Marked point processes in discrete time

Sigman, Karl; Whitt, Ward

doi:10.1007/s11134-019-09612-3

Cited by 6 publications

(5 citation statements)

References 18 publications

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“…This is a time-varying version of the familiar state-space collapse, which goes back to the early HT papers (e.g., Whitt 1971). We remark that the relation is different from the time-varying LL discussed in Bertsimas and Mourtzinou (1997), Fralix and Riano (2010), Sigman and Whitt (2019), and Whitt and Zhang (2019).…”

Section: The Ht Fcltmentioning

confidence: 49%

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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Section: The Ht Fcltmentioning

confidence: 49%

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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“…As noted in Glynn and Whitt (1989a), Whitt (1991), and El-Taha and Stidham (1999), there are many important generalizations of LL, such as the relation H = λG. It remains to establish such results in a periodic setting (Sigman and Whitt 2018).…”

Section: Discussionmentioning

confidence: 90%

“…Related to Section 3, there should be more related good theory to develop associated with discrete time and periodic Palm measures and their application to queues, supplementing Whitt (1983), Miyazawa and Takahashi (1992), section 1.7.4 of Baccelli and Bremaud (1994), and section 1.4 and appendix D of Sigman (1995). Indeed, a contribution is in Sigman and Whitt (2018).…”

Section: Discussionmentioning

confidence: 99%

Periodic Little’s Law

Whitt

Zhang

2019

Operations Research

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Motivated by our recent study of patient flow data from an Israeli emergency department (ED), we establish a sample path periodic Little's law (PLL), which extends the sample path Little's law (LL). The ED data analysis led us to propose a periodic stochastic process to represent the aggregate ED occupancy level, with the length of a periodic cycle being 1 week. Because we conducted the ED data analysis over successive hours, we construct our PLL in discrete time. The PLL helps explain the remarkable similarities between the simulation estimates of the average hourly ED occupancy level over a week using our proposed stochastic model fit to the data and direct estimates of the ED occupancy level from the data. We also establish a steady-state stochastic PLL, similar to the time-varying LL.

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“…This process is well‐defined provided that the associated marginal process defined by

N_{g} (A) = N (A \times ℳ)

for any (measurable)

A \subseteq [0, T]

, known as the ground process, satisfies the finiteness condition for compact sets. Taking

T_{n} \in ℕ

instead leads to a discrete‐time MPP; see, for example, Sigman and Whitt (2019).…”

Section: Point Process Frameworkmentioning

confidence: 99%

A self‐exciting marked point process model for drought analysis

Genest

Jalbert

2021

Environmetrics

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A self‐exciting marked point process approach is proposed to model clustered low‐flow events. It combines a self‐exciting ground process designed to capture the temporal clustering behavior of extreme values and an extended Generalized Pareto mark distribution for the exceedances over a subasymptotic threshold. The model takes into account the dependence between the magnitude and occurrence time of exceedances and allows for closed‐form inference on tail probabilities and large quantiles. It is used to analyze daily water levels from the Rivière des Mille Îles (Québec, Canada) and to characterize drought patterns in the Montréal area. The model is useful to generate short‐term probability forecasts and to estimate the return period of major droughts. This information on the drought events is critical to water resource professionals in planning, designing, building, and managing more efficient water resource systems to hedge against the water shortage in case of extreme droughts.

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Marked point processes in discrete time

Cited by 6 publications

References 18 publications

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

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

Periodic Little’s Law

A self‐exciting marked point process model for drought analysis

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