Customer Arrival Rate Research Articles

Service Quality in a Reduced payroll Environment: Applying Queuing Analysis to Customer Perception Case Study

This study was conducted in a national retail pharmacy company’s stores inWestern North Carolinato examine the impact of the reduction of store staffing, primarily pharmacists and service staff, on customers’ satisfaction with service time. Customer arrival rates and service times for each queue were conducted to determine optimal staffing. A random customer survey in multiple store locations provided customers’ perceptions of service quality. Analysis determined that over 30% of the customers surveyed were dissatisfied with service time. A regression analysis demonstrated a significant linear relationship (σ = 0.05) between total service time and customer satisfaction. Study results indicate that cutting staff could result in an unacceptable loss of a competitive advantage. Payroll cost savings of less than $70,000 per year could result in lost revenue dollars in excess of $1,700,000 per year. Thus reducing staff hours (decreasing payroll) in the short term may negatively impact long-term effectiveness and productivity.

Flow control in capacity-constrained queueing systems with non-stationary arrivals

In this paper, we present a flow control policy to address the problem of service capacity constraints in queueing systems that face non-stationary customer arrival rates. The studied system concerns commercial trucks arriving at international border checkpoints for processing. Based on an extensive range of numerical examples, we quantify how the flow control policy of bringing arrival rates more in line with available capacity affects customer-oriented metrics and measures of efficiency in service capacity resources. We also quantify the expectations that the policy would place on arriving customers. How these quantitative outcomes are influenced by contextual parameters such as the pattern of customer arrivals and the tightness of capacity constraints is also clarified.

학습효과를 고려한 셀프서비스 모델 : 셀프서비스 주유소 분석

In recent years, service delivery systems employing a self-service approach have been rapidly spreading. Since a self-service system provides a lower product price, it attracts more customers. However, some system managers are still hesitant to accept a self-service system, because there is no systematic model to predict its performance. Therefore, this research attempts to provide a systematic and quantitative model to predict the performance of a self-service system, focused specifically on a self-service gas station. Under this model, the traditional queuing theory was adopted to describe the general self-service process, but it is also assumed that some changes occur in both the customer arrival rate and the service performance rate. In particular, the price elasticity was introduced to capture the change in the customer arrival rate, and the existence of learning effect and helpers were assumed to design the changed service performance rate. Under these assumptions, a simulation model for a self-service gas station is established, and three performance measurements, such as average number of customers, average waiting time, and Utilization are observed, depending on the changes in price difference and helper-operating time. In this research, the optimal operation strategy for price differentiation and helper-operating time is proposed in accordance with the level of the customer learning rate. Although this research confines the scope of the study to the self-service gas station model, the results of this research can be applied to any type of self-service system.

Tightness of Invariant Distributions of a Large-scale Flexible Service System Under a Priority Discipline

We consider large-scale service systems with multiple customer classes and multiple server pools; interarrival and service times are exponentially distributed, and mean service times depend both on the customer class and server pool. It is assumed that the allowed activities (routing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a Leaf Activity Priority (LAP) policy, which assigns static priorities to the activities in the order of sequential “elimination” of the tree leaves. We consider the scaling limit of the system as the arrival rate of customers and number of servers in each pool tend to infinity in proportion to a scaling parameter r, while the overall system load remains strictly subcritical. Indexing the systems by parameter r, we show that (a) the system under LAP discipline is stochastically stable for all sufficiently large r and (b) the family of the invariant distributions is tight on scales [Formula: see text] for all ε > 0. (More precisely, the sequence of invariant distributions, centered at the equilibrium point and scaled down by [Formula: see text], is tight.)

Tightness of invariant distributions of a large-scale flexible service system under a priority discipline

We consider large-scale service systems with multiple customer classes and multiple server pools; interarrival and service times are exponentially distributed, and mean service times depend both on the customer class and server pool. It is assumed that the allowed activities (routing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a Leaf Activity Priority (LAP) policy, which assigns static priorities to the activities in the order of sequential "elimination" of the tree leaves. We consider the scaling limit of the system as the arrival rate of customers and number of servers in each pool tend to infinity in proportion to a scaling parameter r, while the overall system load remains strictly subcritical. Indexing the systems by parameter r, we show that (a) the system under LAP discipline is stochastically stable for all sufficiently large r and (b) the family of the invariant distributions is tight on scales $r^{1/2 + \epsilon}$ for all $\epsilon > 0$. (More precisely, the sequence of invariant distributions, centered at the equilibrium point and scaled down by $r^{-(1/2 + \epsilon)}$, is tight.)

