Rates of convergence for queues in heavy traffic. I

1972 ◽  
Vol 4 (2) ◽  
pp. 357-381 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Limit Theorems ◽  
Heavy Traffic ◽  
Rates Of Convergence ◽  
Traffic Intensity ◽  
Channel System ◽  
Multiple Channel ◽  
Skorokhod Representation Theorem ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit ◽  
Quantities Of Interest

Estimates are given for the rates of convergence in functional central limit theorems for quantities of interest in the GI/G/1 queue and a general multiple channel system. The traffic intensity is fixed ≧ 1. The method employed involves expressing the underlying stochastic processes in terms of Brownian motion using the Skorokhod representation theorem.

Rates of convergence for queues in heavy traffic. I

1972 ◽  
Vol 4 (02) ◽  
pp. 357-381 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Limit Theorems ◽  
Heavy Traffic ◽  
Rates Of Convergence ◽  
Traffic Intensity ◽  
Channel System ◽  
Multiple Channel ◽  
Skorokhod Representation Theorem ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit ◽  
Quantities Of Interest

Estimates are given for the rates of convergence in functional central limit theorems for quantities of interest in theGI/G/1 queue and a general multiple channel system. The traffic intensity is fixed ≧ 1. The method employed involves expressing the underlying stochastic processes in terms of Brownian motion using the Skorokhod representation theorem.

Rates of convergence for queues in heavy traffic. II: Sequences of queueing systems

1972 ◽  
Vol 4 (2) ◽  
pp. 382-391 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Central Limit ◽  
Queue Length ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Rates Of Convergence ◽  
Traffic Intensity ◽  
Multiple Channel ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

Estimates are given for the rates of convergence in functional central limit theorems for the queue length process in a sequence of general multiple channel queues. The situation is considered where the traffic intensity in the nth. queue, ρn, tends to ρ ≧ 1 as n approaches infinity. This extends previous work by the author, [6], in which the traffic intensity was fixed ≧ 1.

Rates of convergence for queues in heavy traffic. II: Sequences of queueing systems

1972 ◽  
Vol 4 (02) ◽  
pp. 382-391 ◽  
Author(s):  
Douglas P. Kennedy
Keyword(s):  
Central Limit ◽  
Queue Length ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Rates Of Convergence ◽  
Traffic Intensity ◽  
Multiple Channel ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

Estimates are given for the rates of convergence in functional central limit theorems for the queue length process in a sequence of general multiple channel queues. The situation is considered where the traffic intensity in thenth. queue, ρn, tends to ρ ≧ 1 asnapproaches infinity. This extends previous work by the author, [6], in which the traffic intensity was fixed ≧ 1.

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.

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

1989 ◽  
Vol 21 (2) ◽  
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.

Queues in transportation systems, II: An independently-dispatched system

1974 ◽  
Vol 11 (01) ◽  
pp. 145-158 ◽  
Author(s):  
Michael A. Crane
Keyword(s):  
Limit Theorems ◽  
Transportation Systems ◽  
Linear Network ◽  
Random Functions ◽  
Arrival Processes ◽  
Vehicle Fleet ◽  
Vehicle Capacity ◽  
Service Groups ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

We consider a transportation system consisting of a linear network of N + 1 terminals served by S vehicles of fixed capacity. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to some terminal j, 0 ≦ j ≦ i − 1, and are served as empty units of vehicle capacity become available at i. The vehicle fleet is partitioned into N service groups, with vehicles in the ith group stopping at terminals i, i − 1,···,0. Travel times between terminals and idle times at terminals are stochastic and are independent of the customer arrival processes. Functional central limit theorems are proved for random functions induced by processes of interest, including customer queue size processes. The results are of most interest in cases where the system is unstable. This occurs whenever, at some terminal, the rate of customer arrivals is at least as great as the rate at which vehicle capacity is made available.

The virtual waiting time of the GI/G/1 queue in heavy traffic

1971 ◽  
Vol 3 (2) ◽  
pp. 249-268 ◽  
Author(s):  
E. Kyprianou
Keyword(s):  
Markov Chains ◽  
Waiting Time ◽  
Limit Theorems ◽  
Queueing System ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Double Sequence ◽  
Traffic Intensity ◽  
Virtual Waiting Time

Investigations in the theory of heavy traffic were initiated by Kingman ([5], [6] and [7]) in an effort to obtain approximations for stable queues. He considered the Markov chains {Wni} of a sequence {Qi} of stable GI/G/1 queues, where Wni is the waiting time of the nth customer in the ith queueing system, and by making use of Spitzer's identity obtained limit theorems as first n → ∞ and then ρi ↑ 1 as i → ∞. Here &rHi is the traffic intensity of the ith queueing system. After Kingman the theory of heavy traffic was developed by a number of Russians mainly. Prohorov [10] considered the double sequence of waiting times {Wni} and obtained limit theorems in the three cases when n1/2(ρi-1) approaches (i) - ∞, (ii) -δ and (iii) 0 as n → ∞ and i → ∞ simultaneously. The case (i) includes the result of Kingman. Viskov [12] also studied the double sequence {Wni} and obtained limits in the two cases when n1/2(ρi − 1) approaches + δ and + ∞ as n → ∞ and i → ∞ simultaneously.

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