capacity management: pricing strategy, performance and the role of information

prod&ion economics

Int. J. Production Economics 40 (1995) 89-100 EISEVIER

Capacity management: Pricing strategy, performance and the role of information

Ann van Ackere

London Business School, Sussex Place, Regent’s Park, London. NWI 4SA, United Kingdom

Accepted for publication 25 April 1995

Abstract

A queueing model is used to analyse the interrelation between system performance, pricing strategy and information. Two input control methods are studied: a club approach (controlling the expected arrival rate) and a free-entry system (controlling the maximum backlog), where a new customer may or may not be able to observe the existing backlog.

The club approach generally leads to higher prices, lower utilization and lower profit. Under the free-entry system, customers are better off if they can observe the backlog, but society as a whole is worse off. If there is a shortage of customers, the club approach becomes more attractive, as customers must pre-commit.

Keywords: Queues; Stochastic model applications; Pricing

1. Introduction

Many research efforts have focused on congestion effects and corresponding performance measures such as average throughput time, expected or maximum queue length and makespan [l-3]. Little attention is paid to the fact that an increase in any of these measures implies a delay in job completion, which may cause a decrease in the job’s value. Examples include delayed introduction of new products, payment delays, decreased customers’ willingness to pay, and lateness penalties.

Accepting a project that will compete for com- mon resources may delay completion of already accepted projects. This emphasizes the need to consider lead-time, pricing, and utilization decisions jointly [4]. Issues of project selection have been discussed extensively in the R and D literature, with

a focus on how to rank projects [S] and how to allocate monetary resources [6]. A different part of the literature focuses on determining optimal utilization and pricing schemes for a specific queueing environment, the objective being either social welfare or profit maximization. Stidham [7] provides an overview of this work.

One of the earliest references in the queueing literature is Naor [S]. Customers arrive at a single- server service station (e.g. a post office) according to a Poisson process and decide whether or not to join the queue. Service times are exponential. The optimal toll levels for social welfare and profit maximization differ. Yechiali [9] drops the assumption of Poisson arrivals and assumes individuals can co- operate to achieve a social optimum, i.e., maximize the total benefits accruing to the customers and the facility. Knudsen [lo] considers a multi-server

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90 A. van Ackerellnt. J. Production Economics 40 (1995) 89-100

facility and non-linear waiting costs. Edelson and Hildebrand [ 1 l] assume balking is not allowed; i.e. customers’ decisions on joining the queue depend on their expectations about delay. In this model the revenue-maximizing toll and the socially optimal toll coincide if all customers have the same waiting cost. This is no longer the case for two customer classes with different waiting costs.

Oi [12] discusses the optimality of two-part tariffs in an environment with heterogeneous customers and no congestion effects. Customers pay a lump-sum fee for the right to obtain a good or service, and an agreed price whenever they exercise their rights. If it is possible to discriminate among customers, all consumer surplus (value derived from the service, minus financial and waiting costs) can be extracted by charging a price equal to marginal cost, and a lump-sum fee. Otherwise it may be optimal to charge a high lump-sum fee and a price below marginal cost. Edelson and Hildebrand [ 1 l] consider two-part tariffs in the presence of congestion effects. For a homogeneous customer population this leads to socially optimal pricing, but for heterogeneous customers the expected social welfare can be even lower than in the single-toll case. Barriers to the use of two-part tariffs include resale (if one customer can pay the lump-sum fee, pur- chase a large volume, and resell to other customers, the two-part tariff breaks down) and cost and in- convenience (e.g., the burden of selling tickets at each attraction in an amusement park).

Dewan and Mendelson [13] study the impact of waiting costs in a computer time-sharing environment with heterogeneous users. Customers base the decision to join or not to join the queue on the expected queue length. The authors consider both the short-term pricing issue and the long-term capacity issue. They show that to generate the maximum value for the company (i.e. benefit from the service provided, minus waiting costs and financial costs) the computing facility must operate at a loss.

