A re-examination of experience service offering and regular service pricing under profit maximization

European Journal of Operational Research 254 (2016) 907–915

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastics and Statistics

A re-examination of experience service offering and regular service pricing under profit maximization Zhaotong Lian a, Xinhua Gu a, Jinbiao Wu b,∗ a b

Faculty of Business Administration, University of Macau, Macau SAR, China School of Mathematics and Statistics, Central South University, Changsha 410075, Hunan, China

a r t i c l e

i n f o

Article history: Received 29 August 2015 Accepted 5 May 2016 Available online 19 May 2016 Keywords: Queueing Customer experience Pricing Economic intuitions Managerial insights

a b s t r a c t A firm may offer an experience service free of charge to attract more customers to buy its regular service. This paper provides an economic analysis for interactions between the capacity-constrained firm and its waiting-averse customers, generating certain managerial insights. Free services should be sped up to move customers onto the paid service if it is idle, or slowed down to hold onto customers and avoid exacerbating congestion in the paid service if it is busy. A lower price should be charged for the regular service as compensation to customers for service delay if more of them buy that service. When more customers arrive for experience services, a greater price reduction should be offered to attract them into the regular service if it becomes more congested.

1. Introduction While businesses increasingly treat the improvement of customer experience as a competition differentiator, academics remain inactive about this subject providing little formal research so far. In fact, like “innovation” and “design”, “customer experience” still has no commonly-agreed definition (Richardson, 2010). Moreover, controversies exist among business analyses over the effects of customer experience management. According to a survey study (Allen, Frederick, & Barney, 2005), 80 percent of businesses state that they offer a “great customer experience” but only 8 percent of customers say that they feel the same way. Although some commentators still concern the actual functioning of a product or a service by pointing out that there is no substitute for its acceptable quality or reasonable price, others emphasize the importance of sensory experience in affecting consumer behavior by pointing to the fact that non-monetary burdens (e.g., a disorganized store or a long line at the checkout) can outweigh customers’ consideration of prices. Companies, if insensitive to customer experience while focusing only on low-price competition, may actually decrease the value of their offerings (Berry, Carbone, & Haeckel, 2002). Thus companies are expected to perform best if able to integrate both functional and emotional benefits in their offerings. Along this line, our paper addresses the question of how service systems can strike a proper balance between those two ben∗

Corresponding author. Tel.: +86 15973135021. E-mail address: [email protected] (J. Wu).

http://dx.doi.org/10.1016/j.ejor.2016.05.023 0377-2217/© 2016 Elsevier B.V. All rights reserved.

© 2016 Elsevier B.V. All rights reserved.

efits offered to customers. Providing free experience service is an increasingly popular strategy in practice. Examples include online games, data communications, call centers etc. (Chu & Zhang, 2011; Edvardsson, Enquist, & Johnston, 2005; Froehle & Roth, 2004). The functionality of services may be observably embodied in their prices or has been known to experienced customers (who have tried these services before). Information on the unobserved quality of services can be conveyed to inexperienced customers through free experience service. This business strategy is often useful in managing the emotional component of customers’ experience and enhancing their perception of service received as they learn more about quality (Crawford & Shum, 2005; Osborne, 2005). Firms with more unobserved characteristics offer more such experience service, and customers rightfully interpret it as a signal of their higher quality. It is observed that informative advertising or experience service is unlikely to affect experienced consumers but can impact inexperienced consumers (Ackerberg, 2003). Our work considers a typical service firm that provides potential customers with paid regular service and free experience service. This practice is good for business by attracting more customers, especially when there are a limited number of informed (or experienced) customers. This paper presents a queueing-theoretic formulation of service systems with free experience service. In queueing systems, a capacity-constrained service provider is often faced with delaysensitive customers (Afèche & Mendelson, 2004). In our work all customers are assumed to be homogenous in valuation of service offerings, and each customer decides whether to join the queue or balk at it based on such valuation relative to the posted price

