Equilibrium queueing analysis in a ride-hailing service with sharing option

Abstract This paper examines different car-sharing models offered by ride-hailing firms. In the traditional model, customers are served individually. In the sharing-only model, all customers are willing to share the ride with other customers. In the hybrid model, customers may choose either individual or shared service provided by the same fleet. Sharing customers incur a hassle cost. We develop a queueing game-theoretic model to (i) determine the arrival rates that maximizes social welfare and (ii) choose admission fees that maximizes firm profits. First, we discover a tipping point in the hybrid model. When hassle cost decreases to this point, (i) the optimal customer behavior immediately switches from less than 80% sharing customers to 100%, (ii) the optimal arrival rate increases dramatically, and (iii) the rate at which a decreasing hassle cost improves the maximum social welfare significantly increases. Second, when hassle cost is higher than the tipping point, the hybrid model may generate higher social welfare than the other two models. Hence, the firm can improve social welfare by assigning customers to differentiated services even when they are not heterogeneous. Finally, we propose a profit-maximizing admission fee structure in the hybrid model to maximize firm profits and demonstrate its effectiveness.


Introduction
Recent years have witnessed a rapid growth of ridehailing firms that provide taxi/car service through smart-phone applications.Examples include Uber in North America, Didi in China, Grab in Southeast Asia, and Ola Cabs in India.Compared with traditional taxi operators, these firms take advantage of the latest development in information technology to match customers with drivers efficiently.As these firms continue to expand and evolve, they introduce various new features to the ride-hailing service.
Motivated by observations of simultaneous rides with substantial route overlaps, which indicate opportunities for resource pooling, Uber first introduced the car-sharing service uberPOOL in August 2014, followed by rivals in the market (e.g., GrabShare, Lyft Line, OLA Share).By choosing carsharing service in the mobile application, customers signify their willingness to share rides with other customers.The firms will then match customers who are traveling in the same direction.Unlike the individual one-on-one ride-hailing service, this carsharing service allows firms to improve their resource utilization and to expand their service capacities without recruiting more drivers.Firms set its fare lower than that of individual service to attract customers to this new service, even if they fail to find another customer to share the ride, letting the current customer take the taxi by himself.
However, the car-sharing service has its drawbacks.The driver needs to take one car-sharing customer on a detour to drop off the other customer, who may take too long to board or cannot appreciate the company of other car-sharing customers.For car-sharing customers, this inevitably leads to hassle cost, which grows exponentially with the number of customers sharing the ride.To limit this hassle cost, Uber, for instance, restricts the number of car-sharing customers in a single trip to two.
At the operational level, there are different carsharing models.For example, Pair Taxi, a startup in Singapore, offers car-sharing service to customers departing from the downtown area during afternoon peak hours.By choosing this service, a customer agrees to share the taxi ride with at least one other customer, and no individual service is provided.It is a sharing-only service model.On the other hand, Uber deploys the same driver pool to serve both individuals and shared service requests (i.e., UberX and UberPOOL), and customers can choose between the two services according to their needs.It is essentially a hybrid service model where both individual and shared services are offered.By forcing everyone to share service, the sharing-only model maximizes service capacity for the same amount of resource and will generate a significantly higher throughput than traditional individual service.However, the total accumulated hassle cost will also be driven to its maximum extent.The hybrid service model may lessen the great hassle cost by letting customers choose whether to use shared service, but it may not fully utilize the service capacity.
Customers are strategic, so they apply a cost-andbenefit analysis to decide whether to use individual or shared service.For them, there are several benefits from the shared service.First, the price is lower than the individual service.For example, UberPOOL is up to 40% cheaper than UberX.Second, the expected waiting time can potentially be reduced for sharing customers.Without the sharing option, one of the customers in a shared ride will have to wait longer for another car.Meanwhile, the downside of shared service is the hassle cost.If a sharing customer requests a trip to a popular destination (e.g., central business district during morning peak hours), there is a high probability that she will share the ride with someone.In this case, her expected hassle cost will be high.
Despite the growing popularity of car-sharing models in the ride-hailing industry, little analysis has been done to guide its operations.In this paper, we take the perspective of a ride-hailing firm who wants to determine the optimal customer arrival rates that create the maximum social welfare, and subsequently, choose admission fees that extract the maximum profits.We compare three models: (i) individual service model, (ii) sharing-only service model, and (iii) hybrid service model, regarding the maximum social welfare and the maximum firm profits, as well as the corresponding optimal customer arrival rates and admission fees.This allows us to develop insights and provide guidelines for the operations of different car-sharing models.
We model a ride-hailing service system using a stylized unobservable single-server queue, where customers cannot observe the real-time queue length at arrivals.When the need arises, a customer will send a service request to the ride-hailing system.If it is an individual service request, it will be added to the end of the (virtual) queue and wait to be processed.If it is a shared service request, the system will search for an unpaired shared service request in the queue.If the search is successful, the new request will be paired with the unpaired shared service request, and the customer will skip part of the queue.However, if there are no unpaired shared service requests in the queue, the new request will be added to the end of the queue as an unpaired shared service request.While there are any service requests in the queue, the ride-hailing system will look for drivers to fulfill these pending requests using the First-In-First-Out discipline.If a shared service request is not paired before the system finds a driver for the request, the customer will take the ride alone.
If all customers choose to be served alone, the hybrid model becomes an M=M=1 model, whose queueing economic analysis is available in Hassin and Haviv (2003).If all customers want to share service, the hybrid model becomes a partial-batch bulk service (M=M ½2 =1) model (see, e.g., Gross et al., 2008).In this paper, we carry out the queueing economic analysis for this M=M ½2 =1 model.More generally, the hybrid model is not easy to analyze if some customers choose shared service while others opt for individual service.Specifically, the average waiting time of sharing customers depends on two factors: (i) the number of customers in the queue at her arrival, and (ii) the position of the unpaired sharing customer in the queue if there is one.This special feature of the hybrid model makes tracking such a system challenging.In this paper, we develop a new Markov chain structure, which tracks the unpaired sharing customer's position in the queue and the number of drivers needed to clear the current queue.This Markov chain enables the derivation of service level measures for both individual and sharing customers, and further the queueing economic analysis of the hybrid model.
Using these results, we generate comparative statics of these three models (i.e., M=M=1, M=M ½2 =1, and hybrid models).In this paper, we focus on the system under excessive demand where demand rate is greater than or equal to the capacity of the ridehailing firm.This focus is consistent with the observation that ride-hailing systems are capacityconstrained (Farrell & Greig, 2016).We compare the maximum social welfare and the corresponding optimal arrival rates across the three models.We observe that the hybrid model provides (weakly) higher social welfare and is more robust than the other two models.It is because the hybrid model offers both individual and shared services to customers, providing more flexibility to both customers and the firm.
Interestingly, we discover a tipping point in the hybrid model under the optimal customer behavior.Naturally, the maximum social welfare in the hybrid model increases when the hassle cost decreases.However, when the hassle cost decreases to this tipping point, (i) the optimal customer behavior immediately switches from less than 80% customers using shared service to all using it, (ii) the optimal arrival rate increases dramatically, and (iii) the rate at which a decreasing hassle cost improves the maximum social welfare in the hybrid model significantly increases.
Conventional wisdom says that offering differentiated services is beneficial only when customers are heterogeneous.However, we show that when the hassle cost is higher than the tipping point, the hybrid model, with homogeneous customers using both individual and shared services, generates higher maximum social welfare than the other two models.In other words, the ride-hailing firm can assign homogeneous customers to differentiated services (i.e., individual and shared services) to increase total social welfare.Few papers in the literature observe this phenomenon.One example is Hassin (1985) who finds that a priority regime can be used to improve welfare or profit when customers are homogeneous.Our result extends this observation from priorities to differentiated services.
Finally, we investigate the profit-maximizing admission fees in the three models.We characterize customers' symmetric equilibrium behavior in the hybrid model under any admission fees.Then, we propose a profit-maximizing admission fee structure and demonstrate that it can correctly induce the maximum profit in the hybrid model.When the hassle cost is relatively large, the individual service is priced higher than the shared service.When the hassle cost decreases, the profit-maximizing admission fee of individual service increases while the fee of shared service decreases so that more customers will select shared service.When the hassle cost drops to the tipping point, the profit-maximizing admission fee for shared service in the hybrid model becomes identical to the optimal admission fee in the M=M ½2 =1 model, and the profit-maximizing admission fee for the individual service can be set arbitrarily high so that no customers will be interested.
The paper proceeds as follows.In Sec. 2, we review the relevant literature.Section 3 introduces the M=M ½2 =1 and hybrid models, and derive service level measures such as expected waiting time and probability of sharing.In Sec. 4, for each model, we derive the maximum social welfare and the corresponding optimal arrival rates.Then, in Sec. 5, we discuss profit maximization through admission fees.Finally, Sec. 6 concludes.

