The snowball effect of customer slowdown in critical many-server systems

Selen, Jori; Adan, I.J.B.F.; Kulkarni, Vidyadhar G.; Leeuwaarden, Johan S. H. van

doi:10.1080/15326349.2015.1136221

Cited by 6 publications

(5 citation statements)

References 28 publications

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“…Aksin and Harker's (2001) work is motivated by a customer contact centre in which service rates decrease with load because agents require access to a shared information system with limited capacity, whereas Dong et al (2013) extend the Erlang A model to incorporate slowdown effects. Slowdown could also occur if customers who wait longer have longer average service times, as modeled by Chan et al (2015) and Selen et al (2015). Selen et al (2015) formulate their model as a QBD, as we do, but their phase variable is defined differently from ours.…”

Section: Analytical and Simulation Studiesmentioning

confidence: 99%

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Modeling Load and Overwork Effects in Queueing Systems with Adaptive Service Rates

Delasay

Ingolfsson

Kolfal

2016

Operations Research

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Servers in many real queueing systems do not work at a constant speed. They adapt to the system state by speeding up when the system is highly loaded or slowing down when load has been high for an extended time period. Their speed can also be constrained by other factors, such as geography or a downstream blockage. We develop a state-dependent queueing model in which the service rate depends on the system "load" and "overwork." Overwork refers to a situation where the system has been under a heavy load for an extended time period. We quantify load as the number of users in the system and we operationalize overwork with a state variable that is incremented with each service completion in a high-load period and decremented at a rate that is proportional to the number of idle servers during low-load periods. Our model is a quasi-birth-and-death process with a special structure that we exploit to develop efficient and easy-toimplement algorithms to compute system performance measures. We use the analytical model and simulation to demonstrate how using models that ignore adaptive server behavior can result in inconsistencies between planned and realized performance and can lead to suboptimal, unstable, or oscillatory staffing decisions.

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Section: Analytical and Simulation Studiesmentioning

confidence: 99%

“…Slowdown could also occur if customers who wait longer have longer average service times, as modeled by Chan et al (2015) and Selen et al (2015). Selen et al (2015) formulate their model as a QBD, as we do, but their phase variable is defined differently from ours.…”

Section: Analytical and Simulation Studiesmentioning

confidence: 99%

Modeling Load and Overwork Effects in Queueing Systems with Adaptive Service Rates

Delasay

Ingolfsson

Kolfal

2016

Operations Research

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“…A Markov process with a structure in which there are no upward transitions is amenable to the same solution approaches. The single-server priority system is one of many models that possesses this structure; a few others can be found in [41,42,99,105]. Due to the upward structure the balance equations could be solved recursively by treating them as second-order difference equations.…”

Section: Takeawaysmentioning

confidence: 99%

Analysis of structured Markov processes

Adan¹,

Leeuwaarden²,

Selen³

2017

Preprint

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Markov processes are popular mathematical models, studied by theoreticians for their intriguing properties, and applied by practitioners for their flexible structure. With this book we teach how to model and analyze Markov processes. We classify Markov processes based on their structural properties, which in turn determine which analytic methods are required for solving them. In doing so, we start in each chapter with specific examples that naturally lead up to general theory and general methods. In this way the reader learns about Markov processes on the job.By studying this book, the reader becomes acquainted with the basic analytic methods that come into play when systems are modeled as structured Markov processes. These basic methods will likely prove useful, in real-time when studying the examples at hand, but more importantly for future encounters with Markov processes not covered in this book. Methods are more important than examples. The methods have a large scope of application, even outside the scope of Markov processes, in areas like probability theory, industrial engineering, mechanical engineering, physics and financial mathematics.This book arose from various courses taught in the last decade at master level and postgraduate level. We thank the students and colleagues that participated in these courses for their valuable feedback.iii Contents 4 Birth-and-death processes 4.

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“…Of course, there are many systems beyond those covered in this section that can be modeled by class M Markov chains. For example, class M chains were recently used to model medical service systems in [8], [5], and [25].…”

Section: Examples Of Class M Markov Chainsmentioning

confidence: 99%

Clearing analysis on phases: Exact limiting probabilities for skip-free, unidirectional, quasi-birth-death processes

Doroudi¹,

Fralix²

2016

Stoch. Syst.

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Many problems in computing, service, and manufacturing systems can be modeled via infinite repeating Markov chains with an infinite number of levels and a finite number of phases. Many such chains are quasi-birth-death processes (QBDs) with transitions that are skip-free in level, in that one can only transition between consecutive levels, and unidirectional in phase, in that one can only transition from lower-numbered phases to higher-numbered phases. We present a procedure, which we call Clearing Analysis on Phases (CAP), for determining the limiting probabilities of such Markov chains exactly. The CAP method yields the limiting probability of each state in the repeating portion of the chain as a linear combination of scalar bases raised to a power corresponding to the level of the state. The weights in these linear combinations can be determined by solving a finite system of linear equations.1. Introduction. This paper studies the stationary distribution of Class M Markov chains, which are continuous time Markov chains (CTMCs) 1 having the following properties (see Fig. 1 and Fig. 2):• The Markov chain has a state space, E, that can be decomposed as E = R ∪ N , where R represents the infinite repeating portion of the chain, and N represents the finite nonrepeating portion of the chain. 2 ‡

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The snowball effect of customer slowdown in critical many-server systems

Cited by 6 publications

References 28 publications

Modeling Load and Overwork Effects in Queueing Systems with Adaptive Service Rates

Modeling Load and Overwork Effects in Queueing Systems with Adaptive Service Rates

Analysis of structured Markov processes

Clearing analysis on phases: Exact limiting probabilities for skip-free, unidirectional, quasi-birth-death processes

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