Unobservable Queues Research Articles

Equilibrium balking strategies in unobservable queues with multiple vacations and an optional service

This paper examines equilibrium mixed strategies in unobservable Markovian queues featuring a second optional service with server vacations, where arriving customers may choose to join or balk the system. All customers arriving at the system receive the essential service, and some customers opt for the second service after the first service has been completed. Once all customers in the system have been served, the server takes the first of multiple vacations. If no customers are waiting upon from the vacation, then the server takes another vacation. In unobservable queues, arriving customers cannot know the queue length; however, the information pertaining to the server state may be available. By examining unobservable queues (fully unobservable and almost unobservable cases), it is possible to formulate an equilibrium joining strategy as well as the socially optimal probability of joining a fully unobservable queue. This paper also presents numerical examples illustrating how system parameters affect mixed equilibrium and socially optimal balking strategies.

Optimal pricing and capacity management in service systems with delay-sensitive mixed-risk customers

In service systems with delay-sensitive customers, customer decisions to join the queue or balk influence demand, impacting service provider profits. Studies show that customers can be risk-seeking for short waits and risk-averse when waits exceed expectations. Prior research on this mixed-risk attitude is limited, focussing on optimal pricing policies from a social welfare perspective. This study fills this gap by investigating the impact of this behaviour on customer joining strategies and the provider's pricing and capacity policies. We analyse a profit-maximising firm managing an unobservable queue where customers display mixed-risk behaviour towards delay parameterised by a risk-switching point and a degree of risk propensity. Customers join or balk based on expected utility, while the firm determines the service price and rate to maximise expected profit. We derive optimal strategies for both customers and the firm under scenarios with concave or convex capacity costs. Computational experiments show that optimal pricing, capacity, and profit peak at intermediate risk-switching points. For smaller points, optimal price and capacity rise with the customers' degree of risk propensity, while larger points have minimal impact. Numerical exploration emphasises the significance of capacity cost and the effectiveness of leveraging price over controlling capacity for maximising firm profitability.

Analysis of strategic customer behavior in observable M/G/1 queues with impatience

ABSTRACT In this paper, we propose a queueing-game-theoretic model and analyze the strategic behavior of customers and social optimization in an observable M / G / 1 queue, in which arriving customers decide whether to join the system or balk based on a new binary and random reward-cost structure. Each incoming customer to the queue has a relative tolerance time. If the customer’s service does not begin (or end) within his or her relative tolerance time, the customer will incur a cost for his or her waiting. We first derive closed-form solutions for customers’ equilibrium and socially optimal joining strategies using the technique of the Laplace-Stieltjes transform. Furthermore, some representative numerical experiments are performed to visualize the theoretical results. The numerical scenarios illustrate the influence of the relative tolerance time on equilibrium strategy and socially optimal strategy. Finally, we compare the effect of relative tolerance time on social welfare in observable and unobservable queues. The numerical results show that observable queues lead to higher social welfare. This study provides guidance for system providers in designing more economical and sustainable public service systems.

Strategic shield against external shocks in a Markovian queue with vulnerable server

<p style='text-indent:20px;'>A threshold-type queue-length control strategy is proposed in the paper to investigate the influence of external shocks to a queueing system where customers are strategic and the server is vulnerable. An empty system is required to initiate service only when the number of waiting customers reaches a given threshold. External shocks occur according to a Poisson process, and once occur, the server breaks down and the customer being served is forced to leave the system. Arriving customers have to decide whether to join the system or not based on a reward-cost structure under different levels of information. The focus is on examining the equilibrium performance of the system under the interaction between the server's states and customers' joining decisions for different information levels. The equilibrium threshold in the observable queue and mixed joining probability in the unobservable queue are obtained. The optimal value of threshold <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> is discussed by taking strategic customer behavior and vulnerability of the server into consideration. These findings have important managerial implications on the evaluation of the shield threshold for the system with external attacks and unreliability factor, and also on optimal operation management of the system.</p>

Equilibrium and social optimality in queues with service rate and customers’ joining decisions

ABSTRACT This paper treats a single server Markovian queueing system with changeable service rates and customer choice. When a customer arrives at an empty system, he is served by the server at a low service rate. This low service rate continues until the queue length reaches a threshold. Then the server switches to a high service rate to serve customers who join the system. The high service rate continues until the system becomes empty and a low service rate starts again. Under a service reward and waiting cost structure, customers decide to join or to balk based on their net utility of joining the system. The customers’ equilibrium and socially optimal joining strategies are studied under two information scenarios which are observable and unobservable queues, respectively. Such a model fits a stochastic service system where the server can make his service rate decision and customers can make their joining or baking decisions and thus has wide applications in service industries. Our focus is on examining the equilibrium performance of the queue under the interaction between the server’s service rate and customers’ joining decisions. We develop the steady-state distribution and performance measures of the system by formulating two-dimensional continuous-time Markov chains.