Dynamic server allocation at parallel queues

This article explores whether dynamically reassigning servers to parallel queues in response to queue imbalances can reduce average waiting time in those queues. Approximate dynamic programming methods are used to determine when servers should be switched, and the performance of such dynamic allocations is compared to that of a pre-scheduled deterministic allocation. The proposed method is tested on both synthetic data and data from airport security checkpoints at Boston Logan International Airport. It is found that in situations where the uncertainty in customer arrival rates is significant, dynamically reallocating servers can substantially reduce waiting time. Moreover, it is found that intuitive switching strategies that are optimal for queues with homogeneous entry rates are not optimal in this setting.

In This Issue

In This Issue

An explicit solution for a tandem queue with retrials and losses

We consider a retrial tandem queueing system with two servers whose service times follow two exponential distributions. There are two types of customers: type one and type two. Customers of type one arrive at the first server according to a Poisson process. An arriving customer of type one that finds the first server busy joins an orbit and retries to enter the server after some time. We assume that the arrival rate of customers from the orbit is a linear function of the number of retrial customers. After being served at the first server, a customer of type one moves to the second server. Customers of type two directly arrive at the second server according to another Poisson process. Customers of both types one and two are lost if the second server is busy upon arrival. For this model, we derive explicit expressions of the joint stationary distribution between the number of customers in the orbit and the states of the servers. We prove that the stationary distribution is computed by a numerically stable algorithm. Numerical examples are provided to show the influence of parameters on the performance of the system.

On the Accuracy of Fluid Models for Capacity Sizing in Queueing Systems with Impatient Customers

We consider queueing systems in which customers arrive according to a Poisson process and have exponentially distributed service requirements. The customers are impatient and may abandon the system while waiting for service after a generally distributed amount of time. The system incurs customer-related costs that consist of waiting and abandonment penalty costs. We study capacity sizing in such systems to minimize the sum of the long-term average customer-related costs and capacity costs. We use fluid models to derive prescriptions that are asymptotically optimal for large customer arrival rates. Although these prescriptions are easy to characterize, they depend intricately upon the distribution of the customers' time to abandon and may prescribe operating in a regime with offered load (the ratio of the arrival rate to the capacity) greater than 1. In such cases, we demonstrate that the fluid prescription is optimal up to O(1). That is, as the customer arrival rate increases, the optimality gap of the prescription remains bounded.

Modeling content availability in peer-to-peer swarming systems

BitTorrent, the immensely popular file swarming system, has a fundamental problem: availability. Swarming systems are peer-to-peer systems where users simultaneously interested in the same content cooperate with each other. Although swarming in BitTorrent scales impressively to tolerate massive flash crowds for popular content, it does little to disseminate unpopular content as they are limited by the availability of a seed or publisher. The extent of publisher unavailability is severe, e.g., our measurement study shows that 40% of the swarms have no publishers available more than 50% of the time. In this paper we develop a model to quantify content availability in BitTorrent. An M/G/∞ queueing system is used to capture BitTorrent’s scale-free property, i.e., more peers bring in more capacity to the system. The busy period of the queue captures an uninterrupted duration for which the content is available. The busy period increases exponentially with the arrival rate of customers as well as the time spent by each customer in the system. Armed with this model, we ask how content availability can be improved. Our model suggests an intriguing answer, namely, bundling builds availability in BitTorrent. Bundling is a growing trend today among BitTorrent publishers; instead of disseminating individual files via isolated swarms, a publisher packages a number of related files and disseminates it via a single larger swarm. To appreciate why bundling improves content availability, consider a bundle of K files. The popularity of the bundle is roughly K times the popularity of an individual file as a peer requesting any file requests the entire bundle. The size of the bundle is roughly K times the size of an individual file, so each peer stays K times as long in the system. Our model suggests that the busy period of the bundled swarm is a factor e ) larger than that of an individual swarm. If the busy period lasts until the publisher reappears, the content will be available throughout. In related work, [2] modeled the performance of peer-topeer systems. In contrast, we present a model for availability. [3] points out that peers who want a file may have to wait indefinitely to obtain certain chunks that are not available, a problem referred to as the blocked leecher problem (BLP). Here we show that bundling is one solution to mitigate this availability problem.

