◽  
Author(s):  
Donald L. Iglehart ◽  
Ward Whitt
Keyword(s):  
Stochastic Processes ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
General Setting ◽  
Critical Value ◽  
Multiple Channel ◽  
Heavy Traffic Limit ◽  
Service Channels ◽  
Traffic Behavior

This paper is a sequel to [7], in which heavy traffic limit theorems were proved for various stochastic processes arising in a single queueing facility with r arrival channels and s service channels. Here we prove similar theorems for sequences of such queueing facilities. The same heavy traffic behavior prevails in many cases in this more general setting, but new heavy traffic behavior is observed when the sequence of traffic intensities associated with the sequence of queueing facilities approaches the critical value (ρ = 1) at appropriate rates.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (4) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U(∊)n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M∊ = max {U(∊)n, n ≧ 0}, v0 = min {n : U(∊)n = M∊}, v1 = max {n : U(∊)n = M∊}. The joint limiting distribution of ∊2σ∊–2v0 and ∊σ∊–2M∊ is determined. It is the same as for ∊2σ∊–2v1 and ∊σ–2∊M∊. The marginal ∊σ–2∊M∊ gives Kingman's heavy traffic theorem. Also lim ∊–1P(M∊ = 0) and lim ∊–1P(M∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U(∊)n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

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