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and on the expected cost of waiting (Cachon & Harker, 2002; Van Mieghem, 20 0 0). A customer may try experience service first or buy regular service directly. After experiencing the free service, the customer with more information about the service offered may move on to the paid service or just leave the service system altogether. These decisions of customers are affected by their sensory experience and their aversion to waiting, among other things. Given the limited capacity, supplying experience service to some customers is likely to prolong the waiting time for other customers who want to purchase regular service. While longer waiting may drive away some customers, free experience service or discounted service prices can act as an attraction to keep other customers in waiting rooms. It is then natural to ask under what conditions the free experience service should be offered, what price should be charged for the regular service, and what rate of experience service should be maintained in order to maximize profits. This paper addresses those questions by analyzing equilibrium strategies of a monopolistic service firm. Such market structure is also assumed in Chen and Frank (2004). Our analysis is conducted for a Markovian single server queueing system (i.e., the M/M/1 queue) as the most suitable model for our purposes. Since our paper is an application of queueing theory to service firms, it is necessary to briefly review the related literature on queueing issues and other applications. This literature was initiated by Naor (1969) and Edelson and Hildebrand (1975) using a reward-cost structure to capture customers’ desire for service and their dislike for waiting. Private decisions are made not only by servers to maximize revenue or profit but also by customers to determine whether to join or balk. A growing number of papers have arisen later on as variants of early M/M/1 models to examine various queueing characteristics. Those subsequent extensions include: priorities (Gilland & Warsing, 2009; Hassin & Haviv, 1997), consumer heterogeneity (Mendelson & Whang, 1990), reneging and jockeying (Hassin & Haviv, 1994), schedules and retrials (Economou & Kanta, 2011), setup and closedown times (Burnetas & Economou, 2007; Sun, Guo, & Tian, 2010), unreliable servers and delaying repairs (Wang & Zhang, 2011, 2013), customer intensity and congestion (Anand, Fazil Pac, & Veeraraghavan, 2011), information and uncertainty (Guo & Zipkin, 2007; Hassin, 2007), etc. Analyses of queueing systems have also emerged in the form of monographs (see (Hassin & Haviv, 2003) for pricing with queues and (Stidham, 2009) for main approaches and associated results in broader areas). Our paper analyzes the optimal pricing of service firms by incorporating into queueing systems the free experience service, which may increase the service load and affect the joining-balking decision due to the resulting congestion and delay. Our queueing-theoretic treatment of customer experience is related to a recent study in the literature (Zhou, Lian, & Wu, 2014), but differs from it in three important aspects. First, providing free experience service is costly to firms, but such costliness is missing from the formulation of the firm decision problem in the previous study that maximizes revenue rather than profit. Our work considers the cost of free experience service to the firm as well as the cost of waiting to its customers, thus making analyses more realistic and precise. Second, distinguishing between informed and uninformed customers with different Poisson arrival rates, the previous study has to deal with the separate effects of such a distinction on firm pricing. To examine the combined effects of different types of customers on the same capacity-constrained server, we introduce the ratio of customers interested in experience service to all arriving customers with an overall Poisson rate. This new treatment offers us more flexibility in analyzing the impacts of changes in the ratio on firm pricing. Third, the previous study considers priorities given to informed customers to examine optimal firm decisions in various cases. In contrast, with this ratio in our work, there is no need to derive main results through priorities. Instead,

we can now flexibly treat the ratio as an underlying parameter to shed new light on related management issues: whether to offer free experience service and by how much; how to effectively promote regular service and at what price. Obviously, our extension of the previous study allows us to generate more insights into service pricing issues in a more precise manner. The rest of the paper proceeds as follows. Section 2 incorporates free experience service into a queueing model. Section 3 examines the steady-state distribution of the queueing system and associated measures of stationary queueing performance. Section 4 derives equilibrium strategies for the unobservable case. Section 5 analyzes the optimal price of regular service and the optimal rate of experience service offering. Section 6 presents a motivational example with numerical experiments. Section 7 contains concluding remarks, followed by a technical appendix. 2. Model formulation Consider a monopolistic firm that provides consumers with two kinds of services (paid regular service and free experience service). This setting of services operated by the firm is depicted in Fig. 1, and modeled as a Markovian single-server queueing system, with customers arriving at the rate of λ according to a Poisson process. The time of servicing is assumed to be exponentially distributed at the rate of μ for experience service and at the rate of μ1 for regular service, where μ ≥ μ1 . This kind of distribution describes the time between service events, which occur continuously and independently at a constant average rate (of μ or μ1 per unit of time). The mean duration of a service event is therefore shorter for the free service (with 1/μ units of time) than for the paid service (with 1/μ1 units of time). The rate μ is treated in our modeling as a choice variable for the firm: the greater this rate, the faster a customer is handled for free service and the shorter this type of service will last. If an arriving customer finds the server idle, he may try the experience service with probability β or instead buy the regular service directly with probability β (≡ 1 − β ). There are two interpretations of this probability distribution. First, β (or β ) can be viewed as the proportion of new (or member) customers to all arriving customers. Second, if allowing for possible free-riding by member customers, β (or β ) can be alternatively regarded as the proportion of customers using (or not using) the experience service to all arriving customers.1 After experiencing the free service, the customer may either buy the regular service with probability α or leave the service system with probability α (≡ 1 − α ). Such probability distribution can be thought of as the customer’s choice variable based on his perception of the experience service. The role of learning about service quality for consumption decision is not the focus in this paper, and can be further analyzed in future research. Arriving customers, if interested in services offered, will have to wait for service on an FCFS basis. For simplicity we assume that the Poisson process for customer arrivals and the two exponential

1 Since experience service is enjoyable but free of charge, some of informed customers can consume it again to increase their utility if at no cost. However, such free riding is costly not only to the firm (see Carlton and Chevalier, 2001 and Antia, Bergen, and Dutta, 2004 for other kinds of free riding) but also to those customers since their utility increase carries an implicit cost (i.e., waiting) even under no explicit cost (i.e., money). Since experience service is offered in a limited amount and since regular service, though not free of charge, may be supplied with no congestion, those informed customers are effectively faced with a tradeoff between the value of experience service and the cost of waiting for it. They may choose to buy regular service directly when seeing the free service section is too busy. Thus the capacity constraint on experience service can actually act as a device to reduce the problem of free riding, and we will not deal explicitly with this problem given that customers are usually impatient and averse to waiting.