This paper follows the approach of explicitly including the trade-off between utilization and the value of each job in the analysis. It differs from the above literature by its focus on how the facility is managed and the role of information. Two pricing strategies are considered: either a lump-sum fee or a price per unit, but not both. Price is not allowed

to depend on actual queue length. If this alternative were feasible, it would be optimal. Peak-load pricing is an attempt to approximate this situation (e.g. amusement parks charge a higher fee at weekends) but the fee is still independent of the actual queue.

A situation is considered where the manager of a facility must decide on the number of customers (or jobs) to accept. All jobs are identical. The manager can control the load directly by turning away jobs whenever the backlog (or queue length) exceeds a critical level. This approach is labelled the ‘free-entry system’. If arriving customers can observe the existing queue (typically the case at a service facility) the manager can select the fee such that customers do not join if the backlog is too high, as in Naor [S]. If the customer cannot assess the backlog accurately (as in a jobshop environment) the manager has the option to reject the job if the backlog is judged to be too high. Customers base their behaviour on expected queue length, as in Dewan and Mendelson [13]. Customers, however, are aware that the manager may reject jobs, and factor this element into their expectations. Com- paring these two situations raises an information issue: if the queue is not visible to customers, should queue-related information be made available to them? For instance, should airline passengers be informed of over-bookings (an invisible queue) which may force passengers to ‘wait’ (i.e. take the next flight)?

Alternatively, the manager can use a club approach: potential customers pay a subscription fee to be allowed to join the queue at a later time. The manager accepts a limited number of customers (‘members’) and guarantees they will receive service, effectively controlling the expected arrival rate. It is assumed that each customer is entitled to one service during a specific time period. In a production environment the club approach is more likely to be encountered internally: the decision to pursue a specific number of projects is made, and all jobs deriving from these projects must be performed. A free-entry system is more likely to occur in ex- ternal situations, where any customer can ask for a job to be performed, but acceptance depends on the backlog. The membership system offers several advantages. There may be a feeling of exclusiveness (passers-by cannot join in), waiting times become

A. van Ackerellnt. J. Production Economics 40 (1995) 89-100 91

more predictable as membership is known and limited and, most importantly, the customer is guaranteed access to the facility. Therefore individuals may have a higher willingness to pay for a service delivered in a club context.

The choice of models is driven by the aim of identifying the simplest possible models which cap- ture the impact of differences in the time at which a customer commits himself, and the information available to him at that stage. For reasons of tracta- bility the model assumes negative exponential in- terarrival and service times. It is also assumed that the potential arrival rate equals the service rate. The potential arrival rate is the arrival rate which would materialize if there were no congestion nor financial costs. To obtain closed-form solutions, linear waiting costs are assumed.

This paper shows that, under these assumptions, the club approach leads to longer delays, despite lower utilization. Therefore, this can only be optimal if customers are willing to pay a premium for guaranteed service. Another result is that in the free-entry system, customers are better off if they can observe the queue, but society as a whole is worse OR the increase in consumer surplus is less than the decrease in profit. If the potential arrival rate is considerably less than the service rate and the backlog is visible, a club approach may be more profitable to the manager, as customers are forced to pre-commit.

Section 2 presents the model and the assumptions. Sections 3 and 4 analyse this model for the control situations described above. Section 5 compares the models and draws conclusions. Section 6 contains some suggestions for further research.

2. The model

The manager of a facility is faced with a large number of identical jobs that compete for the facility’s resources. Let p denote the service rate; i.e. the average number of jobs that could be carried out- during a specific time period (say, a year). It is assumed that the service rate is fixed, and that the facility incurs only fixed costs. These can be assumed to equal zero without loss of generality.

The manager selects a fee, F, as well as 1 (the number of jobs to be performed during the period of membership) or K (the maximum number of jobs present, which indirectly determines the effective arrival rate A). The choices of F and 2 or K are interdependent as the fee level influences the arrival rate and the maximum queue length a customer will tolerate. We are slightly abusing notation by using 1 for the arrival rate in the club model, and the effective arrival rate (i.e. customers who arrive and enter) in the free-entry model. In the first case a customer is indifferent between either paying the fee and getting service, or not joining. In the latter case, if the queue is observable, a customer will only join if the number of jobs present is less than K. If the queue is not observable, customers will be turned away if the backlog equals K.

