Unobservable Queues Research Articles

The Impact of Inspection Cost on Equilibrium, Revenue, and Social Welfare in a Single-Server Queue

Classical models of customer decision making in unobservable queues assume acquiring queue length information is too costly. However, due to recent advancements in communication technology, various services now make this kind of information accessible to customers at a reasonable cost. In our model, which reflects this new opportunity, customers choose among three options: join the queue, balk, or inspect the queue length before deciding whether to join. Inspection is associated with a cost. We compute the equilibrium in this model and prove its existence and uniqueness. Based on two normalized parameters—congestion and service valuation—we map all possible input parameter sets into three scenarios. Each scenario is characterized by a different impact of inspection cost on equilibrium and revenue-maximization queue disclosure policy: fully observable (when inspection cost is very low), fully unobservable (when inspection cost is too high), or observable by demand (when inspection cost is at an intermediate level). We show that when maximizing social welfare, the optimal disclosure policy is zero inspection cost. We show the structure remains the same when a fraction of the customers are considered urgent, that is, they always join, whereas the others are nonurgent and therefore join according to their equilibrium strategy.

Optimal Signaling Mechanisms in Unobservable Queues

In many systems with limited service capacity, customers must wait in a queue for service. When customers cannot observe the queue, how should a revenue-maximizing service provider convey information about wait times? In “Optimal Signaling Mechanisms in Unobservable Queues,” D. Lingenbrink and K. Iyer study this problem and characterize the structure of the optimal signaling mechanism. To signal optimally, the service provider uses two possible signals, “short” and “long,” to tell customers the queue length is short when below a threshold and long when above it. For the specific case of linear waiting costs, the authors explicitly compute this threshold. Furthermore, they show that for an optimally chosen fixed service price, optimal signaling produces the same expected revenue as a pricing mechanism that sets prices based on the number of customers waiting. This suggests that in settings where one cannot dynamically update prices, signaling can be effective in generating revenue.

Self-Regulation of an Unobservable Queue

We consider an unobservable M/M/1 queue where customers are homogeneous with respect to service valuation and cost per unit time of waiting. It is well known that left to themselves, in equilibrium, customers join the queue at a rate higher than is socially optimal. Hence, regulation schemes, under which the resulting equilibrium joining rate coincides with the socially optimal one, should be considered. We suggest a classification of regulation schemes based on a few desired properties and use it to classify schemes from the existing literature. To the best of our knowledge, none of the existing schemes possesses all of the properties, and in this paper we suggest such a scheme. Its novelty is in assigning random priorities to customers, prior to their decision whether to join or balk. We also introduce variations of this regulation scheme as well as others that are also based on randomization. The e-companion is available at https://doi.org/10.1287/mnsc.2017.2728 . This paper was accepted by Gad Allon, operations management.

Equilibrium strategic behavior of customers in the M/M/1 queue with partial failures and repairs

This paper analyzes the customers’ equilibrium strategic behavior in the M/M/1 queue subject to Poisson generated partial failures and repairs. In such a queueing system, whenever a partial failure occurs, the system does not admit any customer and continues the service at a lower rate instead of stopping working completely. This goes on till the system becomes empty and, at that point, an exponential repair time is sent on. The arriving customer’s dilemma is whether to enter or balk based on a linear reward-cost structure. We discuss the fully observable and fully unobservable queues respectively. For each type of queue, the corresponding equilibrium balking strategies of customers and the expected social benefits per time unit are derived. Finally, we illustrate the theoretical results by presenting some numerical examples.

Equilibrium and optimal balking strategies of customers in unobservable queues with double adaptive working vacations

This paper studies the customers’ equilibrium and socially optimal balking behaviour in some single-server Markovian queues with double adaptive working vacations. Once the system becomes empty, the server takes a working vacation with a low service rate. If there are customer arrivals during this period, a regular busy period begins as soon as he finishes the vacation. Otherwise, he takes another working vacation with a much lower service rate than that in the first working vacation. After completing the second vacation, the server either stays idle or begins a busy period. We discuss two types of unobservable queues: the almost unobservable queues and the fully unobservable queues, respectively, and arriving customers can’t observe system occupancy in both cases. For each type of queues, we get both the customers’ equilibrium and socially optimal balking strategies and make numerical comparisons between them. We observe that their positive and stable equilibrium strategy and optimal strategy are unique, and especially, the customers’ actual arrival rates in equilibrium in vacation states are not necessarily smaller than that in busy state in the almost unobservable queues. Moreover, we also find that whether the system information should be revealed to customers depends on the potential demand arrivals.

