State Dependent Queueing Model Research Articles

Performance analysis and cost investigations for state-dependent single unreliable server finite queue under F-policy using GA and QNM

State-dependent queues are essential as they accurately capture the dynamic behavior of systems, enabling more realistic modeling and analysis of various real-world scenarios. The present investigation deals with a state-dependent single unreliable server finite queueing model under admission control F-policy. The admission control F-policy is a valuable tool to control the congestion problem of customers where the queue forms. In the proposed model, the server is prone to breakdown and can fail at any time. There is a two-phase repair facility available to address server failures. Chapman–Kolmogorov (C–K) steady-state equations based on the birth–death process are formed and solved using the matrix analytic method to formulate a mathematical model. By setting suitable values for state-dependent parameters, some particular models, such as the machine repair problem (MRP) and queueing model with customers’ balking and feedback, can also be deduced. Furthermore, various queueing indices are established to predict the validity of state-dependent queues. A total cost function for the system is constructed to enhance customer service while minimizing costs. This cost function is then optimized using a genetic algorithm (GA) and quasi-Newton method (QNM). The cost optimization in the state-dependent queueing model enables businesses to optimize resource allocation and achieve cost-efficient service delivery, leading to improved profitability and customer satisfaction.

A traffic queueing model for exit lanes for left-turn intersections

The unconventional design of exit-lanes for left-turn (EFL) has been confirmed to be an effective way to increase intersection capacity when left-turn demand is high. However, because of the unique feature of EFL control, there would be a strong oscillation in the left-turn queues at signalized intersections when the EFL strategy is used, which makes the analytical evaluation of the EFL strategy very difficult. In particular, the random nature of traffic demand would make it even harder to estimate the queues. To this end, in this paper, we propose a signal state-dependent queueing model to quantify the dynamic behavior of left-turn queues at signalized intersections under EFL control. The obtained results will contribute to establishing an analytical numerical framework that gives us the possibility to study the effect of various factors on the left-turn queues at the EFL intersections and provide insight into the unique queueing behavior that is most likely to manifest this strategy. We validated the proposed model using field data, and it shows a good agreement between the estimated results and the field results. In addition, the numerical experimental results show that the obtained performance measures are practically beneficial to provide a numerical tool to quantify the uncertainty in the queues of left-turn movements with EFL control and thus enrich the guidelines for further improving the use of EFL design.

A Queuing Model for Mixed Traffic Flows on Highways considering Fluctuations in Traffic Demand

Modelling the mixed traffic flows of autonomous vehicles (AVs) and human-driven vehicles (HVs) on highways is challenging. Randomness, fluctuations, and congestion exist in the mixed traffic flows. This paper extends the current literature by proposing an M/G(n)/c/c state-dependent queuing model operating in a random environment. The fluctuating traffic demand is addressed by arrival rates modulated by the random environment. Meanwhile, a Markovian arrival process (MAP) is incorporated to describe the platoons. We investigate the performance of the mixed traffic flow under the I policy (AVs and HVs travel together in all lanes) and the D policy (one lane is designated to AVs). Numerical experiments reveal the following interesting findings: (1) the fluctuation degree of traffic demand, the traffic intensity, and the penetration rate of AVs play essential roles in determining the performance of mixed traffic flows. (2) The I policy should always be adopted if the travel time is more valuable. In terms of output rate, the choice between the I and the D policies depends on the traffic intensity, SCV of arrival rates and penetration rate. (3) A larger penetration rate is required to completely eliminate congestion on a longer highway segment.

Spatial Capacity Planning

We study the relationship between capacity and performance for a service firm with spatial operations, in the sense that requests arrive with origin-destination pairs. An example of such a system is a ride-hailing platform in which each customer arrives in the system with the need to travel from an origin to a destination. We propose a parsimonious representation of a spatial multiserver system through a state-dependent queueing model that captures spatial frictions as well as spatial economies of scale through the service rate. In a classical [Formula: see text] queueing model, the square root safety (SRS) staffing rule is known to balance server utilization and customer wait times. By contrast, we find that the SRS rule does not lead to such a balance in spatial systems. In a spatial environment, pick-up times increase the load in the system; furthermore, they are an endogenous source of extra workload that leads the system to only operate efficiently if there is sufficient imbalance between supply and demand. In heavy traffic, we derive the mapping from load to operating regimes and establish implications on various metrics of interest. In particular, to obtain a balance of utilization and wait times, the service firm should use a higher safety factor, proportional to the offered load to the power of [Formula: see text]. We also discuss implications of these results for general systems.

