Diffusion Approximation for Fair Resource Control—Interchange of Limits Under a Moment Condition

2021 ◽  
Author(s):  
Heng-Qing Ye ◽  
David D. Yao
Keyword(s):  
Diffusion Approximation ◽  
Resource Control ◽  
Moment Condition ◽  
Diffusion Limit ◽  
Free Process ◽  
Stochastic Processing ◽  
Phase Type ◽  
Arrival Processes ◽  
Service Times ◽  
Processing Network

In a prior study [Ye HQ, Yao DD (2016) Diffusion limit of fair resource control–Stationary and interchange of limits. Math. Oper. Res. 41(4):1161–1207.] focusing on a class of stochastic processing network with fair resource control, we justified the diffusion approximation (in the context of the interchange of limits) provided that the pth moment of the workloads are bounded. To this end, we introduced the so-called bounded workload condition, which requires the workload process be bounded by a free process plus the initial workload. This condition is for a derived process, the workload, as opposed to primitives such as arrival processes and service requirements; as such, it could be difficult to verify. In this paper, we establish the interchange of limits under a moment condition of suitable order on the primitives directly: the required order is [Formula: see text] on the moments of the primitive processes so as to bound the pth moment of the workload. This moment condition is trivial to verify, and indeed automatically holds in networks where the primitives have moments of all orders, for instance, renewal arrivals with phase-type interarrival times and independent and identically distributed phase-type service times.

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.

scholarly journals Fitting traffic traces with discrete canonical phase type distributions and Markov arrival processes

2014 ◽  
Vol 24 (3) ◽  
pp. 453-470 ◽  
Author(s):  
András Meszáros ◽  
János Papp ◽  
Miklós Telek
Keyword(s):  
Stochastic Models ◽  
Model Fitting ◽  
Canonical Representation ◽  
Distance Measures ◽  
Moment Matching ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Canonical Representations ◽  
Arrival Processes ◽  
Markov Arrival

Abstract Recent developments of matrix analytic methods make phase type distributions (PHs) and Markov Arrival Processes (MAPs) promising stochastic model candidates for capturing traffic trace behaviour and for efficient usage in queueing analysis. After introducing basics of these sets of stochastic models, the paper discusses the following subjects in detail: (i) PHs and MAPs have different representations. For efficient use of these models, sparse (defined by a minimal number of parameters) and unique representations of discrete time PHs and MAPs are needed, which are commonly referred to as canonical representations. The paper presents new results on the canonical representation of discrete PHs and MAPs. (ii) The canonical representation allows a direct mapping between experimental moments and the stochastic models, referred to as moment matching. Explicit procedures are provided for this mapping. (iii) Moment matching is not always the best way to model the behavior of traffic traces. Model fitting based on appropriately chosen distance measures might result in better performing stochastic models. We also demonstrate the efficiency of fitting procedures with experimental results

scholarly journals Detailed state probability distribution of infinite servers queues with phase-type distributed service times

2015 ◽  
Vol 3 (1) ◽  
pp. 29
Author(s):  
Ivan Mura
Keyword(s):  
Probability Distribution ◽  
State Probability ◽  
Phase Type ◽  
Distributed Service ◽  
Service Times

ONTARE. REVISTA DE INVESTIGACIÓN DE LA FACULTAD DE INGENIERÍAEste documento presenta un análisis detallado de la cola M/PH/∞, la cual permite determinar de una forma analítica, tanto para un estado transitorio como para uno estacionario, la distribución de probabilidad de los clientes en las distintas fases del servicio. El análisis se basa en la correspondencia que se puede encontrar entre el proceso estocástico que representa el número de clientes en servicioen las diferentes fases de la distribución de PH, y en un proceso estocástico que representa la evolución del número de clientes en los nodos de una red Jackson en la que todos los centros de servicio son colas M/M/∞.

scholarly journals Radiation diffusion in a ultra-relativistic expanding shell in relation to gamma-ray bursts

2021 ◽  
Vol 57 (1) ◽  
pp. 85-98
Author(s):  
I. A. Siutsou ◽  
А. Е. Kurguzava
Keyword(s):  
Initial Data ◽  
Diffusion Approximation ◽  
Thin Shell ◽  
Gamma Ray ◽  
Diffusion Time ◽  
Gamma Ray Bursts ◽  
Diffusion Limit ◽  
Time Interval ◽  
Radiation Diffusion ◽  
Initial Stage

 The present-day observational data obtained by satellite  observatories cover seven decades of gamma-ray energy, and there is no universal general model describing the formation of the spectrum. Therefore, it is important to describe the initial stages of radiation propagation in an ultrarelativistically expanding shell. The aim of this study was to obtain equations describing the propagation of radiation in a relativistically expanding shell in the diffusion limit, solve them for natural initial data, and apply the results obtained to the initial radiation of gamma-ray bursts. The following results were obtained: the initial stage of the gamma-ray burst in a photon-thin case can be described by radiation diffusion in an ultrarelativistically expanding shell; the time interval at which it is still possible to use the diffusion approximation increases with increasing the depth inside the shell quadratically; the value of the depth beyond which the diffusion approximation can be used increases, and the value of the radiation intensity decreases in diffusion time approaches; during the main radiation of the photon-thin shell, the diffusion approximation is suitable for most of the jet. The parameters of emission are close to the ones of short gamma-ray bursts.

Overflow processes for multiserver queues with finite waiting room

1984 ◽  
Vol 16 (1) ◽  
pp. 8-8
Author(s):  
Jos H. A. De Smit
Keyword(s):  
Renewal Process ◽  
Numerical Results ◽  
Markov Renewal Process ◽  
Waiting Room ◽  
Multiserver Queues ◽  
Phase Type ◽  
Service Times ◽  
Overflow Process ◽  
Explicit Expressions

The overflow process of the multiserver queue with phase-type service times and finite waiting room is a Markov renewal process. The solution for this process is obtained. If the service times are exponential the overflow process reduces to a renewal process. For the latter case explicit expressions and numerical results are given.

A versatile Markovian point process

1979 ◽  
Vol 16 (4) ◽  
pp. 764-779 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Point Processes ◽  
Probability Distributions ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Probability Models ◽  
Phase Type ◽  
Finite State ◽  
Markov Point Processes ◽  
Arrival Processes ◽  
Markov Modulated

We introduce a versatile class of point processes on the real line, which are closely related to finite-state Markov processes. Many relevant probability distributions, moment and correlation formulas are given in forms which are computationally tractable. Several point processes, such as renewal processes of phase type, Markov-modulated Poisson processes and certain semi-Markov point processes appear as particular cases. The treatment of a substantial number of existing probability models can be generalized in a systematic manner to arrival processes of the type discussed in this paper.Several qualitative features of point processes, such as certain types of fluctuations, grouping, interruptions and the inhibition of arrivals by bunch inputs can be modelled in a way which remains computationally tractable.

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