Z. Lian et al. / European Journal of Operational Research 254 (2016) 907–915

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Fig. 1. Structure of the service system.

processes for the experience and regular service times are mutually independent. Also, we assume 0 < α ≤ 1 and 0 < β ≤ 1 to avoid trivial cases. 3. Steady state probabilities ln this section we present the stability condition of the queueing system, derive the steady state distribution of customers in the system, and provide some measures of service performance. Denote by τ n the epoch when the nth customer departs from the system after the completion of his service consumption. The generalized service time χ n is defined as the period from the epoch when his consumption starts to the epoch τ n of his departure. Note that χ n may represent i) the time of regular service he buys directly after his arrival, ii) the time of experience service he tries without moving on to regular service, or iii) the total times of experience service he tries first and regular service he then buys. If assuming χ n to be i.i.d. for n = {1, 2, }, we can use χ to denote χ n , whose distribution function is denoted by  (t) = P{χ ≤ t } and presented in the following proposition. Proposition 3.1. The distribution function of the generalized service time is given by μ t

β +αβ − αβ1+β¯ β − μβt −μ e μ1 e (t ) = 1 − β +αβ β μ1 − μ

(1)

The proof of this proposition is given in the Appendix. The residual waiting time ξ (i.e., the length of time between arriving at the server and starting to receive some service) is assumed to be exponentially distributed with the mean 1/λ. Then, the random variable ξ + χ measures the entire interval of time during which a typical customer stays in the server, with its distribution function (t) obtained below

(t ) = P{ξ + χ ≤ t } =

0

0

x

αβ β Define ρ = λ( β + μ1 + μ ) as an index for server congestion, which is positively affected by both the expected rate of customer αβ β arrivals λ and the mean length of the two services ( β + μ1 + μ ) . This length is a weighted sum of the average duration 1/μ1 of regular service and the average duration 1/μ of experience service, with the weights equal to the probability (αβ + β ) of regular service being sold to customers and the probability β of experience service being consumed for free by them.

Proposition 3.2. The queueing system is stable if and only if server congestion is not too serious in the sense of ρ < 1. The proof of this proposition is given in the Appendix. We will work with ρ < 1 in what follows. Naturally, the more congested the server, the less likely it is to have idle seats for customers. Hence, lim P {Nn = 0} is directly related to ρ , with n→∞

lim P {Nn ≥ 1} = 1 − ρ defined as the complementary likelihood of

n→∞

the opposite event. From these two probabilities and the two more others (1) and (2), it follows that

F (t ) = lim P {Tn ≤ t } = ρ (t ) + (1 − ρ )(t )

μμ1 − λ[(αβ + β¯ )μ + βμ1 ] −λt e (μ − λβ )[μ1 − λ(αβ + β¯ )] μ t λβμ1 (αβ + β¯ ) − 1 − e αβ +β¯ ¯ ¯ [μ(αβ + β ) − μ1 β ][μ1 − λ(αβ + β )] μt λβμ(αβ + β¯ ) + e− β . ¯ ¯ [μ(αβ + β ) − μ1 β ][μ1 − λ(αβ + β )] 

=

(2)

Define Tn = τn+1 − τn as the inter-departure time between customers n and n + 1, and Nn as the number of available seats under

(3)

Thus the expectation of the generalized inter-departure time T is calculated as

E[T ] =

λe−λ(x−y) d(y )dx

μμ1 = 1− e−λt (μ − λβ )[μ1 − λ(αβ + β¯ )] μ t λμ (αβ + β¯ )2 − 1 + e αβ +β¯ μ(αβ + β¯ ) − μ1 β μ1 − λ(αβ + β¯ ) μt β2 λμ1 − e− β . μ(αβ + β¯ ) − μ1 β μ − λβ

= P {Nn = 0}P {ξ + χ ≤ t } + P {Nn ≥ 1}P {χ ≤ t }.

= 1−

β¯ + αβ β + . μ1 μ

 t

P {Tn ≤ t } = P {Nn = 0}P {Tn ≤ t |Nn =0} + P {Nn ≥ 1}P {Tn ≤t |Nn ≥1}

n→∞

with the expected duration of service

E[χ ] =

idle capacity when customer n arrives. We have

0

+∞

t dF (t )

μμ1 − λ[(αβ + β¯ )μ + βμ1 ] λ(μ − λβ )[μ1 − λ(αβ + β¯ )] λβ (αβ + β¯ )(β¯ − β α¯ ) + . [μ(αβ + β¯ ) − μ1 β ][μ1 − λ(αβ + β¯ )]

(4)

The expected sojourn time of an arriving customer is the amount of time he is expected to spend in the service system before leaving. To better examine this variable, we describe the evolution of the system by means of a continuous-time Markov chain {C(t), N(t), t ≥ 0} with the state space S = {(0 )} ∪ {(0, n ), n ≥

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1, } ∪ {(1, n ), n ≥ 1}, where C(t) is defined below as a binary variable for a certain type of service provided at time t,



C (t ) =

0, 1,

if the server is providing experience service at time t , if the server is providing regular service at time t ,

and N(t) is the number of customers in the system (including those who have been receiving certain services) at time t. The steady-state distribution of the Markov chain and the related probability generating functions for |z| ≤ 1 are described in the Appendix. The resulting joint distribution is given below. Theorem 1. The stationary joint distribution between the server state and the queue length at an arbitrary moment has the following generating functions

