Transient and periodic solution to the time-inhomogeneous quasi-birth death process

Margolius, Barbara

doi:10.1007/s11134-007-9027-8

Cited by 14 publications

(13 citation statements)

References 19 publications

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“…The gray area indicates the 95% confidence intervals of Q N t , t ∈ (0, 1). They match pretty well with the analytical results derived by Margolius [41].…”

Section: An Examplesupporting

confidence: 87%

“…697]). As noted by Margolius [41], computational methods and approximation techniques involved in time-varying queueing problems have long been regarded as challenging. The time-varying ingredients can exist in the arrival processes, service durations, or the number of servers, as mentioned by Alfa and Margolius [4].…”

Section: Time-varying Queuesmentioning

confidence: 99%

“…Zeifman et al [60] approximated the limiting mean value (expected queue length at some given time) of the single-server model with the transient distribution by restricting their difference to some controllable extent. The asymptotic periodic solutions for single-server and multi-server models were achieved in Margolius [41], where distributions and moments were presented in terms of integral equations.…”

Section: Time-varying Queuesmentioning

confidence: 99%

“…The parameters are the same as those used by Margolius [41] and Zeifman et al [60]: Since only Q N 0 is generated in each trial, to get the samples at different times in a periodic cycle (whose length is 1), we only need to change the phases of the sinusoid functions in each run.…”

Section: An Examplementioning

confidence: 99%

See 3 more Smart Citations

Perfect and nearly perfect sampling of work-conserving queues

2015

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We present sampling-based methods to treat work-conserving queueing systems. A variety of models are studied. Besides the First Come First Served (FCFS) queues, many efforts are putted on the accumulating priority queue (APQ), where a customer accumulates priority linearly while waiting. APQs have Poisson arrivals, multi-class customers with corresponding service durations, and single or multiple servers.Perfect sampling is an approach to draw a sample directly from the steady-state distribution of a Markov chain without explicitly solving for it. Statistical inference can be conducted without initialization bias. If an error can be tolerated within some limit, i.e. the total variation distance between the simulated draw and the stationary distribution can be bounded by a specified number, then we get a so called "nearly" perfect sampling.Coupling from the past (CFTP) is one approach to perfect sampling, but it usually requires a bounded state space. One strategy for perfect sampling on unbounded state spaces relies on construction of a reversible dominating process. If only the dominating property is guaranteed, then regenerative method (RM) becomes an alternative choice. In the case where neither the reversibility nor dominance hold, a nearly perfect sampling method will be proposed. It is a variant of dominated CFTP that we call the CFTP Block Absorption (CFTP-BA) method.Time-varying queues with periodic Poisson arrivals are being considered in this thesis. It has been shown that a particular limiting distribution can be obtained for each point in time in the periodic cycle. Because there are no analytical solutions in closed forms, we explore perfect (or nearly perfect) sampling of these systems.

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“…The gray area indicates the 95% confidence intervals of Q N t , t ∈ (0, 1). They match pretty well with the analytical results derived by Margolius [41].…”

Section: An Examplesupporting

confidence: 87%

Section: Time-varying Queuesmentioning

confidence: 99%

Section: Time-varying Queuesmentioning

confidence: 99%

Section: An Examplementioning

confidence: 99%

See 2 more Smart Citations

Perfect and nearly perfect sampling of work-conserving queues

2015

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“…This is a BDP with the birth and death rates λ n (t) = λ(t) and μ n (t) = min (n, S) μ(t), respectively. There are a number of investigations of this model in a general situation, and, especially, in the case of periodic intensities and for the simplest M t /M t /1 model (see Di Crescenzo and Nobile, 1995;Knessl, 2000;Knessl and Yang, 2002;Mandelbaum and Massey, 1995;Margolius, 2007a;2007b;Massey and Whitt, 1994;Massey, 2002;Zhang and Coyle, 1991).…”

Section: T /M T /S Queuementioning

confidence: 99%

On truncations for weakly ergodic inhomogeneous birth and death processes

Zeifman

Satin

Korolev

et al. 2014

International Journal of Applied Mathematics and Computer Science

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We investigate a class of exponentially weakly ergodic inhomogeneous birth and death processes. We consider special transformations of the reduced intensity matrix of the process and obtain uniform (in time) error bounds of truncations. Our approach also guarantees that we can find limiting characteristics approximately with an arbitrarily fixed error. As an example, we obtain the respective bounds of the truncation error for an Mt/Mt/S queue for any number of servers S. Arbitrary intensity functions instead of periodic ones can be considered in the same manner.

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From the Birth-and-Death Process to Structured Markov Chains

2013

Fundamentals of Matrix-Analytic Methods

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Transient and periodic solution to the time-inhomogeneous quasi-birth death process

Cited by 14 publications

References 19 publications

Perfect and nearly perfect sampling of work-conserving queues

Perfect and nearly perfect sampling of work-conserving queues

On truncations for weakly ergodic inhomogeneous birth and death processes

From the Birth-and-Death Process to Structured Markov Chains

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