Diffusion Limits for Open Networks of Finite-Buffer Queues

1996 ◽  
Vol 10 (3) ◽  
pp. 341-361
Author(s):  
Indrajit Bardhan
Keyword(s):  
Brownian Motion ◽  
Communication Systems ◽  
Heavy Traffic ◽  
Finite Buffer ◽  
Diffusion Approximations ◽  
Loss Networks ◽  
Positive Orthant ◽  
System Process ◽  
Open Networks ◽  
Diffusion Limits

This paper presents diffusion limits for congestion in networks of finite-buffer queues. We consider both loss networks, such as those in communication systems, and networks with manufacturing blocking. In both cases, the number in system process, under conditions of approximate balance under heavy traffic and appropriate scaling of buffers, is shown to behave like a multidimensional Brownian motion reflected to stay within a rectangle in the positive orthant. The two limits differ in directions of reflection off the faces representing full buffers. The limits suggest possible diffusion approximations for finitebuffer networks.

scholarly journals On the Return Time for a Reflected Fractional Brownian Motion Process on the Positive Orthant

2011 ◽  
Vol 48 (01) ◽  
pp. 145-153 ◽  
Author(s):  
Chihoon Lee
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Network Models ◽  
Return Time ◽  
Brownian Motion Process ◽  
Positive Orthant ◽  
Time Result ◽  
Heavy Tailed

We consider a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = R + d , with drift r 0 ∈ R d and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r 0 and reflection directions, we establish a return time result for the RFBM process Z; that is, for some δ, κ > 0, sup x∈B E x [τ B (δ)] < ∞, where B = {x ∈ S : |x| ≤ κ} and τ B (δ) = inf{t ≥ δ : Z(t) ∈ B}. Similar results are known for reflected processes driven by standard Brownian motions, and our result can be viewed as their FBM counterpart. Our motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

scholarly journals A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant

2011 ◽  
Vol 48 (03) ◽  
pp. 820-831
Author(s):  
Chihoon Lee
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Network Models ◽  
Compact Set ◽  
Brownian Motion Process ◽  
Positive Orthant ◽  
Heavy Tailed ◽  
Necessary And Sufficient

We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = ℝ+ d , with drift r 0 ∈ ℝ d and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r 0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= E x [V(Ž̆(1))] − V(x) ≤ −βV(x) + b 1 C (x), x ∈ S, for an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

scholarly journals On the Return Time for a Reflected Fractional Brownian Motion Process on the Positive Orthant

2011 ◽  
Vol 48 (1) ◽  
pp. 145-153 ◽  
Author(s):  
Chihoon Lee
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Network Models ◽  
Return Time ◽  
Brownian Motion Process ◽  
Positive Orthant ◽  
Time Result ◽  
Heavy Tailed

We consider a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = R+d, with drift r0 ∈ Rd and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r0 and reflection directions, we establish a return time result for the RFBM process Z; that is, for some δ, κ > 0, supx∈BEx[τB(δ)] < ∞, where B = {x ∈ S : |x| ≤ κ} and τB(δ) = inf{t ≥ δ : Z(t) ∈ B}. Similar results are known for reflected processes driven by standard Brownian motions, and our result can be viewed as their FBM counterpart. Our motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

scholarly journals A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant

2011 ◽  
Vol 48 (3) ◽  
pp. 820-831 ◽  
Author(s):  
Chihoon Lee
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Network Models ◽  
Compact Set ◽  
Brownian Motion Process ◽  
Positive Orthant ◽  
Heavy Tailed ◽  
Necessary And Sufficient

We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = ℝ+d, with drift r0 ∈ ℝd and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= Ex[V(Ž̆(1))] − V(x) ≤ −βV(x) + b1C(x), x ∈ S, for an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

scholarly journals Homogeneous row-continuous bivariate markov chains with boundaries

1988 ◽  
Vol 25 (A) ◽  
pp. 237-256
Author(s):  
J. Keilson ◽  
M. Zachmann
Keyword(s):  
Data Transmission ◽  
Communication Systems ◽  
Buffer Capacity ◽  
Passage Time ◽  
Finite Buffer ◽  
Transition Rates ◽  
Boundary Problems ◽  
Ergodic Distribution ◽  
The Matrix ◽  
Voice Data

The matrix-geometric results of M. Neuts are extended to ergodic row-continuous bivariate Markov processes [J(t), N(t)] on state space B = {(j, n)} for which: (a) there is a boundary level N for N(t) associated with finite buffer capacity; (b) transition rates to adjacent rows and columns are independent of row level n in the interior of B. Such processes are of interest in the modelling of queue-length for voice-data transmission in communication systems. One finds that the ergodic distribution consists of two decaying components of matrix-geometric form, the second induced by the finite buffer capacity. The results are obtained via Green's function methods and compensation. Passage-time distributions for the two boundary problems are also made available algorithmically.

scholarly journals High Capacitive Secure Image Transmission Over Wireless Channels