Literature review
The bulk of the progress in queueing economics literature can be attributed to the seminal work of Naor (1969), who delineated the various aspects and parameters of an observable queue.Edelson and Hilderbrand (1975) modify some of the assumptions to model unobservable queues.Furthermore, they conclude that the equilibrium profit-maximizing price imposed by the monopoly induces the maximum social welfare (i.e., social welfare in this article).Chen and Frank (2004) prove that the equivalence between social welfare and profit maximization only holds when customers have linear preferences.This conclusion can be extended to other theoretical queueing models, as long as customers are homogenous (see, e.g., Chen & Frank, 2001;Hassin, 1986).
We next focus our attention on the queueing optimization literature that deals with revenue and social welfare maximization, especially when more than one service class is on offer.The goal of this endeavor is to scrutinize customer choices (through self-selection) when different services are offered at varying prices.Knudsen (1972) studies the social welfare maximization problem in the context of a multi-server (M=M=k) queue.Mendelson (1985) examines the pricing and capacity decisions of a queueing model in the framework of a computing system.Mendelson and Whang (1990) derive incentive compatible pricing strategies for an M=M=1 queue with multiple priority classes that ensure optimality at the individual and societal levels.Maglaras and Zeevi (2005) investigate a system with two nonsubstitutable services serving heterogeneous customers.These customers are price-and delay-sensitive, and the authors design a revenue-maximizing policy for the system.Pangburn and Stavrulaki (2008) juxtapose segmentation and pooling policies for revenue maximizing firms offering different services.Sainathan (2018) considers the strategies of prioritization and strategic delays within the realm of ancillary firms serving different classes of customers (patient and impatient).Some other papers on consumers' self-selection in the service and operations literature include Lederer and Li (1997), Af eche (2013), Af eche and Pavlin (2016), and Maglaras et al. (2018).For a comprehensive and in-depth account of the queueing optimization literature, readers are directed to Hassin (1986).
While the popular notion is that priority regime can be used to improve social welfare or profit when customers are heterogeneous, Hassin (1985) shows that priorities can also be used for these purposes when customers are homogeneous.Our paper compliments Hassin (1985) by showing that differentiated services, like the individual and sharing services in the hybrid model, can be used for similar purposes in the homogeneous customer case.Moreover, we propose an admission fee structure in the hybrid model to induce the desired customer behavior and verify that it works.
We segue into the marketing literature on the economics of differential pricing and consumer choice when different products, instead of services, are purchased by customers.We are primarily concerned with papers on price discrimination and customer self-selection in monopolies.Moorthy (1984) expounds on the idea of market segmentation based on customer self-selection; it is presented as an extension of the third degree Pigouvian (see, e.g., Pigou, 1920) price discrimination model.He shows that a monopolist incurs a deadweight loss when inducing self-selection, while a social planner is always able to optimize social welfare under selfselection.Efforts to explain price discrimination, quality and allocation variation, and customer selfselection in monopolistic settings are noted in Cooper (1984), Salant (1989), Varian (1989), and Anderson and Dana (2009), amongst others.The aforementioned marketing literature deals with monopolist firms that sell different products across product lines; instead, our model deals with services offered by a monopolist.While this literature models self-selection and shows that price discrimination is beneficial with heterogeneous customers, we show that service differentiation and self-selection can be beneficial even with homogeneous customers.
We model the sharing-only service through a bulk service queueing system.The first significant paper on this topic is Bailey (1954), who derives the mean queue length and waiting time from the server's point of view.Complementary to this, Downton (1955) derives these measures under steady state from a customer's angle.Deb and Serfozo (1973) propose an optimal size for customer batches to minimize the costs incurred while servicing customers at one location, while Barnett (1973) and Deb (1978) study the optimal control policies of shuttle services operating between two terminals.Different arrival strategies in bulk service queues under steady state are examined in Glazer and Hassin (1987) and Stein et al. (2007).The Israeli Queue also services customers in groups, where each new arrival may (or may not) join an existing group in its queue position according to some predetermined probability (see Boxma et al. 2008;Perel andYechiali 2013, 2015).Our work, however, endogenizes the joining and sharing probabilities such that each customer optimizes his expected utility while taking into account other customers' decisions.Other notable works in the domain of bulk service queues include those by Chaudhry and Templeton (1983) and Hassin and Haviv (2003).No prior work, to the best of our knowledge, discusses the equilibrium queueing behavior of a partial batch service queue, which is one of the features of our paper.
Our paper also contributes to the steadily growing literature on the sharing economy.This trend can be attributed to the growth of peer-to-peer (P2P) markets recently, which collectively comprise the sharing economy.Einav et al. (2016) provide a comprehensive discussion on the topic by studying the past, present, and future of P2P markets.Benjaafar et al. (2019) use the example of car-sharing to expound on sharing within a P2P economy.They show that collaborative consumption is always advantageous to consumers.Ibrahim (2018), Gurvich et al. (2016), andCachon et al. (2017) specifically study service platforms with self-scheduling capacities, i.e., platforms where firms (drivers) have the freedom to decide their periods of service.Another stream of literature (see, e.g., Bai et al., 2019;Guda and Subramanian 2019;Gurvich et al., 2016;Taylor, 2018) closely examines the on-demand service economy, which encompasses the exchanges between waitingtime sensitive customers and independent firms.A recent paper by Kostami et al. (2017) explores the optimal pricing and capacity allocation decisions in facilities offering shared services, e.g., gyms and nightclubs, where the same space is shared by different groups of people like men and women.
As far as the ride-sharing economy is concerned, numerous papers in recent years model this system as a two-sided market and study the (dynamic) pricing and matching mechanisms at play-these include works by Bimpikis et al. (2019), Banerjee et al. (2016), Ozkan and Ward (2020), Fang et al. (2017), Cachon et al. (2017), andJacob andRoet-Green (2021).The papers by Banerjee et al. (2016) and Jacob and Roet-Green (2021) come close to our paper.Banerjee et al. (2016) model drivers and passengers as jobs and servers of a strategic queueing model.Since the jobs and servers are themselves not stationary, they are studied under the framework of matching queues.The optimal pricing strategies are formulated to ensure revenue maximization and social welfare optimization, although the authors consider both static and dynamic pricing models and model the overall system as a two-sided platform.While the above paper models customers as heterogeneous and having only the option of choosing a solo ride, we model customers as homogeneous entities who self-select between two service classes based on their utilities.Similar to our paper, Jacob and Roet-Green (2021) consider both the individual and the shared services.However, they focus on designing menus of price-service offerings to elicit private information from two different types of customers.Our paper complements this nascent yet burgeoning literature regarding ride-sharing specifically and the sharing economy in general, with a focus on new queueing models to capture the customers' sharing behavior and its implication on social welfare and the monopolist's profits.

