The two-moment three-parameter decomposition approximation of queueing networks with exponential residual renewal processes

Kim, Sun-Kyo

doi:10.1007/s11134-011-9226-1

Cited by 4 publications

(5 citation statements)

References 23 publications

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“…Horváth et al [23] approximated each station by a MAP/MAP/1 model. Kim [28,29] approximated each queue by a MMPP(2)/GI/1 model, where the arrival process is a Markov-modulated Poisson process with two states (a special MAP).…”

Section: Decomposition Approximationsmentioning

confidence: 99%

“…Remark 8 (The Kim [28,29] MMPP(2) decomposition.) In Kim [28,29], a decomposition approximation of queueing networks based on MMPP(2)/GI/1 queues was investigated.…”

Section: The Idc Equation Systemmentioning

confidence: 99%

“…The general framework here allows different choices of (i) the correction terms α i,j in §3.3.2 and β i in §3.3.3 and (ii) the feedback elimination procedure. The default correction terms are given in (25) and (29). For the feedback elimination procedure, we apply nearimmediate feedback elimination to all stations.…”

Section: The Full Rqna Algorithmmentioning

confidence: 99%

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A Robust Queueing Network Analyzer Based on Indices of Dispersion

Whitt

You

2020

Preprint

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We develop a robust queueing network analyzer algorithm to approximate the steady-state performance of a single-class open queueing network of single-server queues with Markovian routing. The algorithm allows non-renewal external arrival processes, general service-time distributions and customer feedback. We focus on the customer flows, defined as the continuous-time processes counting customers flowing into or out of the network, or flowing from one queue to another. Each flow is partially characterized by its rate and a continuous function that measures the stochastic variability over time. This function is a scaled version of the variance-time curve, called the index of dispersion for counts (IDC). The required IDC functions for the flows can be calculated from the model primitives, estimated from data or approximated by solving a set of linear equations. A robust queueing technique is used to generate approximations of the mean steady-state performance at each queue from the IDC of the total arrival flow and the service specification at that queue. The algorithm effectiveness is supported by extensive simulation studies and heavy-traffic limits.

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Section: Decomposition Approximationsmentioning

confidence: 99%

“…Remark 8 (The Kim [28,29] MMPP(2) decomposition.) In Kim [28,29], a decomposition approximation of queueing networks based on MMPP(2)/GI/1 queues was investigated.…”

Section: The Idc Equation Systemmentioning

confidence: 99%

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A Robust Queueing Network Analyzer Based on Indices of Dispersion

Whitt

You

2020

Preprint

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“…A MAP can model the dependence among interarrival times (or service times) because a MAP is not a renewal process. Horváth et al (2010) approximated each station by a

MAP / MAP / 1

model, while Kim (2011a, 2011b) approximated each queue by a

MMPP false(2 false) / GI / 1

model, where the arrival process is a Markov‐modulated Poisson process with two states (a special MAP).…”

Section: Introductionmentioning

confidence: 99%

A robust queueing network analyzer based on indices of dispersion

Whitt

You

2021

Naval Research Logistics

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This dissertation concludes five years of doctoral study and research. I am deeply indebted to many exceptional scholars and inspiring individuals, who guide me through an voyage of discovery that never meant to be a downwind sailing.First and foremost, I would like to thank my advisor, Professor Ward Whitt, for the enlightening guidance throughout my life as a Ph.D. student. He is always a walking encyclopedia to me with his accurate pointers to the literature and his prompt and constructive feedback. His generosity in sharing his knowledge from both the mathematics and engineering point of view helped me greatly in understanding the big picture of the subject matter of this work. As a great educator and scholar, he has been and will always be a role model that I look up to.

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“…As a simple and quick approximation for queueing network performance analysis, Poisson processes can be used when interarrivals and service times are considered as relatively close to the exponential distribution. As one of the most popular approaches in queueing network analysis, the decomposition method approximates arrival and departure processes as renewal processes; see Kuehn [13], Shanthikumar and Buzacott [15], Whitt [17], and Kim [10].…”

Section: Introductionmentioning

confidence: 99%

Minimal LST representations of MAP(n)s: Moment fittings and queueing approximations

Kim

2016

Naval Research Logistics

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A Markovian arrival process of order n, MAP(n), is typically described by two n × n transition rate matrices in terms of 2n 2 − n rate parameters. While it is straightforward and intuitive, the Markovian representation is redundant since the minimal number of parameters is n 2 for non-redundant MAP(n). It is well known that the redundancy complicates exact moment fittings. In this article, we present a minimal and unique Laplace-Stieltjes transform (LST) representations for MAP(n)s. Even though the LST coefficients vector itself is not a minimal representation, we show that the joint LST of stationary intervals can be represented with the minimum number of parameters. We also propose another minimal representation for MAP(3)s based on coefficients of the characteristic polynomial equations of the two transition rate matrices. An exact moment fitting procedure is presented for MAP(3)s based on two proposed minimal representations. We also discuss how MAP(3)/G/1 departure process can be approximated as a MAP(3). A simple tandem queueing network example is presented to show that the MAP(3) performs better than the MAP(2) in queueing approximations especially under moderate traffic intensities. of (n − 1) × (n − 1) principal minors of matrix Q, then 1 =q 1 +q 2 + · · · +q n . Let π be the stationary probability vector for Q, that is, π Q = 0 and π e = 1. Then, π = q/ 1 and the arrival rate isAlso, if we denote the stationary probability vector for the embedded Markov chain by p, that is, p = pP and pe = 1, then p = π D 1 /λ A .Note that π Q = π (D 0 + D 1 ) = 0 ⇒ π (−D 0 ) = π D 1 and that (D 0 + D 1 )e = Qe = 0 ⇒ −D 0 e = D 1 e.

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The two-moment three-parameter decomposition approximation of queueing networks with exponential residual renewal processes

Cited by 4 publications

References 23 publications

A Robust Queueing Network Analyzer Based on Indices of Dispersion

A Robust Queueing Network Analyzer Based on Indices of Dispersion

A robust queueing network analyzer based on indices of dispersion

Minimal LST representations of MAP(n)s: Moment fittings and queueing approximations

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