State dependent queueing models under admission control F-policy: a survey

The optimal F-policy can be used to control the admission of customers in queueing scenarios having limited waiting space. According to F-policy, as soon as the capacity of the system becomes full, the arriving customers are restricted to join the system until there is a drop in the number of customers in the system to a threshold level ‘F’ due to departure of the served customers. In spite of enormous applications of F-policy models in several queueing systems, a limited number of papers has appeared in the queueing literature. The main objective of this article is to provide a state-of-the-art and survey of literature on the state dependent queueing models operating under F-policy. In this article, we describe the mathematical formulation of queueing models operating under F-policy in two categories namely, finite capacity and finite population models. The methodological aspect of non-Markov models with F-policy is presented. The exhaustive list of work done on F-policy is also provided in tabular form.

Computational Performance of a Single Server State Dependent Queue

In this article we consider a single server state dependent queuing system with the service rate varying according to types of customers like normal, tagged and heavy tailed. We develop a state dependent queuing model in which the service rate depends on customer type arrive to the system with quasi-birth-death environment structure and using matrix geometric method we develop the system performance measures. Also, we use these performance measures by utilizing the maximum potential of server and perform sensitivity analysis with numerical illustrations.

Admission control for finite capacity queueing model with general retrial times and state-dependent rates

The finite state dependent queueing model with \begin{document}$ F $\end{document} -policy is investigated by considering the general retrial attempts. On arrival in the system, if the job finds the server engaged, it is forced to enter into the retrial orbit. After a random period of time, the job from the retrial orbit re-attempts for the service. According to \begin{document}$ F $\end{document} -policy, as the system attains its full capacity, the arrivals are restricted to join the system until the number of jobs comes down to the prefixed threshold value ' \begin{document}$ F $\end{document} '. The supplementary variable corresponding to the remaining retrial time is used to frame the governing equations which are solved by using Laplace-Stieltjes transform and then applying the recursive method. Special models for machine repair and time-sharing queue are deduced by setting the state dependent rates. Several system indices are obtained explicitly which are further used to facilitate the sensitivity analysis by considering a numerical illustration. A cost function is constructed and minimized for evaluating the optimal threshold parameter and optimal service rate.

Stability and busy periods in a multiclass queue with state-dependent arrival rates

We introduce a multiclass single-server queueing system in which the arrival rates depend on the current job in service. The system is characterized by a matrix of arrival rates in lieu of a vector of arrival rates. Our proposed model departs from existing state-dependent queueing models in which the parameters depend primarily on the number of jobs in the system rather than on the job in service. We formulate the queueing model and its corresponding fluid model and proceed to obtain necessary and sufficient conditions for stability via fluid models. Utilizing the natural connection with the multitype Galton–Watson processes, the Laplace–Stieltjes transform of busy periods in the system is given. We conclude with tail asymptotics for the busy period for heavy-tailed service time distributions for the regularly varying case.

Spatial Capacity Planning

We study the relationship between capacity and performance for a service firm with spatial operations, in the sense that requests arrive with origin-destination pairs. An example of such a system is a ride-hailing platform in which each customer arrives in the system with the need to travel from an origin to a destination. We propose a state-dependent queueing model that captures spatial frictions as well as spatial economies of scale through the service rate. In a classical M/M/n queueing model, the square root safety (SRS) staffing rule is known to balance server utilization and customer wait times. By contrast, we find that the SRS rule does not lead to such a balance in spatial systems. In a spatial environment, pickup times increase the load in the system; furthermore, they are an endogenous source of extra workload that leads the system to only operate efficiently if there is sufficient imbalance between supply and demand. In heavy traffic, we derive the mapping from load to regimes of interest and establish that to obtain a balance of utilization and wait times, the service firm should use a higher safety factor, proportional to the offered load to the power of 2/3. We discuss implications of these results for general systems.

M/G/c/c state dependent queuing model for a road traffic system of two sections in tandem

We propose in this article a M/G/c/c state dependent queuing model for road traffic flow. The model is based on finite capacity queuing theory which captures the stationary density-flow relationships. It is also inspired from the deterministic Godunov scheme for the road traffic simulation. We first present a reformulation of the existing linear case of M/G/c/c state dependent model, in order to use flow rather than speed variables. We then extend this model in order to consider upstream traffic demand and downstream traffic supply. After that, we propose the model for two road sections in tandem where both sections influence each other. In order to deal with this mutual dependence, we solve an implicit system given by an algebraic equation. Finally, we derive some performance measures (throughput and expected travel time). A comparison with results predicted by the M/G/c/c state dependent queuing networks shows that the model we propose here captures really the dynamics of the road traffic.