Coordinating replenishment mechanisms in supply chain: From the collaborative supplier and store-level retailer perspective

Supply chain integration has affected the replenishment operations between suppliers and store-level retailers. This research focuses on the store-level retailer's replenishment problems and proposes three collaborative replenishment mechanism models in the collaborative supplier and store-level retailer environment. The simulation models are developed based on a case grocery company and explore the impacts of different scenarios (such as customer arrival rates, inventory control policies, and periodic review times) on the system performance. The results can provide the implications for suppliers and store-level retailers to coordinate their replenishment mechanisms. Finally, practical replenishment issues and future directions are discussed.

Lévy processes with adaptable exponent

In this paper we consider Lévy processes without negative jumps, reflected at the origin. Feedback information about the level of the Lévy process (‘workload level’) may lead to adaptation of the Lévy exponent. Examples of such models are queueing models in which the service speed or customer arrival rate changes depending on the workload level, and dam models in which the release rate depends on the buffer content. We first consider a class of models where information about the workload level is continuously available. In particular, we consider dam processes with a two-step release rule and M/G/1 queues in which the arrival rate, service speed, and/or jump size distribution may be adapted depending on whether the workload is above or below some levelK. Secondly, we consider a class of models in which the workload can only be observed at Poisson instants. At these Poisson instants, the Lévy exponent may be adapted based on the amount of work present. For both classes of models, we determine the steady-state workload distribution.

Lévy processes with adaptable exponent

In this paper we consider Lévy processes without negative jumps, reflected at the origin. Feedback information about the level of the Lévy process (‘workload level’) may lead to adaptation of the Lévy exponent. Examples of such models are queueing models in which the service speed or customer arrival rate changes depending on the workload level, and dam models in which the release rate depends on the buffer content. We first consider a class of models where information about the workload level is continuously available. In particular, we consider dam processes with a two-step release rule and M/G/1 queues in which the arrival rate, service speed, and/or jump size distribution may be adapted depending on whether the workload is above or below some level K. Secondly, we consider a class of models in which the workload can only be observed at Poisson instants. At these Poisson instants, the Lévy exponent may be adapted based on the amount of work present. For both classes of models, we determine the steady-state workload distribution.

Sensitivity of Optimal Capacity to Customer Impatience in an Unobservable M/M/S Queue (Why You Shouldn't Shout at the DMV)

This paper employs sample path arguments to derive the following convexity properties and comparative statics for an M/M/S queue with impatient customers. If the rate at which customers balk and renege is an increasing, concave function of the number of customers in the system (head count), then the head-count process and the expected rate of lost sales are decreasing and convex in the capacity (service rate or number of servers). This result applies when customers cannot observe the head count, so that the balking probability is zero and the reneging rate increases linearly with the head count. Then the optimal capacity increases with the customer arrival rate but is not monotonic in the reneging rate per customer. When capacity is expensive or the reneging rate is high, the optimal capacity decreases with any further increase in the reneging rate. Therefore, managers must understand customers' impatience to avoid building too much capacity, but customers have an incentive to conceal their impatience, to avoid a degradation in service quality. If the system manager can prevent customers from reneging during service (by requiring advance payment or training employees to establish rapport with customers), the system's convexity properties are qualitatively different, but its comparative statics remain the same. Most important, the prevention of reneging during service can substantially reduce the total expected cost of lost sales and capacity. It increases the optimal capacity (service rate or number of servers) when capacity is expensive and reduces the optimal capacity when capacity is cheap.

Dynamic Pricing of Perishable Assets Under Competition

We study dynamic price competition in an oligopolistic market with a mix of substitutable and complementary perishable assets. Each firm has a fixed initial stock of items and competes in setting prices to sell them over a finite sales horizon. Customers sequentially arrive at the market, make a purchase choice, and then leave immediately with some likelihood of no purchase. The purchase likelihood depends on the time of purchase, product attributes, and current prices. The demand structure includes time-variant linear and multinomial logit demand models as special cases. Assuming deterministic customer arrival rates, we show that any equilibrium strategy has a simple structure, involving a finite set of shadow prices measuring capacity externalities that firms exert on each other: equilibrium prices can be solved from a one-shot price competition game under the current-time demand structure, taking into account capacity externalities through the time-invariant shadow prices. The former reflects the transient demand side at every moment, and the latter captures the aggregate supply constraints over the sales horizon. This simple structure sheds light on dynamic revenue management problems under competition, which helps capture the essence of the problems under demand uncertainty. We show that the equilibrium solutions from the deterministic game provide precommitted and contingent heuristic policies that are asymptotic equilibria for its stochastic counterpart, when demand and supply are sufficiently large. This paper was accepted by Yossi Aviv, operations management.