The time T that each job spends in the facility is a random variable, whose distribution depends on 1. The expected time in the system is an increasing and convex function of 1 (see e.g. [l, Chapter 23). Let V(t) denote the monetary value of a job completed in t time units. This value is a decreasing function of t, whose shape depends on specific assumptions regarding the cost of delay. For instance, if the only losses incurred are interest costs due to the late perception of revenues, V(t) is convex in t. It will be assumed that V(t) is a continuous, twice differentiable function, that customers arrive according to a Poisson process, and that service times are exponential.

To obtain closed-form solutions the following assumptions are made:

(a) V(t) is linear in t; i.e. V(t) = v - apt, where u is a customer’s maximum willingness to pay (i.e. the value of the job if it were completed instantly), and the parameter a measures a customer’s impatience: the higher a, the more sensitive the value of the job to delay. This function is used (in preference to u - at) to ensure that the results do not depend on the choice of time unit.

(b) In the free-entry system it is assumed that the rate at which potential customers show up equals the service rate. This assumption greatly simplifies the algebra, and allows the optimal K to be ex- pressed as a simple function of the parameters p, a and u. The impact of this assumption is discussed when the various models are compared.


3. The club approach

The manager sets the fee F to control the expected arrival rate. He maximizes expected profit per period, subject to the constraint that customers are willing to join. Customers will only join if E[Y(T)II] - F 2 0, where E [ ] denotes expected value and ( indicates that the expectation is condi- tional on the decision variable J.. Any pair (F, A) satisfying this equation with a strict inequality would be suboptimal from the manager’s point of view, as he could increase F, without reducing membership. Therefore, for any 2, the optimal fee level will satisfy E[V(T)lA] - F = 0, and consumer surplus equals zero. For heterogeneous customers, the condition F = E [ I/ (T) 1 A] would only apply to the value function of the marginal customer, and only this customer would have zero consumer surplus. This yields the following optimization problem:

max AF a

s.t. E[V(T)IA] -F = 0. (1)

Substituting the constraint into the objective function yields

my AE[V(T)I,I]. (2)

At the optimum, no new customers want to join, and none of the members wants to leave. The first- and second-order conditions are:

FOC: E[V(T)lTJ = - AaEc’;~)“‘, (3)

sot. A ~2~C~V)14 + 2 W~CUl4 < o aA2 an . . (4)

The first-order condition states that the marginal profit from an additional customer (F = E[ V(T)lA]) should equal the loss of profit from the existing customers; i.e., the expected arrival rate 1, multiplied by the externality imposed on each customer (aE[V(T)l12]/aA). The second term of the second-order condition is negative, but the sign of the first term depends on the shape of V(t)

and on the arrival and service processes. The second-order condition is satisfied if E[V(T)l A] is concave in 1. Dewan and Mendelson [ 13) discuss conditions for the existence and uniqueness of an interior solution for a similar problem.

This model assumes that customers do not balk. This assumption can be motivated as follows. If arriving members are unable to assess the queue length accurately, they have no reason to balk. If they can observe the queue length, they balk only if the observed queue q is such that E [ V(T)lA, q] < 0, as the fee is a sunk cost when they arrive. The numerical examples illustrate that this situation occurs with very low probability.

It would be more accurate to model this case using a finite calling population for the membership. Although there exists an analytical solution for the expected waiting time for this model, it does involve the sum of two series. Consequently, we cannot derive an analytical optimal membership size.

4. The free-entry system

This section considers the case where the manager controls congestion by managing the backlog. Two cases are distinguished, depending on whether or not a new customer is able to observe the existing backlog. Although K is by definition an integer-valued variable, it is treated as being continuous for this analysis, the exact optimal value being the integer part of either the resulting maximum backlog or the resulting maximum backlog plus one, with the corresponding fee.

