On the asymptotic relationship between the overflow probability and the loss ratio

2001 ◽  
Vol 33 (04) ◽  
pp. 836-863 ◽  
Author(s):  
Han S. Kim ◽  
Ness B. Shroff
Keyword(s):  
Length Distribution ◽  
Upper And Lower Bounds ◽  
Log P ◽  
Buffer System ◽  
Finite Buffer ◽  
Loss Ratio ◽  
Fluid Queue ◽  
Queue Length Distribution ◽  
Overflow Probability ◽  
Asymptotic Relationship

In this paper we study the asymptotic relationship between the loss ratio in a finite buffer system and the overflow probability (the tail of the queue length distribution) in the corresponding infinite buffer system. We model the system by a fluid queue which consists of a server with constant rate c and a fluid input. We provide asymptotic upper and lower bounds on the difference between log P{Q > x} and logP L (x) under different conditions. The conditions for the upper bound are simple and are satisfied by a very large class of input processes. The conditions on the lower bound are more complex but we show that various classes of processes such as Markov modulated and ARMA type Gaussian input processes satisfy them.

On the asymptotic relationship between the overflow probability and the loss ratio

2001 ◽  
Vol 33 (4) ◽  
pp. 836-863 ◽  
Author(s):  
Han S. Kim ◽  
Ness B. Shroff
Keyword(s):  
Length Distribution ◽  
Upper And Lower Bounds ◽  
Log P ◽  
Buffer System ◽  
Finite Buffer ◽  
Loss Ratio ◽  
Fluid Queue ◽  
Queue Length Distribution ◽  
Overflow Probability ◽  
Asymptotic Relationship

In this paper we study the asymptotic relationship between the loss ratio in a finite buffer system and the overflow probability (the tail of the queue length distribution) in the corresponding infinite buffer system. We model the system by a fluid queue which consists of a server with constant rate c and a fluid input. We provide asymptotic upper and lower bounds on the difference between log P{Q > x} and logPL(x) under different conditions. The conditions for the upper bound are simple and are satisfied by a very large class of input processes. The conditions on the lower bound are more complex but we show that various classes of processes such as Markov modulated and ARMA type Gaussian input processes satisfy them.

Infinite- and finite-buffer Markov fluid queues: a unified analysis

2004 ◽  
Vol 41 (02) ◽  
pp. 557-569 ◽  
Author(s):  
Nail Akar ◽  
Khosrow Sohraby
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Length Distribution ◽  
Matrix Exponential ◽  
Divide And Conquer ◽  
Buffer System ◽  
Finite Buffer ◽  
Fluid Queues ◽  
Queue Length Distribution ◽  
Markov Fluid Queues

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite- and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrix-exponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrix-exponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.

scholarly journals Infinite- and finite-buffer Markov fluid queues: a unified analysis

2004 ◽  
Vol 41 (2) ◽  
pp. 557-569 ◽  
Author(s):  
Nail Akar ◽  
Khosrow Sohraby
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Length Distribution ◽  
Matrix Exponential ◽  
Divide And Conquer ◽  
Buffer System ◽  
Finite Buffer ◽  
Fluid Queues ◽  
Queue Length Distribution ◽  
Markov Fluid Queues

In this paper, we study Markov fluid queues where the net fluid rate to a single-buffer system varies with respect to the state of an underlying continuous-time Markov chain. We present a novel algorithmic approach to solve numerically for the steady-state solution of such queues. Using this approach, both infinite- and finite-buffer cases are studied. We show that the solution of the infinite-buffer case is reduced to the solution of a generalized spectral divide-and-conquer (SDC) problem applied on a certain matrix pencil. Moreover, this SDC problem does not require the individual computation of any eigenvalues and eigenvectors. Via the solution for the SDC problem, a matrix-exponential representation for the steady-state queue-length distribution is obtained. The finite-buffer case, on the other hand, requires a similar but different decomposition, the so-called additive decomposition (AD). Using the AD, we obtain a modified matrix-exponential representation for the steady-state queue-length distribution. The proposed approach for the finite-buffer case is shown not to have the numerical stability problems reported in the literature.

scholarly journals On the Effect of Finite Buffer Truncation in a Two-Node Jackson Network

2005 ◽  
Vol 42 (01) ◽  
pp. 199-222 ◽  
Author(s):  
Yutaka Sakuma ◽  
Masakiyo Miyazawa
Keyword(s):  
Stability Condition ◽  
Queue Length ◽  
Length Distribution ◽  
Buffer Size ◽  
Finite Buffer ◽  
Jackson Network ◽  
Numerical Examples ◽  
Queue Length Distribution ◽  
Special Cases ◽  
The Stability