λβ z2 (z − 1 )[λ(1 − z ) + μ1 ] π0 , f0 (z ) f1 (z ) − μμ1 β (α z + α¯ β¯ ) λz2 (z − 1 )[λβ¯ (1 − z ) + μ(αβ + β¯ )] P1 (z ) = π0 , f0 (z ) f1 (z ) − μμ1 β (α z + α¯ β¯ ) P0 (z ) =

(5)

where

λz(1 − z ) + μ(z − αβ ¯ ), π0 = 1 − ρ .

f 0 (z ) =

f1 (z ) = λz(1 − z ) + μ1 (z − β¯ ),

The proof of Theorem 1 is provided in the Appendix. By using the well-known matrix-geometric method (see Neuts, 1981), we obtain the joint distribution of customers and states in the system. The recursive structure of our model is derived in the Appendix, and some measures of queueing performance are given in the following. Corollary 1. (i) The probability of the system providing experience service is

P0 (1 ) =

βλ . μ

(ii) The probability that the system provides regular service is

P1 (1 ) =

(β¯ + αβ )λ . μ1

(iii) The expected queue length is

L = P0 (1 ) + P1 (1 ) =



  β αβ + β¯ λ + μ μ1

W = L/λ = −

λ . μμ1



2 W (2) ∂∂λ > 0, ∂∂λW2 > 0. 2 W (3) ∂∂α ≥ 0, ∂∂αW2 ≥ 0.

μ W W W (4) ∂∂β = 0 if α = αo (≡ 1 − μ1 ); ∂∂β < 0 if 0 ≤ α < α o ; ∂∂β > 0 if 2W ∂ α o < α ≤ 1. In addition, ∂β 2 ≥ 0.

This proposition is quite intuitive. Service delay W should be shorter with a greater service rate μ (or shorter service length 1/μ) but will become longer with a higher arrival rate λ. A longer delay W in service can also be caused by a higher proportion α of customers who buy regular service after trying free service. Moreover, there will be a longer delay if more customers try experience service first (with a higher β ) and if many of them subsequently buy regular service (α > α o ); the reverse occurs if α < α o . Furthermore, the delay function W is convex in λ, μ, α , and β , implying that service delay will be prolonged at an increasing rate with respect to customer arrivals, regular service proportions, and experience service proportions if the regular server is sufficiently busy. Service delay will be shortened yet at a decreasing rate with respect to the rates and proportions of experience service if the regular server is not too busy.

4. Equilibrium analysis In this section we identify equilibrium strategies of risk neutral customers. They maximize expected net benefits from service consumption by deciding on whether to join or balk. We assume that customers cannot observe the queue length upon arrival, and that a firm provides monopolistic services for those customers. While some customers may directly purchase regular service at a unified price p upon arrival, the firm offers experience service free of charge to attract more customers. They are allowed to try experience service before buying regular service, so they have a choice to make at the arriving epoch. Their choice is assumed to be irrevocable, in the sense that balking customers are prohibited from retrials while customers after entering regular service cannot renege on their purchase. Each customer has an equal valuation of consumed service amounting to R dollars, but pays a price p for ¯

β that R > p + c ( μ + αβμ+β ). 1 Suppose customers arrive according to an i.d.d. Poisson process with the rate of , which is lower than μ + μ1 to make the system stable (recall Proposition 3.3). The decision problem of a typical customer is a symmetric game, where his decision affects the net benefit of service consumption accruing to other customers via service congestion or system delay. Such strategic interaction is analyzed through Nash equilibrium strategies. There are two pure strategies: join or balk. One may not join the system if all others ¯

Using Little’s law, we derive the expected sojourn time

β αβ + β¯ + μ μ1

2 W (1) ∂∂μ < 0, ∂∂μW2 > 0.

β regular service and incurs a cost of sojourn at c ( μ + αβμ+β ), where 1 c (> 0) is the waiting cost per unit of time. Reasonably, we assume

 λ ( μ + μ1 − λ ) λ2 × 1+ − . μμ1 μμ1 − λβμ1 − λ(αβ + β¯ )μ 

Proposition 3.4.

λ ( μ + μ1 − λ ) 1+ μμ1 − λβμ1 − λ(αβ + β¯ )μ



(6)

Remark 1. When β = 0 as a special case, our setting becomes a classic M/M/1 queueing system. In this case, W = μ 1−λ is consis1

tent with the well-known feature of the M/M/1 system.

The proof of Propositions 3.3 and 3.4 is provided in the Appendix, and they will be used later to derive important results. Proposition 3.3. If the system is stable, then λ < μ + μ1 . The interpretation of this proposition is straightforward. The system is likely to be stable if customer arrivals are not too heavy compared to the service rates (μ, μ1 ).