M/g/c/c State Dependent Queueing Model for Road Traffic Simulation

In this paper, we present a stochastic queuing model for the road traffic, which captures the stationary density-flow relationships in both uncongested and congestion conditions. The proposed model is based on the $M/g/c/c$ state dependent queuing model of Jain and Smith, and is inspired from the deterministic Godunov scheme for the road traffic simulation. We first propose a reformulation of the $M/g/c/c$ state dependent model that works with density-flow fundamental diagrams rather than density-speed relationships. We then extend this model in order to consider upstream traffic demand as well as downstream traffic supply. Finally, we calculate the speed and travel time distributions for the $M/g/c/c$ state dependent queuing model and for the proposed model, and derive stationary performance measures (expected number of cars, blocking probability, expected travel time, and throughput). A comparison with results predicted by the $M/g/c/c$ state dependent queuing model shows that the proposed model correctly represents the dynamics of traffic and gives good performances measures. The results illustrate the good accuracy of the proposed model.

A Queuing Network Based Optimization Model for Calculating Capacity of Subway Station

Capacity of subway station is an important factor to ensure the safety and improve the transportation efficiency. In this paper, based on the M/G/C/C state-dependent queuing model, a probabilistic selection optimization model is proposed to assess the capacity of the station. The goal of the model is to maximize the output rate of the station, and the decision variables of the model are the selection results of the passengers. Finally, this paper takes a subway station of Shanghai Metro as a case study and calculates the optimal selection probability. The proposed model could be used to analyze the average waiting time, congestion probability, and other evaluation indexes; at the same time, it verifies the validity and practicability of the model.

Modeling Load and Overwork Effects in Queueing Systems with Adaptive Service Rates

Servers in many real queueing systems do not work at a constant speed. They adapt to the system state by speeding up when the system is highly loaded or slowing down when load has been high for an extended time period. Their speed can also be constrained by other factors, such as geography or a downstream blockage. We develop a state-dependent queueing model in which the service rate depends on the system “load” and “overwork.” Overwork refers to a situation where the system has been under a heavy load for an extended time period. We quantify load as the number of users in the system, and we operationalize overwork with a state variable that is incremented with each service completion in a high-load period and decremented at a rate that is proportional to the number of idle servers during low-load periods. Our model is a quasi-birth-and-death process with a special structure that we exploit to develop efficient and easy-to-implement algorithms to compute system performance measures. We use the analytical model and simulation to demonstrate how using models that ignore adaptive server behavior can result in inconsistencies between planned and realized performance and can lead to suboptimal, unstable, or oscillatory staffing decisions.

A PH/PH[formula omitted] state-dependent queuing model for metro station corridor width design

Metro station corridor width design considering demand fluctuation as well as the randomness and state-dependence of service time is an urgent concern and a complicated random planning issue. This paper confirms the accuracy of phase-type distribution (PH) fitting for passenger arrival intervals and service times with randomness and state-dependence in metro station corridors. A PH/PH(n)/C/C state-dependent queuing model is thus established by a finite level-dependent quasi-birth–death (QBD) process. The existing M/G(n)/C/C, M/G/1/C, and D/D/1/C models are proved to be special cases of the PH/PH(n)/C/C model through theoretical derivation and the precision of the proposed model is analyzed through simulation tests. The quantitative relationship between the level of service (LOS) and the corridor width is established based on the proposed model. A total of 81 experiments are designed to compare the calculations between the proposed model and the M/G(n)/C/C, M/G/1/C, and D/D/1/C models. Comparison results demonstrate that (1) the value of effective width of the PH/PH(n)/C/C queuing model is higher than those of the M/G(n)/C/C, M/G/1/C, and D/D/1/C models; (2) the real area occupied per person in the corridor of the PH/PH(n)/C/C queuing model is mostly proximate to the designed LOS, whereas those of the M/G(n)/C/C, M/G/1/C, and D/D/1/C models fail to meet the designed LOS; and (3) the performance measures of the PH/PH(n)/C/C queuing model enjoy high performance-width elasticity and are significantly improved compared with those of the M/G(n)/C/C, M/G/1/C, and D/D/1/C models.

