State-dependent Service Research Articles

Classical and Bayesian estimates of traffic intensity for M/M/1 queue with state-dependent service

This paper presents the estimates of the parameters of a Markovian queuing model that considers a Poisson arrival rate and a state-dependent service rate based on the system’s condition. State-dependent service implies that the server’s performance varies according to the number of customers waiting for service. If there’s a lengthy queue, the server operates more efficiently, whereas if there are fewer customers, the server works at a slower pace. The primary goal of this paper is to compute estimates for the traffic intensity ( ρ ), using two prominent statistical approaches: maximum likelihood (ML) and Bayesian estimation. The central premise of our study revolves around these estimation techniques under the assumption of various prior distributions, including the beta prior of first and second kind, gamma, truncated gamma, and uniform prior. To achieve our objectives, we employ the powerful method of Markov Chain Monte Carlo (MCMC) simulations. Here MCMC simulations serve as a crucial tool in generating estimates for traffic intensity, encompassing both ML and Bayesian perspectives. Moreover, we go beyond point estimates and obtained confidence intervals and credible intervals for the parameters. Also, we calculate the posterior risk which takes into consideration not only the estimates themselves but also the associated uncertainties.

Finite MX/M/C Queue with State Dependent Service, Two-Class Arrivals and Impatient Customers

Objectives: 1. To compute the transition probabilities for the queueing system to be in different states. 2. To calculate the average system length and mean waiting time under the specified parameters. 3. To investigate and test the impact of system characteristics on the anticipated system length as well as mean waiting times. Method: MATLAB software is used for the Runge-Kutta method to calculate probabilities and system constants. Findings: An increase in Type-I arrival rate λ1 and Type-II arrival rate λ2 will cause both mean waiting times and queue length to increase, and they are reduced with an increase in service rates µ1 and µ2 of Type-I and Type-II classes. A rise in parameters such as special service rate µ3, probability of balking and reneging will result in a reduction in both mean waiting times and queue length. Novelty: This article covers a batch arrival finite Markovian queueing system with multi-server support and consumer impatience. In real time, all arrivals need not enter the queue at the same arrival rates, as it is based on the needs of the customer. In view of this, we have assumed two types of customers along with two different rates of arrival and state-dependent service. We did our work in transient mode, as time can influence the waiting lines. Also determined the system's performance indices and demonstrated the impact of the input parameters on the system's constants. Keywords: Batch Arrival, Customer Impatience, Two Class Customers, State Dependent Service, Multi Server Facility

User-centric Markov reward model for state-dependent Erlang loss systems

Markov reward models are commonly used in the analysis of systems by integrating a reward rate to each system state. Typically, rewards are defined based on system states and reflect the system’s perspective. From a user’s point of view, it is important to consider the changing system conditions and dynamics while the user consumes a service. The key contributions of this paper are proper definitions for (i) system-centric reward and (ii) user-centric reward of the Erlang loss model M/M/n-0 and M/M(x)/n with state-dependent service rates, as well as (iii) the analysis of the relationships between those metrics. Our key result allows a simple computation of the user-centric rewards. The differences between the system-centric and the user-centric rewards are demonstrated for a real-world cloud gaming use case. To the best of our knowledge, this is the first analysis showing the relationship between user-centric rewards and system-centric rewards. This work gives relevant and important insights in how to integrate the user’s perspective in the analysis of Markov reward models and is a blueprint for the analysis of other services beyond cloud gaming while also considering user engagement.

Parallel and series queueing model with state and time dependent service

Parallel and series queueing model with state and time dependent service

Asymptotic Diffusion Method for Retrial Queues with State-Dependent Service Rate

In this paper, we consider a retrial queue with a state-dependent service rate as a mathematical model of a node of FANET communications. We suppose that the arrival process is Poisson, the delay duration is exponentially distributed, the orbit is unlimited, and there is multiple random access from the orbit. There is one server, and the service time of every call is distributed exponentially with a variable parameter depending on the number of calls in the orbit. The service rate has an infinite number of values. We propose the asymptotic diffusion method for the model study. The asymptotic diffusion approximation of the probability distribution of the number of calls in the orbit is derived. Some numerical examples are demonstrated.

