Canonical Form of Order-2 Non-stationary Markov Arrival Processes

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

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Transient Markov arrival processes

The Annals of Applied Probability ◽

10.1214/aoap/1050689597 ◽

2003 ◽

Vol 13 (2) ◽

pp. 628-640 ◽

Cited By ~ 19

Author(s):

Peter Taylor ◽

Marie-Ange Remiche ◽

Guy Latouche

Keyword(s):

Arrival Processes ◽

Markov Arrival

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A minimal representation of Markov arrival processes and a moments matching method

Performance Evaluation ◽

10.1016/j.peva.2007.06.001 ◽

2007 ◽

Vol 64 (9-12) ◽

pp. 1153-1168 ◽

Cited By ~ 95

Author(s):

M. Telek ◽

G. Horváth

Keyword(s):

Minimal Representation ◽

Matching Method ◽

Arrival Processes ◽

Markov Arrival

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Queues with marked customers

Advances in Applied Probability ◽

10.1017/s000186780004862x ◽

1996 ◽

Vol 28 (02) ◽

pp. 567-587 ◽

Cited By ~ 9

Author(s):

Qi-Ming He

Keyword(s):

Detailed Analysis ◽

Waiting Times ◽

Queueing Systems ◽

Fundamental Period ◽

Different Types ◽

Arrival Processes ◽

Queue Lengths ◽

Points Of View ◽

Markov Arrival

Queueing systems with distinguished arrivals are described on the basis of Markov arrival processes with marked transitions. Customers are distinguished by their types of arrival. Usually, the queues observed by customers of different types are different, especially for queueing systems with bursty arrival processes. We study queueing systems from the points of view of customers of different types. A detailed analysis of the fundamental period, queue lengths and waiting times at the epochs of arrivals is given. The results obtained are the generalizations of the results of theMAP/G/1 queue.

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Strong convergence of a class of non-homogeneous Markov arrival processes to a Poisson process

Statistics & Probability Letters ◽

10.1016/j.spl.2007.07.018 ◽

2008 ◽

Vol 78 (4) ◽

pp. 445-455 ◽

Cited By ~ 1

Author(s):

James Ledoux

Keyword(s):

Strong Convergence ◽

Poisson Process ◽

Arrival Processes ◽

Markov Arrival

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COMPARISON OF QUEUES WITH DIFFERENT DISCRETE-TIME ARRIVAL PROCESSES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964801151016 ◽

2001 ◽

Vol 15 (1) ◽

pp. 1-14 ◽

Cited By ~ 3

Author(s):

Arie Hordijk

Keyword(s):

Lower Bound ◽

Discrete Time ◽

Sojourn Time ◽

Arrival Process ◽

Convex Ordering ◽

Arrival Processes ◽

Markov Arrival ◽

Markov Arrival Process ◽

Special Case ◽

Stochastic Event

Traveling times in a FIFO-stochastic event graph are compared in increasing convex ordering for different arrival processes. As a special case, a stochastic lower bound is obtained for the sojourn time in a tandem network of FIFO queues with a Markov arrival process. A counterexample shows that the extended Ross conjecture is not true for discrete-time arrival processes.

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