A Rare-Event Simulation Algorithm for Periodic Single-Server Queues

Ma, Ni; Whitt, Ward

doi:10.1287/ijoc.2017.0766

Cited by 6 publications

(23 citation statements)

References 37 publications

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“…It can be shown by induction on the number of arrivals in (θ ′ , θ] (see also (Ma and Whitt 2017)) that, if the system is empty at time θ ′ ∈ [0, θ],…”

Section: Proof Of Proposition 44mentioning

confidence: 99%

Randomized Dimension Reduction for Monte Carlo Simulations

Kahale

2020

Management Science

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We present a new unbiased algorithm that estimates the expected value of f (U ) via Monte Carlo simulation, where U is a vector of d independent random variables, and f is a function of d variables. We assume that f does not depend equally on all its arguments. Under certain conditions we prove that, for the same computational cost, the variance of our estimator is lower than the variance of the standard Monte Carlo estimator by a factor of order d. Our method can be used to obtain a low-variance unbiased estimator for the expectation of a function of the state of a Markov chain at a given time-step. We study applications to volatility forecasting and time-varying queues. Numerical experiments show that our algorithm dramatically improves upon the standard Monte Carlo method for large values of d, and is highly resilient to discontinuities.

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“…It can be shown by induction on the number of arrivals in (θ ′ , θ] (see also (Ma and Whitt 2017)) that, if the system is empty at time θ ′ ∈ [0, θ],…”

Section: Proof Of Proposition 44mentioning

confidence: 99%

Randomized Dimension Reduction for Monte Carlo Simulations

Kahale

2020

Management Science

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“…We start in Section 2 by presenting two simulation examples to illustrate the effectiveness of our new algorithm. Then, in Section 3, we introduce the two technical tools we will apply: (1) an extension of the rareevent simulation algorithm for the GI t /GI/1 model from Ma and Whitt (2018) to the GI t /GI t /1 model with a general service-rate control and (2) heavy-traffic limits involving scaling of the underlying deterministic periodic arrival-rate function. We start in earnest in Section 4.…”

Section: Organization Of This Papermentioning

confidence: 99%

“…Theorem 1 shows that the rate-matching control stabilizes the workload process as well as the queue-length process. We discuss the extension of the rare-event simulation algorithm from Ma and Whitt (2018) to our setting and its application to perform simulation search in Section 5. In both Sections 4 and 5, we will be brief because we can draw on Whitt (2015) and Ma and Whitt (2018).…”

Section: Organization Of This Papermentioning

confidence: 99%

“…Our primary tool for finding good (η, ξ) controls is a simulation search algorithm. For that purpose, we extend the rare-event simulation algorithm for the time-varying workload process in the periodic GI t /GI/1 model in Ma and Whitt (2018) to the GI t /GI t /1 model, where the service rate is time varying as well. (The notation GI t means that the process is a deterministic time transformation of a renewal process; see Section 4.)…”

Section: The Primary Tool: a Simulation Search Algorithmmentioning

confidence: 99%

“…The waiting time W(t) coincides with the workload L(t) when μ(t) 1 for all t, but not otherwise. As in Ma and Whitt (2018), the rare-event simulation algorithm calculates the periodic steady-state workload L y and waiting time W y , starting at time yc within a cycle of length c, 0 ≤ y < 1. We use a search over the parameters (η, ξ), as discussed in Section 5, to solve the optimization problems [Equations (9) and 10].…”

Section: The Primary Tool: a Simulation Search Algorithmmentioning

confidence: 99%

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Minimizing the Maximum Expected Waiting Time in a Periodic Single-Server Queue with a Service-Rate Control

Whitt

2019

Stochastic Systems

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We consider a single-server queue with unlimited waiting space, the first-come, first-served discipline, a periodic arrival-rate function, and independent and identically distributed service requirements, where the service-rate function is subject to control. We previously showed that a rate-matching control, whereby the service rate is made proportional to the arrival rate, stabilizes the queue-length process but not the (virtual) waitingtime process. To minimize the maximum expected waiting time (and stabilize the expected waiting time), we now consider a modification of the service-rate control involving two parameters: a time lag and a damping factor. We develop an efficient simulation search algorithm to find the best time lag and damping factor. That simulation algorithm is an extension of our recent rare-event simulation algorithm for the GI t /GI/1 queue to the GI t /GI t /1 queue, allowing the time-varying service rate. To gain insight into these controls, we establish a heavy-traffic limit with periodicity in the fluid scale. This produces a diffusion control problem for the stabilization, which we solve numerically by the simulation search in the scaled family of systems with ρ ↑ 1. The state space collapse in that theorem shows that there is a time-varying Little's law in heavy traffic, implying that the queue length and waiting time cannot be simultaneously stabilized in this limit. We conduct simulation experiments showing that the new control is effective for stabilizing the expected waiting time for a wide range of model parameters, but we also show that it cannot stabilize the expected waiting time perfectly. History: Former designation of this paper was SSy-2018-012.R2.

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Stabilizing the virtual response time in single-server processor sharing queues with slowly time-varying arrival rates

Cho

2020

Ann Oper Res

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Motivated by the work of Whitt [1], who studied stabilization of the mean virtual waiting time (excluding service time) in a GI t /GI t /1/F CF S queue, this paper investigates the stabilization of the mean virtual response time in a single-server processor sharing (PS) queueing system with a time-varying arrival rate and a service rate control (a GI t /GI t /1/P S queue). We propose and compare a modified square-root (SR) control and a difference-matching (DM) control to stabilize the mean virtual response time of a GI t /GI t /1/P S queue. Extensive simulation studies with various settings of arrival processes and service times show that the DM control outperforms the SR control for heavy-traffic conditions, and that the SR control performs better for lighttraffic conditions.

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A Rare-Event Simulation Algorithm for Periodic Single-Server Queues

Cited by 6 publications

References 37 publications

Randomized Dimension Reduction for Monte Carlo Simulations

Randomized Dimension Reduction for Monte Carlo Simulations

Minimizing the Maximum Expected Waiting Time in a Periodic Single-Server Queue with a Service-Rate Control

Stabilizing the virtual response time in single-server processor sharing queues with slowly time-varying arrival rates

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