Queues with time-dependent arrival rates. II — The maximum queue and the return to equilibrium

1968 ◽  
Vol 5 (03) ◽  
pp. 579-590 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Fixed Amount ◽  
Deterministic Theory ◽  
Return To Equilibrium ◽  
Maximum Value ◽  
Equilibrium Distributions ◽  
Truncated Normal

During a rush hour, the arrival rateλ(t) of customers to a service facility is assumed to increase to a maximum value exceeding the service rateμ, and then decrease again. In Part I it was shown that, afterλ(t) has passedμ, the expected queueE{X(t)} exceeds that given by the deterministic theory by a fixed amount, (0.95)L, which is proportional to the (–1/3) power ofa(t) =dλ(t)/dtevaluated at timet= 0 whenλ(t) =μ.The maximum ofE{X(t)}, therefore, occurs whenλ(t) again is equal toμat timet1as predicted by deterministic queueing theory, but is larger than given by the deterministic theory by this same constant (0.95)L(providedt1is sufficiently large). It is shown here that the maximum queue, suptX(t) is, approximately normally distributed with a mean (0.95) (L+L1) larger than predicted by deterministic theory whereL, is proportional to the (–1/3) power ofa(t1). We also investigate the distribution ofX(t) at the end of the rush hour when the queue distribution returns to equilibrium. During the transition, the queue distribution is approximately a mixture of a truncated normal and the equilibrium distributions. These results are applied to a case whereλ(t) is quadratic int.

Queues with time-dependent arrival rates. II — The maximum queue and the return to equilibrium

1968 ◽  
Vol 5 (3) ◽  
pp. 579-590 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Fixed Amount ◽  
Deterministic Theory ◽  
Return To Equilibrium ◽  
Maximum Value ◽  
Equilibrium Distributions ◽  
Truncated Normal

During a rush hour, the arrival rate λ(t) of customers to a service facility is assumed to increase to a maximum value exceeding the service rate μ, and then decrease again. In Part I it was shown that, after λ(t) has passed μ, the expected queue E{X(t)} exceeds that given by the deterministic theory by a fixed amount, (0.95)L, which is proportional to the (–1/3) power of a(t) = dλ(t)/dt evaluated at time t = 0 when λ(t) = μ. The maximum of E{X(t)}, therefore, occurs when λ(t) again is equal to μ at time t1 as predicted by deterministic queueing theory, but is larger than given by the deterministic theory by this same constant (0.95)L (provided t1 is sufficiently large). It is shown here that the maximum queue, suptX(t) is, approximately normally distributed with a mean (0.95) (L + L1) larger than predicted by deterministic theory where L, is proportional to the (–1/3) power of a(t1). We also investigate the distribution of X(t) at the end of the rush hour when the queue distribution returns to equilibrium. During the transition, the queue distribution is approximately a mixture of a truncated normal and the equilibrium distributions. These results are applied to a case where λ(t) is quadratic in t.

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (3) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the form λ(t) = λ(0) — βt2 for some constant β. Diffusion approximations show that for λ(0) sufficiently close to the service rate μ, the mean queue length at time 0 is proportional to β–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for all λ(0) and β. Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

Queues with time-dependent arrival rates. III — A mild rush hour

1968 ◽  
Vol 5 (03) ◽  
pp. 591-606 ◽  
Author(s):  
G. F. Newell
Keyword(s):  
Queueing Theory ◽  
Queue Length ◽  
Arrival Rate ◽  
Time Dependent ◽  
Service Rate ◽  
Diffusion Approximations ◽  
Dimensionless Form ◽  
Queue Lengths ◽  
The Mean ◽  
Mean Queue Length

The arrival rate of customers to a service facility is assumed to have the formλ(t) =λ(0) —βt2for some constantβ.Diffusion approximations show that forλ(0) sufficiently close to the service rateμ, the mean queue length at time 0 is proportional toβ–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for allλ(0) andβ.Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.

