State-dependent Arrival Rates Research Articles

Exponential Single Server Queues in an Interactive Random Environment

We consider exponential single server queues with state-dependent arrival and service rates that evolve under influences of external environments. The transitions of the queues are influenced by the environment’s state and the movements of the environment depend on the status of the queues (bidirectional interaction). The environment is constructed in a way to encompass various models from the recent Operations Research literature, where a queue is coupled with an inventory or with reliability issues. With a Markovian joint queueing-environment process, we prove separability for a large class of such interactive systems; that is, the steady state distribution is of product form and explicitly given. The queue and the environment processes decouple asymptotically and in steady state. For nonseparable systems, we develop ergodicity and exponential ergodicity criteria via Lyapunov functions. By examples we explain principles for bounding departure rates of served customers (throughputs) of nonseparable systems by throughputs of related separable systems as upper and lower bound.

Analysis of classical retrial queue with differentiated vacation and state dependent arrival rate.

In present paper we have introduced the concept of differentiated vacations in a retrial queueing model with state dependent arrival rates of customers. The arrival rate of customers is different in various states of the server. The vacation types are differentiated by means of their durations as well as the previous state of the server. In type I vacation, server goes just after providing service to at least one customer whereas in type II, it comes after remaining free for some time. In steady state, we have obtained the system size probabilities and other system performance measures. Finally, sensitivity and cost analysis of the proposed model is also performed. The probability generating function technique, parabolic method and MATLAB is used for the purpose.

Computing the conditional entry-state distribution in Erlang loss systems

We consider an Erlang loss system with state-dependent arrival rates. Given the system is in steady-state and there are j customers being served, the system operator may wish to know about the distribution of arrival states for the j customers in service. Specifically, they may want the steady-state probability that any given customer entered when the system had i servers busy, given j customers are currently being served. We term this metric the conditional entry-state distribution and develop an algorithm to compute it.

Dimensioning a queue with state-dependent arrival rates

In an observable queue, customers joining decisions may be influenced by wait-aversion and crowd-attraction. These opposing phenomena and the diversity of arriving customers lead to an arrival process that depends on the number of present customers. For the system manager, having more customers may be beneficial as it can increase future arrivals due to the attraction generated. It may also saturate the system, resulting in long waits. Rejection at arrival may then be employed as a way to obtain a trade-off between these conflicting objectives. With this in mind, we developed a Markov decision process approach to determine how to optimally reject customers in a queueing system with state-dependent arrivals.When the arrival rate is bounded, we compute the optimal policy from a value iteration approach. When the arrival rate is decreasing and convex, we prove that it has a threshold form. When the arrival rate is increasing and potentially unbounded, uniformization may not apply. In dealing with this case, we restrict the analysis to stationary policies and prove the optimality of threshold policies from a computational approach. In addition, we show how to compute the optimal threshold within a finite number of iterations and prove that the long-run expected cost is decreasing and convex in the number of servers. We finally illustrate the applicability of our results through the analysis of a linearly increasing arrival rate, determining the main drivers of control decisions.

Service Rate and Admission Control in an M/G/1 Queue with State-Dependent Arrival Rates

Admission and service rate control problems in queueing systems have been studied in the literature. For an exponential service time distribution, the optimality of the threshold-type policy has been proved. However, in production systems, the production time follows a general distribution, not an exponential one. In this paper, control of the service speed according to the number of customers is considered. The analytical results of an M/G/1 queue with arrival and service rates that depend on the number of customers in the system, which is called an Mn/Gn/1 queue, are used to compute the performance measure of service rate control. In particular, for the case in which the arrival rates are the same among queue-length intervals, a computation method for deriving stationary distributions is developed. Constant, uniform, exponential, and Bernoulli distributions on the service time are considered via numerical experiments. The results show that the optimal threshold depends on the type of distribution, even if the mean value of the service time is the same. In addition, when the reward rate is small, a case in which a non-threshold-type service rate control policy outperforms all threshold-type policies is identified.

Analysis of a two-critical-number production policy in an M/G-type production system with waiting time information

ABSTRACT In this paper, a production and inventory system with state-dependent arrival rates is considered under a two-critical-number policy. If the number of products in the inventory reaches the value of a parameter of this policy, the production stops, and if the number of customers who are waiting for products reaches the value of another parameter, the production restarts. In this production and inventory system, the probabilities for the number of products in the inventory and the number of waiting customers in a steady state are developed by using the analytical results of an M/G/1/K queue with state-dependent arrival rates. In addition, the waiting time information policy is investigated. Short waiting time information may decrease the probability that the actual waiting time is shorter than the informed waiting time, which is regarded as a measure of customers’ satisfaction. Sensitivity analysis is conducted via numerical experiments. The numerical examples show that the critical numbers for maximizing the average profit are insensitive to the processing time distributions. They also show that the use of a suitable waiting time information policy is important for the satisfaction of customers, especially if the variance in the processing time is small.

Queueing analysis for operations modeling in port logistics

Purpose The use of queueing network models was stimulated by the appearance (1975) of the exact product form solution of a class of open, closed and mixed queueing networks obeying the local balance principle and solved, a few years later, by the popular mean value analysis algorithm (1980). Since then, research efforts have been produced to approximate solutions for non-exponential services and non-pure random mechanisms in customer processing and routing. The purpose of this paper is to examine the suitability of modeling choices and solution approaches consolidated in other domains with respect to two key logistic processes in container terminals. Design/methodology/approach In particular, the analytical solution of queueing networks is assessed for the vessel arrival-departure process and the container internal transfer process with respect to a real terminal of pure transshipment. Findings Numerical experiments show the extent to which a decomposition-based approximation, under fixed or state-dependent arrival rates, may be suitable for the approximate analysis of the queueing network models. Research limitations/implications The limitation of adopting exponential service time distributions and Poisson flows is highlighted. Practical implications Comparisons with a simulation-based solution deliver numerical evidence on the companion use of simulation in the daily practice of managing operations in a finite-time horizon under complex policies. Originality/value Discussion of some open modeling issues and encouraging results provide some guidelines on future research efforts and/or suitable adaption to container terminal logistics of the large body of techniques and algorithms available nowadays for supporting long-run decisions.