β join, or go ahead for service under R − p > c ( μ + αβμ+β ) if none 1 of them join. However, a dominant strategy does not exist in the case with imperfect information. Hence, a mixed strategy should be considered in the equilibrium analysis. This strategy q ∈ [0, 1] is defined as the probability of joining the system, with which the effective arrival rate λ = q of customers is used for business decision by the firm. Correspondingly, an equilibrium arrival rate is λe = qe , with qe denoting the equilibrium mixed strategy. Using the standard equilibrium analysis for unobservable queueing models (Hassin and Haviv, 2003, pp. 46–47), we derive ¯

Z. Lian et al. / European Journal of Operational Research 254 (2016) 907–915

the equilibrium arrival rate as follows:

⎧ ⎪ ⎨0,

, λe ( p, μ, α , β ) = ⎪λ0 ( p, μ, α , β ), ⎩

if if if ≤

firm’s overall profit function is specified as

¯ 1 − G. V ( p, μ ) = (β¯ + βα ) pλe ( p, μ ) − d0 βαμ − d1 βμ

p > R − cW (0, μ, α , β ), p < R − cW ( , μ, α , β ), R − cW ( , μ, α , β ) p ≤ R − cW (0, μ, α , β )

Profit maximization can be expressed as a two-step process



max V ( p, μ ) = max

(7) where λ0 (p, μ, α , β ) is the solution to p = R − cW (λ0 , μ, α , β ). Identifying the equilibrium other than the extreme rate λe = {0 or } is equivalent to finding the solution to the equation p + cW (λ0 , μ, α , β ) = R. Solving this equation yields

λ0 ( p, μ, α , β ) = μμ1

(R − p)μμ1 − c [(R − p)μμ1 + c(μ + μ1 ) − c] − cμμ1 (8)

μ≥μ1

μ≥μ1 ,p>0

c

μμ1

< R − p ≤ cW ( , μ, α , β ).

Proposition 4.1. With the equilibrium arrival rate defined above, we obtain the following results: (i) If not all the arriving customers decide to join the queue, then

∂λe > 0, ∂μ

∂λe < 0, ∂p

∂ 2 λe < 0, ∂ p2

∂λe ≤ 0. ∂α

∂λe e If 0 ≤ α < α o , then ∂λ ∂β > 0. If α = αo, then ∂β = 0. If α o < ∂λ α ≤ 1, then ∂βe < 0.

(ii) If all the arriving customers decide to join the queue, then

∂λe ∂λe ∂λe ∂λe = = = = 0. ∂p ∂μ ∂α ∂β The proof of Proposition 4.1 is presented in the Appendix. This proposition is briefly explained by using the equilibrium mixed strategy qe . If not all the arriving customers join the queue, then a typical customer is more likely to decide to join under a greater rate of experience service, under a lower proportion of other customers who buy regular service, or under a lower price of this paid service. Moreover, he/she will less likely join the system if more of others try experience service and if many of them buy regular service subsequently. However, he/she will more likely join the system even though more of others try experience service, as long as not many of them move on to regular service. These results imply that the probability of a potential customer joining the system hinges on the service price and server congestion in a negative manner as expected.

max{(β¯ + βα ) pλe ( p, μ ) p≥0



¯ 1 − G} . − d0 βαμ − d1 βμ

(9)

In the first step, the (partially) optimal price p∗ of regular service is derived for any given rate μ of experience service, i.e., p∗ = p∗ (μ ). In the second step, the optimal rate μ∗ of experience service is established on the basis of p∗ (μ), and then the fully optimized price p∗∗ = p∗ (μ∗ ) can be determined through substitution. The first-order condition for step-one optimization is

λe ( p, μ ) + p

where

 = βμ1 + (αβ + β¯ )μ,

911

∂λe ( p, μ ) = 0. ∂p

The second-order sufficient Proposition 4.1, that is,

2

condition

is

satisfied

under

∂λe ( p, μ ) ∂ 2 λe ( p, μ ) +p ≤ 0. ∂p ∂ p2

Hence, there exists a globally unique optimal price p∗ (μ) given the experience service rate μ, as presented in the following proposition whose proof is given in the Appendix. Proposition 5.1. For any given experience service rate μ, the optimal price posted by the service provider is



p (μ ) = ∗

R − cW ( , μ ), −c R−  μμ1 ,

if ϒ (μ ) ≥ , if ϒ (μ ) <

(10)

where

  μμ1 μμ1 − (μ + μ1 ) ϒ (μ ) = 1+c ,    = (μ + μ1 ) − 2 − μμ1 , =

c2  2 + c[Rμμ1 (2 + ) + c ].

The corresponding equilibrium arrival rate equals



λe ( p∗ (μ )) =

, ϒ ( μ ),

if ϒ (μ ) ≥ , if ϒ (μ ) < .

(11)

The resulting equilibrium profit function is

V ( p∗ (μ ), μ ) = (β¯ + βα ) p∗ (μ )λe ( p∗ (μ ), μ ) − d0 βαμ ¯ 1−G − d1 βμ

⎧ ¯ ¯ 1 (β + βα )[R − cW ( , μ )] −d0 βαμ − d1 βμ ⎪ ⎨ −G, if ϒ (μ ) ≥ , = −c ¯ ¯ ⎪ ⎩(β + βα )(R − μμ1 )ϒ (μ ) − d0 βαμ − d1 βμ1 −G, if ϒ (μ ) < . (12)

5. Regular service pricing and experience service offerings Customers’ equilibrium strategies derived above are incorporated into the firm’s profit maximization discussed below. The firm maximizes its expected profit by choosing the optimal price p of regular service and the optimal rate μ of experience service offering, with customers’ choice (α , β ) treated as underlying parameters. For simplicity, the firm is assumed to operate its business under a constant marginal cost at d0 for experience service and d1 for regular service, with G denoting the firm-wide fixed cost. The

Remark 2. Proposition 5.1 offers a threshold Y(μ) for business decision making. If the arrival rate is high (> Y(μ)), the firm may serve only a portion of arriving customers by adjusting the price p∗ (μ) of regular service given the rate μ of experience service. If the arrival rate is low (≤ Y(μ)), the firm must serve all potential customers. The optimal rate μ∗ of experience service can be analyzed through step-two optimization as depicted and analyzed in Section 6. A direct computing yields the following result.