The potential arrival rate is assumed to equal the service rate 6 which implies among others that for Poisson arrivals and exponential service times, the fraction of jobs accepted equals K/(K + 1) (i.e. the effective arrival rate, A, equals pK/(K + 1)) and the expected time in the system E(T) equals

(K + l)/(U VI*

4.1. New customers observe the queue length

Here the manager decides on a maximum backlog K, and selects the corresponding fee F, such

A. van Ackerellnt. J. Production Economics 40 (1995) 89-100 93

that an arriving customer only joins the queue if fewer than K customers are present. Let E[ V( Tk)l K] denote the expected value of a job for a customer who observes a backlog k < K, where k includes his own job. The optimal fee is determined by the expectations of the ‘marginal customers’; i.e. customers facing a backlog of K. The manager’s problem can be formulated as:

pK F “2” K+l

s.t. W’t~dIKl = F,

or equivalently

(5)

The first- and second-order conditions equal:

(6)

- PK ~W’(~dl~l =K-l aK *

(7)

- 2P SOC: tK + 1I3 W’Vdl~l

2P aJwt~K)l~l +(K + 1)2 aK

PK ~2.U~t~dl~l < o +K+l aK2 ”

The left-hand side of the first-order condition equals the marginal profit of an increase in K (change in effective arrival rate times revenue per customer), the right-hand side equals the marginal cost (expected effective arrival rate times the externality imposed on each customer by a new customer). The first two terms of the second-order condition are negative, while the sign of the third term depends on the shape of V(t). A sufficient (but not necessary) condition for Eq. (8) to hold is that E[V(T,)IK] be concave.

4.2. New customers do not observe the queue length

The manager again decides on a maximum backlog K and arriving customers are not allowed to join the queue if the existing backlog equals K. The corresponding fee is determined by the expected value of a job, rather than the expected value of a job facing the maximum backlog. The manager’s problem can be formulated as:

PK F “p K+l

s.t. EC~(WKl= F,

or equivalently:

(9)

rnp & E[V(T)IK]. (10)

The first- and second-order conditions are as in the previous section, with E [ V( T) I K] replacing

~W’(~dl~l:

FOC: (K ! 1)2’ W’GV~I

- 6 W~G’-)l~l =ET-l dK ’

ax~)l~l aK

(11)

PK a2EWG91Kl <o +K+l dK2 ”

5. Comparison of the various models

(12)

First, general results are formulated for the visible and invisible queue models. Then explicit results for the case V(t) = v - apt are stated. The subscripts c, s and i refer to the club approach, the visible-queue case and the invisible-queue case respectively.

5.1. Visible versus invisible queues

Let w denote the expected time in the system and p the utilization rate. When the queue is visible,

94 A. van Ackerelht. J. Production Economics 40 (1995) 89-100

a shorter maximum queue length is expected. Prop- osition 1 states sufficient conditions to this effect.

Proposition 1. Let K, and Ki denote the optimal solutions for the visible and invisible queue models respectively, and assume that the second-order conditions are satisjied. lf

l~W’V,)I-F;1/~KI > lJW’U’)IW~KI, (13)

then K, < Ki, & < ;ii, W, < wi and ps < pi.

Proof. The fact E[V(T)IK] 3 E[V(T,)(K], to- gether with Eqs. (7) and (13) implies

- PK WYG’-dl~l =--- KS+ 1 aK KS

- PK, WJ’G’YKI ‘K, + 1 8K KS’

where IK. means that the expression is evaluated at K = K,. But ,u/(K + l)‘E[V(T)jK] decreases with K, and is steeper than (- pK/K + 1) (aE[ V( T)] K]/aK) by the second order condition (12). Therefore, Eq. (11) implies K, < Ki. This implies that more customers arebeing turned away in the visible-queue case, and thus A, < &, w, < wi and

PS < Pi-

Condition (13) states that a change in the maximum backlog has less impact on the expected value of the average job, than on the expected value of a job facing the maximum queue length, which is

Table 1 Closed-form solutions for v(t) = v - apf

Club approach

quite intuitive. A similar result cannot be stated in general terms for the club approach. To establish A, < ;i,, we need to show that E[V(T)lL] < E [T/(TK)( K], where K yields an effective arrival rate equal to ;1, i.e. 1 = pK/(K + 1). Some simple algebra yields E(TI A) < E(TKI K), indicating that for many ‘reasonable’ functions V( .), the result 2, < & should hold.