Models and analysis
A monopoly firm with an unobservable single-server facility, where the real-time queue length information is unavailable to customers, serves a stream of customers.This is the case because most ride-hailing platforms do not provide customers with information on their positions in the queue.We assume customer arrivals follow a homogeneous Poisson process with rate K > 0 to keep our analysis tractable and focus on first-order insights.We further assume service times are exponentially distributed with a service rate l.When the individual service is the only option for customers, one service completion takes away one customer from the head of the queue, and it is an M=M=1 model.Although a ride-hailing service typically has many drivers providing service, we approximate the service system as a single server queue with l representing the combined service rate from all the drivers, which is the common approach in the ondemand and queueing economics literature (e.g., Af eche 2013; Banerjee et al. 2016;Hassin 2016;Mendelson and Whang 1990).When only the shared service is offered to customers, all customers need to share service with other customers if there are other customers in the queue.Following the practice of realworld ride-hailing firms, we assume that one service completion takes away up to two customers from the head of the queue.Then, the system is a partial-batch bulk service (M=M ½2 =1) model (see Gross et al., 2008).We assume customer paths are quite similar so that sharing of customers has little impact on the time taken for a ride and thereby l remains unchanged.In a hybrid model, both individual and shared services, are offered to customers.Customers using the individual service want to be served alone, and we call them individual customers.Customers using the shared service are willing to share the service with at most one other customer, and we call them sharing customers.
When an individual customer arrives, she will be allocated to the end of the queue and wait for her turn.When a sharing customer arrives, the firm will look for a sharing customer who has not been paired (with sharing customers) in the queue.If there is an unpaired sharing customer, the new sharing customer will be paired with this one and jump the queue.Otherwise, if there are no unpaired sharing customers in the queue, the new sharing customer joins the queue as an unpaired sharing customer.At any time, there is at most one unpaired sharing customer in the queue.At a service completion, the service is delivered to the customer at the head of the queue, who may be an individual customer, an unpaired sharing customer, or two paired sharing customers.Moreover, other than the sharing customers who get paired with other sharing customers in the queue upon arrival, all customers are served using the First-In-First-Out (FIFO) discipline.The FIFO principle is a common practice among service systems and it is perceived to be fair and easy to implement.We further note that the hybrid model contains M=M=1 and M=M ½2 =1 models as special cases.If there are only individual (or sharing) customers, the hybrid model works as an M=M=1 (or M=M ½2 =1) model.
At the service completion, a customer receives a service reward R.During the waiting time (from arrival until departure) in the system, customers incur a linear waiting cost with marginal rate c.When two customers share service, each of them incurs a hassle cost, h.If a customer chooses to balk, she receives zero utility.All the parameters, K, l, R, c and h, are assumed to be common knowledge.Note that once customers join, they do not abandon their positions in the queue.This is because customers' past waiting time becomes a sunk cost to them, and their expected waiting time in the future stays the same as the long-run average due to the Markovian system.Thus, once customers join, it is their rational decision to not abandon.
Customer expected waiting time in an M=M=1 model is 1 lÀK ; see, e.g., Adan and Resing (2002).We next look into the service level measures of the M=M ½2 =1 and hybrid models.

M=M ½2 =1 model
For the M=M ½2 =1 model, we first characterize the expected waiting time W B and the probability of sharing service p share as follows.
Proposition 1.In a partial-batch M=M ½2 =1 queue with arrival rate k < 2l, (i) The steady state probability distribution of the queue length is 2 ð0, 1Þ: (ii) The expected waiting time for any customers is (iii) The probability of sharing service among customers is We see from Proposition 1 that all measures of interest in the M=M ½2 =1 model can be expressed as functions of h.Moreover, there is a one-to-one correspondence between h 2 0, 1Þ ½ and k 2 0, 2lÞ ½ : In what follows and for the ease of exposition, we focus on h and then use (1) to retrieve k if necessary.
Proposition 1(iii) shows that the probability of any customer sharing service with a fellow customer increases in k, i.e., the higher the demand rate, the higher probability customers share service.

