scholarly journals A fluid queue with a finite buffer and subexponential input

2000 ◽  
Vol 32 (1) ◽  
pp. 221-243 ◽  
Author(s):  
A. P. Zwart
Keyword(s):  
Stationary Distribution ◽  
Fluid Model ◽  
Buffer Size ◽  
Finite Buffer ◽  
Asymptotic Results ◽  
Fluid Queue ◽  
Buffer Content ◽  
The Mean ◽  
Heavy Tailed ◽  
Finite Dam

We consider a fluid model similar to that of Kella and Whitt [32], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established by Kella and Whitt are extended to the finite buffer case: it is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N On-Off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue.

scholarly journals A fluid queue with a finite buffer and subexponential input

2000 ◽  
Vol 32 (01) ◽  
pp. 221-243 ◽  
Author(s):  
A. P. Zwart
Keyword(s):  
Stationary Distribution ◽  
Fluid Model ◽  
Buffer Size ◽  
Finite Buffer ◽  
Asymptotic Results ◽  
Fluid Queue ◽  
Buffer Content ◽  
The Mean ◽  
Heavy Tailed ◽  
Finite Dam

We consider a fluid model similar to that of Kella and Whitt [32], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established by Kella and Whitt are extended to the finite buffer case: it is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N On-Off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue.

Fluid Model Driven by an M/M/1 Queue with Set-Up and Close-Down Period

2014 ◽  
Vol 513-517 ◽  
pp. 3377-3380
Author(s):  
Fu Wei Wang ◽  
Bing Wei Mao
Keyword(s):  
Stationary Distribution ◽  
Solution Method ◽  
Fluid Model ◽  
Model Driven ◽  
Buffer Content ◽  
The Laplace Transform ◽  
Matrix Geometric Solution ◽  
The Mean ◽  
Joint Stationary Distribution ◽  
Set Up

The fluid model driven by an M/M/1 queue with set-up and close-down period is studied. The Laplace transform of the joint stationary distribution of the fluid model is of matrix geometric structure. With matrix geometric solution method, the Laplace-Stieltjes transformation of the stationary distribution of the buffer content is obtained, as well as the mean buffer content. Finally, with some numerical examples, the effect of the parameters on mean buffer content is presented.

A fluid model with data message discarding

2002 ◽  
Vol 34 (02) ◽  
pp. 329-348
Author(s):  
Bin Liu ◽  
Attahiru Sule Alfa
Keyword(s):  
Stationary Distribution ◽  
Performance Measures ◽  
Fluid Model ◽  
Threshold Value ◽  
Numerical Results ◽  
Finite Buffer ◽  
Control Parameters ◽  
Level Crossing ◽  
Data Message ◽  
Buffer Content

In this paper, we study a fluid model with partial message discarding and early message discarding, in which a finite buffer receives data (or information) from N independent on/off sources. All data generated by a source during one of its on periods is considered as a complete message. Our discarding scheme consists of two parts: (i) whenever some data belonging to a message has been lost due to overflow of the buffer, the remaining portion of this message will be discarded, and (ii) as long as the buffer content surpasses a certain threshold value at the instant an on period starts, all information generated during this on period will be discarded. By applying level-crossing techniques, we derive the equations for determining the system's stationary distribution. Further, two important performance measures, the probability of messages being transmitted successfully and the goodput of the system, are obtained. Numerical results are provided to demonstrate the effect of control parameters on the performance of the system.

A fluid model with data message discarding

2002 ◽  
Vol 34 (2) ◽  
pp. 329-348 ◽  
Author(s):  
Bin Liu ◽  
Attahiru Sule Alfa
Keyword(s):  
Stationary Distribution ◽  
Performance Measures ◽  
Fluid Model ◽  
Threshold Value ◽  
Numerical Results ◽  
Finite Buffer ◽  
Control Parameters ◽  
Level Crossing ◽  
Data Message ◽  
Buffer Content

In this paper, we study a fluid model with partial message discarding and early message discarding, in which a finite buffer receives data (or information) from N independent on/off sources. All data generated by a source during one of its on periods is considered as a complete message. Our discarding scheme consists of two parts: (i) whenever some data belonging to a message has been lost due to overflow of the buffer, the remaining portion of this message will be discarded, and (ii) as long as the buffer content surpasses a certain threshold value at the instant an on period starts, all information generated during this on period will be discarded. By applying level-crossing techniques, we derive the equations for determining the system's stationary distribution. Further, two important performance measures, the probability of messages being transmitted successfully and the goodput of the system, are obtained. Numerical results are provided to demonstrate the effect of control parameters on the performance of the system.

