The Curse of Reliability: Outsourcing Restoration Services for

Infrequent, High-Impact Equipment Failures�

Sang-Hyun Kim, Morris A. Cohen, Serguei Netessine, and Senthil Veeraraghavany

The Wharton School, University of Pennsylvania

Philadelphia, PA 19104

Abstract

Firms in industries such as aerospace and defense, high-tech manufacturing, and telecommu-

nications rely on functioning mission-critical equipment and cannot a¤ord signi�cant operational

downtime due to equipment failures. Although resolving problems quickly is essential to min-

imizing the impact of disruptions, the additional challenge in a decentralized supply chain is

incentivizing the suppliers to provide prompt restoration services. A key obstacle is that equip-

ment failures may occur rarely, making it too expensive for a supplier to commit the necessary

resources since they will be idle most of the time.

A widely adopted incentive mechanism is performance-based contracting (PBC), in which

suppliers receive compensation based on realized equipment downtime. It is believed to be

an e¤ective means to incentivize speedy restoration services. Using a model that builds upon

the principal-agent framework, in this paper we show that designing a successful PBC creates

nontrivial challenges that are unique to this environment. Namely, due to the infrequent and

random nature of equipment failures, a seemingly innocuous choice of performance measures

used in contracts may create unexpected incentives, resulting in counterintuitive optimal con-

tract structures. We �nd that the de�nition of the performance metric becomes critical, i.e., in

general a contract based on sample-average downtime is superior to a contract based on cumu-

lative downtime. We also �nd that, paradoxically, a �rm that values a consistent level of service

delivery would �nd PBC most costly to implement when a product is most reliable. In such

cases, high equipment reliability may be accompanied by excessive contracting costs.

�The previous title was �Incentives in Low-Demand Service Systems�.yEmail: {shkim, cohen, netessine, senthilv}@wharton.upenn.edu

1

1 Introduction

Unexpected disruptions of mission-critical operations can lead to dramatic consequences. For exam-

ple, in August 2007 Samsung Electronics, the world�s largest manufacturer of NAND �ash memory

chips, experienced a brief but detrimental power outage that shut down its six production lines.

The outage, which lasted less than 24 hours, cost Samsung an estimated $40M and caused 3.4

percent drop in the stock price of Apple, Inc., as investors feared that shortage of NAND chips

could delay Apple�s production of the iPod and the iPhone (PC World 2007). While �rms put

much e¤ort in preventing such events from happening, perfect prevention is impossible to achieve.

Therefore, contingency planning is essential; however severe an initial disruption may be, its impact

can be signi�cantly reduced if an a¤ected system is quickly restored to its normal condition through

carefully thought-out recovery action plans and prior deployment of resources.

When a supply chain is decentralized, however, implementing a contingency plan requires co-

ordinated e¤orts by distinct organizations. Intel faces similarly high �nancial losses when its semi-

conductor manufacturing equipment goes down. As a result, the company requires its equipment

suppliers to respond within 15 minutes to a failure (Harrington 2006). How can Intel make sure

that a promise to do so is actually ful�lled? After all, it is Intel, not the supplier, who bears the

direct consequences of the failure and has more urgency; on the other hand, it is challenging and

costly for the supplier to dedicate resources for fast diagnosis and repair at every customer location

because the supplier has limited capacity and because equipment failures are rare events. For a

contingency plan to work in a decentralized supply chain a proper incentive has to be provided to

the supplier.

Across commercial and government supply chains, performance-based contracting (PBC) is gain-

ing wide acceptance as an e¤ective instrument for providing such incentives. Under PBC, compen-

sation to a supplier is based on realized service outcomes such as equipment uptime or downtime

which is directly related to the value created by the customer through the use of the equipment.

For example, many service level agreements that Internet service providers o¤er stipulate �nancial

remedies for a failure to deliver promised network uptime, which depends on how fast the problem

can be resolved after it is detected and how frequently such problems arise (Stanbury 2004). PBC

is even more pervasive at major federal government agencies including the Department of Defense

2

(DoD), General Services Administration, Department of Transportation, and the National Aero-

nautics and Space Administration (NASA), because of a major push by the White House in recent

years (see Government Accountability O¢ ce 2002 and O¢ ce of Management and Budget 2003).

The DoD, in particular, has initiated a policy called Performance-Based Logistics that mandates

all of its outsourced logistics and services be performance-based (DoD 2003). As in commercial

scenarios government contracts specify performance goals and the ensuing �nancial consequences

of achieving or missing these goals. The following is an excerpt from a service contract for data

processing equipment owned by the U.S. Army (Army Material Command 2006) illustrating how

a typical contract is set up:

If a complete critical system remains inoperative and cannot perform the scheduled

workload due to a product malfunction (system downtime) through no fault or negli-

gence of the Government for a period of 24 contiguous hours, the Contractor shall grant

a credit to the Government for each half-hour of downtime.

Although there is some evidence suggesting that PBC successfully incentivizes suppliers to

meet required performance goals (Geary 2006), there is a �ip side: �nancial risks that accompany

such contracts. Since service outcomes are inherently random (making precise resource planning

and predictable problem resolutions di¢ cult to achieve), �uctuations in the suppliers�contractual

income streams that depend on such outcomes are inevitable. Indeed, a principal motivation for

the customer to adopt PBC is to transfer the risk of output uncertainty to the supplier in the form

of contract payment uncertainty. To many suppliers, this uncertainty is a source of great concern,

as they prefer predictable cash �ow. For example, it is possible under PBC for a supplier to make a

negative pro�t if an unforeseen problems such as a spare parts shortage contributes to a long delay

in completing a required restoration service. Such aversion to revenue risk makes implementing

PBC ine¢ cient, because suppliers demand a risk premium as a condition for entering into a PBC

arrangement. In the airline industry, for instance, high risk premiums demanded by maintenance

service providers have become an important sticking point (Sobie 2007).