5.2. Comparison of the three models for V(t) = v - apt

The results are summarized in Table 1. (See [l, Chapter 23 for the evaluation of E(T IA).) The second-order conditions are satisfied for the three models. F, n and CS, respectively, denote the optimal fee, expected profit and expected consumer surplus. For the visible-queue case K, > 1 if and only if v/a > 3, i.e. the gross value of a job (v) exceeds the cost of waiting for the duration of a service time (a) by a factor of 3. When v/a < 1, it is not worth operating the facility. If 1 < v/a < 3, a corner solution obtains: K = 1; i.e. no backlog is allowed. It is assumed that the inequality v/a 2 3 is satisfied. For the invisible-queue case Ki exceeds 1 if and only if v/a > 2.

For all three models, the higher the value of jobs (v large), the higher the optimal arrival rate, maximum backlog, utilization, expected waiting time, fee and expected profit, while an increase in the customers’ impatience (large a) causes all of these to decrease. Expected consumer surplus is non-zero only in the visible-queue case. It increases with v,

K

I

P w

F

II

CS

Visible queue

,/l +u/u- 1

Al - l/J=73 (1 - l/Jl+vla)

45-G/(2/4

v+a(l-JG)

.Uu + 241 - J_)

(ap/2) (Jw - 3 + 2/J-)

Invisible queue

Jz;7;;-1 Al - &a) 1-m J;ir;;iP V-&i@

c& -&a)’ 0

A. van Ackerellnt. J. Production Economics 40 (199.5) 89-100 95

but is not monotone in a. For a given K, expected consumer surplus increases with the level of impatience a (the higher the cost of waiting, the larger the consumer surplus of customers facing a short queue) and for a given level of impatience, the expected consumer surplus increases with K (more customers are allowed to join and experience consumer surplus). However, as customers become less patient (a increases), the optimal value of K decreases. For patient customers (u/a large) the first effect dominates, leading to an increase in consumer surplus as the customers become less patient (a increases). For impatient customers the second effect dominates, resulting in consumer surplus decreasing with a. Proposition 2 summarizes the comparison of the three models.

Proposition 2. For all v, a, ,u, such that v/u 2 3:

2, < & < A, PC < Ps < Pi, Ks < Ki,

Fc < Fiy Fc < Fs, WC > Wi > W,,

71, < 7T, < 71, + CS, < Ri.

If V/U < 8 then Fi < F,, wK < Wi, with reversed inequalities for v/u > 8.

Proof. These results follow from algebraic manip- ulations, and using the following equations: 1, = pK,/(K, + 1) and li = pKi/(Ki + 1).

Proposition 2 is illustrated in Table 2 and Figs. l-4 for p = 1, u = 50 and the customers’ impatience parameter a varying from 1 to 10. ‘Expected delay (K)’ denotes the expected time in the system for the marginal customer. For the club approach, the probability that a customer would prefer to balk is very small (below 5% for all examples).

Utilization is lowest for the club model, and highest for the invisible-queue case. The difference increases as customers become less patient. The expected time in the system (Fig. 1) is largest for the club model and smallest for the visible-queue case. The expected time in the system for the marginal customer in the visible-queue case exceeds that of the average customer in the invisible-queue case for impatient customers and is lower otherwise. Profit in the invisible-queue case exceeds total welfare in the visible-queue case (Fig. 2). The fee (Fig. 3) is

lowest for the club model. The fee in the invisible- queue model exceeds the fee in the visible-queue model for impatient customers: a lower fee is required to induce impatient customers to join when they observe a long queue. This figure also shows the fee for the club model when customers are willing to pay a premium for guaranteed service, as discussed below.