The solution of an infinite set of differential-difference equations occurring in polymerization and queueing problems

1963 ◽  
Vol 59 (1) ◽  
pp. 117-124 ◽  
Author(s):  
A. Wragg
Keyword(s):  
Difference Equations ◽  
Queueing Theory ◽  
Bessel Functions ◽  
Formal Solution ◽  
Monomer Concentration ◽  
Arrival Rate ◽  
Time Dependent ◽  
Type Integral ◽  
Dependent Parameter ◽  
Infinite Set

AbstractThe time-dependent solutions of an infinite set of differential-difference equations arising from queueing theory and models of ‘living’ polymer are expressed in terms of modified Bessel functions. Explicit solutions are available for constant values of a parameter describing the arrival rate or monomer concentration; for time-dependent parameter a formal solution is obtained in terms of a function which satisfies a Volterra type integral equation of the second kind. These results are used as the basis of a numerical method of solving the infinite set of differential equations when the time-dependent parameter itself satisfies a differential equation.

Evaluation of Vehicle Parking Queueing in a National Park: Case Study of the Laurance S. Rockefeller Preserve in Grand Teton National Park

2017 ◽  
Vol 2654 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Antonio Fuentes ◽  
Kevin Heaslip ◽  
Ashley D’Antonio ◽  
Majid Khalilikhah ◽  
Ali Soltani-Sobh
Keyword(s):  
National Parks ◽  
Performance Measures ◽  
Queueing Theory ◽  
National Park ◽  
Arrival Rate ◽  
Service Rate ◽  
Evaluation Of Performance ◽  
Visitor Attractions ◽  
Measures Of Performance

In national parks, there is a tension between providing areas for vehicles and accommodating other visitor activities in the park. This tension often means there is more demand for vehicle parking than the supply can accommodate. This study examined one of Grand Teton National Park’s (GRTE) visitor attractions with visitor parking provided, the Laurance S. Rockefeller Preserve (LSR) in Jackson, Wyoming. With approximately 54 designated parking spots available, a study to determine the queueing theory measures of performance was conducted under observed and higher-value parameters to evaluate the system. It was determined that under the observed average arrival rate of 25 vehicles per hour (vph) and 35 vph with an average service rate (time a vehicle was parked) of 1 h and 20 min, the system resulted in good performance with a facility utilization value of 61% and 84%, respectively. However, under the higher-value parameters, the results were poorer, with facility utilization values greater than 88%. This study provides a reference for the evaluation of performance measures and can be applicable to future changes in the LSR at GRTE, other national parks parking, or in general parking areas where queueing may be anticipated.

scholarly journals Performance Modelling of Health-care Service Delivery in Adekunle Ajasin University, Akungba-Akoko, Nigeria Using Queuing Theory

2020 ◽  
pp. 119-127
Author(s):  
Orimoloye Segun Michael
Keyword(s):  
Health Care ◽  
Service Delivery ◽  
Queuing Theory ◽  
Health Care Service ◽  
Health Centre ◽  
Arrival Rate ◽  
Queuing System ◽  
Service Rate ◽  
Suitable Model ◽  
The Mean

The queuing theory is the mathematical approach to the analysis of waiting lines in any setting where arrivals rate of the subject is faster than the system can handle. It is applicable to the health care setting where the systems have excess capacity to accommodate random variation. Therefore, the purpose of this study was to determine the waiting, arrival and service times of patients at AAUA Health- setting and to model a suitable queuing system by using simulation technique to validate the model. This study was conducted at AAUA Health- Centre Akungba Akoko. It employed analytical and simulation methods to develop a suitable model. The collection of waiting time for this study was based on the arrival rate and service rate of patients at the Outpatient Centre. The data was calculated and analyzed using Microsoft Excel. Based on the analyzed data, the queuing system of the patient current situation was modelled and simulated using the PYTHON software. The result obtained from the simulation model showed that the mean arrival rate of patients on Friday week1 was lesser than the mean service rate of patients (i.e. 5.33> 5.625 (λ > µ). What this means is that the waiting line would be formed which would increase indefinitely; the service facility would always be busy. The analysis of the entire system of the AAUA health centre showed that queue length increases when the system is very busy. This work therefore evaluated and predicted the system performance of AAUA Health-Centre in terms of service delivery and propose solutions on needed resources to improve the quality of service offered to the patients visiting this health centre.

scholarly journals Large Deviations for the Single-Server Queue and the Reneging Paradox

2021 ◽  
Author(s):  
Rami Atar ◽  
Amarjit Budhiraja ◽  
Paul Dupuis ◽  
Ruoyu Wu
Keyword(s):  
Large Deviations ◽  
Decay Rate ◽  
Sample Path ◽  
Arrival Rate ◽  
Rate Function ◽  
Service Rate ◽  
Single Server ◽  
Large Deviations Principle ◽  
Long Run ◽  
The Individual