In this paper we use a stylized principal-agent contracting framework to model an outsourcing

environment in which equipment failures occur infrequently and where the customer cannot write

the supplier�s capacity investment decision directly into the contract but instead must incentivize

3

the supplier using PBC. We show that there are some unique challenges in such an environment.

Because equipment failures do not occur very often, any signal about the supplier�s capacity in-

vestment that the customer might intuit from the supplier�s delivery of service (which is the basis

of contract payments) is likely to be highly variable. The main theme we explore in this paper

is this interaction between an external condition (imperfect equipment reliability which translates

into rare and random equipment failures) and internal uncertainty (inherently random service out-

comes), as well as their combined e¤ect on the e¢ ciency of PBC. In particular, we are interested

in answering the following questions. What challenges does PBC present for service outsourcing in

an environment where the demands for service, i.e., equipment failures, arrive infrequently? Under

what conditions is PBC most e¤ective? How e¢ cient are commonly observed contracts, and which

is typically preferable and under what circumstances?

To this end, we consider two types of common PBC: a contract based on cumulative downtime

and a contract based on the sample-average of downtime. Under the former, the supplier is penalized

for the total equipment downtime within a contract period, whereas under the latter the supplier

is penalized for the total downtime divided by the number of equipment failure incidents. Both

are designed to achieve the same goal, namely, to incentivize the supplier to reserve a high level of

service capacity so that restoration can be completed quickly upon equipment failure. Despite the

common goal, however, we �nd that the two performance measures used in these contracts create

completely di¤erent risk pro�les, and thus may induce very di¤erent responses from the supplier in

terms of his capacity decision. For example, one would expect that, for a given downtime penalty,

the supplier will build more capacity when the equipment is more prone to failure. This is indeed

the case for the contract based on cumulative downtime, but when the contract is based on average

downtime, the supplier�s optimal capacity investment is non-monotone in equipment failure rate (it

actually increases and then decreases). We identify conditions under which one contract is more

e¢ cient than the other, and show that the contract based on average downtime is superior when

service time variability is not too high. Finally, we �nd that both PBC types are most costly

to implement when the equipment is most reliable. This paradoxical situation arises because the

quality of information about the supplier�s performance is worst when there are fewest opportunities

to perform, i.e., when the equipment almost never fails. Therefore, high reliability does not always

mean good news �when fast restoration is as important as high reliability, infrequent failures make

4

it challenging to implement PBC successfully and other contracting means might become necessary.

Our �ndings are consistent with industry reports indicating disagreement among customers and

suppliers on the appropriate risk premium required for PBC.

The rest of the paper is organized as follows. After a brief survey of related literature, we

lay out modeling assumptions in Section 3. In the next section we consider two benchmark cases:

the �rst-best and the deterministic equipment failures case. In Section 5, we present an in-depth

analysis of the two types of contracts assuming that a standard on expected service time after each

failure incident has been established. In Section 6 we study alternative objectives for the customer.

Finally, in Section 7, we summarize the major �ndings and discuss future directions for research.

2 Literature Review

Our model applies the principal-agent theory to a service outsourcing environment. The most

closely related service operations literature concerned with modeling delays and waiting times has

traditionally focused on queuing problems, i.e., settings in which server utilization is relatively high.

There, economic decisions are made on how to manage congestion, either by adding more servers

or by changing the rate of service. Examples of articles that consider principal-agent relationships

with signi�cant queuing e¤ects include Gilbert and Weng (1998), Plambeck and Zenios (2003), Lu

et al. (2007), Ren and Zhou (2007), and Hasija et al. (2007). Some of these and related papers

model system behaviors in the heavy-tra¢ c regime, i.e., when server utilization approaches one. In

contrast, our model considers the opposite end of the spectrum, namely, where utilization is close

to zero; the service capacity in our model is generally idle except when responding to infrequent

disruptions, and hence queuing for service is not an issue. The �demand�in our problem context

means infrequent but high-impact disruptions that incur large opportunity costs. As a result, a

high level of service capacity must be maintained to reduce the impact of the disruptions.

Service parts inventory management is an area in which rare equipment failures drive managerial

decisions, just as in our model. Sherbrooke (1992), Muckstadt (2005), and Cohen et al. (1990) give

an overview of theory and applications of this stream of research. For the most part this literature

does not consider contracting or incentives. Exceptions occur in our prior papers (Kim et al.,

2007a,b), which study contracting issues in after-sales product support outsourcing. The current

5

paper is very distinct from our earlier work since, here, we highlight the challenges associated with

PBC that arise due to the infrequent nature of equipment failures and the role of performance

metric speci�cations, a topic that has not been explored before. Furthermore, in the current paper

we focus on the restoration service aspect rather than on inventory management.

Principal-agent theory has become a foundation on which many important issues in operations

management can be analyzed. Our model uses the moral hazard (�hidden action�) framework rely-

ing on the assumption that it is prohibitively expensive to monitor an agent�s action, and therefore

the parties contract on a commonly observable performance measure which is a noisy indicator of

the agent�s action. The classical principal-agent model (see Bolton and Dewatripont 2005) includes

an assumption that the agent�s action is the sole contributor to changing the distribution of the

performance outcome. Such a framework di¤ers from our model in that we have two sources of un-

certainty that determine the performance outcome: random equipment failures and random service

times. While the latter is impacted by the agent�s action, the former is not. Moreover, the classical

framework does not explicitly account for rare performance signals due to high equipment reliability

(for example, it does not allow for no signal at all if the equipment does not fail). There are very few

papers in economics that consider an environment in which performance outcomes are intermittent

and randomly realized. In this respect, the paper that is closest to ours is Abreu et al. (1991). To

investigate the role of information in a repeated partnership game, they construct a model in which

the signal quality of the agents�actions depends on the length of a review period (see also Bolton

and Dewatripont 2005, pp. 436-441, where the same model is interpreted as a machine maintenance

problem). By varying the period length, they show that a feasible incentive-compatible equilibrium

does not exist if the period length is too short, as the performance signal becomes too noisy. We

explore this exact issue in our paper. However, our paper di¤ers from theirs in several important

ways. First, we study a one-period PBC, whereas they consider a repeated game with a �xed-price

contract that is similar to the e¢ ciency wage model of Shapiro and Stiglitz (1984). Second, while

many of our results and insights are driven by the complex interaction between an exogenous un-

certainty (random equipment failures) and an endogenous uncertainty (random service times), no

such interaction exists in the model of Abreu et al. (1991). In particular, our model allows for a

situation in which an agent�s performance signal may not materialize (no equipment failures) which

creates nontrivial measurement issues that, to the best of our knowledge, have not been explored

6

in the classical principal-agent literature.

