Queues with Choice via Delay Differential Equations

2017 ◽  
Vol 27 (04) ◽  
pp. 1730016 ◽  
Author(s):  
Jamol Pender ◽  
Richard H. Rand ◽  
Elizabeth Wesson
Keyword(s):  
Queue Length ◽  
Fluid Model ◽  
Moving Average ◽  
Queueing Systems ◽  
Multinomial Logit Model ◽  
Customer Choice ◽  
Customer Choice Behavior ◽  
Delay Differential ◽  
First Time ◽  
Average Information

Delay or queue length information has the potential to influence the decision of a customer to join a queue. Thus, it is imperative for managers of queueing systems to understand how the information that they provide will affect the performance of the system. To this end, we construct and analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information. In the first fluid model, customers join each queue according to a Multinomial Logit Model, however, the queue length information the customer receives is delayed by a constant [Formula: see text]. We show that the delay can cause oscillations or asynchronous behavior in the model based on the value of [Formula: see text]. In the second model, customers receive information about the queue length through a moving average of the queue length. Although it has been shown empirically that giving patients moving average information causes oscillations and asynchronous behavior to occur in U.S. hospitals, we analytically and mathematically show for the first time that the moving average fluid model can exhibit oscillations and determine their dependence on the moving average window. Thus, our analysis provides new insight on how operators of service systems should report queue length information to customers and how delayed information can produce unwanted system dynamics.

scholarly journals Breaking the Symmetry in Queues with Delayed Information

2021 ◽  
Vol 31 (09) ◽  
pp. 2130027
Author(s):  
Philip Doldo ◽  
Jamol Pender ◽  
Richard Rand
Keyword(s):  
Queue Length ◽  
Choice Behavior ◽  
Multiple Scales ◽  
Choice Model ◽  
Fluid Model ◽  
Queueing Systems ◽  
Model Parameters ◽  
Customer Choice ◽  
Service Rates ◽  
Customer Choice Behavior

Giving customers queue length information about a service system has the potential to influence the decision of a customer to join a queue. Thus, it is imperative for managers of queueing systems to understand how the information that they provide will affect the performance of the system. To this end, we construct and analyze a two-dimensional deterministic fluid model that incorporates customer choice behavior based on delayed queue length information. Reports in the existing literature always assume that all queues have identical parameters and the underlying dynamical system is symmetric. However, in this paper, we relax this symmetry assumption by allowing the arrival rates, service rates, and the choice model parameters to be different for each queue. Our methodology exploits the method of multiple scales and asymptotic analysis to understand how to break the symmetry. We find that the asymmetry can have a large impact on the underlying dynamics of the queueing system.

A Stochastic Analysis of Queues with Customer Choice and Delayed Information

2020 ◽  
Vol 45 (3) ◽  
pp. 1104-1126
Author(s):  
Jamol Pender ◽  
Richard Rand ◽  
Elizabeth Wesson
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Queue Length ◽  
Fluid Model ◽  
Waiting Times ◽  
Queueing Process ◽  
Delayed Information ◽  
Delay Differential ◽  
The Impact ◽  
Bifurcation Chaos

Many service systems provide queue length information to customers, thereby allowing customers to choose among many options of service. However, queue length information is often delayed, and it is often not provided in real time. Recent work by Dong et al. [Dong J, Yom-Tov E, Yom-Tov GB (2018) The impact of delay announcements on hospital network coordination and waiting times. Management Sci. 65(5):1969–1994.] explores the impact of these delays in an empirical study in U.S. hospitals. Work by Pender et al. [Pender J, Rand RH, Wesson E (2017) Queues with choice via delay differential equations. Internat. J. Bifurcation Chaos Appl. Sci. Engrg. 27(4):1730016-1–1730016-20.] uses a two-dimensional fluid model to study the impact of delayed information and determine the exact threshold under which delayed information can cause oscillations in the dynamics of the queue length. In this work, we confirm that the fluid model analyzed by Pender et al. [Pender J, Rand RH, Wesson E (2017) Queues with choice via delay differential equations. Internat. J. Bifurcation Chaos Appl. Sci. Engrg. 27(4):1730016-1–1730016-20.] can be rigorously obtained as a functional law of large numbers limit of a stochastic queueing process, and we generalize their threshold analysis to arbitrary dimensions. Moreover, we prove a functional central limit theorem for the queue length process and show that the scaled queue length converges to a stochastic delay differential equation. Thus, our analysis sheds new insight on how delayed information can produce unexpected system dynamics.

