STAFFING A SERVICE SYSTEM WITH NON-POISSON NON-STATIONARY ARRIVALS

2016 ◽  
Vol 30 (4) ◽  
pp. 593-621 ◽  
Author(s):  
Beixiang He ◽  
Yunan Liu ◽  
Ward Whitt
Keyword(s):  
Service System ◽  
Heavy Traffic ◽  
Arrival Rate ◽  
Service Level ◽  
Arrival Process ◽  
Performance Impact ◽  
Poisson Arrival ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
The Impact

Motivated by non-Poisson stochastic variability found in service system arrival data, we extend established service system staffing algorithms using the square-root staffing formula to allow for non-Poisson arrival processes. We develop a general model of the non-Poisson non-stationary arrival process that includes as a special case the non-stationary Cox process (a modification of a Poisson process in which the rate itself is a non-stationary stochastic process), which has been advocated in the literature. We characterize the impact of the non-Poisson stochastic variability upon the staffing through the heavy-traffic limit of the peakedness (ratio of the variance to the mean in an associated stationary infinite-server queueing model), which depends on the arrival process through its central limit theorem behavior. We provide simple formulas to quantify the performance impact of the non-Poisson arrivals upon the staffing decisions, in order to achieve the desired service level. We conduct simulation experiments with non-stationary Markov-modulated Poisson arrival processes with sinusoidal arrival rate functions to demonstrate that the staffing algorithm is effective in stabilizing the time-varying probability of delay at designated targets.

INFINITE-SERVER QUEUES WITH BATCH ARRIVALS AND DEPENDENT SERVICE TIMES

2012 ◽  
Vol 26 (2) ◽  
pp. 197-220 ◽  
Author(s):  
Guodong Pang ◽  
Ward Whitt
Keyword(s):  
Large Scale ◽  
Heavy Traffic ◽  
Arrival Rate ◽  
Arrival Process ◽  
Regularity Conditions ◽  
Time Varying ◽  
Performance Impact ◽  
Batch Arrivals ◽  
Infinite Server ◽  
Service Times

Motivated by large-scale service systems, we consider an infinite-server queue with batch arrivals, where the service times are dependent within each batch. We allow the arrival rate of batches to be time varying as well as constant. As regularity conditions, we require that the batch sizes be i.i.d. and independent of the arrival process of batches, and we require that the service times within different batches be independent. We exploit a recently established heavy-traffic limit for the number of busy servers to determine the performance impact of the dependence among the service times. The number of busy servers is approximately a Gaussian process. The dependence among the service times does not affect the mean number of busy servers, but it does affect the variance of the number of busy servers. Our approximations quantify the performance impact upon the variance. We conduct simulations to evaluate the heavy-traffic approximations for the stationary model and the model with a time-varying arrival rate. In the simulation experiments, we use the Marshall–Olkin multivariate exponential distribution to model dependent exponential service times within a batch. We also introduce a class of Marshall–Olkin multivariate hyperexponential distributions to model dependent hyper-exponential service times within a batch.

Tollbooth tandem queues with infinite homogeneous servers

2015 ◽  
Vol 52 (4) ◽  
pp. 941-961 ◽  
Author(s):  
Xiuli Chao ◽  
Qi-Ming He ◽  
Sheldon Ross
Keyword(s):  
Performance Measures ◽  
System Performance ◽  
Stochastic Ordering ◽  
Tandem Queues ◽  
Poisson Arrival ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Steady State Solutions ◽  
Transient And Steady State

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

Average delay in queues with non-stationary Poisson arrivals

1978 ◽  
Vol 15 (03) ◽  
pp. 602-609 ◽  
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Intensity Function ◽  
Arrival Process ◽  
Single Server ◽  
Actual Process ◽  
Average Delay ◽  
Average Value ◽  
Poisson Arrival ◽  
Poisson Arrivals ◽  
Customer Delay

One of the major difficulties in attempting to apply known queueing theory results to real problems is that almost always these results assume a time-stationary Poisson arrival process, whereas in practice the actual process is almost invariably non-stationary. In this paper we consider single-server infinite-capacity queueing models in which the arrival process is a non-stationary process with an intensity function ∧(t), t ≧ 0, which is itself a random process. We suppose that the average value of the intensity function exists and is equal to some constant, call it λ, with probability 1. We make a conjecture to the effect that ‘the closer {∧(t), t ≧ 0} is to the stationary Poisson process with rate λ ' then the smaller is the average customer delay, and then we verify the conjecture in the special case where the arrival process is an interrupted Poisson process.