Finally, our paper is related to the literature on supply chain disruption management. Kleindor-

fer and Saad (2005) provide an excellent overview of issues and studies in this stream of research,

with a list of ten principles of supply chain risk assessment and mitigation. Many examples of dis-

ruption management can be found in She¢ (2005). Similarly, Tomlin (2006) considers dual sourcing

as a mitigation strategy for supply chain disruption risks. While much of the literature focuses on

preventing disruptions, contingency planning in a decentralized supply chain is also recognized as

one of the most important aspects of risk management. Our paper contributes to the latter stream.

To summarize, we believe that our paper is the �rst to address the issue of outsourcing equipment

restoration services to mitigate low-probability, high-impact disruptions.

3 Model Assumptions

A risk-neutral customer (�she�) derives utility from continued usage of equipment, which is subject

to random breakdowns. When a breakdown occurs, the equipment needs to be restored to working

condition as quickly as possible. As the customer lacks technical expertise, she delegates the control

of all restoration activities (which we call a service) including diagnostics, parts replacement, and

repairs and testing, to a single supplier (�he�), who is risk-averse. The customer and the supplier

establish a contractual relationship for the duration of one time period (e.g., a year) whose length

is normalized to one. In practice the length of the contracting period is typically tied to the annual

budgeting process and cannot be easily extended. In the beginning of the period the customer o¤ers

a contract to the supplier. In response, the supplier decides how much he should invest in service

capacity. The supplier�s investment is unobservable and non-contractible. Over the length of the

contract period, random equipment failures occur, triggering service activities by the supplier. At

the end of the period, the customer assesses the supplier�s performance based on service completion

times, and payments are made according to the agreed contract terms.

3.1 Equipment Failure Process and Capacity Decision

Equipment failures occur according to a Poisson process with a rate �; which is common knowledge.

While it may be hard to estimate � for new equipment, we assume that there is su¢ cient experience

7

with it to allow for accurate estimation.1 Let N be the random variable representing the number

of equipment failures within the period. In this paper we consider low � values, i.e., � near zero

up to a single digit. In our work with government suppliers and telecommunications companies we

routinely observe such low failure rates for mission-critical equipment. Each incident of equipment

failure triggers a service process performed by the supplier which may include traveling to the

customer�s site, diagnosing the failure, shipping the necessary part and/or repairing failed parts,

installing replacement parts, and testing the equipment. Let Si be the service completion time or,

equivalently, the equipment downtime for the ith failure. For consistency, we refer to it throughout

the paper as the service time. We assume that fSig are i.i.d. with a rate 1=�, where � is the

service capacity set by the supplier at the beginning of the period. We assume that � remains

unchanged throughout the period. Although situations arise in which the capacity level can be

dynamically altered, we focus on the cases in which it is too costly or impractical to do so. Repair

facility purchase/lease, employee training, and process design are some of the examples in which

capacity decisions have a lasting impact. We use the superscript � to denote the supplier�s optimal

response (i.e., capacity choice) to a contract. Since fSig are i.i.d., we drop the subscript i whenever

no confusion arises. We use the convention that � is bounded below by �, which we interpret

as the default level of capacity that the supplier already possesses. We assume that 1=� � 1=�,

i.e., the maximum expected service completion time is much shorter than the mean time between

equipment failures.2 For example, while a network server may go down once or twice a year, repair

standards as low as several hours are common (Cohen et al. 2006).

We assume that expanding capacity beyond � comes at an extra cost c(�� �), i.e., this cost is

linear in � with a unit cost c. Other papers on service operations frequently assume linear service

capacity cost (for example, see Allon and Federgruen 2007). While costly, increased capacity brings

the bene�t of reduced expected service time. We further assume that the variability of service time,

1 In many situations estimates of equipment failure rates at a particular customer location are based on combiningfailure rate observations based on the global equipment population with local information on the size of the installedbase of equipment at the site.

2Note that, in general, � is not constant, since the product failure interarrival times depend on how fast servicesare completed. However, assuming 1=� � 1=� allows us to approximate � to be constant. In addition, the sameassumption makes it unlikely to encounter a situation in which an initiated service is not completed by the end ofthe period, since the probability that a failure occurs within a time interval of order 1=� before the end of the periodis negligible. Similar assumptions on scale di¤erences are frequently made in the service parts management literature(see Muckstadt 2005).

8

de�ned as the coe¢ cient of variation v(�) of S, possesses the following three properties: it does

not increase when capacity increases (v0(�) � 0), there is a decreasing rate of variability reduction

(v00(�) � 0), and �nally that jv0(�)j is bounded above. Despite these restrictions on the shape of

v(�), our assumptions are quite general in that they presuppose no speci�c form of service time

distribution. If the distribution of S is assumed to stay within the same parametric distribution

family after � is changed, such well-known distribution families as exponential and gamma with

a constant shape parameter satisfy these properties (as v(�) is constant for both). However, our

assumptions allow for more general cases in which the distribution is altered from one family to

another. In the following discussions, we will frequently revisit the special case in which v(�) is

constant since it allows for tractable analysis and also coincides with the exponential service time

assumption that is commonly found in the literature.