Strategic joining rules in unobservable queues with dynamic service rate

Abstract This paper considers an unobservable queue with adjustable service rate and strategic customers. A queueing-game-theoretic model is built to capture the interaction between the server’s service rate and customers’ joining decisions. Iterative and recursive methods are used to derive the steady-state distribution and the expected sojourn time in the queue. We obtain customers’ equilibrium and socially optimal joining strategies under two information scenarios which are unobservable and partially unobservable queues, separately. It is found that there are four equilibrium joining strategies at most in the fully unobservable queue and two equilibrium joining strategies in the almost unobservable queue. Interestingly, the social optimal arrival rate is between the minimum and maximum stable equilibria. Thus, in most cases, managers need to charge a price to induce the social optimal customers’ behaviour. However, if the minimum equilibrium is achieved, managers are required to provide a subsidy to maximize the social welfare. unobservable queue; $(m, N)$ policy; equilibrium strategy; social welfare.

On equilibrium threshold strategies when choosing between observable and unobservable queues

We consider threshold equilibrium strategies in a two-server memoryless queueing system where customers inspect one queue before deciding which queue to join. We show that such an equilibrium may not exist. A numerical study indicates that if threshold equilibria exist, the effective arrival rate to the unobserved queue is higher when the regime there is last-come first-served rather than first-come first-served.

Game-theoretic analysis of the single vacation queue with negative customers

ABSTRACT We present a game-theoretic analysis of an M/M/1 queueing system with negative customers and single server vacation. Both positive and negative customers arrive according to a Poisson process and the server stars a vacation when the system is empty. Whenever a negative customer arrives, the positive customer being served (if any) is forced to abandon the system and the server suffers a breakdown, immediately after, a repair is required. During the repair process, positive customers are not allowed to join the system. Besides, they decide whether to join or to balk the system based on a reward-cost structure under four cases of different levels of information. We derive the equilibrium joining strategies of positive customers in each case. Specifically, we obtain the equilibrium threshold in the observable queue and mixed joining probability in the unobservable queue. Finally, the effects of different information levels and several parameters on the equilibrium threshold and mixed joining probabilities are illustrated by numerical examples.

Pricing in Service Systems with Rational Balking and Abandonment of Time‐Sensitive Customers

The current literature on pricing in service systems with time‐sensitive customers predominately ignores the rational abandonment of customers with mixed‐risk attitude. The goal of this study is to address this gap. We consider an unobservable queueing system with a nonlinear waiting cost function, which is concave up to a certain point and then becomes convex, capturing the mixed‐risk attitude of customers observed in empirical studies. We assume that customers are sensitive with respect to waiting time (delay) and strategic regarding their balking and abandonment decisions. We characterize the optimal pricing policy that maximizes the service provider's revenue. We show that the pricing policies studied in the literature, including the joint service and cancellation (entrance) fee policy, are suboptimal and cannot induce the socially optimal behavior. We demonstrate that while the cancellation fee can regulate a customer's balking strategy, the service fee cannot effectively control a customer's abandonment decision. We then provide conditions under which the joint service and cancellation fee policy is optimal. We finally prove that the service provider should compensate customers for their waiting in order to efficiently control the abandonment of customers. We propose a pricing policy, which includes entrance, service, and wait time (delay) fees, that maximizes the provider's revenue. We derive the optimal fees and show that, under the proposed optimal pricing policy, customers pay service and cancellation fees while they are partially compensated for the time spent waiting for service.

The service pricing strategies and the strategic behavior of customers in an unobservable Markovian queue with unreliable server

A Stackelberg game is used to study the service pricing and the strategic behavior of customers in an unreliable and totally unobservable M/M/1 queue under a reward-cost structure. At the first stage, the server manager, acting as a leader, chooses a service price and, at the second stage, a customer, arriving at the system and acting as a follower, chooses to join the system or an outside opportunity, knowing only the service price imposed by the server manager and the system parameters. We show that the constructed game admits an equilibrium and we give explicit forms of server manager and customers equilibrium behavioral strategies. The results of the proposed model show that the assumption that customers are risk-neutral is fundamental for the standard approach usually used. Moreover, we determine the socially optimal price that maximizes the social welfare and we compare it to the Stackelberg equilibrium. We illustrate, by numerical examples, the effect of some system parameters on the equilibrium service price and the revenue of the server manager.

