Rate of convergence of the fluid approximation for generalized Jackson networks

1996 ◽  
Vol 33 (3) ◽  
pp. 804-814 ◽  
Author(s):  
Hong Chen
Keyword(s):  
Rate Of Convergence ◽  
Queue Length ◽  
Queueing Network ◽  
Exponential Rate ◽  
Fluid Level ◽  
Fluid Approximation ◽  
Uniform Topology ◽  
Jackson Networks ◽  
Fluid Network ◽  
Level Process

It is known that a generalized open Jackson queueing network after appropriate scaling (in both time and space) converges almost surely to a fluid network under the uniform topology. Under the same topology, we show that the distance between the scaled queue length process of the queueing network and the fluid level process of the corresponding fluid network converges to zero in probability at an exponential rate.

Rate of convergence of the fluid approximation for generalized Jackson networks

1996 ◽  
Vol 33 (03) ◽  
pp. 804-814 ◽  
Author(s):  
Hong Chen
Keyword(s):  
Rate Of Convergence ◽  
Queue Length ◽  
Queueing Network ◽  
Exponential Rate ◽  
Fluid Level ◽  
Fluid Approximation ◽  
Uniform Topology ◽  
Jackson Networks ◽  
Fluid Network ◽  
Level Process

It is known that a generalized open Jackson queueing network after appropriate scaling (in both time and space) converges almost surely to a fluid network under the uniform topology. Under the same topology, we show that the distance between the scaled queue length process of the queueing network and the fluid level process of the corresponding fluid network converges to zero in probability at an exponential rate.

RATE OF CONVERGENCE OF FLUID APPROXIMATION FOR RE-ENTRANT LINES UNDER FBFS DISCIPLINE

2011 ◽  
Vol 28 (03) ◽  
pp. 401-417
Author(s):  
YONGJIANG GUO
Keyword(s):  
Queue Length ◽  
Idle Time ◽  
Arrival Process ◽  
Service Discipline ◽  
Time Process ◽  
Exponential Rate ◽  
Fluid Limit ◽  
Fluid Approximation ◽  
Uniform Topology ◽  
Busy Time

The re-entrant line networks under a first-buffer-first-served (FBFS) service discipline are considered in this paper. Under uniform topology, if the scaled arrival process and the scaled service process converge to the corresponding fluid limit processes with an exponential rate, we prove that the scaled processes characterizing the re-entrant line converge to the corresponding fluid limit processes with the exponential rate. Here the scaled processes include the queue length process, workload process, busy time process and idle time process.

scholarly journals Laws of Little in an open queueing network

2012 ◽  
Vol 17 (3) ◽  
pp. 327-342 ◽  
Author(s):  
Saulius Minkevičius ◽  
Stasys Steišūnas
Keyword(s):  
Waiting Time ◽  
Queueing Theory ◽  
Queue Length ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Fluid Limit ◽  
Fluid Approximation ◽  
Large Numbers ◽  
Virtual Waiting Time ◽  
Open Queueing Network

The object of this research in the queueing theory is theorems about the functional strong laws of large numbers (FSLLN) under the conditions of heavy traffic in an open queueing network (OQN). The FSLLN is known as a fluid limit or fluid approximation. In this paper, FSLLN are proved for the values of important probabilistic characteristics of the OQN investigated as well as the virtual waiting time of a customer and the queue length of customers. As applications of the proved theorems laws of Little in OQN are presented.

scholarly journals Tail Asymptotics for Monotone-Separable Networks

2007 ◽  
Vol 44 (02) ◽  
pp. 306-320
Author(s):  
Marc Lelarge
Keyword(s):  
Queueing Network ◽  
Network Models ◽  
Moment Generating Function ◽  
Stochastic Petri Nets ◽  
Arrival Process ◽  
Polling Systems ◽  
Tail Asymptotics ◽  
State Variables ◽  
Monotone Functions ◽  
Jackson Networks

A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework contains several classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. We use comparison relationships between networks of this class with independent and identically distributed driving sequences and the GI/GI/1/1 queue to obtain the tail asymptotics of the stationary maximal dater under light-tailed assumptions for service times. The exponential rate of decay is given as a function of a logarithmic moment generating function. We exemplify an explicit computation of this rate for the case of queues in tandem under various stochastic assumptions.

The exponential rate of convergence of the distribution of the maximum of a random walk. Part II

1976 ◽  
Vol 13 (04) ◽  
pp. 733-740
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Random Walk ◽  
Rate Of Convergence ◽  
Real Line ◽  
Exponential Convergence ◽  
Exponential Rate ◽  
The Real ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence ◽  
Successive Maximum

Let Gn (x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn (x) – limn→∞ Gn (x) is asymptotically equal to H(x) γn n–3/2 as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G ∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].

The exponential rate of convergence of the distribution of the maximum of a random walk. Part II

1976 ◽  
Vol 13 (4) ◽  
pp. 733-740 ◽  
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Random Walk ◽  
Rate Of Convergence ◽  
Real Line ◽  
Exponential Convergence ◽  
Exponential Rate ◽  
The Real ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence ◽  
Successive Maximum

Let Gn (x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn (x) – limn→∞ Gn(x) is asymptotically equal to H(x) γn n–3/2 as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].

scholarly journals A M/M/1 Queueing Network with Classical Retrial Policy

2021 ◽  
Vol 12 (1S) ◽  
pp. 329-337
Author(s):  
S. Shanmugasundaram, Et. al.
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Queueing Network ◽  
State Probability ◽  
Queueing Model ◽  
Steady State Probability ◽  
Numerical Examples ◽  
System Length ◽  
Number Of Customers ◽  
Little's Formula

In this paper we study the M/M/1 queueing model with retrial on network. We derive the steady state probability of customers in the network, the average number of customers in the all the three nodes in the system, the queue length, system length using little’s formula. The particular case is derived (no retrial). The numerical examples are given to test the correctness of the model.

The exponential rate of convergence of the distribution of the maximum of a random walk

1975 ◽  
Vol 12 (02) ◽  
pp. 279-288 ◽  
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Rate Of Convergence ◽  
Exponential Convergence ◽  
Random Variables ◽  
Limiting Distribution ◽  
Partial Sums ◽  
Exponential Rate ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence

Let Gn (x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn (x) — G(x) is asymptotically equal to c.H(x)n −3/2 γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.

scholarly journals An Adaptive Observer with Exponential Rate of Convergence

1982 ◽  
Vol 18 (4) ◽  
pp. 343-348 ◽  
Author(s):  
Zenta IWAI ◽  
Makoto SATO ◽  
Akira INOUE ◽  
Kazuo MANO
Keyword(s):  
Rate Of Convergence ◽  
Adaptive Observer ◽  
Exponential Rate ◽  
Exponential Rate Of Convergence
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