3.2 Service Requirement and Customer Objective

The customer�s goal is to minimize total service outsourcing cost subject to a service constraint

de�ned as

E[S j��] = 1=�� � sm: (1)

Service constraints are routinely found in practice, as well as in the literature (in particular, most

service parts inventory management problem formulations include such constraints; see Sherbrooke

1992 and Muckstadt 2005). The constraint (1) expresses the customer�s desire to ensure that the

expected downtime after each equipment failure incident is less than or equal to a target threshold

value sm. By de�ning its reciprocal as �m � 1=sm, we can rewrite (1) as �� � �m. We assume

that �m > � so that (1) can be satis�ed only if the supplier expands capacity beyond the default

level �. The service constraint of the form (1) is used by organizations that put a great weight on

individual service outcomes. Even in cases where (1) is not explicitly imposed, it can be interpreted

to represent the utility function of the customer who assigns a large cost when the expected service

time standard is violated. (The constraint�s shadow price represents that cost.)

In some situations, the customer has a clear estimate of the opportunity cost incurred while

equipment is down and chooses contract terms that minimize her expected lost value plus the

expected contracting cost, without the service constraint of the form (1). This case will be analyzed

9

in Section 6. We believe, however, that employing (1) and focusing on the problem of contracting

cost minimization subject to a per-incident downtime standard makes practical sense because (a) it

is often very di¢ cult to quantify the opportunity cost, and (b) a pro�t-maximizing customer may

face poor service deliveries when the failures are infrequent, as we will show in Section 6. Indeed,

a customer organization such as the U.S. Air Force uses a service constraint similar to (1) in order

to ensure a high level of mission-readiness when called into action at a random point in time (Slay

et al. 1998).3

3.3 Contracting

We assume that the customer cannot directly observe the supplier�s capacity choice �. This is a

reasonable assumption since most suppliers exert a multitude of discretionary e¤orts (such as deci-

sions on spare parts inventory investment, repair depot sta¢ ng, training, transportation methods,

etc.) that are di¢ cult or costly to monitor but that ultimately in�uence how fast the customer�s

equipment can be restored. In the concluding section we will address the issue of when such moni-

toring, though very costly, might still be desirable. Because she cannot contract on �, the customer

enforces her service requirement via a performance-based contract such that the compensation T to

the supplier is tied to an agreed-upon performance measure (e.g., equipment downtime), generically

denoted by X. In this paper we focus on linear contracts that have the payment form T = w�pX,

in which w is the �xed payment independent of the realized performance X, and p is the penalty

rate for each unit of X. The linear form is motivated by practice, in which nonlinear contracts are

rarely observed.4

We consider two types of linear contracts in this paper. The �rst is based on cumulative

downtimePNi=1 Si and is referred to as a CC (cumulative-performance contract). The second

is based on average downtime (PNi=1 Si)=N and is referred to as an AC (average-performance

contract). Subscripts c and a will be used to distinguish between the two. The precise expression

3The U.S. Air Force in fact uses a con�dence interval constraint, i.e., constraint of the form Pr(S > s� j��) � �.Using this constraint, however, does not change any of our results since it can also be converted to a constraint ofthe form �� � �� , where �� is determind from the distribution of S and the value of s� .

4We acknowledge that sometimes linear contracts are modi�ed by a ceiling and a �oor (i.e., maximum and minimumpayments), essentially making them nonlinear contracts. While it would be an interesting exercise to study the impactof these modi�cations, we choose not to since doing so would divert attention from the main tradeo¤s we wish toanalyze.

10

of T will be de�ned in subsequent sections for each case we consider there. We are interested in

these two contract types because both are widely used in practice. Contracts such as the Army

example in the introduction are of the CC type, but AC-type contracts are quite common as

well since many organizations keep track of average downtime as a key performance indicator (see

Gibson 2004). However, there is little understanding as to which one should be used under which

circumstance. A main theme of this paper is to compare the consequences of implementing these

two contracts.

It should be noted that �average� in AC refers to the sample-average of fSig, as opposed to

the time-average, which in fact applies to CC. This last statement follows from the fact that we

have normalized the length of the contracting period to one: the time-average of total downtime

is equal to cumulative downtime since the former is obtained from dividing the latter by one. In

this sense, a comparison between CC and AC can also be viewed as a comparison between two

di¤erent methods of averaging the supplier�s performance. As we will �nd out in Section 5, such

seemingly innocuous choices for evaluating the supplier�s performance may lead to surprisingly

di¤erent results.

The risk-averse supplier is assumed to have a mean-variance utility function that depends on

stochastic compensation T that he receives from the customer:

u(�) = E[T j�; �]� �Var[T j�; �]� c(�� �): (2)

The parameter � is the coe¢ cient of risk aversion. A larger � corresponds to higher risk aversion and

� = 0 represents risk neutrality. We employ the risk aversion assumption to capture the supplier�s

desire to avoid revenue risks posed by PBC. The ine¢ ciency arising from such aversion to risk is

expressed in the second term of (2) and is called the risk premium (see Gollier 2001, p. 20), denoted

by � � �Var[T j�; �]. Our close work with managers in the aerospace and defense industry reveals

that revenue risk is an issue that most concerns them about PBC. At the same time, they express

the need to quantify risk premium for contract negotiation purposes. Therefore, we believe that

the assumption of risk aversion re�ects business realities.5 Revenue risk is especially salient when

5On the other hand, we assume that the customer is risk neutral on the grounds that she represents a largerenterprise, such as the DoD, that is less sensitive to cash �ow risk. Note that her concern about performance risk,i.e., long downtimes, is re�ected in the service constraint (1). All our results remain qualitatively unchanged if both

11

equipment failures are infrequent, because information about the supplier�s performance is quite

limited in such an environment. Our discussions below will further explain this point. The mean-

variance function captures the basic expected revenue vs. revenue risk tradeo¤ for the supplier, and

is widely adopted in the recent operations management literature (for example, see Tomlin 2006,

Van Mieghem 2007, and Kim et al. 2007a).

