Limit Theorems for Queueing Systems with Various Service Disciplines in Heavy-Traffic Conditions

2018 ◽  
Vol 22 (4) ◽  
pp. 1529-1538
Author(s):  
S. A. Grishunina
Keyword(s):  
Limit Theorems ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Traffic Conditions ◽  
Service Disciplines ◽  
Heavy Traffic Conditions

scholarly journals A unified method to analyze overtake free queueing systems

1996 ◽  
Vol 28 (2) ◽  
pp. 588-625 ◽  
Author(s):  
Dimitris Bertsimas ◽  
Georgia Mourtzinou
Keyword(s):  
Heavy Traffic ◽  
Complete Solution ◽  
Queueing Systems ◽  
Exact Results ◽  
Traffic Conditions ◽  
Special Cases ◽  
Number Of Customers ◽  
First In First Out ◽  
Heavy Traffic Conditions ◽  
Unified Method

In this paper we demonstrate that the distributional laws that relate the number of customers in the system (queue), L(Q) and the time a customer spends in the system (queue), S(W) under the first-in-first-out (FIFO) discipline are special cases of the H = λG law and lead to a complete solution for the distributions of L, Q, S, W for queueing systems which satisfy distributional laws for both L and Q (overtake free systems). Moreover, in such systems the derivation of the distributions of L, Q, S, W can be done in a unified way. Consequences of the distributional laws include a generalization of PASTA to queueing systems with arbitrary renewal arrivals under heavy traffic conditions, a generalization of the Pollaczek–Khinchine formula to the G//G/1 queue, an extension of the Fuhrmann and Cooper decomposition for queues with generalized vacations under mixed generalized Erlang renewal arrivals, approximate results for the distributions of L, S in a GI/G/∞ queue, and exact results for the distributions of L, Q, S, W in priority queues with mixed generalized Erlang renewal arrivals.

scholarly journals On the Estimation in Multi-server Multi-Core Open Queuing Networks

2020 ◽  
Vol 8 ◽  
pp. 102-105
Author(s):  
Saulius Minkevicius
Keyword(s):  
Queueing Networks ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Functional Limit ◽  
Fluid Approximation ◽  
Traffic Conditions ◽  
Open Queueing Networks ◽  
Communications Theory ◽  
Multi Server ◽  
Heavy Traffic Conditions

The paper is devoted to the analysis of queueing systems in the context of the network and communications theory. We investigate the estimation in a multi-server multi-core open queueing networks and its applications to the theorems in heavy traffic conditions (fluid approximation, functional limit theorem, and law of the iterated logarithm) for a queue of jobs in a multi-server multi-core open queueing networks..

scholarly journals A unified method to analyze overtake free queueing systems

1996 ◽  
Vol 28 (02) ◽  
pp. 588-625 ◽  
Author(s):  
Dimitris Bertsimas ◽  
Georgia Mourtzinou
Keyword(s):  
Heavy Traffic ◽  
Complete Solution ◽  
Queueing Systems ◽  
Exact Results ◽  
Traffic Conditions ◽  
Special Cases ◽  
Number Of Customers ◽  
First In First Out ◽  
Heavy Traffic Conditions ◽  
Unified Method

In this paper we demonstrate that the distributional laws that relate the number of customers in the system (queue), L(Q) and the time a customer spends in the system (queue), S(W) under the first-in-first-out (FIFO) discipline are special cases of the H = λG law and lead to a complete solution for the distributions of L, Q, S, W for queueing systems which satisfy distributional laws for both L and Q (overtake free systems). Moreover, in such systems the derivation of the distributions of L, Q, S, W can be done in a unified way. Consequences of the distributional laws include a generalization of PASTA to queueing systems with arbitrary renewal arrivals under heavy traffic conditions, a generalization of the Pollaczek–Khinchine formula to the G//G/1 queue, an extension of the Fuhrmann and Cooper decomposition for queues with generalized vacations under mixed generalized Erlang renewal arrivals, approximate results for the distributions of L, S in a GI/G/∞ queue, and exact results for the distributions of L, Q, S, W in priority queues with mixed generalized Erlang renewal arrivals.

scholarly journals Heavy traffic analysis of a transportation network model

1996 ◽  
Vol 33 (03) ◽  
pp. 870-885
Author(s):  
William P. Peterson ◽  
Lawrence M. Wein
Keyword(s):  
Queueing Networks ◽  
Transportation Network ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Network Models ◽  
Reflected Brownian Motion ◽  
Traffic Conditions ◽  
Heavy Traffic Analysis ◽  
Functional Central Limit ◽  
Heavy Traffic Conditions

We study a model of a stochastic transportation system introduced by Crane. By adapting constructions of multidimensional reflected Brownian motion (RBM) that have since been developed for feedforward queueing networks, we generalize Crane's original functional central limit theorem results to a full vector setting, giving an explicit development for the case in which all terminals in the model experience heavy traffic conditions. We investigate product form conditions for the stationary distribution of our resulting RBM limit, and contrast our results for transportation networks with those for traditional queueing network models.

scholarly journals Weak Convergence Limits for Sojourn Times in Cyclic Queues Under Heavy Traffic Conditions

2008 ◽  
Vol 45 (2) ◽  
pp. 333-346 ◽  
Author(s):  
Hans Daduna ◽  
Christian Malchin ◽  
Ryszard Szekli
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Central Limit ◽  
Cycle Time ◽  
Heavy Traffic ◽  
Traffic Conditions ◽  
Sojourn Times ◽  
Population Sizes ◽  
Heavy Traffic Conditions

We consider sequences of closed cycles of exponential single-server nodes with a single bottleneck. We study the cycle time and the successive sojourn times of a customer when the population sizes go to infinity. Starting from old results on the mean cycle times under heavy traffic conditions, we prove a central limit theorem for the cycle time distribution. This result is then utilised to prove a weak convergence characteristic of the vector of a customer's successive sojourn times during a cycle for a sequence of networks with population sizes going to infinity. The limiting picture is a composition of a central limit theorem for the bottleneck node and an exponential limit for the unscaled sequences of sojourn times for the nonbottleneck nodes.

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