Limit theorems for queueing systems with doubly stochastic poisson arrivals (Heavy traffic conditions)

2008 ◽  
Vol 44 (4) ◽  
pp. 352-369 ◽  
Author(s):  
L. G. Afanas’eva ◽  
E. E. Bashtova
Keyword(s):  
Limit Theorems ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Doubly Stochastic ◽  
Traffic Conditions ◽  
Poisson Arrivals ◽  
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.

Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture

1991 ◽  
Vol 23 (1) ◽  
pp. 210-228 ◽  
Author(s):  
Cheng-Shang Chang ◽  
Xiu Li Chao ◽  
Michael Pinedo
Keyword(s):  
Queueing Systems ◽  
Loss Probability ◽  
Service Time Distribution ◽  
Doubly Stochastic ◽  
Virtual Waiting Time ◽  
Two Systems ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Monotonicity Results

In this paper, we compare queueing systems that differ only in their arrival processes, which are special forms of doubly stochastic Poisson (DSP) processes. We define a special form of stochastic dominance for DSP processes which is based on the well-known variability or convex ordering for random variables. For two DSP processes that satisfy our comparability condition in such a way that the first process is more ‘regular' than the second process, we show the following three results: (i) If the two systems are DSP/GI/1 queues, then for all f increasing convex, with V(i), i = 1 and 2, representing the workload (virtual waiting time) in system. (ii) If the two systems are DSP/M(k)/1→ /M(k)/l ∞ ·· ·∞ /M(k)/1 tandem systems, with M(k) representing an exponential service time distribution with a rate that is increasing concave in the number of customers, k, present at the station, then for all f increasing convex, with Q(i), i = 1 and 2, being the total number of customers in the two systems. (iii) If the two systems are DSP/M(k)/1/N systems, with N being the size of the buffer, then where denotes the blocking (loss) probability of the two systems. A model considered before by Ross (1978) satisfies our comparability condition; a conjecture stated by him is shown to be true.

Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's conjecture

1991 ◽  
Vol 23 (01) ◽  
pp. 210-228 ◽  
Author(s):  
Cheng-Shang Chang ◽  
Xiu Li Chao ◽  
Michael Pinedo
Keyword(s):  
Queueing Systems ◽  
Loss Probability ◽  
Service Time Distribution ◽  
Doubly Stochastic ◽  
Virtual Waiting Time ◽  
Two Systems ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Monotonicity Results

In this paper, we compare queueing systems that differ only in their arrival processes, which are special forms of doubly stochastic Poisson (DSP) processes. We define a special form of stochastic dominance for DSP processes which is based on the well-known variability or convex ordering for random variables. For two DSP processes that satisfy our comparability condition in such a way that the first process is more ‘regular' than the second process, we show the following three results: (i) If the two systems are DSP/GI/1 queues, then for all f increasing convex, with V (i), i = 1 and 2, representing the workload (virtual waiting time) in system. (ii) If the two systems are DSP/M(k)/1→ /M(k)/l ∞ ·· ·∞ /M(k)/1 tandem systems, with M(k) representing an exponential service time distribution with a rate that is increasing concave in the number of customers, k, present at the station, then for all f increasing convex, with Q (i), i = 1 and 2, being the total number of customers in the two systems. (iii) If the two systems are DSP/M(k)/1/N systems, with N being the size of the buffer, then where denotes the blocking (loss) probability of the two systems. A model considered before by Ross (1978) satisfies our comparability condition; a conjecture stated by him is shown to be true.

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.

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