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.

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

1974 ◽  
Vol 11 (1) ◽  
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.

Queues in transportation systems, I: a Markovian system

1973 ◽  
Vol 10 (03) ◽  
pp. 630-643
Author(s):  
Michael A. Crane
Keyword(s):  
Limit Theorems ◽  
Transportation Systems ◽  
Mild Conditions ◽  
Special Cases ◽  
Arrival Processes ◽  
Random Fluctuations ◽  
Functional Central Limit Theorems ◽  
Passenger Terminals ◽  
Functional Central Limit ◽  
Markovian System

We study a transportation system consisting of S vehicles of unit capacity and N passenger terminals. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to a terminal j, 1 ≦ j ≦ N, with probability Pij . Customers at each terminal are served as vehicles become available. Each vehicle is dispatched from a terminal when loaded, whereupon it travels to the destination of its passenger, according to a stochastic travel time. It is shown under mild conditions that the system is unstable, due to random fluctuations of independent customer arrival processes. We obtain limit theorems, in certain special cases, for the customer queue size processes. Where a steady-state limit exists, this limit is expressed in terms of the corresponding limit in a related GI/G/S queue. In other cases, functional central limit theorems are obtained for appropriately normalized random functions.

Queues in transportation systems, I: a Markovian system

1973 ◽  
Vol 10 (3) ◽  
pp. 630-643 ◽  
Author(s):  
Michael A. Crane
Keyword(s):  
Limit Theorems ◽  
Transportation Systems ◽  
Mild Conditions ◽  
Special Cases ◽  
Arrival Processes ◽  
Random Fluctuations ◽  
Functional Central Limit Theorems ◽  
Passenger Terminals ◽  
Functional Central Limit ◽  
Markovian System

We study a transportation system consisting of S vehicles of unit capacity and N passenger terminals. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to a terminal j, 1 ≦ j ≦ N, with probability Pij. Customers at each terminal are served as vehicles become available. Each vehicle is dispatched from a terminal when loaded, whereupon it travels to the destination of its passenger, according to a stochastic travel time. It is shown under mild conditions that the system is unstable, due to random fluctuations of independent customer arrival processes. We obtain limit theorems, in certain special cases, for the customer queue size processes. Where a steady-state limit exists, this limit is expressed in terms of the corresponding limit in a related GI/G/S queue. In other cases, functional central limit theorems are obtained for appropriately normalized random functions.

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.

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.

CENTRAL LIMIT AND FUNCTIONAL CENTRAL LIMIT THEOREMS FOR HILBERT-VALUED DEPENDENT HETEROGENEOUS ARRAYS WITH APPLICATIONS

1998 ◽  
Vol 14 (2) ◽  
pp. 260-284 ◽  
Author(s):  
Xiaohong Chen ◽  
Halbert White
Keyword(s):  
Central Limit ◽  
Adaptive Learning ◽  
Limit Theorems ◽  
Asymptotic Distributions ◽  
Central Limit Theorems ◽  
Cumulative Distribution ◽  
Density Estimator ◽  
Dependent Observations ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

We obtain new central limit theorems (CLT's) and functional central limit theorems (FCLT's) for Hilbert-valued arrays near epoch dependent on mixing processes, and also new FCLT's for general Hilbert-valued adapted dependent heterogeneous arrays. These theorems are useful in delivering asymptotic distributions for parametric and nonparametric estimators and their functionals in time series econometrics. We give three significant applications for near epoch dependent observations: (1) A new CLT for any plug-in estimator of a cumulative distribution function (c.d.f.) (e.g., an empirical c.d.f., or a c.d.f. estimator based on a kernel density estimator), which can in turn deliver distribution results for many Von Mises functionals; (2) a new limiting distribution result for degenerate U-statistics, which delivers distribution results for Bierens's integrated conditional moment tests; (3) a new functional central limit result for Hilbert-valued stochastic approximation procedures, which delivers distribution results for nonparametric recursive generalized method of moment estimators, including nonparametric adaptive learning models.

Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach

1973 ◽  
Vol 5 (01) ◽  
pp. 119-137 ◽  
Author(s):  
D. J. Scott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorem ◽  
Stationary Increments ◽  
Mixing Sequences ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.

Sign in / Sign up

Export Citation Format

Share Document