Queues with path-dependent arrival processes

2021 ◽  
Vol 58 (2) ◽  
pp. 484-504
Author(s):  
Kerry Fendick ◽  
Ward Whitt
Keyword(s):  
Point Processes ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Arrival Process ◽  
Diffusion Limit ◽  
Limiting Behavior ◽  
Queue Length Distribution ◽  
Long Run ◽  
Arrival Processes ◽  
Path Dependent

AbstractWe study the transient and limiting behavior of a queue with a Pólya arrival process. The Pólya process is interesting because it exhibits path-dependent behavior, e.g. it satisfies a non-ergodic law of large numbers: the average number of arrivals over time [0, t] converges almost surely to a nondegenerate limit as $t \rightarrow \infty$. We establish a heavy-traffic diffusion limit for the $\sum_{i=1}^{n} P_i/GI/1$ queue, with arrivals occurring exogenously according to the superposition of n independent and identically distributed Pólya point processes. That limit yields a tractable approximation for the transient queue-length distribution, because the limiting net input process is a Gaussian Markov process with stationary increments. We also provide insight into the long-run performance of queues with path-dependent arrival processes. We show how Little’s law can be stated in this context, and we provide conditions under which there is stability for a queue with a Pólya arrival process.

Technical Note—Approximating Systems Fed by Poisson Processes with Rapidly Changing Arrival Rates

2021 ◽  
Author(s):  
Zeyu Zheng ◽  
Harsha Honnappa ◽  
Peter W. Glynn
Keyword(s):  
Decision Making ◽  
Performance Prediction ◽  
Operations Management ◽  
Arrival Rate ◽  
Time Of Day ◽  
Technical Note ◽  
Poisson Processes ◽  
Arrival Process ◽  
Stochastic Fluctuations ◽  
Arrival Processes

In many operations management settings, the arrival process to the system exhibits clear nonstationarities. These nonstationarities may arise as a consequence of time-of-day effects, day-of-week effects, seasonalities, or stochastic fluctuations in the arrival rate. Tools that are developed for performance prediction and decision making when systems are stationary may not be valid in the nonstationary environment. This paper provides a generic, closed-form, and simple approximation tool for when the nonstationarities are caused by rapid fluctuations in the arrival processes.

Impact of weather conditions on airport arrival delay and throughput

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Álvaro Rodríguez-Sanz ◽  
Javier Cano ◽  
Beatriz Rubio Fernández
Keyword(s):  
Decision Making ◽  
Weather Conditions ◽  
Weather Data ◽  
Arrival Process ◽  
Data Set ◽  
Content Type ◽  
Weather Events ◽  
Adverse Weather ◽  
Arrival Processes ◽  
The Impact