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different cases for the limit. This generalizes some recent results obtained for the tandem Jackson network. Special cases and some numerical examples are also presented.

scholarly journals An LCFS finite buffer model with finite source batch input

1989 ◽  
Vol 26 (02) ◽  
pp. 372-380
Author(s):  
Nico M. Van Dijk
Keyword(s):  
Queueing Systems ◽  
Length Distribution ◽  
Source Distribution ◽  
Closed Form Expression ◽  
Finite Buffer ◽  
Form Expression ◽  
Batch Arrivals ◽  
Queue Length Distribution ◽  
Recursive Computation ◽  
Batch Input

Queueing systems are studied with a last-come, first-served queueing discipline and batch arrivals generated by a finite number of non-exponential sources. A closed-form expression is derived for the steady-state queue length distribution. This expression has a scaled geometric form and is insensitive to the input distribution. Moreover, an algorithm for the recursive computation of the normalizing constant and the busy source distribution is presented. The results are of both practical and theoretical interest as an extension of the standard Poisson batch input case.

scholarly journals On the Effect of Finite Buffer Truncation in a Two-Node Jackson Network

2005 ◽  
Vol 42 (1) ◽  
pp. 199-222 ◽  
Author(s):  
Yutaka Sakuma ◽  
Masakiyo Miyazawa
Keyword(s):  
Stability Condition ◽  
Queue Length ◽  
Length Distribution ◽  
Buffer Size ◽  
Finite Buffer ◽  
Jackson Network ◽  
Numerical Examples ◽  
Queue Length Distribution ◽  
Special Cases ◽  
The Stability

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different cases for the limit. This generalizes some recent results obtained for the tandem Jackson network. Special cases and some numerical examples are also presented.

scholarly journals On the Queue Length Distribution in BMAP Systems

2017 ◽  
Vol 2 (4) ◽  
pp. 275 ◽  
Author(s):  
Andrzej Chydzinski
Keyword(s):  
Poisson Process ◽  
Queue Length ◽  
Length Distribution ◽  
Compound Poisson Process ◽  
Arrival Process ◽  
Finite Buffer ◽  
Batch Markovian Arrival Process ◽  
Queue Length Distribution ◽  
Markovian Processes ◽  
The Laplace Transform

Batch Markovian Arrival Process – BMAP – is a teletraffic model which combines high ability to imitate complexstatistical behaviour of network traces with relative simplicity in analysis and simulation. It is also a generalization of a wide class of Markovian processes, a class which in particular include the Poisson process, the compound Poisson process, the Markovmodulated Poisson process, the phase-type renewal process and others. In this paper we study the main queueing performance characteristic of a finite-buffer queue fed by the BMAP, namely the queue length distribution. In particular, we show a formula for the Laplace transform of the queue length distribution. The main benefit of this formula is that it may be used to obtain both transient and stationary characteristics. To demonstrate this, several numerical results are presented.

scholarly journals An LCFS finite buffer model with finite source batch input

1989 ◽  
Vol 26 (2) ◽  
pp. 372-380 ◽  
Author(s):  
Nico M. Van Dijk
Keyword(s):  
Queueing Systems ◽  
Length Distribution ◽  
Source Distribution ◽  
Closed Form Expression ◽  
Finite Buffer ◽  
Form Expression ◽  
Batch Arrivals ◽  
Queue Length Distribution ◽  
Recursive Computation ◽  
Batch Input

Queueing systems are studied with a last-come, first-served queueing discipline and batch arrivals generated by a finite number of non-exponential sources. A closed-form expression is derived for the steady-state queue length distribution. This expression has a scaled geometric form and is insensitive to the input distribution. Moreover, an algorithm for the recursive computation of the normalizing constant and the busy source distribution is presented. The results are of both practical and theoretical interest as an extension of the standard Poisson batch input case.

scholarly journals Analysis of General Input State Dependent Working Vacation Queue with Changeover Time

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Vijaya Laxmi Pikkala ◽  
Suchitra Vepada
Keyword(s):  
Queue Length ◽  
Recursive Algorithm ◽  
Length Distribution ◽  
Input State ◽  
Finite Buffer ◽  
Queue Length Distribution ◽  
State Dependent ◽  
Stationary Queue Length Distribution ◽  
The Moment ◽  
Vacation Period

We consider a finite buffer GI/M(n)/1 queue with multiple working vacations and changeover time, where the server can keep on working but at a slower speed during the vacation period. Moreover, the amount of service demanded by a customer is conditioned by the queue length at the moment service is begun for that customer. We provide a recursive algorithm using the supplementary variable technique to numerically compute the stationary queue length distribution of the system. Finally, some numerical results of the model are presented to show the parameter effect on various performance measures.

Sign in / Sign up

Export Citation Format

Share Document