Hybrid model
In a hybrid model, customers can choose between individual and shared services.Let k 1 and k 2 denote the arrival rates of individual and sharing customers, respectively.Clearly, the stationary condition is that the total workload is less than one, i.e., 1 Given the arrival rates k 1 and k 2 , we let denote the respective expected waiting times of individual and sharing customers.We further let p share ðk 1 , k 2 Þ denote the probability for any sharing customer to share service with another customer.For two extreme cases, where almost all customers are individual or sharing customers, the expected waiting times and the sharing probability can be obtained as follows.
Lemma 1. (Two Extreme Cases) In the hybrid model with individual customers' arrival rate k 1 and sharing customers' arrival rate k 2 , , and where We next derive W 1 , W 2 , and p share for any arrival rates k 1 and k 2 .It is not an easy task since the sharing customers' expected waiting time W 2 depends not only on the number of sharing customers in the queue at her arrival but also on the position of the unpaired sharing customer in the queue if there is any.This feature of the hybrid model makes tracking such a system notoriously difficult.To the best of our knowledge, this model has not been considered in the literature.
To track the state of this hybrid model, we introduce the ð/, LÞ Markov chain (MC), where / !0 is the unpaired sharing customer's position in the queue and L is the total number of service completions required to serve all customers in the queue.Recall that there is at most one unpaired sharing customer in the current queue.We use the / ¼ 0 case to represent the absence of any unpaired customers in the queue.Clearly, L is the sum of the number of individual customers, the number of sharing customer pairs, and the unpaired sharing customer if there is any.
When the system is in state (i, j), three possible events may occur in the Markov chain: 1. Arrival of an individual customer with rate k 1 .
The position of the unpaired sharing customer does not change and the total number of service completions to clear all customers in the current queue increases by one, so the MC moves to state ði, j þ 1Þ: 2. Arrival of a sharing customer with rate k 2 .If there are no unpaired sharing customers in the queue, i.e., i ¼ 0, then this arrival joins the end of the queue and the MC moves to state ðj þ 1, j þ 1Þ; otherwise, this sharing customer is paired with the unpaired sharing customer in queue, and the MC moves to state ð0, jÞ: 3. Service Completion with rate l.
3.1.If there are no unpaired sharing customers, i.e., i ¼ 0, the service completion takes away an individual customer or two paired sharing customers.The MC moves from ð0, jÞ to ð0, j À 1Þ: 3.2.If the unpaired sharing customer is at the head of the queue, i.e., i ¼ 1, the service completion takes her away, and there are no unpaired sharing customers in the queue, i.e., i ¼ 0. In this situation, the MC also moves to state ð0, j À 1Þ: 3.3.If the unpaired sharing customer is not at the head of the queue, i.e., i > 1, the unpaired sharing customer moves forward by one position and the MC moves to state ði À 1, j À 1Þ: Figure 1 illustrates the MC of this hybrid model.Let p ij denote the steady state probability distribution of the MC.From the Poisson-Arrival-Sees-Time-Average property, the probability for any customer (individual or sharing) to see state (i, j) at arrival is p ij .Then, P i ¼ P 1 j¼i p ij denotes the probability that the system state is in the i th row of the MC while Q j ¼ P j i¼0 p ij denotes the probability the system state is in the j th column of the MC.
The likelihood that an arriving sharing customer finds an unpaired sharing customer in the queue is given by P 1 i¼1 P i ¼ ð1 À P 0 Þ: If paired, each sharing customer is paired with another sharing customer, so the probability with which any sharing customer gets paired with another sharing customer p share is twice of ð1 À P 0 Þ; i.e., Naturally, the probability that a sharing customer is served alone is 1 À p share ¼ 2P 0 À 1: Furthermore, we note that the service completion rate of unpaired sharing customer can be calculated in two different ways: (i) the arrival rate of sharing customers times the probability that a sharing customer is served alone; or (ii) the service rate times the probability the system state is in the first row of the MC in Figure 1.Hence, we have Let W 1 and W 2 denote the expected waiting times of individual and sharing customers, respectively.By conditioning on the system state seen by an arriving individual or sharing customer, we have At a sharing customer's arrival, there are two possible situations: no unpaired sharing customer or one unpaired sharing customer in the queue.In the former case, this sharing customer's expected waiting time is identical to an individual customer.In the latter case, her expected waiting time is shortened due to pairing, compared with if she is an individual customer.Thus, sharing customers' expected waiting time should be less than that of individual customers, as the following lemma shows.
Lemma 2. The expected waiting time for an individual customer is greater than that of a sharing customer, i.e., W 1 !W 2 .Moreover, the equality holds only when k 1 < l and k 2 ¼ 0: denote the expected waiting times of arriving sharing customers as unpaired and paired customers, respectively, at arrival, so that We have the following properties of W 1 , W 2 , and p 00 .
(i) The expected waiting times of individual customers W 1 can be expressed using W 2s : The expected waiting times of sharing customers W 2 can be expressed using W 2s and P 0 : (iii) The system idle probability p 00 can be expressed using P 0 : We observe from Lemma 3 that the waiting times of individual customers W 1 and sharing customers W 2 as well as the probability of being idle p 00 are determined by W 2s and P 0 , which only depend on p 0j : Hence, the derivation of W 1 , W 2 , and p share is reduced to deriving p 0j for j ¼ 0, 1, :::: If an arriving sharing customer joins the queue with no unpaired sharing customer, she will wait there as an unpaired sharing customer.Later, she either gets paired upon arrival of the next sharing customer, or she receives service alone before the next sharing customer arrives.We call the period this sharing customer waits in the queue as an unpaired sharing customer, pairing period.In Figure 1, in the beginning of this pairing period, the system leaves row 0, say from state ð0, jÞ, and enters state ðj þ 1, j þ 1Þ upon the arrival of a sharing customer.At the end of this pairing period, the system returns to row 0 when the next sharing customer arrives or ðj þ 1Þ service completions occur, whichever happens first.During this paring period, there may be individual arrivals.These individual arrivals are independent of the current system status and have no impact on the duration of this period.They only alter the state in row 0 to which the system returns, in the MC.
Lemma 4. If the pairing period starts from state ð0, jÞ, it ends at state ð0, kÞ with probability We next focus solely on the time periods for which the system has no unpaired sharing customers, i.e., the states ð0, jÞ for j ¼ 0, 1, :::, and ignore the pairing periods.This forms a one-dimensional MC with transition probabilities derived in Lemma 4. Figure 2 illustrates the transition rates from state ð0, jÞ to other states in this one-dimensional MC.Let p j , j ¼ 0, 1, :::, denote the steady state probability distribution of this one-dimensional MC.Note that this transformation maintains the fraction of time the system spends in state ð0, jÞ out of the total time the system spends in row 0. Hence, we have With the transition probabilities in Lemma 4, we can write the balance equations of the one-dimensional MC in Figure 2 for all states ð0, jÞ p k Á q kj for j ¼ 1, 2, :::: where q kj is from (8).Further, we have the normalization condition Then, the expected waiting time of individual and sharing customers and the sharing customers' probability of sharing service (W 1 , W 2 , and P share ) can be derived using the following algorithm.

Insights
In this section, we use Algorithm 1 to generate insights on (i) the sharing customers' expected waiting time W 2 , (ii) the sharing customers' probability of sharing service p share , and (iii) the difference between sharing and individual customers' waiting times W 2 À W 1 , as functions of the proportion of individual customers in the whole customer base k 1 =K where K ¼ k 1 þ k 2 : These insights will help us understand the social welfare maximization problem in Sec. 4.
The average waiting times of individual and sharing customers W 1 and W 2 are determined by the When the total workload grows, individual customers will see a more congested system upon arrivals, so their expected waiting time W 1 will increase.Similarly, the expected waiting time of sharing customers who cannot be paired at arrivals increases, because of a more congested system upon arrivals.Although other sharing customers' expected waiting time can potentially be shortened by sharing service with a fellow sharing customer in the queue, the reduction in waiting time cannot offset the increase caused by the growing total workload.Thus, W 2 increases with k 1 .Figure 3(a), where W 2 is plotted as a function of k 1 =K, confirms the above intuition.Note that W 2 will approach 1 when the total workload ðK þ k 1 Þ=2l approaches one, which may only happen when K ! 1 and k 1 !2l À K: The sharing customers' probability of sharing service p share increases with (i) the arrival rate of sharing customers k 2 , and (ii) the average waiting time of sharing customers W 2 .For the same sharing customers' expected waiting time W 2 , higher sharing customers' arrival rate k 2 leads to higher probability of sharing service p share .Similarly, for the same arrival rate of sharing customers k 2 , the longer average waiting time of sharing customers W 2 leads to a higher probability of sharing p share .When k 1 increases, the arrival rate of sharing customers k 2 ¼ K À k 1 decreases, and the average waiting time of sharing customers W 2 increases, as shown in Figure 3(a).In Figure 3(b), we plot the probability of sharing service p share as a function of k 1 =K: The change in p share depends on whether the decreasing k 2 or the increasing W 2 is the dominant factor as k 1 increases.When K < 1, the decreasing k 2 is the dominant factor, hence the probability of sharing service p share decreases with k 1 .Note that when k 1 increases to K < 1, i.e., k 2 decreases to zero, from Lemma 1(ii), we have p share ¼ 0. However, when K ! 1, the increasing W 2 is the dominant factor, hence the probability of sharing service p share increases with k 1 .Note that, for K ! 1, the total workload ðK þ k 1 Þ=2l may approach one when k 1 !2l À K, where all customers' expected waiting time approaches 1: In this case, the sharing customers' expected waiting time is sufficiently long for the system to pair up all sharing customers, i.e., p share ¼ 1.
The difference between sharing and individual customers' waiting times W 2 À W 1 is also determined by two factors: (i) the total workload ðK þ k 1 Þ=2l, and (ii) the probability of sharing service p share .Recall from Lemma 2 that W 2 À W 1 0: When k 1 increases from zero, the total workload increases from K=2l, and the expected waiting times of both customer types increase.However, the increase in the total workload affects sharing customers' expected waiting time less significantly than individual customers', because sharing service can potentially cut their waiting time and the probability of sharing service p share does not decrease much in the K < 1 case, or even increases in the K ! 1 case; see, e.g., Figure 3(b).Thus, W 2 À W 1 decreases with k 1 at first.In the K < 1 case, when k 1 approaches K, the probability of sharing service p share decreases to zero (see, e.g., Figure 3b), where the sharing customers' expected waiting time becomes the same as the individual customers', i.e., W 2 À W 1 !0: In the K ! 1 case, when k 1 grows large, the probability of sharing service p share increases to one (see, e.g., Figure 3b).In this situation, the waiting time of both types of customers approaches 1, and the reduction in sharing customers' waiting time due to service sharing grows to 1, i.e., lim ðKþk 1 Þ=2l!1 ðW 2 À W 1 Þ ¼ À1: Figure 3(c), where W 2 À W 1 is plotted as a function of k 1 =K, confirms the above intuition.
We summarize the above insights as observations.
(i) The expected waiting time of sharing customers W 2 is increasing in k 1 .(ii) The sharing customers' probability of sharing service p share decreases in k 1 if K < 1, and increases in k 1 if K !1: (iii) The difference between sharing and individual customers' waiting times W 2 À W 1 is decreasing in k 1 when k 1 is small; and it may become an increasing function of k 1 when K < 1 and k 1 is close to K.

