Customer Interarrival Times Research Articles

Instability of SRPT, SERPT and SJF multiclass queueing networks

We provide two examples of strictly subcritical multiclass queueing networks which are unstable under the shortest remaining processing time (SRPT) service protocol. Both of them are reentrant lines with two servers and eight customer classes. The customer service times in our first system are deterministic, yielding an example of an unstable shortest remaining expected processing time (SERPT) network. In the second one, the service times in one customer class are randomized. Both our examples show also system instability under the shortest job first (SJF) discipline. A simulation study of robustness of our results with respect to changes in the customer interarrival and service times is also provided. Our results indicate that size-based service policies may not use the available resources efficiently in a multiserver network setting and in fact cause instability effects. This is in sharp contrast with their satisfactory performance for single-server queues.

Heavy traffic analysis for single-server SRPT and LRPT queues via EDF diffusion limits

Extending the results of Kruk (Queueing theory and network applications. QTNA 2019. Lecture notes in computer science, vol 11688. Springer, Cham, pp 263–275, 2019), we derive heavy traffic limit theorems for a single server, single customer class queue in which the server uses the Shortest Remaining Processing Time (SRPT) policy from heavy traffic limits for the corresponding Earliest Deadline First queueing systems. Our analysis allows for correlated customer inter-arrival and service times and heavy-tailed inter-arrival and service time distributions, as long as the corresponding stochastic primitive processes converge weakly to continuous limits under heavy traffic scaling. Our approach yields simple, concise justifications and new insights for SRPT heavy traffic limit theorems of Gromoll et al. (Stoch Syst 1(1):1–16, 2011). Corresponding results for the longest remaining processing time policy are also provided.

Instability of LAS multiclass queueing networks

We provide an example of a strictly subcritical multiclass queueing network which is unstable under the least attained service (LAS) service protocol. It is a reentrant line with two servers and six customer classes. The customer interarrival times in our system are bounded below and have finite exponential moments, while the corresponding service times are deterministic. As a special case, we obtain a deterministic, strictly subcritical unstable LAS network.

Reinforcement Learning-Based and Parametric Production-Maintenance Control Policies for a Deteriorating Manufacturing System

The model of a stochastic production/inventory system that is subject to deterioration failures is developed and examined in this paper. Customer interarrival times are assumed to be random and backorders are allowed. The system experiences a number of deterioration stages before it ultimately fails and is rendered inoperable. Repair and maintenance activities restore the system to its initial and previous deterioration state, respectively. The duration of both repair and maintenance is assumed to be stochastic. We address the problem of minimizing the expected sum of two conflicting objective functions: the average inventory level and the average number of backorders. The solution to this problem consists of finding the optimal tradeoff between maintaining a high service level and carrying as low inventory as possible. The primary goal of this research is to obtain optimal or near-optimal joint production/maintenance control policies, by means of a novel reinforcement learning-based approach. Furthermore, we examine parametric production and maintenance policies that are often used in practical situations, namely, Kanban, (s, S), threshold-type condition based maintenance and periodic maintenance. The proposed approach is compared with the parametric policies in an extensive series of simulation experiments and it is found to clearly outperform them in all cases. Based on the numerical results obtained by the experiments, the behavior of the parametric policies as well as the structure of the control policies derived by the Reinforcement Learning-based approach is investigated.

Joint inventory control and pricing in a service-inventory system

This study addresses joint inventory control and pricing decisions for a service-inventory system. In such a system both an on-hand inventory item and a positive service time are required to fulfill customer demands. The service-inventory system also captures main features of the classical inventory systems with a positive processing time, e.g., make-to-order systems. In this study the service-inventory system is modeled as an M/M/1 queue in which the customer arrival rate is price dependent. The inventory of an individual item is continuously reviewed under an (r,Q) policy. The replenishment lead times of the inventory are exponentially distributed. Furthermore, customers arriving during stock-out periods are lost. The stochastic customer inter-arrival times, service times, and inventory replenishment lead times cause the high complexity of the problem and the difficulty in solving it. The aim of this study is to formulate the problem and solve it to optimality. We make three main contributions: (1) We integrate inventory control and pricing in the service-inventory system. The problem is formulated and analyzed as a fractional programming problem, and structural properties are explored for the model. (2) Two solution algorithms are proposed. The first one provides optimal solutions, while the second one is more efficient. (3) The impact of the integrated inventory control and pricing decisions on the overall system performance is investigated. We compare the solutions of the models both with and without fill-rate and service-reliability constraints and report the main interesting managerial insights.