Filtering of Markov renewal queues, IV: Flow processes in feedback queues

1985 ◽  
Vol 17 (2) ◽  
pp. 386-407 ◽  
Author(s):  
Jeffrey J. Hunter
Keyword(s):  
Queue Length ◽  
Queueing Systems ◽  
Arrival Process ◽  
Markov Renewal Process ◽  
Renewal Processes ◽  
Flow Process ◽  
Special Cases ◽  
Flow Processes ◽  
Transition Points ◽  
The Times

This paper is a continuation of the study of a class of queueing systems where the queue-length process embedded at basic transition points, which consist of ‘arrivals’, ‘departures’ and ‘feedbacks’, is a Markov renewal process (MRP). The filtering procedure of Çinlar (1969) was used in [12] to show that the queue length process embedded separately at ‘arrivals’, ‘departures’, ‘feedbacks’, ‘inputs’ (arrivals and feedbacks), ‘outputs’ (departures and feedbacks) and ‘external’ transitions (arrivals and departures) are also MRP. In this paper expressions for the elements of each Markov renewal kernel are derived, and thence expressions for the distribution of the times between transitions, under stationary conditions, are found for each of the above flow processes. In particular, it is shown that the inter-event distributions for the arrival process and the departure process are the same, with an equivalent result holding for inputs and outputs. Further, expressions for the stationary joint distributions of successive intervals between events in each flow process are derived and interconnections, using the concept of reversed Markov renewal processes, are explored. Conditions under which any of the flow processes are renewal processes or, more particularly, Poisson processes are also investigated. Special cases including, in particular, the M/M/1/N and M/M/1 model with instantaneous Bernoulli feedback, are examined.

Controlled queues in heavy traffic

1975 ◽  
Vol 7 (3) ◽  
pp. 656-671 ◽  
Author(s):  
John H. Rath
Keyword(s):  
Diffusion Process ◽  
Queue Length ◽  
Queueing System ◽  
Systems Approach ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Control Policy ◽  
Controlled Diffusion ◽  
Controlled Queues

This paper studies a controlled queueing system in which the decisionmaker may change servers according to rules which depend only on the queue length. It is proved that for a given control policy a properly normalised sequence of these controlled queue length processes converges weakly to a controlled diffusion process as the queueing systems approach a state of heavy traffic.

scholarly journals Influence of Parking Price on Parking Garage Users’ Behaviour

1970 ◽  
Vol 24 (5) ◽  
pp. 413-423 ◽  
Author(s):  
Jelena Simićević ◽  
Nada Milosavljević ◽  
Goran Maletić
Keyword(s):  
Traffic Congestion ◽  
Logit Model ◽  
Multinomial Logit Model ◽  
Great Sensitivity ◽  
Multinomial Model ◽  
Parking Garage ◽  
Policy Makers ◽  
Binary Logit ◽  
First Time ◽  
Made In

Parking charge is a powerful tool for solving parking and traffic congestion problems. In order to achieve the expected effects without any adverse impact it is necessary to understand well the users’ responses to this policy. This paper, based on a sample of interviewed parking garage users, has developed binary logit model for identification and quantification of characteristics of users and trips, on which the acceptance of parking price is dependent. In addition, multinomial logit model has been made in order to predict what the users will opt for when faced with an increase in parking price. For the first time the parameter “shorten duration” has been introduced which has shown to be the most significant in making behaviour-related decisions. The results show that the users with the purpose work are the most sensitive to an increase in parking charge, what can be deemed positive for policy makers. However, great sensitivity of the users with the purpose shopping should cause their concern. The results of the multinomial model show that they would not discontinue coming into the area after all.

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