Finite M/M/C Queue with Two-Class Arrivals, State dependent service, Multi-servers and Customers Impatience

Finite M/M/C Queue with Two-Class Arrivals, State dependent service, Multi-servers and Customers Impatience

A Markovian two server queue with state dependent hybrid service discipline

A Markovian two server queue with state dependent hybrid service discipline

Analysis of second order properties of production–inventory systems with lost sales

We consider a single-item production–inventory system under a base stock policy for inventory control. We model the system as a closed Gordon–Newell network. The population size of the network is equal to the base stock level, which is the sum of the finished goods and work-in-process inventory. Each exogenous demand, which follows a Poisson process, releases a production order for a new unit and increases the amount of the work-in-process inventory. When there are no items in the finished goods inventory available, arriving demand is lost. The replenishment network operates with state dependent service rates, which we assume to be increasing and concave. First, we analyze the queue length behavior of a two node system and provide conditions under which the mean queue length at the production server is convex in the number of customers in the system. We prove that this leads to convexity of a standard cost function. Using Norton’s theorem, we are able to generalize our results for arbitrarily large production–inventory systems.

Modified State-Dependent Queuing Model for the Capacity Analysis of Metro Rail Transit Station Corridor during COVID-19

The COVID-19 pandemic policies have had a significant impact on the daily commuter flow at the metro rail transit stations. In this study, we propose a modified state-dependent M(n)/G(n)/C/C queuing model for the analysis of commuter flow in the corridor of metro rail transit stations in the COVID-19 situation in order to ensure safe social distance. The proposed model is a finite capacity queuing system with state-dependent commuter arrivals and state-dependent service rates based on the flow–density relationship. First, a mathematical queuing model is developed by using the birth–death process (BDP) and the expected number of commuters, and average area occupied per commuter and blocking probabilities are computed. Then, the accuracy of the proposed model is verified by a discrete-event simulation (DES) framework. (1) The proposed model’s results are compared to those of the existing M/G(n)/C/C model. The proposed modified model’s sensitivity analysis revealed that the anticipated number of commuters in the corridor remains smaller when the arrival rate is state-dependent. (2) In accordance with COVID-19 protocol, when the facility is congested, commuters are discouraged from entering and a safe social distance is maintained between them. (3) No commuters are impeded, and adequate throughput is ensured from the corridor. The proposed model will assist the metro rail transit station operators in making intelligent decisions regarding the operations in the COVID-19 situation.

A PH i /PH i , n / C / C Queuing Model in Randomly Changing Environments for Traffic Circulation Systems

Robust optimal design of circulation systems (e.g., roads for vehicles or corridors for pedestrians) relies on an accurate steady-state traffic flow model that considers the effect of randomly changing environmental factors (e.g., daily periodicity and weather). Most analytical models assume that the customer interarrival time and service time of circulation facilities follow the exponential distribution with fixed rate parameters, which is unrealistic in most cases. In this paper, we develop a stationary PH i /PH i , n / C / C state-dependent queuing model in a randomly changing environment (RE), which is represented by a Markov chain. The model simultaneously considers the general randomness of arrival and service, the randomly varying rate parameters, and the state-dependent service (the travel time increases with the number of customers). The existing matrix analytic scheme (MAS) algorithm is extended to solve the proposed model because it avoids the explicit calculation of probability distributions. The space complexity of the algorithm is only linear in the number of RE states and is independent of the enormous (four-dimensional) state space of the Markov process. Its time complexity is a linear function of the product of the queue capacity and the number of RE states. Our model is validated versus simulation estimates. The obtained conditional performance measures can accurately capture the queue accumulation and dissipation and reveal the effect of randomly changing environments. Numerical experiments provide some interesting findings. (1) The proposed stationary model coincides with the transient M( t )/G x / C / C fluid queuing model under special conditions. (2) Under high traffic intensities, increasing the randomness in the duration time of the RE state leads to an obvious growth in the conditional queue length. (3) An increase in the facility length leads to an increase or a decrease in the average output rate, depending on whether the congestion dissipates effectively in one cycle. (4) A larger width is required to obtain the maximum average output rate for traffic demand with a greater nonuniformity.