For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.

scholarly journals Simple and Yet Efficient Estimators for Markovian Multiserver Queues

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Emilio Suyama ◽  
Roberto C. Quinino ◽  
Frederico R. B. Cruz
Keyword(s):  
Maximum Likelihood ◽  
Arrival Rate ◽  
Research Area ◽  
Service Rate ◽  
Traffic Intensity ◽  
Moment Estimator ◽  
Likelihood Estimator ◽  
Future Developments ◽  
Multiserver Queues

Estimators for the parameters of the Markovian multiserver queues are presented, from samples that are the number of clients in the system at arbitrary points and their sojourn times. As estimation in queues is a recognizably difficult inferential problem, this study focuses on the estimators for the arrival rate, the service rate, and the ratio of these two rates, which is known as the traffic intensity. Simulations are performed to verify the quality of the estimations for sample sizes up to 400. This research also relates notable new insights, for example, that the maximum likelihood estimator for the traffic intensity is equivalent to its moment estimator. Some limitations of the results are presented along with a detailed numerical example and topics for future developments in this research area.

Minimum carbon dioxide emission based selection of traffic route with unsignalised junctions in tandem network

2019 ◽  
Vol 30 (3) ◽  
pp. 657-675 ◽  
Author(s):  
Anand Jaiswal ◽  
Cherian Samuel ◽  
Chirag Chandan Mishra
Keyword(s):  
Carbon Dioxide ◽  
Queuing Theory ◽  
Mathematical Formulation ◽  
Carbon Dioxide Emission ◽  
Arrival Rate ◽  
Selection Strategy ◽  
Service Rate ◽  
Content Type ◽  
Fuel Consumption Rate ◽  
Route Choices

Purpose The purpose of this paper is to provide a traffic route selection strategy based on minimum carbon dioxide (CO2) emission by vehicles over different route choices. Design/methodology/approach The study used queuing theory for Markovian M/M/1 model over the road junctions to assess total time spent over each of the junctions for a route with junctions in tandem. With parameters of distance, mean service rate at the junction, the number of junctions and fuel consumption rate, which is a function of variable average speed, the CO2 emission is estimated over each of the junction in tandem and collectively over each of the routes. Findings The outcome of the study is a mathematical formulation, using queuing theory to estimate CO2 emissions over different route choices. Research finding estimated total time spent and subsequent CO2 emission for mean arrival rates of vehicles at junctions in tandem. The model is validated with a pilot study, and the result shows the best vehicular route choice with minimum CO2 emissions. Research limitations/implications Proposed study is limited to M/M/1 model at each of the junction, with no defection of vehicles. The study is also limited to a constant mean arrival rate at each of the junction. Practical implications The work can be used to define strategies to route vehicles on different route choices to reduce minimum vehicular CO2 emissions. Originality/value Proposed work gives a solution for minimising carbon emission over routes with unsignalised junctions in the tandem network.

Multiple channel queues in heavy traffic. I

1970 ◽  
Vol 2 (01) ◽  
pp. 150-177 ◽  
Author(s):  
Donald L. Iglehart ◽  
Ward Whitt
Keyword(s):  
Heavy Traffic ◽  
Arrival Rate ◽  
Standard System ◽  
Interarrival Time ◽  
Service Rate ◽  
Service Channel ◽  
Multiple Channel ◽  
System A ◽  
The Mean ◽  
Service Channels

The queueing systems considered in this paper consist of r independent arrival channels and s independent service channels, where as usual the arrival and service channels are independent. Arriving customers form a single queue and are served in the order of their arrival without defections. We shall treat two distinct modes of operation for the service channels. In the standard system a waiting customer is assigned to the first available service channel and the servers (servers ≡ service channels) are shut off when they are idle. Thus the classical GI/G/s system is a special case of our standard system. In the modified system a waiting customer is assigned to the service channel that can complete his service first and the servers are not shut off when they are idle. While the modified system is of some interest in its own right, we introduce it primarily as an analytical tool. Let λ i denote the arrival rate (reciprocal of the mean interarrival time) in the ith arrival channel and μ j the service rate (reciprocal of the mean service time) in the jth service channel. Then is the total arrival rate to the system and is the maximum service rate of the system. As a measure of congestion we define the traffic intensity ρ = λ/μ.

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