If customers are willing to pay for the guarantee of obtaining service, the customers’ valuation of a job, v, depends on the management method. De- note by P, (Pi) the premium required to operate in club-mode when the backlog is visible (invisible). In other words, if the backlog is visible, a manager is indifferent between operating a free-entry system with valuation v and operating a club system with valuation v + P,, and similarly for the invisible backlog. For this example P, = a, Pi = (3u/2 - u$ + 2fi - &) and Pi > P, whenever u/u > 3. These results are illustrated in Table 2 and Figs. 3 and 4. For the club model with valuation equal to u + P,, the fee and utilization equal that of the corresponding free-entry model, while the expected waiting time is longer. For the club model with valuation equal to v + Pi, the fee, the utilization and the expected waiting time exceed those of the corresponding free-entry model.

The free-entry model assumed that the potential arrival rate equals the service rate. If the potential arrival rate (A’) is larger, the manager will achieve higher profits, because the same policy will yield a higher utilization. It may be optimal to increase the fee and/or decrease K. This does not necessarily imply that the free-entry option will become even more attractive to managers: more customers are being turned away, which may increase a customer’s willingness to pay a premium for the club approach. This is compatible with the empirical observation that high-priced sports-clubs are located mainly in densely populated areas, where competition for access to the limited number of public facilities is strongest (A’ large). Table 3 shows a numerical example that illustrates the impact of varying the potential arrival rate. Let p = 1, v = 50, and consider a = 3 and u = 8. These results were obtained by numerical evaluation. The discussion summarizes the observations of a large number of numerical examples.

96

Table 2

A. van Ackerelint. J. Production Economics 40 (1995) 89-100

Sensitivity with respect to the impatience parameter a (u = 50, p = 1)

M/W Utilization 0.86 0.80 0.76 0.72 Fee 42.93 40.00 37.75 35.86 Profit 36.86 32.00 28.51 25.72 Expected delay 7.07 5.00 4.08 3.54 Probability (Balk) 0.00 0.00 0.01 0.02

M/M/l/K (visible) Maximum backlog 6.14 4.10 3.20 2.67 Utilization 0.86 0.80 0.76 0.73 Fee 43.86 41.80 40.39 39.30 Profit 37.72 33.60 30.78 28.61 Expected delay 3.57 2.55 2.10 1.84 Expected delay (K) 6.14 4.10 3.20 2.67 Consumer surplus 2.21 2.49 2.52 2.44 Welfare 39.93 36.10 33.30 31.04

M/M/l/K (invisible) Maximum backlog 9.00 6.07 4.77 4.00 Utilization 0.90 0.86 0.83 0.80 Fee 45.00 42.93 41.34 40.00 Profit 40.50 36.86 34.18 32.00 Expected delay 5.00 3.54 2.89 2.50

M/M/l: required premium for same projit as M/M/l/K (visible) Premium 1.00 2.00 3.00 4.00 Utilization 0.86 0.80 0.76 0.73 Fee 43.86 41.80 40.39 39.30 Profit 37.72 33.60 30.78 28.61 Expected delay 7.14 5.10 4.20 3.67 Probability (Balk) 0.00 0.00 0.01 0.01

M/M/l: required premium for same profit as MIMIIlK (invisible) Premium 4.23 6.03 7.43 8.63 Utilization 0.86 0.81 0.77 0.74 Fee 46.86 45.44 44.31 43.31 Profit 40.50 36.86 34.18 32.00 Expected delay 1.36 5.29 4.38 3.83 Probability (Balk) 0.00 0.00 0.01 0.01