Routing and staffing in emergency departments: A multiclass queueing model with workload dependent service times

Efficient patient flow through an emergency department is a critical factor that contributes to a hospital’s performance, which influences overall patient health outcomes. In this work, we model a multiclass multiserver queueing system where patients of varying acuity receive care from one of several wards, each ward is attended by several nurses who work as a team. Supported by empirical evidence that a patient’s time-in-ward is a function of the nurse-patient ratio in that ward, we incorporate state-dependent service times into our model. Our objective is to reduce patient time in system and to control nurse workload by jointly optimizing patient routing and nurse allocation decisions. Due to the computational challenges in formulating and solving the queueing model representation, we study a corresponding deterministic fluid model which serves as a first-order approximation of the multiclass queueing model. Next, we formulate and solve an optimization model using the first-order control equations and input the results into a discrete-event simulation to estimate performance measures, such as patient length-of-stay and ward workload. Finally, we present a case study using retrospective data from a real hospital which highlights the importance of accounting for nurse workload and service behavior in developing routing and staffing policies.

Analytically simple solution to discrete-time queue with catastrophes, balking and state-dependent service

Analytically simple solution to discrete-time queue with catastrophes, balking and state-dependent service

On state dependent batch service queue with single and multiple vacation under Markovian arrival process

On state dependent batch service queue with single and multiple vacation under Markovian arrival process

Processor Sharing Queues With Impatient Customers and State-Dependent Rates

We study queues with impatient customers and Processor Sharing (PS) discipline as well as other variants of PS discipline, namely, Discriminatory Processor Sharing (DPS) and Generalized Processor Sharing (GPS) disciplines, where customers have deadlines until the end of service (DES). Customers arrive according to a state-dependent Poisson process and have general impatience. Customers have exponential service times with state-dependent service rates. Analytical methods based on simple Markov chains are given for the performance analysis of such queues. The principal measures of performance are the steady-state probability of missing deadline and the steady-state probability of blocking. Similar results are obtained for related queues with Random Order Service (ROS) discipline where customers have deadlines until the beginning of service (DBS). In view of a lack of exact analytical results for First Come First Served (FCFS) queues with state-dependent rates, a highly accurate approximation method is also given for these latter queues. The efficacy and accuracy of the approach are illustrated by some numerical examples and simulation experiments.

Optimal control of a queue under a quality-of-service constraint with bounded and unbounded rates

We consider a simple Markovian queue with Poisson arrivals and exponential service times for jobs. The controller chooses state-dependent service rates from an action space. The queue has a finite buffer, and when full, new jobs get rejected. The controller’s objective is to choose optimal service rates that meet a quality-of-service constraint. We solve this problem analytically and compute it numerically under two cases: When the action space is unbounded and when it is bounded.

Compound intervened poisson distribution through one server queuing model with non-homogeneous compound poisson arrivals in bulk and its service rates

Queuing models are used to determine the pragmatic approach in different fields of engineering, sciences, and allied areas. Most of these, the arrivals are dependent on time and would be described by a Poisson process which is non-homogeneous. In this research paper, a one-server queuing model with model with non-homogeneity compound Poisson bulk arrivals has been studied with Compound Intervention Poisson distribution (CIPD). It is observed that the system performance measures are highly influenced by non-homogeneous bulk arrivals rate and batch size distribution parameters. Simulated studies presented that the intervened and compound intervened parameter increases the service rate in terms of utilization, the mean number of customer who undertakes service and emptiness pattern in the queue. Sensitivity analysis has been carried out for the developed model. Non-Homogeneous Poisson process, bulk arrivals, state dependent service rates, Compound Intervened Poisson distribution

