Asymptotic loss probability in a finite buffer fluid queue with hetergeneous heavy-tailed on--off processes

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.

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A fluid queue with a finite buffer and subexponential input

Advances in Applied Probability ◽

10.1017/s000186780000985x ◽

2000 ◽

Vol 32 (01) ◽

pp. 221-243 ◽

Cited By ~ 4

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.

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Loss Probability of a Burst Arrival Finite Queue with Synchronized Service

Probability in the Engineering and Informational Sciences ◽

10.1017/s026996480000245x ◽

1992 ◽

Vol 6 (2) ◽

pp. 201-216 ◽

Cited By ~ 8

Author(s):

Masakiyo Miyazawa

Keyword(s):

Markov Chain ◽

Loss Probability ◽

Buffer Size ◽

Embedded Markov Chain ◽

Main Concern ◽

Single Server ◽

Finite Buffer ◽

Single Server Queue ◽

Server Queue ◽

Computation Algorithm

We are concerned with a burst arrival single-server queue, where arrivals of cells in a burst are synchronized with a constant service time. The main concern is with the loss probability of cells for the queue with a finite buffer. We analyze an embedded Markov chain at departure instants of cells and get a kind of lumpability for its state space. Based on these results, this paper proposes a computation algorithm for its stationary distribution and the loss probability. Closed formulas are obtained for the first two moments of the numbers of cells and active bursts when the buffer size is infinite.

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Volume and duration of losses in finite buffer fluid queues

Journal of Applied Probability ◽

10.1239/jap/1445543849 ◽

2015 ◽

Vol 52 (3) ◽

pp. 826-840 ◽

Cited By ~ 1

Author(s):

Fabrice Guillemin ◽

Bruno Sericola

Keyword(s):

Busy Period ◽

Buffer Capacity ◽

Exact Expression ◽

Loss Probability ◽

Finite Buffer ◽

Fluid Queues ◽

Buffer Content ◽

Phase Type ◽

Phase Type Distributions ◽

Arrival Processes

We study congestion periods in a finite fluid buffer when the net input rate depends upon a recurrent Markov process; congestion occurs when the buffer content is equal to the buffer capacity. Similarly to O'Reilly and Palmowski (2013), we consider the duration of congestion periods as well as the associated volume of lost information. While these quantities are characterized by their Laplace transforms in that paper, we presently derive their distributions in a typical stationary busy period of the buffer. Our goal is to compute the exact expression of the loss probability in the system, which is usually approximated by the probability that the occupancy of the infinite buffer is greater than the buffer capacity under consideration. Moreover, by using general results of the theory of Markovian arrival processes, we show that the duration of congestion and the volume of lost information have phase-type distributions.

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

Journal of Applied Probability ◽

10.1017/s0021900200018349 ◽

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.

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ON THE WORKLOAD PROCESS IN A FLUID QUEUE WITH A RESPONSIVE BURSTY INPUT AND SELECTIVE DISCARDING

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964803174074 ◽

2003 ◽

Vol 17 (4) ◽

pp. 527-543

Author(s):

Parijat Dube ◽

Eitan Altman

Keyword(s):

Probability Density Function ◽

Probability Density ◽

Feedback System ◽

Moment Generating Function ◽

General Distribution ◽

Finite Buffer ◽

Fluid Queue ◽

The Stability ◽

The Moment ◽

Stationary Workload

We analyze a feedback system consisting of a finite buffer fluid queue and a responsive source. The source alternates between silence periods and active periods. At random epochs of times, the source becomes ready to send a burst of fluid. The length of the bursts (length of the active periods) are independent and identically distributed with some general distribution. The queue employs a threshold discarding policy in the sense that only those bursts at whose commencement epoch (the instant at which the source is ready to send) the workload (i.e., the amount of fluid in the buffer) is less than some preset threshold are accepted. If the burst is rejected then the source backs off from sending. We work within the framework of Poisson counter-driven stochastic differential equations and obtain the moment generating function and hence the probability density function of the stationary workload process. We then comment on the stability of this fluid queue. Our explicit characterizations will further provide useful insights and “engineering” guidelines for better network designing.

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Cell loss asymptotics for buffers fed with a large number of independent stationary sources

Journal of Applied Probability ◽

10.1239/jap/1032374231 ◽

1999 ◽

Vol 36 (1) ◽

pp. 86-96 ◽

Cited By ~ 21

Author(s):

Nikolay Likhanov ◽

Ravi R. Mazumdar

Keyword(s):

Local Limit ◽

Cell Loss ◽

Buffer Overflow ◽

Finite Buffer ◽

Loss Probabilities ◽

Measure Change ◽

Self Similar ◽

Heavy Tailed ◽

Overflow Probabilities ◽

Stationary Sources

In this paper we derive asymptotically exact expressions for buffer overflow probabilities and cell loss probabilities for a finite buffer which is fed by a large number of independent and stationary sources. The technique is based on scaling, measure change and local limit theorems and extends the recent results of Courcoubetis and Weber on buffer overflow asymptotics. We discuss the cases when the buffers are of the same order as the transmission bandwidth as well as the case of small buffers. Moreover we show that the results hold for a wide variety of traffic sources including ON/OFF sources with heavy-tailed distributed ON periods, which are typical candidates for so-called ‘self-similar’ inputs, showing that the asymptotic cell loss probability behaves in much the same manner for such sources as for the Markovian type of sources, which has important implications for statistical multiplexing. Numerical validation of the results against simulations are also reported.

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