Purpose Weather events have a significant impact on airport arrival performance and may cause delays in operations and/or constraints in airport capacity. In Europe, almost half of all regulated airport traffic delay is due to adverse weather conditions. Moreover, the closer airports operate to their maximum capacity, the more severe is the impact of a capacity loss due to external events such as weather. Various weather uncertainties occurring during airport operations can significantly delay some arrival processes and cause network-wide effects on the overall air traffic management (ATM) system. Quantifying the impact of weather is, therefore, a key feature to improve the decision-making process that enhances airport performance. It would allow airport operators to identify the relevant weather information needed, and help them decide on the appropriate actions to mitigate the consequences of adverse weather events. Therefore, this research aims to understand and quantify the impact of weather conditions on airport arrival processes, so it can be properly predicted and managed. Design/methodology/approach This study presents a methodology to evaluate the impact of adverse weather events on airport arrival performance (delay and throughput) and to define operational thresholds for significant weather conditions. This study uses a Bayesian Network approach to relate weather data from meteorological reports and airport arrival performance data with scheduled and actual movements, as well as arrival delays. This allows us to understand the relationships between weather phenomena and their impacts on arrival delay and throughput. The proposed model also provides us with the values of the explanatory variables (weather events) that lead to certain operational thresholds in the target variables (arrival delay and throughput). This study then presents a quantification of the airport performance with regard to an aggregated weather-performance metric. Specific weather phenomena are categorized through a synthetic index, which aims to quantify weather conditions at a given airport, based on aviation routine meteorological reports. This helps us to manage uncertainty at airport arrival operations by relating index levels with airport performance results. Findings The results are computed from a data set of over 750,000 flights on a major European hub and from local weather data during the period 2015–2018. This study combines delay and capacity metrics at different airport operational stages for the arrival process (final approach, taxi-in and in-block). Therefore, the spatial boundary of this study is not only the airport but also its surrounding airspace, to take both the arrival sequencing and metering area and potential holding patterns into consideration. Originality/value This study introduces a new approach for modeling causal relationships between airport arrival performance indicators and meteorological events, which can be used to quantify the impact of weather in airport arrival conditions, predict the evolution of airport operational scenarios and support airport decision-making processes.

scholarly journals Determining the optimal piecewise constant approximation for the nonhomogeneous Poisson process rate of Emergency Department patient arrivals

2021 ◽  
Author(s):  
Alberto De Santis ◽  
Tommaso Giovannelli ◽  
Stefano Lucidi ◽  
Mauro Messedaglia ◽  
Massimo Roma
Keyword(s):  
Emergency Department ◽  
Optimization Problem ◽  
Discrete Event ◽  
Arrival Rate ◽  
Black Box ◽  
Arrival Process ◽  
Accurate Estimation ◽  
Piecewise Constant ◽  
Piecewise Constant Approximation ◽  
The Impact

AbstractModeling the arrival process to an Emergency Department (ED) is the first step of all studies dealing with the patient flow within the ED. Many of them focus on the increasing phenomenon of ED overcrowding, which is afflicting hospitals all over the world. Since Discrete Event Simulation models are often adopted to assess solutions for reducing the impact of this problem, proper nonstationary processes are taken into account to reproduce time–dependent arrivals. Accordingly, an accurate estimation of the unknown arrival rate is required to guarantee the reliability of results. In this work, an integer nonlinear black–box optimization problem is solved to determine the best piecewise constant approximation of the time-varying arrival rate function, by finding the optimal partition of the 24 h into a suitable number of not equally spaced intervals. The black-box constraints of the optimization problem make the feasible solutions satisfy proper statistical hypotheses; these ensure the validity of the nonhomogeneous Poisson assumption about the arrival process, commonly adopted in the literature, and prevent mixing overdispersed data for model estimation. The cost function of the optimization problem includes a fit error term for the solution accuracy and a penalty term to select an adequate degree of regularity of the optimal solution. To show the effectiveness of this methodology, real data from one of the largest Italian hospital EDs are used.

EVALUASI KINERJA JALAN NASIONAL TERHADAP KARAKTERISTIK LALU LINTAS PADA RUAS JALAN NASIONAL KOTA PALEMBANG

2020 ◽  
Vol 9 (1) ◽  
Author(s):  
Sartika Nisumanti ◽  
Evina Krisna
Keyword(s):  
Traffic Congestion ◽  
Traffic Accidents ◽  
Road Traffic ◽  
Heavy Traffic ◽  
Service Level ◽  
Traffic Volume ◽  
The Road ◽  
Road Capacity ◽  
On The Road ◽  
The Impact