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Proposition 5.2. Y(μ) is an increasing function in μ, that is

d ϒ (μ ) > 0. dμ Two managerial insights can be derived from the above analytical results. (a) If ≤ Y(μ), we know p∗ (μ) > 0 since W p∗ (μ) = R − cW ( , μ ) and ∂∂μ < 0. (b) The situation of ≤ Y(μ) given (referred to as demand shortage) can be mitigated by reducing μ under Y (μ) > 0. Intuitively, when the arrival rate is low (≤ Y(μ)), the firm may extend the duration 1/μ of free experience service to attract more customers. On the one hand, such lower rate μ of experience service necessitates a corresponding reduction in the price of regular service. On the other hand, this lower rate μ tends to make the demand shortage less serious as the firm is soliciting business by serving all potential customers. It follows from Proposition 5.2 that

⎧ ¯ ¯ 1 (β + βα )[R − cW ( , μ )] − d0 βαμ − d1 βμ ⎪ ⎨ −G, if μ ≥ max { μ , μ } 0 1 V ( p∗ (μ ), μ ) = , −c ¯ ¯ ⎪ ⎩(β + βα )(R − μμ1 )ϒ (μ ) − d0 βαμ − d1 βμ1 −G, if μ1 ≤ μ < μ0 where μ0 is the unique solution to ϒ (μ0 ) = . The concavity of V(p∗ (μ), μ) with respect to μ is analytically unclear due to the complexity of this function. Fortunately, numerical simulation can be conducted to show that the function is strictly concave. We can then compute the optimal value μ∗ of experience service rate using an algorithm. To do this, take the derivative of the function:

⎧ ∂W ( ,μ) + ¯ d0 , ⎪ ∂μ )c ⎪ ⎨if μ ≥ max({βμ+βα, μ },

∂ V ( p∗ (μ ), μ ) 0 1 −c = ∂ μμ 1 ∂μ ⎪ ϒ ( μ ) + ( R − ⎪ ⎩ ∂μ if μ1 ≤ μ < μ0 .

−c ∂ ϒ (μ ) μμ1 ) ∂μ

+

d0

βα +β¯

,

Let U1 be the set of solutions to the following equation:

d0 ∂ W ( , μ ) =− , ¯ ∂μ (β + βα )c

s.t.

μ ≥ max{μ0 , μ1 },

(13)

and U2 be the set of solutions to the equation below:

−

−c ∂ μμ  − c ∂ ϒ (μ ) d0 1 · ϒ (μ∗2 ) + (R − ) = , ∂μ μ∗2 μ1 ∂μ βα + β¯ s.t. μ1 ≤ μ < μ0 .

(14)

Algorithm 1. Step 1: Find μ0 by solving ϒ (μ0 ) = ; Step 2: Find all elements of U1 and U2 by solving (13) and (14), respectively; Step 3: Find the optimal value of μ by comparing the values of V(p∗ (μ), μ) for all critical points μ ∈ {μ0 } ∪ U1 ∪ U2 . We then have: μ∗ = arg maxμ∈{μ0 }∪U1 ∪U2 {V ( p∗ (μ ), μ )}, which leads to the fully optimized price p∗∗ = p∗ (μ∗ ) and the fully maximized profit V∗∗ = V ( p∗ (μ∗ ), μ∗ ). In Section 6 we resort to numerical examples for a calibration analysis of full optimization. This analysis will be devoted to exploring the impacts of underlying parameters (α , β , μ1 ) on firm choice variables (μ∗ , p∗∗ ). 6. Numerical analysis An example presented in this section involves a popular kind of health care center observed in China, where medical massage service is provided as a traditional Chinese therapy. There is one professional masseur in each therapy room, who provides some massage service with certain healing effects. As a way of on-site advertising, free experience service is offered to customers in these