After being o¤ered contract terms, the supplier chooses capacity �� to maximize his utility

(2). Anticipating the supplier�s choice of capacity, the customer decides on contract terms that

(a) minimize her total compensation to the supplier, (b) induce the supplier to voluntarily choose

�� that satis�es the service constraint (1), and (c) ensure that the supplier participates in the

contractual relationship. The last requirement is expressed in terms of the individual rationality

(IR) constraint u(��) � u, where u denotes the supplier�s reservation utility, i.e., the level of utility

that he obtains if he opts out of the contract. We normalize u to zero throughout this paper,

because doing so changes no qualitative insights. As is typical in most principal-agent models, the

IR constraint will turn out to be binding in all cases we analyze. As a consequence, the customer�s

optimal cost, denoted by , will equal the supply chain�s total cost, because u = 0.

4 Benchmark Cases

Before analyzing the model we have laid out thus far, we consider two cases whose results serve as

benchmarks and useful building blocks. After presenting the solution under the �rst-best scenario,

we analyze a special case in which there is no uncertainty in equipment reliability, i.e., the equipment

failure process is deterministic. Although this assumption is clearly unrealistic (e.g., �the equipment

breaks down exactly once every 120 days�), we examine its rami�cations because they highlight

the special role of exogenous uncertainty (due to equipment failures) in our model. As we will

�nd out in Section 5, the uncertainty in the failure process and the uncertainty in the supplier�s

performance interact to create interesting dynamics, which can be contrasted to the results derived

in this section.

parties are risk averse but if the supplier is more so.

12

4.1 The First-Best Solution

Under the �rst-best solution, the customer can contract directly on the supplier�s capacity choice

� so that no e¢ ciency is lost. She can o¤er a �xed payment T = w that guarantees the supplier�s

participation and solves the problem

minw;�

E[T j�; �] subject to E[S j�] � 1=�m and u(�) � 0:

The solution is straightforward. The customer o¤ers wFB = c(�m��) that reimburses the supplier�s

capacity cost and sets �FB = �m. As a result, the customer incurs the expected total cost FB =

wFB = c(�m � �), which also equals the supply chain�s cost.

4.2 Deterministic Equipment Failure Process

Let us assume that the interarrival times of equipment failures are constant and equal to 1=�. For

simplicity�s sake, let us also assume that the �rst failure occurs 1=� time units after time zero.

Then the number of failures within the period is n = b�c, where the bracket denotes the greatest

integer less than or equal to �.6 We consider only situations in which � � 1, since restoration

services are not needed when the equipment does not fail and there will be no contracting if � < 1.

The following lemma speci�es how the supplier chooses his capacity given contract terms (w; p)

under CC and AC.

Lemma 1 The supplier�s utilities are concave under both contracts. De�ne

�(�) � v(�)2 � �v(�)v0(�) � 0: (3)

Under CC and AC, the supplier chooses ��dc > �m and ��da > �m, respectively, that satisfy the

following optimality conditions

[CC]p

�2+2�p2

�3�(�) =

c

b�c and [AC]p

�2+2�p2

b�c�3 �(�) = c,

6The consideration of continuous values of � in a deterministic setting forces us to encounter situations in whichan initiated service is not completed within the period, as we can always choose � so that the last arrival is arbitrarilyclose to the end of the period. We will not deal explicitly with these special cases, however, because they areinconsequential to our analysis.

13

provided that p�s are su¢ ciently large to admit interior solutions. In both cases, @��j=@c < 0,

@��j=@p > 0, and @��j=@� > 0, for j 2 fdc; dag. Moreover, ��dc is nondecreasing in � whereas ��da

is nonincreasing in �.

Intuitively, the quantity �(�) in (3) is the normalized rate at which the supplier�s revenue risk

is reduced (i.e., changes in Var[T j�; �]) as a result of increasing � by one unit. It turns out

that identical values of �(�) apply to both CC and AC. The �rst term v(�)2 in �(�) represents

the portion of risk reduction rate due to lower expected service time E[S j�], whereas the second

term ��v(�)v0(�) � 0 represents a similar quantity due to reduced variability in S. Under the

assumptions made regarding the shape of v(�), it can be easily veri�ed that �0(�) � 0, which implies

that it becomes more di¢ cult to reduce the revenue risk as capacity grows larger.

We can explain the supplier�s optimal capacity choice ��j in response to variations in parameters

c, p, and � as follows. The supplier�s incentive to invest in capacity is higher when (a) the unit

capacity cost is lower (@��j=@c < 0), (b) the performance incentive is higher (@��=@p > 0), or

(c) the supplier is more risk-averse (@��j=@� > 0). The �rst two results are intuitive. The third

result arises from the fact that increased capacity leads to reduced service time uncertainty, as

Var[S j�] = v(�)2(E[S j�])2 decreases in �. In other words, a risk-averse supplier can hedge against

revenue risk by taking a preventive measure, namely, by increasing capacity �. This e¤ect becomes

more pronounced the more risk-averse the supplier.

The key result from Lemma 1 is that ��dc and ��da move in opposite directions as more failures

occur (i.e., for larger n = b�c). Under CC, there are two reasons for ��dc to be a nondecreasing

function of �. First, given a �xed penalty rate p, more equipment failures imply greater expected

total penalties for the supplier, as the contract stipulates that he loses money each minute the

equipment is down. To avoid such losses, the supplier increases capacity �. Second, since the

revenue risk increases with more failures (since Var[Pni=1 Si j�; �] = nv(�)2=�2 is proportional to

n), the risk-averse supplier strives to avoid this uncertainty by increasing capacity even further,

because doing so reduces the uncertainty in his performance. Combined, these two e¤ects induce

the supplier to choose higher capacity with more equipment failures.

When AC is used, however, the supplier reacts in a completely di¤erent and counterintuitive

manner to a change in �. Since the supplier is compensated based on the average downtime

14

(Pni=1 Si)=n, his expected contract payment is independent of how many failures occur, which is in

sharp contrast to what we observed under CC conditions. At the same time, the supplier bene�ts

from sampling variance reduction: the more failures occur (larger n), the less uncertain his average

performance is. Therefore, the supplier becomes less concerned about his revenue risk and is more

willing to gamble by choosing lower capacity. From this discussion, we witness the �rst evidence

that CC and AC can lead to very di¤erent consequences, depending on equipment characteristics

(represented by �), even though both contracts are designed to achieve the same goal: to give the

supplier incentives to reserve a high level of capacity.