0.68 0.65 0.63 0.60 0.58 0.55

34.19 32.68 31.29 30.00 28.79 27.64 23.38 21.36 19.58 18.00 16.57 15.28

3.16 2.89 2.67 2.50 2.36 2.24 0.02 0.03 0.04 0.04 0.05 0.05

2.32 2.06 1.85 1.69 1.56 1.45 0.70 0.67 0.65 0.63 0.61 0.59

38.42 37.67 37.03 36.46 35.96 35.51 26.83 25.34 24.05 22.92 21.91 21.01

1.66 1.53 1.43 1.35 1.28 1.22 2.32 2.06 1.85 1.69 1.56 1.45 2.30 2.13 1.94 1.74 1.54 1.33

29.13 21.47 25.99 24.66 23.45 22.34

3.47 3.08 2.78 2.54 2.33 2.16 0.78 0.76 0.74 0.72 0.70 0.68

38.82 37.75 36.17 35.86 35.00 34.19 30.14 28.51 27.04 25.72 24.50 23.38

2.24 2.04 1.89 1.77 1.67 1.58

5.00 6.00 7.00 8.00 9.00 10.00 0.70 0.67 0.65 0.63 0.61 0.59

38.42 37.67 37.03 36.46 35.96 35.51

26.83 25.34 24.05 22.92 21.91 21.01

3.32 3.06 2.85 2.69 2.56 2.45

0.02 0.02 0.03 0.03 0.04 0.04

9.69 10.66 11.56 12.40 13.20 13.96 0.7 1 0.69 0.66 0.64 0.62 0.60

42.42 41.58 40.80 40.06 39.35 38.67 30.14 28.51 27.04 25.72 24.50 23.38

3.46 3.18 2.91 2.79 2.65 2.53 0.02 0.02 0.03 0.03 0.04 0.04

When the potential arrival rate 1’ exceeds the of 0.6). One may wonder why it is not optimal to service rate p, increased profits are achieved by have an infinite maximum queue length. The increasing fees and decreasing the maximum queue explanation lies in the timing of customers’ com- length. The case A’ < p is less clear cut. In the mitment: in the club model, customers will occa- visible-backlog case, it is optimal to increase fees, sionally face a long queue because they commit which results in a smaller maximum queue length. ahead of time and fees are sunk, while in the vis- In the case a = 8, 2’ = 0.6, profits are less than in ible-queue case customers only commit after ob- the club system (which has an optimal arrival rate serving the queue. In the club case it is possible to

A. van Ackere flnt. J. Production Economics 40 (1995) 89-100 97

0’ ’ I I I I I I I I I I 1 2 3 4 5 6 7 8 9 10

Customers’ impatience parameter a

0 Club t Visible 0 Invisible A Visible (K)

Fig. 1. Time in system.

45

40 -

; “a 35 -

? 5 30 -

E : 25 -

E!

+” 20 -

Y c 15 - e a

d 10 3 8 a 5

B I

01 ’ I I I I I I I I I I 1 2 3 4 5 6 7 8 9 10


C Club t Visible 0 Invisible A Visible + CS

Fig. 2. Profit and total welfare.


50 -5

40 -

30 -

10 -

o- ’ I I I I I I I I I _ 1 2 3 4 5 6 7' 8 9 10


q Club + Visible = Club (Ps) 0 Invisible A Club (Pi)

Fig. 3. Fees.

70 r 1

6o ;11_:_:T1__1_TIII-1 50 : ====”

E .? 40

E : a 30 i

1 2 3 4 5 6 7 8 9 10


0 ” + ” + P(S) 0 v + p(i)

Fig. 4. Premium needed for club mode.