Comparison of socially optimal and profit maximizing prices in an unobservable queue with heterogeneous waiting costs

We consider a general unobservable queueing model in which customers are allowed to join or balk upon arrival. The service provider charges the same admission fee to all joining customers. All joining customers receive the same reward and incur heterogeneous waiting cost rates. We show that the socially optimal arrival rate is greater than or equal to the profit maximizing arrival rate. Equivalently, the socially optimal admission fee is smaller than or equal to the profit maximizing admission fee.

ON MARKOVIAN QUEUES WITH SINGLE WORKING VACATION AND BERNOULLI INTERRUPTIONS

This paper considers the customers’ equilibrium and socially optimal joining–balking behavior in a single-server Markovian queue with a single working vacation and Bernoulli interruptions. The model is motivated by practical service systems where the service rate can be adjusted according to whether or not the system is empty. Specifically, we focus on a single-server queue in which the server's service rate is reduced from a regular to a lower one when the system becomes empty. This lower rate period is called a working vacation for the server which may represent that part of the service facility is under a maintenance process or works on other non-queueing job, or simply for saving the energy (for a machine server case). In this paper, we assume that the working vacation period is terminated after a random period or with probability p after serving a customer in a non-empty system. Such a system is called a queue with single working vacation and Bernoulli interruptions. Customers are strategic and can make choice of joining or balking based on different levels of system information. We consider four scenarios: fully observable, almost observable, almost unobservable, and fully unobservable queue cases. Under a reward-cost structure, we analyze the customer's equilibrium and social-optimal strategies. In addition, the effects of system parameters on optimal strategies are illustrated by numerical examples.

Service pricing and customer behaviour strategies of Stackelberg's equilibrium in an unobservable Markovian queue with unreliable server and delayed repairs

Service pricing and customer behaviour strategies of Stackelberg's equilibrium in an unobservable Markovian queue with unreliable server and delayed repairs

Optimal strategies and pricing analysis in M/M/1 queues with a single working vacation and multiple vacations

This paper considers the customers’ equilibrium and socially optimal joining-balking behavior in single-server Markovian queues with a single working vacation and multiple vacations. Arriving customers decide whether to join the system or balk based on the system states and a linear reward-cost structure, which incorporates the desire of customers for service and their dislike to wait. We consider that the system states are almost unobservable and fully unobservable, respectively. For these two cases, we first analyze the stationary behavior of the system, and get the equilibrium strategies of the customers and compare them to socially optimal balking strategies using numerical examples. We also study the pricing problem that maximizes the server’s profit and derive the optimal pricing strategy. Finally, the social benefits of the almost and fully unobservable queues are compared by numerical examples.

Comparisons of customer behavior in Markovian queues with vacation policies and geometric abandonments

This paper mainly studies customers’ equilibrium balking behavior in Markovian queues with single vacation and geometric abandonments. Whenever the system becomes empty, the server begins a vacation. If it is still empty when the vacation ends, the server stays idle and waits for new arrivals. During a vacation, abandonment opportunities occur according to a Poisson process, and at an abandonment epoch, customers decide sequentially whether they renege and leave the system or not. We consider four information levels: the fully/almost observable cases and the almost/fully unobservable cases, and get the customers’ equilibrium balking strategies, respectively. Then we also get their optimal balking strategies for the almost observable and the almost/fully unobservable cases, and make comparisons of customer strategies and social welfare for the almost observable and the almost/fully unobservable queues with single vacation and multiple vacations. Because of abandonment, we find that the customers’ equilibrium threshold in a vacation may exceed the one in a busy period in the fully observable queues. However, it has little effect on their equilibrium threshold in the almost observable queues, although frequent abandonment opportunity arrival inhibits their optimal threshold. Interestingly, for the almost unobservable queues, customers who arrive in a busy period are not affected by reneging that happened in the previous vacation when they make decisions of joining or balking, whereas the social planner expects that the customers can take it into consideration for social optimization. In the fully unobservable queues, because of no information, possible reneging surely influences customers’ equilibrium and optimal balking behavior. For the almost observable and the almost/fully unobservable queues, the optimal social welfare is greater in the queues with single vacation than that in the queues with multiple vacations.

The potentially negative effects of cooperation in service systems

AbstractThe ‘Price of Anarchy’ states that the performance of multi-agent service systems degrades with the agents’ selfishness (anarchy). We investigate a service model in which both customers and the firm are strategic. We find that, for a Stackelberg game in which the server invests in capacity before customers decide whether or not to join, there can be a ‘Benefit of Anarchy’, that is, customers acting selfishly can have a greater overall utility than customers who are coordinated to maximize their overall utility. We also show that customer anarchy can be socially beneficial, resulting in a ‘Social Benefit of Anarchy’. We show that such phenomena are rather general and can arise in multiple settings (e.g. in both profit-maximizing and welfare-maximizing firms, in both capacity-setting and price-setting firms, and in both observable and unobservable queues). However, the underlying mechanism leading to the Benefit of Anarchy can differ significantly from one setting to another.