Social welfare maximization
In this section, we carry out queueing economic analysis of our service models with a focus on the social welfare, which is defined as the total service reward less the sum of total waiting and hassle costs.We aim to compare the three models (i.e., M=M=1, M=M ½2 =1 and hybrid) regarding the maximum social welfare and the corresponding optimal arrival rates under the assumption that the firm can fully control customer behavior.Note that the maximum social welfare provides an upper bound on the firm's profitability, and the firm cannot extract more profit than that from customers.In Sec. 5, we will discuss how to induce the maximum profit using admission fees.
The queueing economic analysis of the M=M=1 model is available in Hassin and Haviv (2003).To be self-contained, we summarize related results in Appendix OA1.We next look into the M=M ½2 =1 and hybrid models.

M=M ½2 =1 model
To derive the maximum social welfare and the optimal arrival rate in the M=M ½2 =1 model, we first investigate a restricted M=M ½2 =1 model where we assume all arrivals join.Then, the total social welfare of a restricted M=M ½2 =1 model, denoted by S B ðkÞ, can be calculated as the arrival rate multiplied by individual social welfare, which is service reward R less the expected waiting cost cW B ðkÞ less the expected hassle cost hp share ðkÞ: Using results from Proposition 1, we derive the following.
When the arrival rate k increases from zero, the system is not congested, so the increase in total service reward dominates the increase in the expected waiting and hassle costs, and the social welfare S B ðkÞ tends to increase.When k grows larger, the increase in the expected waiting and hassle costs will eventually dominate the increase in total service reward, and the social welfare S B ðkÞ will decrease.Thus, S B ðkÞ is an unimodal function of k 2 ð0, 2lÞ, which can be verified using (13).
We next examine the original M=M ½2 =1 model under excessive arrival rate, especially when k approaches 2l: The firm may not admit all but the right number of customers to maximize the social welfare in this model because of the unimodality of S B ðkÞ: The following proposition summarizes this result.
Proposition 2. (Social Welfare Maximization-M=M ½2 =1) In the M=M ½2 =1 model, there exists an arrival rate cutoff where h B is the unique solution of ð2R À 4hÞh , so that (i) the optimal arrival rate is (

Hybrid model
Let S M ðk 1 , k 2 Þ denote the social welfare in the hybrid model with arrival rates ðk 1 , k 2 Þ: Given arrival rates ðk 1 , k 2 Þ, the total social welfare can be expressed as the sum of (i) the individual customers' total value, which is the individual customers' arrival rate times the difference between the service reward and the individual customers' expected waiting cost, and (ii) the sharing customers' total value, which can be derived similarly as sharing customers' total value in the M=M ½2 =1 model in ( 13).
For any total arrival rate K, the maximum social welfare is which is a two-variable optimization problem and may become difficult to solve directly.Similar to the M=M ½2 =1 model in Sec.4.1, we simplify this problem by first deriving the maximum social welfare under the condition that all customers use either individual or shared service and they do not balk; i.e., k 1 þ k 2 ¼ K: Under this condition, the two-variable optimization problem is reduced to a one-variable optimization problem.We refer to it as the restricted hybrid model.
Under the assumption that no customers balk, the social welfare S M ðk 1 , K À k 1 Þ as a function of k 1 can be written as the total arrival rate K multiplied by the service reward less the expected hassle cost less the expected waiting cost for any customer.
By a Golden-section search on k 1 2 0, K ½ , we can identify the restricted hybrid model's maximum social welfare and the corresponding optimal individual customers' arrival rate as follows.
S M ðKÞ ¼ max for any total arrival rate K > 0, respectively.
When the total workload K increases from zero, the system experiences light traffic, so the expected waiting times and the probability of sharing are relatively low.In this case, the arrival rate K determines the magnitude of the maximum social welfare S M ðKÞ: Thus, S M ðKÞ increases with K when K is small.When K is large, and the system is heavily loaded, the negative externality each extra customer brings to the system, i.e., waiting and hassle costs, dominates the increase in service reward.Then, no matter how customers are assigned to the two service types, the sum of expected waiting and hassle costs will increase.Thus, when K grows large, the maximum social welfare S M ðKÞ decreases in K. Figure 4(a), where S M ðKÞ is plotted as a function of arrival rate K for different hassle costs and representative R, l, and c, confirms the above intuition.We summarize it as an observation.
Observation 2. The restricted hybrid model's maximum social welfare S M ðKÞ is an unimodal function of the arrival rate K.
Using the maximum social welfare of the restricted hybrid model S M ðKÞ, we next derive the maximum social welfare of the original hybrid model S Ã M ðKÞ: Let K H denote the maximum point of the restricted hybrid model's social welfare S M ðkÞ on ð0, 2lÞ, i.e., K H ¼ argmax 0<k<2l S M ðkÞ: If the arrival rate is K < K H and because S M ðKÞ is an increasing function of K on ð0, KÞ, it is optimal to admit all customers.Then, S Ã M ðKÞ ¼ S M ðKÞ, and ðk Ã 1 ðKÞ, k Ã 2 ðKÞÞ ¼ ð k 1 ðKÞ, K À k 1 ðKÞÞ: If the arrival rate is K !K H , it may not be optimal for the firm to admit all customers, but only sufficient to obtain effective arrival rate Note that, under excessive arrival rate, i.e., K !K H , the optimal arrival rate is K H : Thus, we have In other words, to derive the maximum social welfare of the hybrid model, we maintain S M ðKÞ for K 2 ð0, K H Þ, and replace S M ðKÞ for We next compare the maximum social welfares of the M=M=1, M=M ½2 =1 and hybrid models, under the excessive arrival rate, i.e., K !2l: This focus matches several real-world situations.For example, the service requests overwhelm the capacity of ridehailing firms during afternoon peak hour, on rainy days, after popular sports events, etc.We note that although the comparison is made for excessive demand, our analytical procedure is also applicable to more general cases.
In Figure 5, we plot the maximum social welfare and the corresponding optimal arrival rates in all three models, and the optimal customer composition in the hybrid model, as functions of the hassle cost h, for the case R ¼ 10, l ¼ 1, and c ¼ 1. (The values of R, l, and c are representative and the insights and guidelines we develop below hold for other values of R, l, and c.)Note that in the M=M=1 model, customers do not share service, so the measures in M=M=1 model stay constant regarding the hassle cost h.