Analysis of an N-policy MX/M/1 Two-phase Queueing System with State-dependent Arrival Rates and Unreliable Server

Analysis of an N-policy MX/M/1 Two-phase Queueing System with State-dependent Arrival Rates and Unreliable Server

A Markovian single server working vacation queue with server state dependent arrival rate, balking and with breakdown

A Markovian single server working vacation queue with server state dependent arrival rate, balking and with breakdown

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.

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

A Markovian Working Vacation Queue with Server State Dependent Arrival Rate and with Unreliable Server

A Markovian Working Vacation Queue with Server State Dependent Arrival Rate and with Unreliable Server

Multilevel rationing policy for spare parts when demand is state dependent

The multilevel rationing (MR) policy is the optimal inventory control policy for single-item M / M / 1 make-to-stock queues serving different priority classes when demand rate is constant and backlogging is allowed. Make-to-repair queues serving different fleets differ from make-to-stock queues because in the setting of the former, each fleet comprises finitely many machines. This renders the characterization of the optimal control policy of the spare part inventory system difficult. In this paper, we implement the MR policy for such a repair shop/spare part inventory system. The state-dependent arrival rates of broken components at the repair shop necessitate a different queueing-based solution for applying the MR policy from that used for make-to-stock queues. We find the optimal control parameters and the cost of the MR policy; we, then compare its performance to that of the hybrid FCFS and hybrid priority policies described in the literature. We find that the MR policy performs close to the optimal policy and outperforms the hybrid policies.

Optimal control of M/M/1 two-phase queueing system with state-dependent arrival rate, server breakdowns, delayed repair, and N-policy

In this paper, we examine a two-stage queueing system where the arrivals are Poisson with rate depends on the condition of the server to be specific: vacation, pre-service, operational or breakdown state. The service station is liable to breakdowns and deferral in repair because of non-accessibility of the repair facility. The service is in two basic stages, the first being bulk service to every one of the customers holding up on the line and the second stage is individual to each of them. The server works under N-policy. The server needs preliminary time (startup time) to begin batch service after a vacation period. Startup times, uninterrupted service times, the length of each vacation period, delay times and service times follows an exponential distribution. The closed form of expressions for the mean system size at different conditions of the server is determined. Numerical investigations are directed to concentrate the impact of the system parameters on the ideal limit N and the minimum base expected unit cost.

<i>M</i><sup align="right"><i>X</i></sup>/<i>G</i>/1 state dependent arrival queue with optional service and vacation under randomised policy

The present study deals with the performance analysis of a single server queues with state dependent arrival rates in batches. It is considerable that the units join the system for essential and optional services available in the system. The server can avail the at most m-vacations under randomised vacations policy. The system is analysed by using the supplementary variable technique with the assumptions of probability reasoning for study the steady state behaviour of the system and to obtain various performance indices of queue size distribution. The maximum entropy principle is used to find the approximate results of performance measures and comparative study of exact and approximate waiting times of the units is discussed. The numerical illustration is also carried out to facilitate the minimum cost for optimal parameters and validate the other results of the system.

<i>M</i><sup align="right"><i>X</i></sup>/<i>G</i>/1 state dependent arrival queue with optional service and vacation under randomised policy

The present study deals with the performance analysis of a single server queues with state dependent arrival rates in batches. It is considerable that the units join the system for essential and optional services available in the system. The server can avail the at most m-vacations under randomised vacations policy. The system is analysed by using the supplementary variable technique with the assumptions of probability reasoning for study the steady state behaviour of the system and to obtain various performance indices of queue size distribution. The maximum entropy principle is used to find the approximate results of performance measures and comparative study of exact and approximate waiting times of the units is discussed. The numerical illustration is also carried out to facilitate the minimum cost for optimal parameters and validate the other results of the system.

ANALYSIS OF MULTI-RESOURCE LOSS SYSTEM WITH STATE-DEPENDENT ARRIVAL AND SERVICE RATES

In this paper, we study a generalization of the classical multi-dimensional Erlang loss model with state-dependent arrival and service rates, in which customers at arrival occupy random quantities of various resources and release them at departure. Total amount of resources allocated to customers cannot exceed predefined maximum levels. There can be two types of customers: positive customers, which occupy positive quantities of resources, and negative customers, which occupy negative quantities of resources. Negative customers increase the amount of resources available to positive customers and therefore decrease blocking of positive customers caused by lack of resources.

Analysis of Batch Arrival Bulk Service Queue with Two Fluctuating Modes of Service Closedown Multiple Vacation and State Dependent Arrival Rate

Analysis of Batch Arrival Bulk Service Queue with Two Fluctuating Modes of Service Closedown Multiple Vacation and State Dependent Arrival Rate

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

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

We consider the M/M/1 → •, M/M/1/0 type retrial queueing system with state dependent parameters. The intensities of input, service (at both stations) and retrials depend in an arbitrary way on the current number of customers in the orbit. Because operation of the first station of this tandem does not depend on the state of the second station, analysis of the system is implemented in two steps. At the first step, we compute the marginal distribution of the states of the first station in a simple form. At the second step, analytical formulas are presented for computation of the steady state-distribution of the tandem. Formulas for computing the most important performance measures of the system are derived. Results can be used for solving various problems of optimal control by the system operation.