On Optimal Ordering of Service Parameters of a Coxian Queueing Model with Three Phases

We analyze a Coxian stochastic queueing model with three phases. The Kolmogorov equations of this model are constructed, and limit probabilities and the stationary probabilities of customer numbers in the system are found. The performance measures of this model are obtained and in addition the optimal order of service parameters is given with a theorem by obtaining the loss probabilities of customers in the system. That is, putting the greatest service parameter at first phase and the second greatest service parameter at second phase and the smallest service parameter at third phase makes the loss probability and means waiting time minimum. We also give the loss probability in terms of mean waiting time in the system. is the transition probability from j-th phase?to??phase . In this manner while and this system turns into queueing model and while the system turns into Cox(2) queueing model. In addition, loss probabilities are graphically given in a 3D graph for corresponding system parameters and phase transient probabilities. Finally it is shown with a numeric example that this theorem holds.

Spectrum of the Operator Corresponding to the M/M/1 Queueing Model with Vacations and Multiple Phases of Operation and Application

We describe the point spectrum of the operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operation.Then by using this result we prove that the essential growth bound of the C0-semigroup generated by the operator is 0,the C0-semigroup is not compact,not eventually compact,even not quasi-compact.Moreover,we verify that it is impossible that the time-dependent solution of the M/M/1 queueing model with vacations and multiple phases of operation exponentially converges to its steady-state solution.In addition,we obtain the spectral radius and essential spectral radius of the C0-semigroup.Lastly,we discuss other spectrum of the operator and obtain a set which belongs to the union of its continuous spectrum and residual spectrum.

Passengers gathering-scattering analysis at the corners of subway stations

During both daily operation and emergency evacuation,the corners of walking facilities in subway stations play an important role in efficient circulation.However,the effectiveness of the corner is difficult to assess.In this paper,a method of passenger gathering and scattering analysis based on queueing models was proposed to investigate the corner performance in subway stations.Firstly,we constructed a set of state spaces of passenger flow according to passenger density and proposed the state transition model of passenger flow.Moreover,the model of passenger flow blocking and unblocking probability were also presented.Then,to illustrate the validity of the method and model,several passenger gathering-scattering scenarios and were simulated to verify the influence of passenger distribution and facility width on passenger walking,and the blocking probability,throughput,and expected time were also analyzed under various widths of the target corridor and arrival rates.Results showed that the proposed model can reproduce the trend of walking parameters changing and the self-organizing phenomenon of'faster is lower'.With the increase of arrival rates of passengers,walking speeds of passengers decrease and the expected walking time is prolonged,and the blocking probability sharply increased when the arrival rate exceeded 7 peds/s.In addition,with change of width of the target facility,efficiency of capacity of walking circulation facility fluctuated.With the width of the target corridor enlarged by 10%,the steady state of passenger flow was less crowded.Therefore,corridor width is critical to the circulation efficiency of passengers in subway stations.The conclusions will help to develop reasonable passenger flow control plans to ease the jam and keep passengers walking safely.

A multi-class time-dependent model for the analysis of waiting phenomena at a motorway tollgate

The planning, design and operational management of motorway toll booths are of great interest in traffic engineering, as these facilities directly influence the quality of the service offered to users. This paper focuses on a time-dependent queue model based on the coordinates transformation criterion for operations assessment at a motorway tollgate. This model allows to face the whole spectrum of situations that may characterize a toll booth,some of which often fall outside the boundaries of the probabilistic theory for stationary queues.The paper proposes an M=G=1 multi-class queue model for the evaluation of evolutionary profiles of waiting times and queue lengths by closed-form equations. The results obtained for three numerical test cases show a good approximation level, compared with the mean values of queue parameters obtained reiterating a discrete-state simulation model.The proposed time-dependent equations will be useful in technical cases, allowing to operate quickly and compactly even when probabilistic queue theory is not applicable or produce unrealistic results, and the burden of complexity of the simulation approach is not conveniently absorbable. The discussion highlights a significant flexibility of the model proposed in addressing situations with conventional vehicles, i.e., with total human control and proposes some considerations for application in future scenarios with the presence of connected vehicles(CVs).

State of the art and computational aspects of time-dependent waiting models for non-signalised intersections

Time-dependent models are of great importance in highway engineering as they are appropriate for evaluating waiting times and queue lengths at intersections,which are integral parts of various activities in planning,verification and decision support for infrastructure.After reviewing the literature of the main time-dependent models based on the coordinate transformation method and a discussion about some computational issues in time-evolution profiles for non-signalised intersections,the paper identifies the requirements these models have to satisfy in order to be used as"basic"cases for analysing complex evolutionary situations.Three"basic"cases are presented with their timedependent equations for vehicle waiting times and vehicle number;they have been completed and dimensionally homogenised in this paper.As they are recursive,these formulas can be applied for sequential intervals in the time domain in both vehicles and passenger car units.The closed-form expressions for state variables show to be mutually equivalent in comparison with discrete event simulation models and imbedded Markov chain results.For all the three models the paper presents a common deterministic simplification for average waiting time,with good approximation results in the tested cases.The proposed time-dependent formulas will contribute to a better adherence to the real phenomena,compared to the extremely simplified and unrealistic methods suggested by the international manuals for level of service assessment.The proposed formulas will be useful for current applications and possible future development in order to meet the emerging needs of road and transport engineering.

A Multi-server Queue in a Multi-phase Random Environment with Waiting Servers and Customers'Impatience Under Synchronous Working Vacation Policy

In this paper,we develop an M/M/c queueing system in a Markovian environment with waiting servers,balking and reneging,under both synchronous single and multiple working vacation policies.When the system is in operative phase j,j=1,K¯,customers are served one by one.Once the system is empty,the servers have to wait a random period of time before leaving,causing the system to move to vacation phase 0 at which new arrivals can be served at lower rate.Using the method of the probability generating functions,we establish the steady-state analysis of the system.Special cases of the queueing model are presented.Then,explicit expressions of the useful system characteristics are derived.In addition,a cost model is constructed to define the optimal values of service rates,simultaneously,to minimize the total expected cost per unit time via a quadratic fit search method.Numerical examples are provided to display the impact of different system characteristics.

Stationary distributions for two-dimensional sticky Brownian motions:Exact tail asymptotics and extreme value distributions

Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions,which find applications in many areas including queueing theory and mathematical finance.In this paper,we focus on stationary distributions for sticky Brownian motions.Main results obtained here include tail asymptotic properties in the marginal distributions and joint distributions.The kernel method,copula concept and extreme value theory are the main tools used in our analysis.