Anticipating these supplier responses, the customer chooses the pair of contract terms (w; p)

that solves her optimization problem

minw;p

E[Tj j�; ��j ] subject to E[S j��j ] � 1=�m and u(��j ) � 0;

whose solution is as follows. The optimal solutions are denoted by the superscripts DCC and DAC

for CC and AC, respectively.

Proposition 1 (Optimal contracts under deterministic failure process) Suppose that

�(�) + ��0(�) � 0 for � � �m. (4)

Then the service constraint �� � �m binds at the optimal solution. De�ne vm � v(�m), �m � �(�m),

and

Qjm(�) � 1 +q1 + 8�c�mV

jm(�); (5)

where V jm(�) = �m= b�c for j 2 fDCC;DACg. Under CC and AC, the customer o¤ers penalty

rates pDCC and pDAC , respectively, that satisfy pDCC = pDAC= b�c = 2c�2m�b�cQDCCm (�)

��1. In

equilibrium, the supplier chooses �DCC = �DAC = �m and is left with zero utility under both

contracts. The resulting expected costs for the customer are identical with the same risk premiums

�j =c�m2

�v2m�m

� 1� 2

Qjm(�)

!, j 2 fDCC;DACg: (6)

15

The condition (4) is trivially satis�ed when v(�) is constant, as is the case if the distribution of

service time is exponential or gamma with a constant shape parameter and stays within the same

family when � changes. The literature makes such an assumption frequently (for example, see Ata

and Shneorson 2006). Note that (4) is a su¢ cient condition that is not tight, in that the service

constraint will continue to bind in many instances in which the condition is violated.

As is well known from the principal-agent literature, the �rst-best solution cannot be achieved

with PBC when risk aversion is present. The deviation from the �rst-best solution results in inef-

�ciency captured in the risk premiums �DCC and �DAC . Surprisingly, they are identical according

to (6) despite di¤erent supplier reactions to the two contracts that we found above. The key to

understanding this result is revealed by examining the coe¢ cients of variation (CV) of the two

performance measuresPni=1 Si and (

Pni=1 Si)=n:s

Var [Pni=1 Si j�; �]

(E [Pni=1 Si j�; �])2

=

sVar[(

Pni=1 Si)=n j�; �]

(E[(Pni=1 Si)=n j�; �])2

=v(�)pb�c

: (7)

That is, the two contracts result in exactly the same variability, measured by CV. After optimal

decisions are made, (7) is modi�ed toqV jm(�) =

p�m= b�c, j 2 fDCC;DACg, which remains

identical for both contracts because the same capacity �m is chosen in equilibrium. Risk premiums

�DCC and �DAC are the same too since they are completely determined by the common value of

V jm(�) (see (6)) despite the di¤erence in the optimal penalty rates: pDCC = pDAC= b�c. Thus, we

conclude that CC and AC result in the same supply chain ine¢ ciencies when the equipment failure

process is deterministic. Although other interesting observations can be made from Proposition 1,

we postpone discussion of them to Section 5.3, where we make thorough comparison between CC

and AC.

5 Model Analysis

Having analyzed benchmark cases, we now return to the model that we outlined in Section 3. As

will be described below, many of the solution features of Section 4.2 will carry over to situations

in which the equipment fails randomly. However, there are some surprising deviations, which we

highlight in this section. We present derivations of optimal contract parameters for CC and AC

16

in separate subsections (Sections 5.1 and 5.2) as they require substantially distinct analyses. In

Section 5.3, we compare the solutions of the two contracting problems.

5.1 Cumulative-Performance Contract

When the equipment failures occur randomly, there is always a positive probability that the equip-

ment fails at least once even if � < 1. Hence, the customer contracts with the supplier whenever

� > 0, in contrast to the deterministic case in which the contractual relationship is established

only when � � 1. Accordingly, the domain of � that we analyze in this section is enlarged. The

performance measure that the supplier is evaluated against under CC is the cumulative downtimePNi=1 Si. Since the number of arrivals N is Poisson-distributed, it is a compound Poisson random

variable (see Ross 1996, pp. 82-89) with the mean and the variance

E[PNi=1 Si j�; �] = E[N j�]E[S j�] = �=�; (8)

Var[PNi=1 Si j�; �] = Var[N j�](E[S j�])2 + E[N j�]Var[S j�] = �

�1 + v(�)2

�=�2: (9)

Both the mean and the variance are proportional to �, which is a direct consequence of the Poisson

failure rate assumption. Notice that the variance has two components: the variance of the service

time S (scaled by E[N j�]) and the variance of the number of equipment failures N (scaled by

(E[S j�])2). The latter was absent in the deterministic failure process case. As we will see later,

this additional variance term plays a key role in di¤erentiating the e¤ectiveness of CC from that of

AC. The solution to the supplier�s capacity decision problem is similar to that speci�ed in Lemma

1.

Lemma 2 The supplier�s utility is concave under CC. The supplier chooses ��c > �m that satis�es

the optimality conditionp

�2+2�p2

�3(1 + �(�)) =

c

�; (10)

provided that p is su¢ ciently large to admit interior solutions. Moreover, @��c=@c < 0, @��c=@p > 0,

@��c=@� > 0, and @��c=@� > 0.

Comparing the �rst-order condition (10) to its deterministic counterpart in Lemma 1, we see

that there are two changes: b�c is replaced by � and �(�) is replaced by 1+�(�). The additional term

17

1 in front of �(�) comes from Var[N j�], which was absent in the deterministic case. Comparing

the two optimality conditions, it is apparent that ��c > ��dc for � � 1 (the domain on which ��dc

is de�ned), as illustrated in Figure 1(a). In other words, the supplier chooses higher capacity

level than he would if the failures occurred deterministically, because he wants to hedge against

an additional uncertainty originating from stochastic equipment failures. The sensitivity analyses

listed in Lemma 2 are analogous to those in Lemma 1, and the same intuitions underlie them.