Matrix-geometric solution of multi-server queueing systems with Bernoulli scheduled modified vacation and retention of reneged customers: A meta-heuristic approach

ABSTRACT In the present article, we deal with the multi-server finite capacity queueing system with Bernoulli’s scheduled modified vacation policy and the realistic retaining policy of reneged customers. After completion of the service of any customer, the server decides whether to go for the vacation of random duration or to continue facilitating the service to the waiting customer, if any, present in the queue. All servers, homogeneous in nature, provide the state-dependent service following the threshold policy to reduce the overload of the system. The impatience behavior of customers like balking and reneging is also considered in the stochastic modeling of the studied problem. The matrix analytic approach is employed to obtain the steady-state probabilities with which various system performance measures are also developed with practical justification. Finally, the expected cost minimization problem is formulated and dealt with the meta-heuristic approach: particle swarm optimization (PSO). All results of numerical simulation and optimal analysis are summarized in tables and graphs to provide quick insight. The concluding remarks and future scopes have also been discussed.

Prediction of Two-Node Tandem Queue with Feedback Having State and Time Dependent Service Rates

Queuing networks are current area of great research and application interest, in view of their increased applicability in - modelling manufacturing facilities and computer/communication networks, production and assembly lines, hospitals, transport systems, banks and so forth. The present paper develops and analyses a tandem queuing model having two nodes with feedback. Such type of model finds its application in many fields, e.g. telecommunication, inventory, hospitals, traffic control and so on. For example, in hospitals, a patient first visits the doctor and goes through a series of examinations, after which he revisits the doctor for medication. The inter-arrival time is exponentially distributed, whereas the service rate for each node depends on the aggregate of customers in the respective queues and thus follows non-homogeneous Poisson process. After constructing the difference –differential equations, and using PGF technique, we have obtained the joint probability distribution for queue size. Some key measures such as the “average number of customers in queue”, “utilization time of each queue”, “average waiting time in each queue” are computed.

State-dependent service rates in make-to-order shops: an assessment by simulation

Most literature on make-to-order shops assumes that service rates are independent of the system state. In practice however, the service rate is often dependent on the workload level experienced by the worker. While a body of knowledge on state-dependent service rates exists, the available literature has not given sufficient attention to make-to-order shops, which are often characterized by complex routings and defined due dates, which means delivery performance becomes a major concern. This study uses simulation to assess the performance impact of state-dependent service rates under different degrees of routing directedness. We show that including information on the load upstream of a station when making service rate adjustments has the potential to improve performance compared to considering the load directly queuing at a station only, as has been the case in previous research on state-dependent service rates. Moreover, using the same threshold to trigger service rate adjustments at each station in shops with directed routings leads to higher service rates at upstream stations. This service rate imbalance can be avoided by using different triggering thresholds for upstream and downstream stations. Further, and most importantly, we show that although speeding up behavior during high load periods significantly improves performance, if worker fatigue leads to a decrease in the service rate in response to the initial increase then performance may in fact deteriorate.

Novel Solutions for Closed Queueing Networks with Load-Dependent Stations

Load-dependent closed queueing networks are difficult to approximate since their analysis requires to consider statedependent service demands. Standard evaluation techniques, such as mean-value analysis, are not equally efficient in the load-dependent setting, where mean queue-lengths are insufficient alone to recursively determine the model equilibrium performance. As such, novel exact techniques to address this class of models can benefit performance engineering practice by offering alternative trade-offs between accuracy and computational cost. In this paper, we derive novel exact solutions for the normalizing constant of state probabilities in the load-dependent setting. For single-class load-dependent models, we provide an explicit exact formula for the normalizing constant that is valid for models with arbitrary load-dependent rates. From this result, we derive two novel integral forms for the normalizing constant in multiclass load-dependent models, which involve integration in the real and complex domains, leading to novel numerical approximations. The paper also illustrates through experiments the computational gains and accuracy of the obtained expressions.