Pricing in Service Systems with Rational Balking and Abandonment of Time-Sensitive Customers

The current literature on pricing in service systems with time-sensitive customers predominately ignores the rational abandonment of customers with mixed-risk attitude. The goal of this paper is to address this gap. We consider an unobservable queueing system with a nonlinear waiting cost function, which is concave up to a certain point and then becomes convex, capturing the mixed-risk attitude of customers observed in empirical studies. We assume that customers are sensitive with respect to waiting time (delay) and strategic regarding their balking and abandonment decisions. We first analyze the equilibrium customers' behavior and determine the optimal social behavior. We then characterize the optimal pricing policy that maximizes the service provider's revenue. We show that the pricing policies studied in the literature, including the joint service and cancellation (entrance) fee policy, are suboptimal and cannot induce the socially optimal behavior. We demonstrate that while the cancellation fee can regulate a customer's balking strategy, the service fee cannot effectively control a customer's abandonment decision. We then provide conditions under which the joint service and cancellation fee policy is optimal. We finally prove that the service provider should compensate customers for their waiting in order to efficiently control the abandonment of customers. We propose a pricing policy, which includes entrance, service, and wait time (delay) fees, that maximizes the provider's revenue. We derive the optimal fees and show that, under the proposed optimal pricing policy, customers pay service and cancellation fees while they are partially compensated for the time spent waiting for service.

Social Optimization and Pricing Strategies in Unobservable Queues with Delayed Multiple Vacations

The paper considers the single-server Markovian queues with delayed multiple vacations. Once the system becomes empty, the server will experience a changeover time to begin the vacation. By using the matrix-geometric solution method, we obtain the stationary probability distribution and the mean queue length under almost unobservable and fully unobservable cases. Based on the system status information and a linear reward-cost structure reflecting the desire of customers for service and their unwillingness for waiting, the social welfare function is determined. We also investigate optimal pricing strategies of the server under the ex-postpayment scheme and the ex-antepayment scheme, and we get the customer’s equilibrium strategy under optimal pricing strategy. Finally, we illustrate the effect of several parameters and information levels on socially optimal strategies and optimal social benefit by numerical examples.

Managing operations in customer-intensive services with forward-looking customers

PurposeThis paper aims to investigate the optimal price and service rate decisions in a customer-intensive service, where customers’ perceived service quality decreases in the service speed. Customers are assumed to be forward-looking in purchase decision-making and heterogeneous in their reservation utilities. The purpose of this paper is to study the impact of customers’ forward-looking behavior and the heterogeneity on the operational decisions in a customer-intensive context.Design/methodology/approachThe service is delivered through an M/M/1 queue system with unobservable queues. Customers are forward-looking in queue joining decisions, where the purchase decisions are made when the expected utility is greater than the reservation utility. The optimal price and service rate decisions are analyzed with both homogeneous and heterogeneous customers, where homogenous customers have the same reservation utility in purchase decision-making, while heterogeneous customers have different reservation utilities, which are captured by a random variable.FindingsThe optimal price and service rate decisions with forward-looking customers depend on the customer intensity, potential market size and customers’ reservation utility distribution. The results suggest that customers’ heterogeneity in terms of their reservation utilities affects the optimal decisions, market coverage and the expected revenue. Service providers need to take customers’ heterogeneity and the forward-looking behavior into operational decision-making.Originality/valueThis paper extends previous studies in customer-intensive service and contribute to the service operations management area by explicitly incorporating customers’ forward-looking behavior and heterogeneity in purchase decision-making. Assuming customers are forward-looking and heterogeneous is more realistic and practical. The results highlight that knowing customers’ behavioral characteristics can better improve decision-making in service operations, which is critical for enhancing customers’ satisfaction and loyalty, thus critical to a firm’s success in the market with intensive competition.

Analysis of the equilibrium strategies in the GeoGeo1 queue with multiple working vacations

In this paper, we deal with the customers’ equilibrium strategies in a Geo / Geo / 1 queuing system with multiple working vacations. The arriving customers decide whether to join or balk the queuing system depended on the information of the server’s states and the queue length. Based on the reward-cost structure, we obtain the equilibrium balking strategies and socially optimal strategies in three cases of the fully observable queue, the partially observable queue and the unobservable queue. In addition, we provide some numerical experiments to illustrate the influence of the information level on the equilibrium behavior and to compare the customers’ equilibrium and socially optimal strategies.