It is intriguing to discover a tipping point in the hybrid model, i.e., h ¼ 4.76 in Figure 5.When hassle cost h decreases to the tipping point, there is a significant improvement in both the maximum social welfare's increase rate (e.g., Figure 5a) and the optimal arrival rates (e.g., Figure 5b) in the hybrid model.Furthermore, the optimal customer composition in the hybrid model immediately switches from more than 20% customers using individual service to all using shared service (e.g., Figure 5c).
The tipping point separates two operational regions.When hassle cost is relatively low, e.g., h 4:76 in Figure 5(a), the hybrid model under the optimal customer behavior operates the same as the M=M ½2 =1 model, and generates greater social welfare than the M=M=1 model.We call this region the Bulk-Service (BS) region.When hassle cost is relatively high, e.g., h > 4.76 in Figure 5(a), the hybrid model provides greater social welfare than the other two models.We call this region the Hybrid-Service (HS) region.Overall, the hybrid model provides (weakly) higher social welfare and is more robust than the other two models.The intuition is that the hybrid model offers both individual and shared service options to customers, which provides the firm with more flexibility than the other two models.If the firm only assigns customers to individual (or shared) service option, it is effectively an M=M=1 (or M=M ½2 =1) model.Then, for any hassle cost, the hybrid model should do at least as good as the other two models.
When hassle cost is low, the upside of shortening waiting time by pooling customers dominates the downside of an increasing hassle cost.Therefore, in the hybrid model, it will be optimal to assign all customers to shared service.This makes the hybrid model operate similar to an M=M ½2 =1 model.In the asymptotic case, when hassle cost approaches infinity, the firm may not want customers to share, since even under a small probability of sharing, the expected hassle cost is enormous and the social welfare will become negative.In this case, it would be optimal to assign all customers to individual service in the hybrid model, which makes it run exactly like an M=M=1 model.When the hassle cost is in an intermediate range, the firm prefers the hybrid model, which strikes a delicate balance between the average waiting and hassle costs.Moreover, we discover that in the HS region, the firm assigns homogeneous customers to differentiated services (i.e., individual and sharing services) to increase social welfare, countering the conventional wisdom that offering differentiated services is beneficial only to heterogeneous customers.
In both operational regions, reducing hassle cost improves social welfare.However, one unit deduction in hassle cost leads to different social welfare improvement in the two regions.For example, in Figure 5(a), if we reduce hassle cost by 1.24 from 6 to 4.76 in the HS region, the maximum social welfare of the hybrid model increases by 0.2 from 4.70 to 4.90, but if we reduce hassle cost by 0.76 from 4.76 to 4 in the BS region, the maximum social welfare of the hybrid model increases by 0.87 from 4.90 to 5.77.In the BS region, it is optimal for the firm to offer only the shared service to customers.In this case, all customers are sharing customers, and reducing hassle cost will increase the maximum social welfare to the full extent.In the HS region, it is optimal for the firm to assign only a fraction of customers to the shared service and assign the rest to the individual service (see Figure 5c).In this case, a decreasing hassle cost affects only those sharing customers, so the magnitude of improvement in social welfare is not as high as that in the BS region.
When hassle cost decreases, it is optimal for the firm to serve more customers, i.e., the optimal arrival rates of the hybrid model increases in Figure 5(b).Here is the intuition.When sharing service becomes less costly, the system can afford to handle more customers, and improve social welfare, under the condition that more customers use the shared service (see Figure 5c).It is noteworthy that even under excessive arrival rate, the M=M ½2 =1 model may not be optimal when hassle cost is high.It raises the alarm to firms that only offer the shared service option.It is only when hassle cost is sufficiently low that the firm may resort to M=M ½2 =1 model entirely and avoid operational complexity; otherwise, it is better to implement the hybrid model.
In both operational regions, the firm has incentives to reduce hassle cost by initiatives such as limiting the number of passengers in one shared ride, optimizing the matching algorithm to minimize customers' detour time, and providing customers better instruction on the pickup location.Moreover, it is vital for the firm to reduce the hassle cost to the tipping point to make service sharing a social welfare maximizing "mass action."(It is used in Sociology for situations in which a large number of people behave simultaneously in similar ways individually and without coordination.) To explain the existence of the tipping point, we plot the maximum social welfare of the restricted and original models in Figure 6(a)-(e) and 6(f)-(j), respectively, as functions of the total arrival rate K for five representative hassle costs h ¼ 4, 4.76, 4.77, 5, and 6.Recall from Secs.4.1 and 4.2 that the manner in which the maximum social welfare of the original model SðKÞ depends on that of the corresponding restricted models SðKÞ: For any arrival rate K smaller than the maximum point of the restricted model's maximum social welfare K, i.e., K K, the maximum social welfare of the original model is the same as that of the restricted model, i.e., SðKÞ ¼ SðKÞ: Otherwise, for K > K, we have SðKÞ ¼ Sð KÞ: Then Figure 6(f)-(j) are generated based on Figure 6(a)-(e).
When hassle cost is very high, the hybrid model under the optimal customer behavior operates the same as an M=M=1 model.When hassle cost decreases, for example from h ¼ 6 to 4.77, the optimal restricted hybrid model gradually deviates from the optimal M=M=1 model, and the optimal arrival rate K H in the hybrid model increases continuously from 0.71 to 0.99 (see Figure 6c-e).When hassle cost decreases to h ¼ 4.77 (see, e.g., Figure 6c), the restricted hybrid model's optimal arrival rate is K H ¼ 0:99, where the maximum value is only slightly greater than that of the restricted M=M ½2 =1 model.Moreover, the restricted hybrid model's maximum social welfare becomes almost flat on the interval 0:99, 1:18 ½ : On this interval, when the total arrival rate K increases, the positive surplus brought in by these extra customers balances off the negative externality, they impose on the service system.Once the hassle cost reaches h ¼ 4.76 (see Figure 6b), the optimal arrival rate switches to K H ¼ 1:18 right away, and it is optimal to operate the restricted hybrid model the same as a restricted M=M ½2 =1 model.This switch is immediate, unlike the gradual transformation from M=M=1 model to the hybrid model when the hassle cost decreases from 6 to 4.77 in Figure 6(c)-(e).As the hassle cost continues to decrease below h ¼ 4.76, the restricted hybrid model continues to operate the same as a restricted M=M ½2 =1 model.

