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 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 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 Markovian Queueing System with Discouraged Arrivals and Self-Regulatory Servers

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
K. V. Abdul Rasheed ◽  
M. Manoharan
Keyword(s):  
Queueing System ◽  
Queueing Systems ◽  
Single Server ◽  
Expected Number ◽  
Multiple Servers ◽  
Special Cases ◽  
Service Rates ◽  
Expected Waiting Time ◽  
Number Of Customers ◽  
Steady State Probabilities

We consider discouraged arrival of Markovian queueing systems whose service speed is regulated according to the number of customers in the system. We will reduce the congestion in two ways. First we attempt to reduce the congestion by discouraging the arrivals of customers from joining the queue. Secondly we reduce the congestion by introducing the concept of service switches. First we consider a model in which multiple servers have three service ratesμ1,μ2, andμ(μ1≤μ2<μ), say, slow, medium, and fast rates, respectively. If the number of customers in the system exceeds a particular pointK1orK2, the server switches to the medium or fast rate, respectively. For this adaptive queueing system the steady state probabilities are derived and some performance measures such as expected number in the system/queue and expected waiting time in the system/queue are obtained. Multiple server discouraged arrival model having one service switch and single server discouraged arrival model having one and two service switches are obtained as special cases. A Matlab program of the model is presented and numerical illustrations are given.

scholarly journals A new look at transient versions of Little's law, and M/G/1 preemptive last-come-first-served queues

2010 ◽  
Vol 47 (2) ◽  
pp. 459-473 ◽  
Author(s):  
Brian H. Fralix ◽  
Germán Riaño
Keyword(s):  
Point Processes ◽  
Queueing System ◽  
Queueing Systems ◽  
Time Dependent ◽  
Structural Form ◽  
Preemptive Resume ◽  
Special Cases ◽  
Number Of Customers ◽  
Regulated Brownian Motion ◽  
New Look

We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we show that previous laws (see Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this framework, a new law can be derived as well, which gives higher-moment expressions for very general types of queueing system; in particular, the laws hold for systems that allow customers to overtake one another. What is especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide expressions for all moments of the number of customers in the system in an M/G/1 preemptive last-come-first-served queue at a time t > 0, for any initial condition and any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling) that are analogous to the special cases found in Abate and Whitt (1987c), (1988). These expressions are then used to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see Abate and Whitt (1987a), (1987b)).

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 A new look at transient versions of Little's law, and M/G/1 preemptive last-come-first-served queues

2010 ◽  
Vol 47 (02) ◽  
pp. 459-473 ◽  
Author(s):  
Brian H. Fralix ◽  
Germán Riaño
Keyword(s):  
Point Processes ◽  
Queueing System ◽  
Queueing Systems ◽  
Time Dependent ◽  
Structural Form ◽  
Preemptive Resume ◽  
Special Cases ◽  
Number Of Customers ◽  
Regulated Brownian Motion ◽  
New Look

We take a new look at transient, or time-dependent Little laws for queueing systems. Through the use of Palm measures, we show that previous laws (see Bertsimas and Mourtzinou (1997)) can be generalized. Furthermore, within this framework, a new law can be derived as well, which gives higher-moment expressions for very general types of queueing system; in particular, the laws hold for systems that allow customers to overtake one another. What is especially novel about our approach is the use of Palm measures that are induced by nonstationary point processes, as these measures are not commonly found in the queueing literature. This new higher-moment law is then used to provide expressions for all moments of the number of customers in the system in an M/G/1 preemptive last-come-first-served queue at a time t &gt; 0, for any initial condition and any of the more famous preemptive disciplines (i.e. preemptive-resume, and preemptive-repeat with and without resampling) that are analogous to the special cases found in Abate and Whitt (1987c), (1988). These expressions are then used to derive a nice structural form for all of the time-dependent moments of a regulated Brownian motion (see Abate and Whitt (1987a), (1987b)).

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