Equilibrium strategies of the unobservable M/M/1 queue with balking and delayed repairs

Wang and Zhang (2011) studied the equilibrium threshold balking strategies for the fully observable and partially observable single-server queues with server breakdowns and delayed repairs. Customers decide whether to join or balk the system based on observation of the queue length and status of the server at their arrival instants. The present paper aims to study the corresponding unobservable cases in which the queue length is unknown to arriving customers. The model under consideration can be viewed as an M/M/1 queue in a random environment. Equilibrium mixed strategies are derived for the almost unobservable and fully unobservable queues. Finally, we illustrate the effect of several system parameters on the equilibrium behavior via numerical examples.

Optimal balking strategies for high-priority customers in M/G/1 queues with 2 classes of customers

This paper studies the optimal balking strategies of customers with a higher priority in the single-server M/G/1 queue, where there are two classes of customers who have different priorities. When a low-priority customer is being served, the high-priority customer joins in the system and will be served immediately while the low-priority customer will go back to the head of the queue of its class. Upon arrival, the high-priority customers decide for themselves whether to join or balk based on a linear reward-cost structure that incorporates their desire for service, as well as their unwillingness to wait. We identify equilibrium strategies and socially optimal strategies under one information assumption. The high-priority customers make individual decisions in a unobservable queue. By deriving and solving a set of system equations, we obtain equilibrium strategies and socially optimal strategies. Finally we illustrate our results via numerical experiments.

Equilibrium and optimal balking strategies of customers in Markovian queues with multiple vacations and N-policy

This paper studies the customers’ equilibrium and socially optimal balking strategies in single-server Markovian queues with multiple vacations and N-policy. That is, the server starts taking a vacation time once the system becomes empty, and reactivates in case at least N customers waiting in the queue after completing the vacation. Otherwise, he/she continues taking another vacation. We discuss the fully observable and fully unobservable queues, respectively. For each type of queues, we get both customers’ equilibrium and optimal balking strategies, and make numerical comparisons between them. We observe that the customers’ individual behavior in stable equilibrium always makes the system more congested than the socially optimal one. Moreover, we also make sensitivity analysis of customers’ equilibrium and optimal strategies and social welfare with respect to N and the vacation parameter, respectively. It shows that customers and the social planner all prefer a classical Markovian queue without vacations and N-policy. By comparing social welfare under the two information cases, we find that whether the system information should be revealed to customers depends on the potential arrivals of demand.

Equilibrium balking strategies of customers in Markovian queues with two-stage working vacations

This paper studies the customers’ equilibrium balking behavior in some single-server Markovian queues with two-stage working vacations. That is, the server starts taking two successive working vacations when the system becomes empty, during which he provides low-rate service but maintains different service rates in the two-stage vacations. Based on different precision levels of system information, we discuss observable queues, partially observable queues and unobservable queues, respectively. For each type of queues, we get the customers’ equilibrium balking strategy and equilibrium social welfare per time unit, and numerically observe that their positive equilibrium strategy is unique. Especially, for partially observable queues, the customers’ equilibrium joining probabilities in vacation states are not necessarily smaller than that in busy state.

Regulating an M/G/1 queue when customers know their demand

Customers do not necessarily join a queue at a socially optimal rate. Hence, queueing systems may call for regulation. For customers in an M/G/1 unobservable (not necessarily FCFS) queue and homogeneous with respect to waiting costs and service rewards, we show how queueing systems can be regulated by imposing an entry fee, a holding fee (based on time in the system), or a service fee (based on the required service time) when customers know their service requirements. We start with a unified approach and state the socially optimal fees. We show that customers are always worse off under a flat entry fee in comparison with holding and service fees. As for holding vs. service fees, the answer depends on the queueing regime and/or the service length itself. For example, under FCFS, service fees are preferred by all. Details are given on some common service regimes. We also review the case where customers know only the common distribution of service times, but not their actual requirements.