Ant colony optimisation for impatient customer queue under N-policy and Bernoulli schedule vacation interruption

This paper analyses a finite buffer multiple working vacations queue with balking, reneging and Bernoulli schedule vacation interruption under N-policy. Arriving customer decide either to join the system or to balk. After joining the queue, the customer may renege. The server leaves for working vacation after serving the queue exhaustively. At a service completion instant during working vacation, the server continue the vacation with probability q or if there are at least N customers in the queue the server interrupts the vacation and resumes regular service with probability 1 − q. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a regular service period, during a working vacation period and vacation times are assumed to be exponentially distributed and are mutually independent. Using recursive technique, the steady state system length distributions at various epochs are obtained. Some performance measures of the model and cost analysis using ant colony optimisation are presented. Finally, numerical results showing the effect of model parameters on key performance measures are presented.

Ant colony optimisation for impatient customer queue under N-policy and Bernoulli schedule vacation interruption

This paper analyses a finite buffer multiple working vacations queue with balking, reneging and Bernoulli schedule vacation interruption under N-policy. Arriving customer decide either to join the system or to balk. After joining the queue, the customer may renege. The server leaves for working vacation after serving the queue exhaustively. At a service completion instant during working vacation, the server continue the vacation with probability q or if there are at least N customers in the queue the server interrupts the vacation and resumes regular service with probability 1 − q. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a regular service period, during a working vacation period and vacation times are assumed to be exponentially distributed and are mutually independent. Using recursive technique, the steady state system length distributions at various epochs are obtained. Some performance measures of the model and cost analysis using ant colony optimisation are presented. Finally, numerical results showing the effect of model parameters on key performance measures are presented.

Metaheuristic-based simulation optimization approach to network revenue management with an improved self-adjusting bid price function

ABSTRACTMaking accurate accept/reject decisions on dynamically arriving customer requests for different combinations of resources is a challenging task under uncertainty of competitors' pricing strategies. Because customer demand may be affected by a competitor's pricing action, changes in customer interarrival times should also be considered in capacity control procedures. In this article, a simulation model is developed for a bid price–based capacity control problem of an airline network revenue management system by considering the uncertain nature of booking cancellations and competitors' pricing strategy. An improved bid price function is proposed by considering competitors' different pricing scenarios that occur with different probabilities and their effects on the customers' demands. The classical deterministic linear program (DLP) is reformulated to determine the initial base bid prices that are utilized as control parameters in the proposed self-adjusting bid price function. Furthermore, a simulation optimization approach is applied in order to determine the appropriate values of the coefficients in the bid price function. Different evolutionary computation techniques such as differential evolution (DE), particle swarm optimization (PSO), and seeker optimization algorithm (SOA), are utilized to determine these coefficients along with comparisons. The computational experiments show that promising results can be obtained by making use of the proposed metaheuristic-based simulation optimization approach.

Ant colony optimization for optimum service times in a Bernoulli schedule vacation interruption queue with balking and reneging

This paper analyzes a finite buffer multiple working vacations queue with balking, reneging and Bernoulli schedule vacation interruption. Arriving customers decide either to join the system or to balk. After joining the queue the customers may renege based on their desire for service or their unwillingness for waiting. At a service completion instant during working vacations, the server can decide either to continue the vacation with probability $q$ or interrupt the vacation and resume regular service period with probability $1-q$. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a regular service period, during a working vacation period and vacation times are assumed to be exponentially distributed. Using recursive technique, the steady state system length distributions at various epochs are obtained. Some performance measures of the model and cost analysis using ant colony optimization are presented. Finally, numerical results showing the effect of model parameters on key performance measures are presented.