Dynamic Pricing of Perishable Assets under Competition

We study dynamic price competition in an oligopolistic market with a mix of substitutable and complementary perishable assets. Each firm has a fixed initial stock of items and competes in setting prices to sell them over a finite sales horizon. Customers sequentially arrive at the market, make a purchase choice and then leave immediately with some likelihood of no-purchase. The purchase likelihood depends on the time of purchase, the product attributes and the current prices. The demand structure includes time-variant linear and MultiNomial Logit demand models as special cases. Assuming deterministic customer arrival rates, we show that any equilibrium strategy has a simple structure, involving a finite set of shadow prices measuring capacity externalities that firms exert on each other: equilibrium prices can be resolved from a one-shot price competition game under the current-time demand structure, taking into account capacity externalities through the time-invariant shadow prices. The former reflects the transient demand side at every moment and the latter captures the aggregate supply constraint over the sales horizon. This simple structure sheds light on dynamic revenue management problems under competition, which helps capture the essence of the problems under demand uncertainty. We show that the equilibrium solutions from the deterministic game provide pre-committed and contingent heuristic policies that are asymptotic equilibria for its stochastic counterpart, when demand and supply are sufficiently large.

Allocation of flexible and indivisible resources with decision postponement and demand learning

We consider a firm that uses two perishable resources to satisfy two demand types. Resources are flexible such that each resource can be used to satisfy either demand type. Resources are also indivisible such that the entire resource must be allocated to the same demand type. This type of resource flexibility can be found in different applications such as movie theater complexes, cruise lines, and airlines. In our model, customers arrive according to independent Poisson processes, but the arrival rates are uncertain. Thus, the manager can learn about customer arrival rates from earlier demand figures and potentially increase the sales by postponing the resource allocation decision. We consider two settings, and derive the optimal resource allocation policy for one setting and develop a heuristic policy for the other. Our analysis provides managerial insights into the effectiveness of different resource allocation mechanisms for flexible and indivisible resources.

Optimal timing of switches between product sales for sports and entertainment tickets

AbstractLike airlines and hotels, sports teams and entertainment venues can benefit from revenue management efforts for their ticket sales. Teams and entertainment venues usually offer bundles of tickets early in their selling horizon and put single‐event tickets on sale at a later date; these organizations must determine the best time to offer individual tickets because both types of ticket sales consume the same fixed inventory. We model the optimal a priori timing decision for a seller with a fixed number of identical tickets to switch from selling the tickets as fixed bundles to individual tickets to maximize the revenue realized before the start of the performance season. We assume that bundle and single‐ticket customers each arrive according to independent, nonhomogeneous Markovian death processes with a linear death rate that can vary over time and that the benefit from selling a ticket in a package is higher than from selling the ticket individually. We characterize the circumstances in which it is optimal for the seller to practice mixed bundling and when the seller should only sell bundles or individual tickets, and we establish comparative statics for the optimal timing decision for the special case of constant customer arrival rates. We extend our analytical results to find the optimal time for offering two groups of tickets with high and low demand. Finally, we apply the timing model to a data set obtained from the sports industry. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2008

Incremental discount policy for taxi fare with price-sensitive demand

It takes some time for taxi driver to complete a requested service of passengers and the driver can attend to only one service at a time, and this distinguishes taxi service from ordinary goods. This paper deals with an incremental discount policy on the taxi fare. It is assumed that customers arrive following a Poisson process with price-sensitive arrival rates. With the objective of maximizing the average profit of taxi, a mathematical model is developed based on regenerative process. Through an analysis of the model, we show that the incremental discount policy is beneficial to the taxi company when the customer arrival rate is relatively small. Also, we propose a solution procedure to determine the optimal discount rate and price breakpoint. Finally, test problems are solved for sensitivity studies.

A Monopolistic and Oligopolistic Stochastic Flow Revenue Management Model

This paper studies a one-shot inventory replenishment problem with dynamic pricing. The customer arrival rate is assumed to follow a geometric Brownian motion. Homogeneous customers have an isoelastic demand function and do not behave strategically. We find a closed-form optimal pricing policy, which utilizes current demand information. Under this pricing policy the inventory trajectory is deterministic, and a retailer sells all inventory. We show that dynamic pricing coordinated with the inventory decision achieves significantly higher profits than does static pricing. Furthermore, under oligopolistic competition we establish a weak perfect Bayesian equilibrium for the price and inventory replenishment game. We find the pricing equilibrium to be cooperative even in a noncooperative environment, but that inventory competition results in overstock and damages profits. Finally, we examine the trade-off between dynamic pricing and price precommitment and find that flexible pricing is still beneficial, provided competition is not too intense.