rooms; then, it is their own choice whether to move on to paid regular service. This example motivates us to perform the following numerical analysis, which suggests the usefulness of our theory in application. All assumptions made earlier for the general theorization apply to this specific example as well, with hypothetical values specified for underlying parameters: (R, , c, d0 , d1 , G ) = (100, 8, 1, 10, 5, 20 ). In all the following cases, the parametric values are chosen in such a way that the stability condition holds. Numerical results are depicted in Figs. 2 and 3. Before proceeding it is necessary to look at the effects of service consumption choice (α , β ) on server congestion ρ . Clarifying these effects facilitates subsequent numerical analyses. It is easy ∂ρ ∂ρ to see that ∂β > 0 for α > α o but ∂β < 0 for α < α o due to the faster experience service compared to its regular counterpart ∂ρ (μ > μ1 ). By contrast, ∂α > 0 for all α ∈ (0, 1]. The implications of these derivative signs were already established in Proposition 3.4 for the delay function W and are presented again in the following for greater clarity. If more customers buy regular service after trying experience service, the server with limited capacity will definitely become more congested. However, more customers using experience service can have dichotomous effects on server congestion, depending on whether or not many of them will later buy regular service. Specifically, more customers using experience service may aggravate server congestion if many of them move on to regular service, or lessen it otherwise. Such dichotomity creates certain complicated impacts on the interaction between the firm (faced with constraints on server capacity) and its customers (averse to waiting for delayed service) and hence on their respective optimal actions. Concretely, the dichotomous effects are reflected by relative changes of different curves before and after their crossing points in Figs. 2 and 3. Fig. 2 provides three significant economic intuitions. First, for high values of β (= 0.5 and 0.9, i.e., many arriving customers try ∗ ∂μ∗ experience service), Fig. 2 shows that ∂μ ∂α > 0 if α < α o and ∂α < 0 if α > α o , where α o is equal to 0.5 for β = 0.5 and to 0.6 for β = 0.9 . The interpretation of this result is as follows. If the regular service section is not very busy (α < α o ), a faster handling (μ∗ ↑) of customers in the experience service section is required to induce more of them (α ↑) to move on to the paid service for profit maximization. If regular service becomes very busy (α > α o ), however, a slower experience servicing (μ∗ ↓) for newly arriving customers is needed to accommodate the rising demand (α ↑) of early arriving customers for regular service. Second, if regular service is not busy (α < α o ), more customers trying experience service (higher ∂ρ β ) may alleviate server congestion (smaller ρ due to ∂β < 0 for α < α o ) and free service can be slowed down a little bit (a slightly lower μ∗ for β = 0.9 than for β = 0.5). If regular service is busy (α > α o ), however, more customers trying experience service (higher β ) can turn into a factor contributing to server congestion (greater ∂ρ ρ due to ∂β > 0 for α > α o ) and this free service must be quick-

ened substantially to leave more seats for use in the paid service (a much higher μ∗ for β = 0.9 than for β = 0.5). Third, when most arriving customers are not interested in free-riding (say, a low β at 0.1) but rather buy regular service directly (with high probability β = 0.9), the free service can be made so significantly slow as to become a showcase and its optimal rate μ∗ bears little relation to the consumer choice α of the paid service. Three managerial insights can be derived from Fig. 3 in terms of optimal pricing p∗∗ for regular service based on the choice (α , ∂ρ p∗∗ β ) of waiting-averse consumers. First, ∂∂α < 0 due to ∂α > 0. A lower price level (p∗∗ ↓) of regular service should be maintained as compensation to customers for service delay if the server becomes more congested (ρ ↑) due to more of them buying regular service

Z. Lian et al. / European Journal of Operational Research 254 (2016) 907–915

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μ* versus α

22 20 18

μ*

16 β=0.1 β=0.5 β=0.9

14 12 10 8 0.1

0.2

0.3

0.4

0.5

0.6

α

0.7

0.8

0.9

1

Fig. 2. The impact of consumers’ regular service choice α on the firm’s optimal experience service rate μ∗ for different values of β under μ1 = 8.

p*(μ*) versus α

96.55 96.5

p*(μ*)

96.45 96.4 96.35 96.3 β=0.1 β=0.5 β=0.9

96.25 96.2 0.1

0.2

0.3

0.4

0.5

α

0.6

0.7

0.8

0.9

1

Fig. 3. The relationship between the firm’s optimal price p∗∗ of regular service and consumers’ choice α of this service for different values of β under μ1 = 8.

(α ↑). This effect is manifest systematically by all downward sloping curves in Fig. 3. Second, the price curve p∗∗ for greater β is declining in α more steeply for α > α o (= 0.6), indicating that a greater price reduction ( |p∗∗ | ↑) for busy regular service should be offered to increase its attractiveness to customers when more of them use experience service (β↑) and are also willing to buy regular service (α↑). Third, another threshold α = 0.8 also makes

a difference to the effect of β on price compensation for service delay. If α > α (> α o ), the price curve p∗∗ is lower for higher β ∂ρ (causing greater ρ due to ∂β > 0 for α > α o ). That is, when regular service is already too busy and when more customers are waiting for experience service so as to worsen server congestion, the price of regular service should be lowered further to raise their desire for this service and keep them in waiting rooms.