On the benefits of delayed ordering

Practical experience and scientific research show that there is scope for improving the performance of inventory control systems by delaying a replenishment order that is otherwise triggered by generalised and all too often inappropriate assumptions. This paper presents the first analysis of the most commonly used continuous (s, S) policies with delayed ordering for inventory systems with compound demand. We analyse policies with a constant delay for all orders as well as more flexible policies where the delay depends on the order size. For both classes of policies and general demand processes, we derive optimality conditions for the corresponding delays. In a numerical study with Erlang distributed customer inter-arrival times, we compare the cost performance of the optimal policies with no delay, a constant delay and flexible delays. Sensitivity results provide insights into when the benefit of delaying orders is most pronounced, and when applying flexible delays is essential.

Service Systems with Finite and Heterogeneous Customer Arrivals

We consider service systems with a finite number of customer arrivals, where customer interarrival times and service times are both stochastic and heterogeneous. Applications of such systems are numerous and include systems where arrivals are driven by events or service completions in serial processes as well as systems where servers are subject to learning or fatigue. Using an embedded Markov chain approach, we characterize the waiting time distribution for each customer, from which we obtain various performance measures of interest, including the expected waiting time of a specific customer, the expected waiting time of an arbitrary customer, and the expected completion time of all customers. We carry out extensive numerical experiments to examine the effect of heterogeneity in interarrival and service times. In particular, we examine cases where interarrival and service times increase with each subsequent arrival or service completion, decrease, increase and then decrease, or decrease and then increase. We derive several managerial insights and discuss implications for settings where such features can be induced. We validate the numerical results using a fluid approximation that yields closed-form expressions.

Study of customers' impatience in a GI/M/1/N queue with working vacations

This paper presents a renewal input finite buffer queueing system with balking, reneging and multiple working vacations. Arriving customers decide either to enter the system or to balk, and renege (leave the queue after entering) without being served, based on their desire for service, or their unwillingness for waiting. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a service period, vacation times and service times during vacation are assumed to be exponentially distributed. Employing the supplementary variable technique, we provide a recursive algorithm to compute the stationary system length distributions during vacations and regular busy periods. Based on the various system length distributions, various performance measures of the model and some special cases have been investigated. Numerical results illustrating the effect of the parameters on several performance characteristics are also presented.

Analysis of Discrete-Time Single Server Queue with Balking and Multiple Working Vacations

This paper analyzes a discrete-time single server queueing system with balking and multiple working vacations. The arriving customers balk (that is, do not join the queue) with a probability. The inter-arrival times of customers are assumed to be independent and geometrically distributed. The server works at a different rate rather than completely stopping service during vacations. The service times during a service period, service times during a vacation period and vacation times are assumed to be geometrically distributed. We obtain the closed form expressions for the steady-state probabilities at arbitrary and outside observer’s observation epochs. Computational experiences with a variety of numerical results are discussed in the form of tables and graphs. Moreover, some queueing models discussed in the literature are derived as special cases of our model.

The time to ruin and the number of claims until ruin for phase-type claims

We consider a renewal risk model with phase-type claims, and obtain an explicit expression for the joint transform of the time to ruin and the number of claims until ruin, with a penalty function applied to the deficit at ruin. The approach is via the duality between a risk model with phase-type claims and a particular single server queueing model with phase-type customer interarrival times; see Frostig (2004). This result specializes to one for the probability generating function of the number of claims until ruin. We obtain explicit expressions for the distribution of the number of claims until ruin for exponentially distributed claims when the inter-claim times have an Erlang-n distribution.

Control policies for single-stage production systems with perishable inventory and customer impatience

We consider problems of inventory and admission control for make-to-stock production systems with perishable inventory and impatient customers. Customers may balk upon arrival (refuse to place orders) and renege while waiting (withdraw delayed orders) during stockouts. Item lifetimes and customer patience times are random variables with general distributions. Processing, setup, and customer inter-arrival times are however assumed to be exponential random variables. In particular, the paper studies two models. In the first model, the system suspends its production when its stock reaches a safety level and can resume later without incurring any setup delay or cost. In the second model, the system incurs setup delays and setup costs; during stockouts, all arriving customers are informed about anticipated delays and either balk or place their orders but cannot withdraw them later. Using results from the queueing literature, we derive expressions for the system steady-state probabilities and performance measures, such as profit from sales and costs of inventory, setups, and delays in filling customer orders. We use these expressions to find optimal inventory and admission policies, and investigate the impact of product lifetimes and customer patience times on system performance.