The customer�s contract design problem also proceeds as before and the solutions are analogous

to those in Proposition 1.

Proposition 2 (Optimal cumulative-performance contract) Assume that the condition (4) is sat-

is�ed. The customer o¤ers the penalty rate pCC = 2c�2m��QCCm (�)

��1, where QCCm (�) is de�ned as

(5) with V CCm (�) = (1 + �m) =�. In equilibrium, the supplier chooses �C = �m and is left with zero

utility. The resulting expected cost for the customer is C = c(�m � �) + �CC , where

�CC =c�m2

�1 + v2m1 + �m

��1� 2

QCCm (�)

�: (11)

Notice that the CV under CC,pV CCm (�) =

p(1 + �m) =�, is greater than the one we found

in Proposition 1 for the deterministic case,pV DCCm (�) =

p�m= b�c. Again, this is due to the

additional uncertainty introduced by a stochastic failure process. We will postpone examining of

pCC and �CC to Section 5.3, after we complete the solution speci�cations of the average-performance

contract case.

5.2 Average-Performance Contract

The average downtime, or the sample mean estimator, is de�ned as

bS � (PNi=1 Si)=N jN > 0: (12)

The condition N > 0 is necessary since average downtime is not de�ned when N = 0. The

performance measure under AC is, then, bS1(N > 0), which quanti�es the performance as zero if

N = 0 but (PNi=1 Si)=N otherwise (1(�) is an indicator variable). Its mean and the variance are

evaluated as follows.

18

(a) µ* under the CC for fixed p (b) µ* under the AC for fixed p

v = 1.5

v = 1.5

v = 0.5

v = 0.5

(a) µ* under the CC for fixed p (b) µ* under the AC for fixed p

v = 1.5

v = 1.5

v = 0.5

v = 0.5

Figure 1: Examples showing how ��j , j 2 fdc; dag varies as a function of �. The dotted linesrepresent ��j when service demand process is deterministic. v(�) is assumed to be constant, withthe values speci�ed in the �gures.

Lemma 3

E[bS1(N > 0) j�; �] = Pr(N > 0)E[S j�] = (1� e��)=� and (13)

Var[bS1(N > 0) j�; �] = Pr(N > 0)�Pr(N = 0)(E[S j�])2 +�(�)Var[S j�]

�= (1� e��)

�e�� +�(�)v(�)2

�=�2; (14)

where

�(�) � 1

e� � 1

1Xn=1

�n

n!

1

n(15)

has the following properties:

(i) �0(�) < 0; (ii) lim�!0

�(�) = 1; (iii) lim�!1

�(�) = 0;

(iv) �(�) >1� e���

; and (v)d

d�

�e��

�(�)

�< 0:

Notice that both (13) and (14) are scaled by Pr(N > 0) because bS1(N > 0) is de�ned condition-

ally. In particular, they approach zero as �! 0 since the chance of observing the supplier�s service

delivery performance becomes very small when the equipment is unlikely to fail. The factor �(�)

that appears in (14) is the sampling coe¢ cient of Var[bS j�; �] when sample count N is Poisson-

19

� 1 2 3 4 5 6 7 8 9 10�(�) 0.767 0.577 0.433 0.330 0.258 0.208 0.172 0.147 0.128 0.113

e��=�(�) 0.480 0.235 0.115 0.056 0.026 0.012 0.005 0.002 0.001 4�10�4

Table 1: Numerical values of �(�) and e��=�(�) for � = 1; :::; 10.

distributed.7 As is intuitive, �(�) is a decreasing function since a larger sample size is more likely

with higher � (result (i)). Part (iv) provides the lower bound on �(�), which we will use later to

derive a key insight. Part (v) indicates that �(�) decreases at a slower rate than exponential, a

property that will be useful in later analyses. In fact, the ratio e��=�(�) converges to zero very

quickly as � increases (see Table 1).

Given (13) and (14), the supplier�s capacity choice is speci�ed similar to our earlier result.

Lemma 4 The supplier�s utility is concave under AC. The supplier chooses ��a > �m that satis�es

the �rst-order conditionp

�2+2�p2

�3

�e�� +�(�)�(�)

�=

c

1� e�� ; (16)

provided that p is su¢ ciently large to admit interior solutions. Moreover, @��a=@c < 0, @��a=@p > 0,

and @��a=@� > 0. Also, @��a=@� > 0 for � � 0 but @��a=@� < 0 for su¢ ciently large � for which

e��=�(�) � 0.

The sensitivity results for ��a are similar to those of its deterministic counterpart ��da shown in

Lemma 1, except for the one with respect to � which exhibits non-monotonicity.8 See Figure 1(b)

for examples demonstrating this behavior. Recall from Lemma 1 that, when the equipment failure

process is deterministic, ��da is nonincreasing in � for �xed p. Such behavior was driven by sampling

variance reduction achieved by AC. The same e¤ect is present when the failure process is stochastic,

7Using the identity for exponential integral (see Lebedev 1972)

Ei(�) �Z �

�1

ex

xdx = + ln�+

1Xn=1

�n

n!

1

n;

where = 0:57721::: is the Euler gamma constant, �(�) can be alternatively de�ned as

�(�) � Ei(�)� ln�� e� � 1 :

8Although we are unable to show it analytically due to a complex representation of �(�), extensive numericalexamples indicate that ��a is unimodal in �, i.e., there is a unique �0 for which @�

�a=@� > 0 for � < �0 and @�

�a=@� < 0

for � > �0.

20

but only if the probability of having at least one equipment failure is high; but this insight no longer

holds when � � 0. That is, given that there is a high chance that the equipment never fails, the

supplier reacts in the opposite way, i.e., he increases capacity in response to higher �. The reason

is that there is little bene�t of sampling variance reduction when � � 0. Instead, the supplier under

AC mimics the behavior under CC, as the performance measures become indistinguishable near

� = 0:PNi=1 Si � S11(N = 1) and bS1(N > 0) � S11(N = 1). We call this mimicking behavior

around � = 0 under AC a no-failure e¤ect. Note that this e¤ect was absent in the deterministic

case since contracting occurred only when n = b�c � 1.