Equilibrium and optimal behavior of customers in Markovian queues with multiple working vacations

This paper studies the customers’ equilibrium and socially optimal joining–balking behavior in single-server Markovian queues with multiple working vacations. Different from the classical vacation policies, the server does not completely stop service but maintains a low service rate in vacation state in case there are customer arrivals. Based on different precision levels of the system information, we discuss the observable queues, the partially observable queues, and the unobservable queues, respectively. For each type of queues, we get both the customers’ equilibrium and socially optimal joining–balking strategies and make numerical comparisons between them. We numerically observe that their equilibrium strategy is unique, and especially, the customers’ equilibrium joining probability in vacation state is not necessarily smaller than that in busy state in the partially observable queues. Moreover, we also find that the customers’ individual behavior always deviates from the social expectation and makes the system more congested.

Effect of information, uncertainty and parameter variability on profits in a queue with various pricing strategies

This paper presents an unobservable single-server queueing system with three types of uncertainty, where the service rate, or waiting cost or service quality is random variable that may obtain n(n > 2) values. The information about the realised values of parameters is only known to the server. We are concerned about the server’s behaviour: revealing or concealing the information to customers. The n-value assumption and the server’s behaviour enable us to consider various pricing strategies. In this paper, we analyse the effect of information and uncertainty on profits and make comparisons between the profits under different pricing strategies. Moreover, as for parameter variability reflected by the number of each parameter’s possible choices n, we observe the effect of variable n on all types of profits and find that revealing the parameter information can much more benefit the server with the increase of n.

Equilibrium Mixed Strategies of Customers in an Unobservable Queue with Multiple Vacations

This paper studies and compares the equilibrium mixed strategies of customers with different decision criteria in an unobservable single-server queue with multiple vacations. Arriving customers can’t observe the queue length and the server state, whereas they are informed that the service time is exponentially distributed with unknown parameter, which is either an alternative of two values or belongs to a range. We consider three kinds of customers classified by their different waiting cost functions: linear, quadratic and exponential functions, and find that for the two types of partial information on service time parameter, both the equilibrium threshold expressions of customers’ effective arrival rates based on each criterion are consistent with each other except for the Laplace thresholds.

Pricing and setup/closedown policies in unobservable queues with strategic customers

In this paper we study unobservable Markovian queueing systems with three types of setup/closedown policies: interruptible, skippable and insusceptible setup/closedown policies, respectively. For a system with the interruptible setup/closedown policy, service starts as soon as a customer arrives during a closedown time; However, for a system with the skippable setup/closedown policy, customers arriving in a closedown time (if any) can be served only after the closedown time finishes and the following setup time can be skipped; Then for a system with the insusceptible setup/closedown policy, customers arriving in a closedown time can’t be served until the following setup time finishes. We assume that customers need a price for service, and derive the equilibrium and socially optimal balking strategies for customers as well as the maximal social welfare. Then we make pricing control to motivate customers to adopt the optimal strategies and obtain an appropriate price that also maximizes server’s profit. Moreover, we numerically make some comparisons between the various performance measures.

Optimal Routing Among ⋅/M/1 Queues with Partial Information

When routing dynamically randomly arriving messages, the controller of a high-speed communication network very often gets the information on the congestion state of down stream nodes only after a considerable delay, making that information irrelevant at decision epochs. We consider the situation where jobs arrive according to a Poisson process and must be routed to one of two (parallel) queues with exponential service time distributions (possibly with different means), without knowing the congestion state in one of the queues. However, the (conditional) probability distribution of the state of the unobservable queue can be computed by the router. We derive the joint probability distribution of the congestion states in both queues as a function of the routing policy. This allows us to identify optimal routing schemes for two types of frameworks: global optimization, in which the weighted sum of average queue lengths is minimized, and individual optimization, in which the goal is to minimize the expected delay of individual jobs.