Formula omitted] state dependent travel time models and properties

One of the most important problems in today’s modeling of transportation networks is an accurate estimate of travel time on arterial links, highway, and freeways. There are a number of deterministic formulas that have been developed over the years to achieve a simple and direct way to estimate travel times for this complex task. Realistically, however, travel time is a random variable. These deterministic formula are briefly reviewed and also a new way to compute travel time over arterial links, highway, and freeways, is presented based on an analytical state dependent queueing model. One of the features of the queueing model is that it is analyzed within the context of the theoretical three-phase traffic flow model. We show that the model provides a quantitative foundation alternative to qualitative three-phase traffic flow theory. An important property shown with the model is that the travel time function is not convex, but a sigmoid S-shaped (i.e. logistic curve). Extensive analytical and simulation experiments are shown to verify the S-shaped nature of the travel time function and the use of the model’s method of estimation of travel time over vehicular traffic links as compared with traditional approaches. Finally, it is shown that the point-of-inflection of the S-shaped curve represents the threshold point where the traffic flow volume switches from Free Flow to Congested Flow.

Optimal workload allocation in closed queueing networks with state dependent queues

The problem of optimal workload allocation in closed queueing network models with multi-server exponential infinite capacity workstations and finite capacity state dependent queueing models is examined. The processing rates (i.e. service times) of jobs in the queueing system are the main focus. State dependent queues are appropriate for modeling the transportation and material transfer of the customers and products within the system. By combining these two types of queues, we can model the processing of jobs or customers at service stations and also the material handling transfer of job and customers between these stations. Because the environment is a closed queueing network model, it allows for the dynamic interaction of the jobs and customers in the optimal workload allocation values. This combination of queues and optimal search process provides a broad range of potential applications in manufacturing and service type design operations where the travel time between workstations is important. Conveyors, automated-guided vehicle systems (AGVS), and fork lift trucks are utilized to show the full range of material handling component systems.

A Discrete Event Simulation Model for Evaluating the Performances of an M/G/C/C State Dependent Queuing System

M/G/C/C state dependent queuing networks consider service rates as a function of the number of residing entities (e.g., pedestrians, vehicles, and products). However, modeling such dynamic rates is not supported in modern Discrete Simulation System (DES) software. We designed an approach to cater this limitation and used it to construct the M/G/C/C state-dependent queuing model in Arena software. Using the model, we have evaluated and analyzed the impacts of various arrival rates to the throughput, the blocking probability, the expected service time and the expected number of entities in a complex network topology. Results indicated that there is a range of arrival rates for each network where the simulation results fluctuate drastically across replications and this causes the simulation results and analytical results exhibit discrepancies. Detail results that show how tally the simulation results and the analytical results in both abstract and graphical forms and some scientific justifications for these have been documented and discussed.

Simulating on Passengers Coordination with Distribution Service in Railway Station

Based on the hierarchical characteristics of passengers transfer behavior and the state-dependent queuing model, the D-M/G/C/C queuing network simulation model was put forward for simulating the coordination of subjective passenger behavior and objective distribution environment. Then, the generation method and algorithm of the model is elaborated. In case study, the simulation results are consistent with the actual states of passenger outbound process. So, it is a practical simulation tool for analyzing the dynamic situations of passengers use facilities. DOI: http://dx.doi.org/10.11591/telkomnika.v11i1.1912

Analysis of state dependent bulk arrival unreliable queueing model

The present investigation deals with the single server stochastic model of queueing system, wherein the arrivals of units are in batches and follow Poisson process with state dependent arrival rates. There is a provision of two stages of heterogeneous service with arbitrary distributed different service time. The server may take optional vacation after the completion of the both stages of service of each unit. The server may fail at any instant of service and requires repair. The transient and steady state behaviours of the queue length distribution are studied by using the Laplace transform and probability generating function along with supplementary variable-based methodology, to obtain the performance measures of the system. Some particular cases are discussed by setting the parameters. The maximum entropy principle is applied to find the approximate system state probabilities. The numerical illustrations are also considered to validate the analytical results.

Stationary and transient solution of Markovian queues - an alternate approach

This paper aims at presenting an alternate approach to derive the exact transient solution of Markovian queues. The continued fraction (Jacobi fraction) is expressed as power series and the power series coefficients are connected by a two-dimensional recurrence relation associated with infinite Stieltjes matrix equation. The recurrence relation is solved by using generating functions. A novel state-dependent birth-death queueing model is taken and expressed as continued fractions by employing integral transforms. The stationary probabilities of general state-dependent Markovian queues are obtained from the continued fractions and its associated tridiagonal determinants. As a special case the time-dependent system size probabilities and busy period distribution of a classical single server queue are deduced using continued fraction and its power series. Numerical illustrations are also presented.