Profit maximization
In this section, we investigate how a ride-hailing firm can maximize its profit by setting admission fees.In an unobservable queue, it is a known result that a monopoly firm will set an admission fee that leaves customers no surplus at equilibrium (see, e.g., Chapter 1.3 in Hassin & Haviv, 2003).This enables us to derive the profit-maximizing admission fees by taking the maximum social welfares of the three models (i.e., M=M=1, M=M ½2 =1 and hybrid) in Sec. 4 as the target maximum profits.
Similar to Sec. 4, for the M=M ½2 =1 and hybrid models, we start the discussion from the restricted models.Once the profit-maximizing admission fees in the restricted models are characterized, we use these results to propose admission fees in the original models.The profit-maximizing admission fee in the M=M ½2 =1 model is studied in Sec.5.1.In Sec.5.2, we discuss customers' equilibrium behavior under any admission fees in the restricted hybrid model.Then, in Sec.5.3, we propose a set of admission fees for the original hybrid model and demonstrate that they induce the target maximum profits.Finally, in Sec.5.4, we compare the profit-maximizing admission fees in the three models and provide insights.The M=M=1 model's profit-maximizing fee is in Appendix OA1.

M=M ½2 =1 model
The firm may impose an admission fee f > 0 to maximize its profit.Recall that customers cannot observe the real-time queue length at arrivals, so they rely on the expected queue length in deciding on whether to join or to balk.Customers' expected net utility U B under arrival rate k < 2l and admission fee f is the difference between the service reward R and the sum of three costs: (i) admission fee f, (ii) expected waiting cost, which is the marginal waiting cost c multiplied by the expected waiting time W B ðkÞ, and (iii) expected hassle cost, which can be derived as hassle cost h multiplied by probability of sharing service p share ðkÞ: where the second equation is from Proposition 1 (ii) and (iii).
We first consider the restricted M=M ½2 =1 model where all customers join.Here the firm can set any admission fee to extract profit from customers who cannot balk, even if U B ðK, f Þ < 0: However, to ensure individual rationality in the original M=M ½2 =1 model, which will be discussed next, we only consider admission fees satisfying U B ðK, f Þ !0: Then, the restricted M=M ½2 =1 model's profitmaximizing fee f B is identical to the customers' net utility when they all join and there is no admission fee, i.e., f B ¼ U B ðK, 0Þ, which leaves zero surplus to customers, i.e., U B ðK, f B Þ ¼ 0, and maximizes the firm's gain.The firm's maximum profit is identical to the social welfare in (13), i.e., S B ðKÞ ¼ K f B , which is an unimodal function of K with the maximum point k B given in Proposition 2.
Next, we discuss the customers' equilibrium behavior in the original M=M ½2 =1 model.Because customers are not aware of the real-time queue length, their strategy can be described by each customer's joining probability q 2 0, 1 ½ , which also represents the proportion of customers who will join the queue.Let q e B ðf Þ denote the equilibrium joining probability under admission fee f.Clearly, the equilibrium joining rate k e B ðf Þ ¼ q e B ðf ÞK must not exceed 2l; i.e., k e B ðf Þ < 2l: Proposition 3. (Equilibrium Behavior-M=M ½2 =1) For any admission fee f and hassle cost h, customers' equilibrium arrival rate in the M=M ½2 =1 model is where Finally, we consider the profit-maximizing admission fee in the original M=M ½2 =1 model.From Proposition 3 and the standard arguments as in Hassin and Haviv (2003), we have that: For the arrival rate K < k B , the firm prefers all customers to join, and the profit-maximizing admission fee is the customers' net utility when they all join and there is no admission fee, i.e., f B ¼ U B ðK, 0Þ: For the arrival rate K !k B , the firm charges the customers' net utility when only k B join and there is no admission fee, i.e., f B ¼ U B ð k B , 0Þ, as the profitmaximizing admission fee to induce the optimal equilibrium arrival rate k B : This is summarized as a proposition.
Proposition 4. (Profit-Maximizing Fee-M=M ½2 =1) The firm's profit-maximizing admission fee is where k Ã B and k B are given in Proposition 2.

Restricted hybrid model-equilibrium behavior
Let f 1 and f 2 denote the admission fees imposed for individual and shared services, respectively.Similar to (15) in the M=M ½2 =1 model, individual and sharing customers' expected net utilities can be expressed using and Similar to Sec. 4, we start our discussion with the restricted hybrid model, where all arrivals join.Recall that customers cannot observe the real-time queue length.Because they either use individual or shared service, their strategy can be described by the probability a customer uses individual service q 1 .Then, with probability 1 À q 1 , a customer uses shared service.Let q e 1 denote the equilibrium probability under admission fees f 1 and f 2 .Then, the equilibrium joining rates to individual and shared service are k e 1 ¼ q e 1 K and k e 2 k e 2 !l, W 1 and W 2 are infinitely large and U 1 , U 2 < 0, which cannot hold as an equilibrium.Similar to Sec. 5.1, we only consider f 1 and f 2 such that U 1 , U 2 !0 to ensure individual rationality in the original hybrid model, which will be discussed in Sec.5.3.Proposition 5. (Equilibrium Behavior-Restricted Hybrid Model) The equilibrium arrival rate to individual service is Proposition 5 gives necessary and sufficient conditions of customers' equilibrium behavior in the restricted hybrid model.As we will show next, multiple equilibria may exist.
Proposition 5 shows that customers' equilibrium behavior can be determined by (i) comparing 2 < l and (ii) ensuring non-negativity of the individual utility of the corresponding customer stream.The second criterion can be easily fulfilled by setting small enough (even negative) admission fees f 1 and f 2 .We next look into hp share þ cðW 2 À W 1 Þ as a function of k 1 .
If K < 1, when the hassle cost h is low compared to marginal waiting cost c, the shape of W 2 À W 1 (see, e.g., Observation 1(iii)) determines the shape of hp share þ cðW 2 À W 1 Þ as a function of k 1 : an initially decreasing and then increasing function (see, e.g., the h ¼ 0 curve in Figure 7a).When h is high compared to c, the shape of p share (see, e.g., Observation 1(ii)) determines the shape of hp share þ cðW 2 À W 1 Þ : a decreasing function of k 1 (see, e.g., the h ¼ 6 curve in Figure 7a).Similar intuition applies for the K ! 1 case.When h is low, hp share þ cðW 2 À W 1 Þ is a decreasing function of k 1 (see, e.g., the h ¼ 0 curve in Figure 7b), and when h is high, it may become an unimodal function of k 1 (see, e.g., the h ¼ 6 curve in Figure 7b).Now, the shape of hp share þ cðW 2 À W 1 Þ as a function of k 1 is clear.Recall from Proposition 5 that, when the non-negative utility constraints are lifted by setting small enough (even negative) admission fees, customers' equilibrium behavior is Using this result, we next describe examples of two categories of six possible equilibrium behaviors.

Unique Equilibrium:
1 It is not possible to have three equilibria, nor two equilibria between 0 and K, i.e., 0 < k e1 1 , k e2 1 < K: Note that while we use specific h values (i.e., h ¼ 0 or 6) for demonstration, the above equilibrium behavior patterns likewise hold for other values of h.
There are certain values of k 1 that cannot be induced by any admission fees f 1 and f 2 , especially when hp share þ cðW 2 À W 1 Þ is an increasing function.For example, when K ¼ 1:25 and h ¼ 6 in Figure 7(b), there is no f 1 À f 2 that can induce k e 1 =K 2 ð0, 0:6Þ: When K ¼ 0:75 and h ¼ 0 in Figure 7(a), there is no f 1 À f 2 that can induce k e 1 =K 2 0:8, 1Þ: ½ Furthermore, we note that for K ¼ 0:75 and h ¼ 0 in Figure 7(a), to induce k e 1 =K 2 0, 0:8Þ, ½ we need to use f 1 À f 2 < 0: This may not be realistic, because in practice, ride-hailing firms usually price individual service higher than shared service.

