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.

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.

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.

Derivation of the mean cell delay and cell loss probability for multiple input-queued switches

2000 ◽  
Vol 4 (4) ◽  
pp. 140-142 ◽  
Author(s):  
Hakyong Kim ◽  
Kiseon Kim ◽  
Yongtak Lee
Keyword(s):  
Cell Loss ◽  
Loss Probability ◽  
Multiple Input ◽  
The Mean ◽  
Cell Delay

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.

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.

Characterizing losses during busy periods in finite buffer systems

2003 ◽  
Vol 40 (01) ◽  
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 when ρ > 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, when ρ = 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.

ON AN EQUIVALENCE BETWEEN LOSS RATES AND CYCLE MAXIMA IN QUEUES AND DAMS

2005 ◽  
Vol 19 (2) ◽  
pp. 241-255 ◽  
Author(s):  
René Bekker ◽  
Bert Zwart
Keyword(s):  
Loss Probability ◽  
Single Server ◽  
Finite Buffer ◽  
Single Server Queue ◽  
Release Rates ◽  
Asymptotic Results ◽  
State Dependent ◽  
Tail Distribution ◽  
Server Queue ◽  
Cycle Maximum

We consider the loss probability of a customer in a single-server queue with finite buffer and partial rejection and show that it can be identified with the tail distribution of the cycle maximum of the associated infinite-buffer queue. This equivalence is shown to hold for the GI/G/1 queue and for dams with state-dependent release rates. To prove this equivalence, we use a duality for stochastically monotone recursions, developed by Asmussen and Sigman (1996). As an application, we obtain several exact and asymptotic results for the loss probability and extend Takács' formula for the cycle maximum in the M/G/1 queue to dams with variable release rate.

scholarly journals On the effect of heterogeneity on flocking behavior and systemic risk

2017 ◽  
Vol 34 (3-4) ◽  
Author(s):  
Fei Fang ◽  
Yiwei Sun ◽  
Konstantinos Spiliopoulos
Keyword(s):  
Asymptotic Formula ◽  
Case Studies ◽  
Systemic Risk ◽  
Mean Field ◽  
Loss Probability ◽  
Loss Distribution ◽  
Interacting Diffusions ◽  
The Mean ◽  
Laplace Asymptotics ◽  
Flocking Behavior

AbstractThe goal of this paper is to study organized flocking behavior and systemic risk in heterogeneous mean-field interacting diffusions. We illustrate in a number of case studies the effect of heterogeneity in the behavior of systemic risk in the system, i.e., the risk that several agents default simultaneously as a result of interconnections. We also investigate the effect of heterogeneity on the “flocking behavior” of different agents, i.e., when agents with different dynamics end up following very similar paths and follow closely the mean behavior of the system. Using Laplace asymptotics, we derive an asymptotic formula for the tail of the loss distribution as the number of agents grows to infinity. This characterizes the tail of the loss distribution and the effect of the heterogeneity of the network on the tail loss probability.

Loss Probability of a Burst Arrival Finite Queue with Synchronized Service

1992 ◽  
Vol 6 (2) ◽  
pp. 201-216 ◽  
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.

scholarly journals Volume and duration of losses in finite buffer fluid queues

2015 ◽  
Vol 52 (3) ◽  
pp. 826-840 ◽  
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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