Cell loss probability for M/G/1 and time-slotted queues

2000 ◽  
Vol 37 (04) ◽  
pp. 1149-1156
Author(s):  
David McDonald ◽  
François Théberge
Keyword(s):  
Continuous Time ◽  
Time Slot ◽  
Arrival Rate ◽  
Cell Loss ◽  
Loss Probability ◽  
Finite Buffer ◽  
Asymptotic Results ◽  
Overflow Probability ◽  
The Mean

It is common practice to approximate the cell loss probability (CLP) of cells entering a finite buffer by the overflow probability (OVFL) of a corresponding infinite buffer queue, since the CLP is typically harder to estimate. We obtain exact asymptotic results for CLP and OVFL for time-slotted queues where block arrivals in different time slots are i.i.d. and one cell is served per time slot. In this case the ratio of CLP to OVFL is asymptotically (1-ρ)/ρ, where ρ is the use or, equivalently, the mean arrival rate per time slot. Analogous asymptotic results are obtained for continuous time M/G/1 queues. In this case the ratio of CLP to OVFL is asymptotically 1-ρ.

scholarly journals Asymptotic analysis by the saddle point method of the Anick-Mitra-Sondhi model

2004 ◽  
Vol 2004 (1) ◽  
pp. 19-71 ◽  
Author(s):  
Diego Dominici ◽  
Charles Knessl
Keyword(s):  
Waiting Times ◽  
Saddle Point Method ◽  
Conditional Distributions ◽  
Input Process ◽  
Steady State Distribution ◽  
State Distribution ◽  
Asymptotic Results ◽  
Fluid Queue ◽  
Constant Rate ◽  
Buffer Content

We consider a fluid queue where the input process consists of N identical sources that turn on and off at exponential waiting times. The server works at the constant rate c and an on source generates fluid at unit rate. This model was first formulated and analyzed by Anick et al. (1982). We obtain an alternate representation of the joint steady-state distribution of the buffer content and the number of on sources. This is given as a contour integral that we then analyze in the limit N→∞. We give detailed asymptotic results for the joint distribution as well as the associated marginal and conditional distributions. In particular, simple conditional limits laws are obtained. These show how the buffer content behaves conditioned on the number of active sources and vice versa. Numerical comparisons show that our asymptotic results are very accurate even for N=20.

Cell loss probability for M/G/1 and time-slotted queues

2000 ◽  
Vol 37 (4) ◽  
pp. 1149-1156 ◽  
Author(s):  
David McDonald ◽  
François Théberge
Keyword(s):  
Continuous Time ◽  
Time Slot ◽  
Arrival Rate ◽  
Cell Loss ◽  
Loss Probability ◽  
Finite Buffer ◽  
Asymptotic Results ◽  
Overflow Probability ◽  
The Mean

It is common practice to approximate the cell loss probability (CLP) of cells entering a finite buffer by the overflow probability (OVFL) of a corresponding infinite buffer queue, since the CLP is typically harder to estimate. We obtain exact asymptotic results for CLP and OVFL for time-slotted queues where block arrivals in different time slots are i.i.d. and one cell is served per time slot. In this case the ratio of CLP to OVFL is asymptotically (1-ρ)/ρ, where ρ is the use or, equivalently, the mean arrival rate per time slot. Analogous asymptotic results are obtained for continuous time M/G/1 queues. In this case the ratio of CLP to OVFL is asymptotically 1-ρ.

A TANDEM FLUID QUEUE WITH GRADUAL INPUT

2002 ◽  
Vol 16 (1) ◽  
pp. 29-45 ◽  
Author(s):  
Werner R.W. Scheinhardt ◽  
Bert Zwart
Keyword(s):  
Steady State ◽  
Content Distribution ◽  
Fluid Model ◽  
Sample Path ◽  
Fluid Queue ◽  
Finite Buffers ◽  
Martingale Methods ◽  
Buffer Content

For a two-node tandem fluid model with gradual input, we compute the joint steady-state buffer-content distribution. Our proof exploits martingale methods developed by Kella and Whitt. For the case of finite buffers, we use an insightful sample-path argument to extend an earlier proportionality result of Zwart to the network case.

Characterizing losses during busy periods in finite buffer systems

2003 ◽  
Vol 40 (1) ◽  
pp. 242-249 ◽  
Author(s):  
Erol A. Peköz ◽  
Rhonda Righter ◽  
Cathy H. Xia
Keyword(s):  
Busy Period ◽  
Arrival Rate ◽  
Batch Size ◽  
Buffer Size ◽  
Service Rate ◽  
Finite Buffer ◽  
Initial Number ◽  
Poisson Arrivals ◽  
The Mean ◽  
Number Of Customers

For multiple-server finite-buffer systems with batch Poisson arrivals, we explore how the distribution of the number of losses during a busy period changes with the buffer size and the initial number of customers. We show that when the arrival rate equals the maximal service rate (ρ= 1), as the buffer size increases the number of losses in a busy period increases in the convex sense, and whenρ> 1, as the buffer size increases the number of busy period losses increases in the increasing convex sense. Also, the number of busy period losses is stochastically increasing in the initial number of customers. A consequence of our results is that, whenρ= 1, the mean number of busy period losses equals the mean batch size of arrivals regardless of the buffer size. We show that this invariance does not extend to general arrival processes.

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