State-dependent Service Rates Research Articles

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.

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.

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.

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

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.

Comparison of fluid approximations for service systems with state-dependent service rates and return probabilities

We compare two models of a multi-server queueing system with state-dependent service rates and return probabilities. In both models, upon completing service, customers are delayed prior to possibly returning to service. In one model, the determination of whether a customer will return occurs immediately upon service completion, at the beginning of the delay. In the other, that determination is made at the end of the delay, capturing the idea that it takes time for the customer’s condition and needs to evolve or assess, before it becomes known whether a return to service is needed. Our comparison focuses on fluid approximations of the two models. The fluid approximation for the first model, which has been studied previously, consists of a system of two ordinary differential equations. The fluid approximation for the second model, which is new, consists of a delay differential equation. We find that the two fluid approximations have the same set of equilibrium points, but their transient behavior can differ markedly. Both fluid approximations can exhibit bistability for certain parameter values. We use discrete event simulation to illustrate the extent to which the findings from the fluid approximations carry over to the underlying stochastic models.

Transient Analysis of K-node Tandem Queuing Model with Load Dependent Service Rates

This paper deals with the development and analysis of K-node series and parallel queueing model with load dependent service rates. Here it is assumed that the customers arrive to the initial queue and waiting line for service. After completing the service at first service station they may join one of the (K-1) queues which are parallel and connected to first queue in series. After getting service from the service station they leave the system. Here it is assumed that the service rates in each service station are dependent on number of customers in the queue connected to it .The arrival and service completions in each queue are assumed to follow Poisson processes. Using difference-differential equations the joint probability function of number of customers in each queue are derived. The system performance measures such as average number of customers in each queue ,throughput of each service station ,the probability of idleness of each server ,the waiting time of customer in each queue are derived explicitly. The sensitivity of the model with respect to parameters are analysed through numerical illustration. It is observed that the state dependent service rates has significant influence on performance measures. This model also includes the earlier models as particular cases for specific values of the parameters. This model is useful in analysing the communication networks , transportation systems ,production processes and cargo handling.

Service Rate Optimization of Finite Population Queueing Model with State Dependent Arrival and Service Rates

Providing service immediately after the arrival is rarely been used in practice. But there are some situations for which servers are more than the arrivals and no one has to wait to get served. In this model, arrival rate is

Two Node Tandem Queueing Model with Phase Type State and Time Dependent Service Rates

Two Node Tandem Queueing Model with Phase Type State and Time Dependent Service Rates

A state-dependent queueing system with asymptotic logarithmic distribution

A Markovian single-server queueing model with Poisson arrivals and state-dependent service rates, characterized by a logarithmic steady-state distribution, is considered. The Laplace transforms of the transition probabilities and of the densities of the first-passage time to zero are explicitly evaluated. The performance measures are compared with those ones of the well-known M/M/1 queueing system. Finally, the effect of catastrophes is introduced in the model and the steady-state distribution, the asymptotic moments and the first-visit time density to zero state are determined.

Optimal Control of State-Dependent Service Rates in a MAP/M/1 Queue

In this paper, we study the optimal control of service rates in a queueing system with a Markovian arrival process (MAP) and exponential service times. The service rate is allowed to be state dependent, i.e., we can adjust the service rate according to the queue length and the phase of the MAP. The cost function consists of holding cost and operating cost. The goal is to find the optimal service rates that minimize the long-run average total cost. To achieve that, we use the matrix-analytic methods (MAM) together with the sensitivity-based optimization (SBO) theory. A performance difference formula is derived, which can quantify the difference of the long-run average total cost under any two different settings of service rates. Based on the difference formula, we show that the long-run average total cost is monotone in the service rate and the optimal control is a bang–bang control. We also show that, under some mild conditions, the optimal control policy of service rates is of a quasi-threshold-type. By utilizing the MAM theory, we propose a recursive algorithm to compute the value function related quantities. An iterative algorithm to efficiently find the optimal policy, which is similar to policy iteration, is proposed based on the SBO theory. We further study some special cases of the problem, such as the optimality of the threshold-type policy for the M/M/1 queue. Finally, a number of numerical examples are presented to demonstrate the main results and explore the impact of the phase of the MAP on the optimization in the MAP/M/1 queue.

Mittag - Leffler function distribution - a new generalization of hyper-Poisson distribution

In this paper a new generalization of the hyper-Poisson distribution is proposed using the Mittag-Leffler function. The hyper-Poisson, displaced Poisson, Poisson and geometric distributions among others are seen as particular cases. This Mittag-Leffler function distribution (MLFD) belongs to the generalized hypergeometric and generalized power series families and also arises as weighted Poisson distributions. MLFD is a flexible distribution with varying shapes and has a unique mode at zero or it is unimodal with one/two non-zero modes. It can be under-, equi- or over- dispersed. Various distributional properties like recurrence relation for probability mass function, cumulative distribution function, generating functions, formulas for different type of moments, their recurrence relations, index of dispersion and its classification, log-concavity, reliability properties like survival, increasing failure rate, unimodality, and stochastic ordering with respect to hyper-Poisson distribution are discussed. A particular case of the distribution is shown to arise as the steady state probability of a queuing system under state dependent service rate. The distribution has been found to fare well when compared with the hyper-Poisson and COM-Poisson type negative binomial distributions in its suitability in empirical modeling of differently dispersed count data. It is therefore expected that the proposed MLFD with its interesting features and flexibility will be a useful addition as a model for count data.

Optimal design for a retrial queueing system with state-dependent service rate

This paper considers a single server retrial queue in which a state-dependent service policy is adopted to control the service rate. Customers arrive in the system according to a Poisson process and the service times and inter-retrial times are all exponentially distributed. If the number of customers in orbit is equal to or less than a certain threshold, the service rate is set in a low value and it also can be switched to a high value once this number exceeds the threshold. The stationary distribution and two performance measures are obtained through the partial generating functions. It is shown that this state-dependent service policy degenerates into a classic retrial queueing system without control policy under some conditions. In order to achieve the social optimal strategies, a new reward-cost function is established and the global numerical solutions, obtained by Canonical Particle Swarm Optimization algorithm, demonstrate that the managers can get more benefits if applying this state-dependent service policy compared with the classic model.

Poly-symmetry in processor-sharing systems

We consider a system of processor-sharing queues with state-dependent service rates. These are allocated according to balanced fairness within a polymatroid capacity set. Balanced fairness is known to be both insensitive and Pareto-efficient in such systems, which ensures that the performance metrics, when computable, will provide robust insights into the real performance of the system considered. We first show that these performance metrics can be evaluated with a complexity that is polynomial in the system size if the system is partitioned into a finite number of parts, so that queues are exchangeable within each part and asymmetric across different parts. This in turn allows us to derive stochastic bounds for a larger class of systems which satisfy less restrictive symmetry assumptions. These results are applied to practical examples of tree data networks, such as backhaul networks of Internet service providers, and computer clusters.

A modified Conway–Maxwell–Poisson type binomial distribution and its applications

ABSTRACTThis article proposes a generalized binomial distribution, which is derived from the finite capacity queueing system with state-dependent service and arrival rates. This distribution is also generated from the conditional Conway–Maxwell–Poisson (CMP) distribution given a sum of two CMP variables. In this article, we consider the properties of the probability mass function, indices of dispersion, skewness and kurtosis, and give applications of the proposed distribution. The estimation method and simulation study are also considered.

On a tandem queue with retrials and losses and state dependent arrival, service and retrial rates

On a tandem queue with retrials and losses and state dependent arrival, service and retrial rates