Hybrid model-profit-maximizing admission fees
To maximize profit in the restricted hybrid model with total arrival rate K, we propose to charge admission fees ð f 1 ðKÞ, f 2 ðKÞÞ identical to individual and sharing customers' utilities under the optimal arrival rates and no admission fees, i.e., f 1 ðKÞ ¼ U 1 k 1 ðKÞ, K À k 1 ðKÞ, 0, 0 À Á and f 2 ðKÞ where k 1 ðKÞ ¼ argmax 0 k 1 K S M ðk 1 , K À k 1 Þ is the optimal individual customers' arrival rate in the restricted hybrid model, as obtained in (14).Figure 8 demonstrates our idea for choosing the profit-maximizing admission fees.Figure 8(a) shows that the optimal (equivalently, profit-maximizing) arrival rate is k 1 ¼ 0:26 for the K ¼ 0:9, l ¼ 1, R ¼ 10, c ¼ 1, and h ¼ 4 case.Figure 8(b) illustrates the utilities of individual and sharing customers under no admission fees.At k 1 ¼ 0:26, the utilities of individual and sharing customers are U 1 ¼ 7:52, and U 2 ¼ 5:09: If admission fees ðf 1 , f 2 Þ ¼ ð7:52, 5:09Þ are imposed, as shown in Figure 8(c), from Proposition 5(ii), we know that the profitmaximizing arrival rates ð0:26, 0:64Þ will be induced.Thus, the profit-maximizing admission fees must be ð f 1 ðKÞ, f 2 ðKÞÞ ¼ ð7:52, 5:09Þ: Next, under the proposed profit-maximizing admission fees ð f 1 ðKÞ, f 2 ðKÞÞ, we apply Proposition 5 for any total arrival rate K and numerically check (i) if the desirable joining rates ð k 1 ðKÞ, K À k 1 ðKÞÞ is an equilibrium, and (ii) if there are other equilibria.
The numerical results show that when hassle cost h is low, i.e., h < 5, our proposed admission fees ð f 1 ðKÞ, f 2 ðKÞÞ correctly induce the maximum profit.However, when h is high, i.e., h ! 5, the proposed admission fees may not induce the maximum profit in the restricted hybrid model.For example, Figure 9 shows that for h ¼ 6, our proposed admission fees ð f 1 ðKÞ, f 2 ðKÞÞ, although correctly induce the target maximum profit for a wide range of potential arrival rate K, does not induce the maximum profit for a small range of K 2 1:12, 1:26 ½ , and instead the resulting equilibria have lower profit.This situation reflects the double-equilibria case 2.3 in Sec.5.2.From our exhaustive numerical experiments, we note that this case only happens when K > K H and h is high.Recall that, in the original hybrid model, when K > K H , it is optimal (or profit-maximizing) to admit only K H : Therefore, existence of multiple equilibria will not be a concern in the original hybrid model.
We now formalize our profit-maximizing admission fee structure.Let ðf Ã 1 , f Ã 2 Þ denote the admission fee structure that induces the profit-maximizing arrival rate ðk Ã 1 , k Ã 2 Þ: To induce the profit-maximizing arrival rates in the original hybrid model, we propose the following admission fees based on the proposed admission fees for the restricted hybrid model ð f 1 ðKÞ, f 2 ðKÞÞ : From our understanding of the restricted hybrid model, for K K H , the above proposed admission fees will induce the profit-maximizing arrival rates to individual and shared services.Further, for K > K H , the profit-maximizing admission fees ð f K H ÞÞ induce the arrival rate K H : Then, the situation that the proposed profit-maximizing admission fees may induce another equilibrium other than the desirable one will not happen.
We next show that for K > K H , the admission fees K H ÞÞ, the profit-maximizing arrival rates when the total arrival rate is K H : In Figure 10, for the l ¼ ¼ R ¼ 10, c ¼ 1, and h ¼ 5 case, we plot the utility of individual and sharing customers in the restricted hybrid model for total arrival rate K H ¼ 0:8254 and 0.85.We see that in the restricted hybrid model with K ¼ 0:85, if the system started with all customers using shared service, because individual service provides higher utility than shared service, some customers have incentive to switch to individual service.When the arrival rate to individual service reaches k 1 ¼ 0:57, no more sharing customers are willing to switch to shared service.However, under this situation, all customers have negative utility.Once the assumption that all customers join is lifted, some customers will balk and obtain zero utility, which is still better than negative utility.Thus, the effective total arrival rate will decrease.Through comprehensive numerical tests, we discover that for 8K > K H ¼ 0:8254, the total arrival rate decreases until reaching K H , where the admission fees ð f 1 ð K H Þ, f 2 ð K H ÞÞ correctly induce the profitmaximizing arrival rates ð k 1 ð K H Þ, K H À k 1 ð K H ÞÞ: 5.4.Profit-maximizing admission fees comparison: Hybrid vs. M=M ½2 =1 vs. M=M=1 In this section, we compare the profit-maximizing admission fees of the hybrid model with those of Note that in the M=M=1 model, because customers do not share service, any change in hassle cost does not affect the profit-maximizing admission fee.Similar to Sec. 4, we have two operational regions separated by the tipping point, i.e., h ¼ 4.76.In the Hybrid-Service (HS) region, the individual (or shared) service is priced higher than the M=M=1 (or M=M ½2 =1) model.When the hassle cost h decreases, the profit-maximizing admission fee for individual service increases while that for shared service decreases.This is to make the shared service more attractive, so customers will choose it with a higher probability (equivalently, use individual service with a lower probability); see Figure 5(c).Interestingly, in the HS region, the firm can improve its profitability by inducing homogeneous customers to use differentiated services.It may achieve this by charging different admission fees for different services.
Once the hassle cost h decreases to the tipping point, the system enters the Bulk-Service (BS) region.The profit-maximizing admission fee of the shared service converges to that of the M=M ½2 =1 model, while the profit-maximizing admission fee of the individual service leaps upward.Note that, the dash-dot curve in the BS region in Figure 11 indicates a lower bound for the profit-maximizing admission fee for the individual service.The firm can choose any higher admission fees to prevent customers from selecting the individual service.In this BS region, the firm's operations are greatly simplified.Only shared service is offered, and the profit-maximizing admission fee is the same as that in   the M=M ½2 =1 model.Meanwhile, individual service can either be priced aggressively high or even not be offered to customers at all.

Summary
In this paper, we examine two car-sharing service models offered by ride-hailing firms.For the sharing-only model, we provide closed-form queueing economic analysis.For the hybrid model, which is more challenging to analyze, we establish a new Markov chain structure to track the position of the unpaired sharing customer in the queue and the number of drivers needed to clear the current queue.We provide an efficient exact algorithm to derive all the essential service level measures, which in turn enables the queueing economic analysis of the hybrid model.
We take the perspective of a ride-railing firm that wants to determine the optimal customer arrival rates that create the maximum social welfare, and subsequently, choose admission fees that extract the maximum profits.We compare the two car-sharing models and the no-sharing individual service regarding the maximum social welfare and the corresponding optimal arrival rates.We observe that the hybrid model (weakly) dominates the other two models regarding social welfare, and its maximum social welfare increases when hassle cost decreases.More interestingly, we discover a tipping point in the hybrid model under the optimal customer behavior.When the hassle cost drops to this point, the optimal customer behavior changes dramatically from some customers using shared service to all using it, and the rate at which a decreasing hassle cost increases the maximum social welfare of the hybrid model significantly increases.This result points out the importance for the ride-hailing firm to reduce hassle cost of sharing customers to the tipping point, if it is to make service sharing a social welfare maximizing mass action.
Moreover, when the hassle cost is higher than the tipping point, the ride-hailing firm can obtain greater social welfare by offering differentiated (individual and shared) services to homogeneous customers.This result is counter-intuitive because differentiated services are thought to be advantageous only when customers are heterogeneous.Our result complements Hassin's (1985) observation that priorities can be used to improve social welfare or profit not only for heterogeneous customers but also for homogeneous customers.
Lastly, we study the firm's profit-maximizing admission fees.We propose a profit-maximizing admission fee structure and show that it correctly induces the maximum profit.When the hassle cost is higher than the tipping point, the ride-hailing firm should charge different admission fees for various services to maximize profit.Once hassle cost reaches the tipping point, the firm can significantly simplify its operations by only offering the shared service, while the individual service can either be (i) priced very high so that no customers will use it or (ii) closed as an extreme measure.
Although our research offers interesting and insightful implications for ride-hailing services, it has some limitations.Potential avenues for future research involve considering multiple origins and destinations, examining the effect of customer heterogeneity in terms of willingness to share that is captured by the hassle cost, and modeling multiple servers to consider multiple taxi stands in the same location, e.g., an airport.

Figure 2 .
Figure 2. One-dimensional Markov chain ð0, jÞ of the hybrid model in the time period without unpaired sharing customers.

Figure 5 .
Figure 5. (a) Maximum social welfare S M ðKÞ and (b) corresponding optimal arrival rates for (i) M=M=1, (ii) Bulk service, and (iii) hybrid models; and (c) optimal customer composition for hybrid model, as functions of h, under excessive arrival rate K ¼ 2l and the condition R ¼ 10, l ¼ 1, and c ¼ 1.

Figure 8 .
Figure 8. Target maximum profit, utilities of individual and sharing customers under no and profit-maximizing admission fees, as functions of k 1 , under the condition K ¼ 0:9, l ¼ 1, R ¼ 10, c ¼ 1, and h ¼ 4.

Figure 10 .
Figure 10.Individual utilities of individual and sharing customers under the admission fees ð f 1 ð KÞ, f 2 ð KÞÞ for K !K, as functions of k 1 =K, under the condition l ¼ 1, R ¼ 10, c ¼ 1, and h ¼ 5.

Figure 11 .
Figure 11.Profit-Maximizing Admission Fees for (i) M=M=1, (ii) Bulk service, and (iii) hybrid models, as functions of h, under excessive arrival rate K ¼ 2l and the condition R ¼ 10, l ¼ 1, and c ¼ 1.