2021 ◽  
Vol 10 (2) ◽  
pp. 564-570
Keyword(s):  
Medical Care ◽  
Data Transmission ◽  
Communication Systems ◽  
Wireless Channels ◽  
Clinical Information ◽  
Image Transmission ◽  
Care Practice ◽  
Text Data ◽  
Open Networks ◽  
Security Algorithm

Expansion of internet connectivity and its usage globally has increased various demands of providing security for the data transmission. Telemedicine is a modern way of medical care that can be extended to any remote place across the globe. This medical care practice is a result of the deployment of communication systems and information technology into healthcare system. With this technology the diagnosed data can be shared with physician and take his consultation remarks and also physicians can access to diagnostic archive and share for medical practice and learning. However, this exchange of information is confined with several risks of data theft when they are shared in open networks and hence they are to be protected with high security algorithms. This paper provides a high capacitive security algorithm for protecting the images with hidden confidential information. The approach provides a two-way security by encrypting the clinical information initially and embedding it imperceptibly in the concerned image so that the user on other can obtain both the visual and text data at same instance

Diffusion approximations and models for certain congestion problems

1968 ◽  
Vol 5 (03) ◽  
pp. 607-623 ◽  
Author(s):  
D. P. Gaver
Keyword(s):  
Mathematical Theory ◽  
Review Paper ◽  
Diffusion Processes ◽  
Road Traffic ◽  
Diffusion Theory ◽  
Heavy Traffic ◽  
Integral Transforms ◽  
Diffusion Approximations ◽  
Discrete State ◽  
Traffic Theory

In a variety of the congestion or queueing problems that arise in practice, for example, in studies of the crossing and entry problems of road traffic, (see Evans, Herman, and Weiss [2]), and recently of the service afforded by large centralized and shared computer facilities, (see Scherr [12]), the understanding of system performance furnished by the present mathematical theory is inadequate. The reason is that while the consideration of simple problems typically yields elegant mathematical results, the form of these results—often expressed in terms of integral transforms—is not immediately comprehensible nor useful for simple comparisons. This fact has been remarked upon by Newell, who in [9] has suggested certain more comprehensible but approximate approaches based on diffusion theory; further promising developments and elaborations will be found in [10]. The latter approach is related to the “heavy traffic theory” of J. F. C. Kingman [8], and to some recent work of Iglehart [6]. Of course, the idea of approximating complex discrete-state processes by diffusion processes with continuous paths is not new. It has long been used in genetics, see Feller [3], and the review paper by Kimura [7]. Nonetheless, applications to congestion theory are apparently still rather rare.

The stable M/G/1 queue in heavy traffic and its covariance function

1977 ◽  
Vol 9 (01) ◽  
pp. 169-186 ◽  
Author(s):  
Teunis J. Ott
Keyword(s):  
Brownian Motion ◽  
Waiting Time ◽  
Covariance Function ◽  
Busy Period ◽  
Stationary Process ◽  
Heavy Traffic ◽  
Time Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

Let X(t) be the virtual waiting-time process of a stable M/G/1 queue. Let R(t) be the covariance function of the stationary process X(t), B(t) the busy-period distribution of X(t); and let E(t) = P{X(t) = 0|X(0) = 0}. For X(t) some heavy-traffic results are given, among which are limiting expressions for R(t) and its derivatives and for B(t) and E(t). These results are used to find the covariance function of stationary Brownian motion on [0, ∞).

scholarly journals Heavy-Traffic Limits for a Many-Server Queueing Network with Switchover

2013 ◽  
Vol 45 (3) ◽  
pp. 645-672 ◽  
Author(s):  
Guodong Pang ◽  
David D. Yao
Keyword(s):  
Load Balancing ◽  
Differential Equations ◽  
Piecewise Linear ◽  
Heavy Traffic ◽  
Queueing Network ◽  
State Dependent ◽  
Diffusion Limits ◽  
Qed Regime ◽  
The Many ◽  
And Diffusion

We study a multiclass Markovian queueing network with switchover across a set of many-server stations. New arrivals to each station follow a nonstationary Poisson process. Each job waiting in queue may, after some exponentially distributed patience time, switch over to another station or leave the network following a probabilistic and state-dependent mechanism. We analyze the performance of such networks under the many-server heavy-traffic limiting regimes, including the critically loaded quality-and-efficiency-driven (QED) regime, and the overloaded efficiency-driven (ED) regime. We also study the limits corresponding to mixing the underloaded quality-driven (QD) regime with the QED and ED regimes. We establish fluid and diffusion limits of the queue-length processes in all regimes. The fluid limits are characterized by ordinary differential equations. The diffusion limits are characterized by stochastic differential equations, with a piecewise-linear drift term and a constant (QED) or time-varying (ED) covariance matrix. We investigate the load balancing effect of switchover in the mixed regimes, demonstrating the migration of workload from overloaded stations to underloaded stations and quantifying the load balancing impact of switchover probabilities.

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