A. van Ackere/Int. J. Production Economics 40 (1995) 89-100 99

Table 3 Sensitivity with respect to the potential arrival rate (v = 50, p = 1)

Potential Arrival Rate

M/M/l/K (visible) a=3 Maximum Backlog

Fee profit

a=8 Maximum Backlog

Fee

Profit

M/M/l/K (invisible) a=3 Maximum Backlog

Fe.e

Profit a=8 Maximum Backlog

Fee

Profit

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

2.84 3.12 3.20 3.14 3.02 2.89 2.77 2.65 41.50 40.60 40.40 40.60 41.00 41.30 41.70 42.10 22.20 27.10 30.80 33.40 35.20 36.60 37.60 38.40

1.62 1.68 1.69 1.68 1.66 1.63 1.60 I.56 37.10 40.40 34.50 36.60 36.70 37.00 37.20 37.50 17.00 20.30 22.90 24.90 26.60 27.90 29.00 29.90

7.60 5.83 4.77 4.11 3.67 3.35 3.12 2.94 43.00 41.50 41.30 41.60 42.00 42.30 42.60 42.90 25.60 30.90 34.20 36.20 31.60 38.60 39.30 39.90

3.23 2.82 2.54 2.33 2.17 2.03 1.95 1.87 36.10 35.70 35.90 36.10 36.50 36.90 37.10 37.40 19.80 23.30 25.70 27.50 28.90 29.90 30.80 31.50

have a utilization (and the ensuing congestion) of 0.6 (i.e. all potential customers join the club), and a fee equal to 30. In the visible-queue case with a fee of 30 some customers will observe a long queue and elect not to join. Consequently, when there is a shortage of customers and they have a large waiting cost, a club model allows the manager to lock-in customers by forcing them to pay the fee before observing the actual queue length.

When customers are unable to see the queue, the optimal maximum backlog is considerably higher: given the low arrival rate, the manager is reluctant to turn customers away. As the potential arrival rate decreases below the service rate, the optimal fee first decreases, and then increases. The level of 1’ at which this turn occurs depends on customers’ impatience: the larger a, the lower the turning point. The lowest fee occurs approximately at 1’ = 0.95 for a = 3, and at L’ = 0.85 for a = 8.

5.3. Conclusions

From a managerial point of view, it is unlikely that customers can be turned away without incurring any costs. It was indicated that customers may be willing to pay more for guaranteed service (i.e. the club approach) and the implications were discussed. Turning away customers may also lower the potential arrival rate via a negative word-of- mouth effect (see e.g. [14]).

Under the assumptions of this paper, the club A customer may incur costs when showing up, approach is the least profitable, unless customers but receiving no service. This affects the club model are willing to pay a premium for guaranteed ser- and the free-entry model differently: in the latter vice. If there is a shortage of customers and the case this cost is sunk when the customer decides backlog is visible, a club approach may be more whether or not to pay the fee and join the queue, profitable, as customers are forced to pre-commit. while in the former case the customer decides to Under the free-entry system, customers are better join the club before incurring this cost.

off if they can observe the existing backlog, but society as a whole is worse off.

6. Suggestions for further research

This paper analyses the effects of different load- control methods on optimal utilization and pricing, and discusses the impact of the availability of information about the existing backlog on the manager’s optimal strategy. Several issues deserve further analysis. For instance, do the results still hold if the assumptions of a single server, exponential service times, and Poisson arrivals are dropped?

loo A. van Ackerellnt. J. Production Economics 40 (1995) 89-100

What if club members are entitled to more than one service? If demand per customer can be esti- mated, the general analysis carries through, with the arrival rate equal to the number of customers, multiplied by the average number of services per customer, but arrivals are no longer independent, and the assumption of Poisson arrivals breaks down. Alternatively, one can focus on shorter time periods during which customers request only one service.

References

[l] Gross, D. and Harris, C.M., 1985. Fundamentals of Quen- tals of Queueing Theory, 2nd ed. Wiley, New York. Second Edition.

[2] Van Der Heyden, L., 1981. Scheduling jobs with exponential processing and arrival times on identical processors so as to minimize the expected makespan. Math. OR, 6: 305-312.

[3] Jackson, J.R., 1963. Jobshop-like queueing systems. Mgmt. sci., lo(l): 131-143.

[4] van Ackere, A., 1989. Scheduling resources in project planning - avoiding delays by acquiring resources early may not be cost effective. OR Insight, 2(4): 6-l 1.

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capacity management: pricing strategy, performance and the role of information

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