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7. Concluding remarks This paper employs a queueing-theoretic approach to modeling service systems, with free service offerings aimed at enhancing customer experience. Yet such offerings are costly to the firm and may cause congestion to regular service paid by customers, so that the firm should strike a proper balance between the two types of services. Our model provides a theoretical account of strategic interactions between the capacity-constrained firm and its waitingaverse customers. Our modeling is followed by a numerical analysis to derive economic intuitions and managerial insights. Applying the queueing theory to service issues is useful in facilitating economic analysis. We show that if more customers buy regular service after trying experience service, the server with limited capacity will become more congested. However, more customers using experience service may have dichotomous effects on server congestion, depending on whether many of them will later purchase regular service. The probability of their joining the queue hinges on service prices and server delays in a negative way. We find that the firm’s decision is contingent on market conditions as well as customer strategies. If the market demand is strong, the firm may serve only a portion of arriving customers by adjusting regular service prices and experience service offerings. If customer arrivals are too low, the firm must extend its experience service offerings and lower its regular service charges to attract potential customers. Our numerical analysis of service system generates certain managerial insights for firm decision making. First, if the regular service section is not very busy, a faster handling of customers in the experience service section is required to induce more of them to move on to the paid service for maximal profitability. If regular service becomes very busy, however, it is then necessary to slow down experience service to hold onto customers and avoid aggravating congestion in regular service. Second, more customer arrivals for free service drive up the optimal rate of this service if regular service is busy, but have no large impact on this rate otherwise. Thus, if more customers are trying experience service when regular service is congested, the free service should be quickened to keep them waiting for the paid service. Third, a lower price of regular service should be charged as compensation to customers for service delay if the server becomes more congested due to more of them buying regular service. Finally, a greater price reduction for busy regular service should be offered to increase its attractiveness to customers when more of them arrive for experience service and are also willing to buy regular service. Our work in this paper can be extended to have a better understanding of free service as a new form of advertising to improve customer experience and solicit business for regular service. An extension discussed below lays a sensible foundation for a more in-depth study in the future. Three points are worth mentioning here.2 First, the feature of quality uncertainty among customers and learning quality through experience service should be addressed in a new model (Hassin, 2007; Sahib & Gu, 2013). This model may start with a service system with uncertain quality, and free experience service is provided for uninformed customers to learn about the true quality. After trying the experience service, they can use the newly obtained information to decide whether or not to join the queue for the regular service. Second, while experience service is meant to be a learning opportunity for customers to

2 We gratefully acknowledge that these interesting points are adapted from the critical comments and valuable suggestions offered by one of the referees for our manuscript. Potentially, the new study discussed below may have to deal with three information problems. Two problems facing customers include the unobservability of queue length and the uncertainty about the quality of service offering. One problem facing the firm is the imperfect information about customer types.

resolve quality uncertainty, they may have different rates of time preferences or different preferences over the same service offerings. Thus customer heterogeneity among the uninformed is important to model (Mendelson & Whang, 1990; Sahib & Gu, 2002) since this factor can determine, via the experience service, whether or not a specific service offered is valuable to a particular type of customers. Naturally, some types of customers will join the queue for the regular service but other types may not. The choice-making process of heterogeneous customers will have to be modeled explicitly, so that the model can incorporate the role of customer types in queueing decisions after the information updating via experience service. Third, the probabilities (α , β ) can no longer be treated as exogenous as in the present paper, but should be endogenized via server congestion and delay aversion in the new model. It is expected to see that the customer net benefit should determine the two probabilities rather than the other way around. They may also be affected by customer preferences and learning processes. Admittedly, it is a challenging task to weave these three points into a tractable model that involves various probability distributions, Markov chains, and probability generating functions. We present these points here to call attention to a better modeling for service systems with customer experience considerations. Acknowledgments This research is funded by the University of Macau under MYRG163-FBA11-LIANZT and MYRG2014-0 0 036-FBA and is partially funded by the project of Mathematics and Interdisciplinary Science, Innovation-Driven project and the Yu Ying project of Central South University. Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.ejor.2016.05.023. References Ackerberg, D. A. (2003). Advertising, learning, and consumer choice in experience good markets: An empirical examination. International Economic Review, 44, 1007–1040. Afèche, P., & Mendelson, H. (2004). Pricing and priority auctions in queueing systems with a generalized delay cost structure. Management Science, 50, 869–882. Allen, J. R., Frederick, F. H., & Barney, H. (2005). The three Ds’ of customer service. Technical Report. Harvard Business School. Anand, K. S., Fazil Pac, M., & Veeraraghavan, S. (2011). Quality-speed conundrum: Trade-offs in customer-intensive services. Management Science, 57, 40–56. Antia, K. D., Bergen, M., & Dutta, S. (2004). Competing with gray markets. MIT Sloan Management Review, 46, 63–69. Berry, L. L., Carbone, L. P., & Haeckel, S. H. (2002). Managing the total customer experience. MIT Sloan Management Review, 43, 85–89. Burnetas, A., & Economou, A. (2007). Equilibrium customer strategies in a single server markovian queue with setup times. Queueing Systems, 56, 213–228. Cachon, G. P., & Harker, P. T. (2002). Competition and outsourcing with scale economies. Management Science, 48(10), 1314–1333. Carlton, D. W., & Chevalier, J. A. (2001). Free-riding and sales strategies for the internet. Journal of Industrial Economics, 49, 441–461. Chen, H., & Frank, M. (2004). Monopoly pricing when customers queue. IIE Transactions, 36, 569–581. Chu, L., & Zhang, H. (2011). Optimal pre-order strategy with endogenous information control. Management Science, 57, 1055–1077. Crawford, G. S., & Shum, M. (2005). Uncertainty and learning in pharmaceutical demand. Econometrica, 73, 1137–1173. Economou, A., & Kanta, S. (2011). Equilibrium customer strategies and social-profit maximization in the single-server constant retrial queue. Naval Research Logistics, 58, 107–122. Edelson, N. M., & Hildebrand, D. K. (1975). Congestion tolls for Poisson queueing processes. Econometrica, 43, 81–92. Edvardsson, B., Enquist, B., & Johnston, R. (2005). Cocreating customer value through hyperreality in the prepurchase service experience. Journal of Service Research, 8, 149–161. Froehle, C. M., & Roth, A. V. (2004). New measurement scales for evaluating perceptions of the technology-mediated customer service experience. Journal of Operations Management, 22, 1–21.

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