A count model based on Mittag-Leffler interarrival times

In this paper, a new generalized counting process with Mittag-Leffler inter-arrival time distribution is introduced. This new model is a generalization of the Poisson process. The computational intractability is overcome by deriving the Mittag-Leffler count model using polynomial expansion. The hazard function of this new model is a decreasing function of time, so that the distribution displays negative duration dependence. The model is applied to a data on interarrival times of customers in a bank counter. This new count model can be simulated by Markov Chain Monte-Carlo (MCMC) methods, using Metropolis- Hastings algorithm. Our new model has many nice features such as its closed form nature, computational simplicity, ability to nest Poisson, existence of moments and autocorrelation and can be used for both equi-dispersed and over-dispersed data.

Discrete-time renewal input multiple vacation queue with accessible and non-accessible batches

This paper analyzes a discrete-time finite buffer single server accessible and non-accessible batch service queue with multiple vacations. The inter-arrival times of customers are arbitrarily distributed. The service times and vacation times are both geometrically distributed. The server starts service in batches of maximum size b with a minimum threshold value a, otherwise it goes for vacation. There is a provision to include the late arriving customers in the ongoing service batch if the size of the batch being served is less than a threshold value d (a ≤ d ≤ b) and the service times are not affected by such inclusions. Using the supplementary variable and the embedded Markov chain techniques, we obtain the steady state probabilities at pre-arrival and arbitrary epochs. Some performance measures, waiting time distribution in the queue, and special cases of the model have also been discussed. Finally, sensitivity analysis is carried out in the form of table and graphs.

The Time-Dependent Mean and Variance of the Non-Stationary Markovian Infinite Server System

Problem statement: In many queuing situations the average arrival and service rates vary over time. In those situations a transient solution for the state probabilities and mean and variance must be obtained. Approach: The mean and the variance of a particular infinite server model will be obtained using the state differential-difference equations and the factorial moment generating function. The average arrival and service rates will be taken to be dependent on time. The individual customer interarrival times and service time are assumed to be exponentially distributed. This is known as the Markovian system. Results: The mean and variance of the system will be established as solutions to two sequential linear ordinary differential equations. A comparison is also made to a previously known result for the corresponding system with a finite number of servers. Conclusion: Simple closed-form equations for the mean and variance of the system are presented.

Discrete-time bulk-service queue with two heterogeneous servers

This paper analyzes a discrete-time bulk-service queueing system with two heterogeneous servers, i.e., two batch servers working with different service rates. The interarrival times of customers and service times of batches are assumed to be independent and geometrically distributed. Applications can be found in a wide variety of real systems including servers formed with different types of processors as a consequence of system updates, nodes in telecommunications network with links of different capacities, nodes in wireless systems serving different mobile users. We obtain closed-form expressions for the steady-state probabilities at arbitrary epoch with the help of displacement operator method and derive the outside observer’s observation epoch probabilities and waiting time distribution measured in slots. Computational experiences with a variety of numerical results in the form of tables and graphs are discussed. Moreover, some queueing models discussed in the literature are derived as special cases of our model. Finally, it is shown that in the limiting case the results obtained in this paper tend to the continuous-time counterpart.

ANALYSIS OF A GI/G Y /1 SYSTEM IN DISCRETE-TIME

ABSTRACT We consider a class of single server queueing systems in which customers arrive singly and service is provided in batches, depending on the number of customers waiting when the server becomes free. Service is independent of the batch size. This system could also be considered as a batch service queue in which a server visits the queue at arbitrary times and collects a batch of waiting customers for service, or waits for a customer to arrive if there are no waiting customers. A waiting server immediately collects and processes the first arriving customer. The system is considered in discrete time. The interarrival times of customers and the inter-visit times of the server, which we call the service time, have general distributions and are represented as remaining time Markov chains. We analyze this system using the matrix-geometric method and show that the resulting R matrix can be determined explicitly in some special cases and the stationary distributions are known semi-explicitly in some other special cases.