<p><em>The roads</em><em> </em><em>transportation is an </em><em>important infrastructure as one of the land transportation infrastructures for the movement of social activities and to support economic development, specifically in Palembang City. </em><em>The population growth of Palembang City has resulted in an increase in the number of vehicles and highway users. As a result</em><em>, transportation activities in Palembang, especially at Parameswara </em><em>roads are increasing. </em><em>The impact of this, there will be heavy traffic volume, resulting in conflicts on the road, which lead to traffic accidents. </em><em>Therefore</em><em>, there will be congestion and a decrease in the performance of the road speed.</em></p><p><em>The research is conducted at Parameswara Road in Palembang that visually diminished the ability to accommodate the road traffic volume per day, accordingly the effect of traffic that occurs due to the lack of road capacity as the sequence of large volume traffic. The purpose of this study is to determine the capacity and level of road services to carried out the performance value on this road. The method used in the analysis is the Greenshield model, Greenberg, and Underwood. This study explains the maximum volume at peak hour that develop on Monday is between 1561 smp/hour and 1549 smp/hour. Whereas the lowest is around 1225 smp/hour and 1008 smp/hour that occurs on Sunday. Therefore, the analysis of service level on the research years at Parameswara Road depicts the saturated traffic conditions and low starting speed with D service index category and service level analysis at 10 years of planning time projections, the lpda result is from 2022 to 2026, The Parameswara road conditions at E and F service index categories are the traffic jam circumstances and slight speed. Hence, it necessitates constructing a non-plot way at Parameswara Road intersection to tackle this traffic congestion.</em></p><p><strong><em>Keywords</em></strong><em>:<strong> </strong></em><em>Greenshield, Greenberg, Underwood</em>, <em>Road Capacity</em><em>.</em></p>

Tollbooth tandem queues with infinite homogeneous servers

2015 ◽  
Vol 52 (04) ◽  
pp. 941-961 ◽  
Author(s):  
Xiuli Chao ◽  
Qi-Ming He ◽  
Sheldon Ross
Keyword(s):  
Performance Measures ◽  
System Performance ◽  
Stochastic Ordering ◽  
Tandem Queues ◽  
Poisson Arrival ◽  
Arrival Processes ◽  
Poisson Arrivals ◽  
Number Of Customers ◽  
Steady State Solutions ◽  
Transient And Steady State

In this paper we analyze a tollbooth tandem queueing problem with an infinite number of servers. A customer starts service immediately upon arrival but cannot leave the system before all customers who arrived before him/her have left, i.e. customers depart the system in the same order as they arrive. Distributions of the total number of customers in the system, the number of departure-delayed customers in the system, and the number of customers in service at time t are obtained in closed form. Distributions of the sojourn times and departure delays of customers are also obtained explicitly. Both transient and steady state solutions are derived first for Poisson arrivals, and then extended to cases with batch Poisson and nonstationary Poisson arrival processes. Finally, we report several stochastic ordering results on how system performance measures are affected by arrival and service processes.

Average delay in queues with non-stationary Poisson arrivals

1978 ◽  
Vol 15 (3) ◽  
pp. 602-609 ◽  
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Intensity Function ◽  
Arrival Process ◽  
Single Server ◽  
Actual Process ◽  
Average Delay ◽  
Average Value ◽  
Poisson Arrival ◽  
Poisson Arrivals ◽  
Customer Delay

One of the major difficulties in attempting to apply known queueing theory results to real problems is that almost always these results assume a time-stationary Poisson arrival process, whereas in practice the actual process is almost invariably non-stationary. In this paper we consider single-server infinite-capacity queueing models in which the arrival process is a non-stationary process with an intensity function ∧(t), t ≧ 0, which is itself a random process. We suppose that the average value of the intensity function exists and is equal to some constant, call it λ, with probability 1.We make a conjecture to the effect that ‘the closer {∧(t), t ≧ 0} is to the stationary Poisson process with rate λ ' then the smaller is the average customer delay, and then we verify the conjecture in the special case where the arrival process is an interrupted Poisson process.

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