We also observe from Figure 1(b) that ��a is not always greater than ��da, the deterministic

counterpart, unlike under CC for which we have found ��c > ��dc. In particular, we see from the

�gure that ��a < ��da is encountered at the lower end of � (around � = 1, the lower bound in the

deterministic case). This is another manifestation of the no-failure e¤ect: facing the high likelihood

that the equipment never fails, the supplier�s incentive to increase capacity is not strong since he

may never be evaluated at all. Such a situation never arose in the deterministic case.

The next proposition shows the equilibrium solutions after we solve the customer�s cost mini-

mization problem.

Proposition 3 (Optimal average-performance contract) Assume that the condition (4) is satis�ed.

The customer o¤ers the penalty rate pAC = 2c�2m�(1� e��)QACm (�)

��1, where QACm (�) is de�ned

as (5) with V ACm (�) = e��+�(�)�m1�e�� . In equilibrium, the supplier chooses �A = �m and is left with

zero utility. The resulting expected cost for the customer is A = c(�m � �) + �AC , where

�AC =c�m2

�e�� +�(�)v(�m)

2

e�� +�(�)�(�m)

��1� 2

QACm (�)

�: (17)

Recall from Proposition 1 that, in the deterministic failure process case, contract parameters

under CC and AC di¤ered at most by a scale factor, as pDCC = pDAC= b�c and �DCC = �DAC .

When the failure process is stochastic, however, the relationships are more complex, as apparent

from the expressions for pAC and �AC in Proposition 3 compared to pCC and �CC in Proposition

2. In particular, �CC 6= �AC ; the e¢ ciencies of CC and AC cease to be the same. In the next

subsection, we investigate what contributes to these di¤erences.

21

5.3 The Comparison of Optimal Contracts

The changes in pj and �j , j 2 fCC;ACg, with respect to parameters c, �m, and � are qualitatively

the same across the contracting scenarios. Under both contracts, (i) @pj=@c > 0, @pj=@�m > 0,

@pj=@� < 0, and (ii) @�j=@c > 0, @�j=@�m > 0, @�j=@� > 0. In other words, the customer has

to o¤er the higher incentive (pj) to induce the target capacity �m as (a) the unit capacity cost

goes up, (b) the service constraint is tightened, and (c) the supplier becomes less risk-averse. The

last result stems from the fact that a less risk-averse supplier is more willing to take a chance on a

fortuitous performance outcome, i.e., realization of shorter service time. Similarly, e¢ ciency loss in

the supply chain, as measured by the risk premium �j , increases as (a) the unit capacity cost goes

up, (b) the service constraint is tightened, and (c) the supplier becomes more risk-averse. These

results are in line with intuition.

Next, we investigate how optimal contract parameters behave as a function of the equipment

failure rate �. We focus on this sensitivity analysis because one of the primary goals of this paper

is to compare the performance of these incentive schemes under varying degrees of equipment

characteristics, represented by equipment reliability 1=�. Unlike with other parameters, varying �

brings solution behaviors that are quite di¤erent across CC and AC. One of them, namely, the

opposite signs of @pj=@�, was alluded to in the discussion of Proposition 1, when a deterministic

failure process was considered. If equipment failures are random, however, the di¤erences are even

more pronounced. The comparisons are summarized in the following proposition.

Proposition 4 (Comparison of optimal contracts with respect to changes in �)

(i) @pCC=@� < 0, whereas @pAC=@� < 0 for � � 0 but @pAC=@� > 0 for su¢ ciently large � for

which e��=�(�) � 0.

(ii) lim�!0 pCC = lim�!0 pAC =1.

(iii) �CC � �AC if V CCm (�) � V ACm (�).

(iv) @�CC=@� < 0 and @�AC=@� < 0.

(v) lim�!0 �CC = lim�!0 �AC =c�m2

�1+v2m1+�m

�.

22

(a) pCC under the CC (b) pAC under the AC

v = 1.5v = 1.5

v = 0.5 v = 0.5

(a) pCC under the CC (b) pAC under the AC

v = 1.5v = 1.5

v = 0.5 v = 0.5

Figure 2: Examples showing how pC and pA vary as a function of �. The dotted lines representpDC and pDA, the deterministic counterparts of pC and pA. As in Figure 1, v(�) is assumed to beconstant in these examples.

Let us examine these results one by one.

Changes in optimal penalty rates as a function of �. We observe that, while pCC is

monotonically decreasing in �, pAC decreases initially but goes up as failures occur more frequently.

See Figure 2 for illustrations. Such behaviors re�ect the supplier�s di¤ering responses to changes

in � under the two contracts, as discussed in Section 5.2. Under CC, as we found out, the supplier

tends to choose higher capacity when more equipment failures are likely; the customer can then

utilize this voluntary action to reach the target capacity �m without providing a strong contractual

incentive, i.e., without o¤ering a high penalty rate p. However, the same logic does not hold when

AC is used, precisely because the supplier�s capacity choice ��a in response to � exhibits non-

monotonicity, as explained in Section 5.2. Therefore, @pAC=@� is non-monotonic as was @��a=@�

but in the opposite direction, since, again, the customer�s goal is to induce the target capacity �m.

This non-intuitive solution behavior is a direct consequence of the no-failure e¤ect, for which

there is no equivalent in the classical principal-agent literature. Virtually all principal-agent models

assume the existence of at least one observation of the agent�s performance. However, in our model,

the supplier�s performance may or may not be observed, because it takes place only when an

exogenous event (i.e., equipment failure) occurs. When it is likely that no performance is realized,

the usual insights may be reversed, as we have shown here.

23

Figure 3: Risk premiums when v = 1.

πCC

πAC

πDCC = πDAC

v = 1 CC moreefficient

AC moreefficient

Figure 4: Regions on the (λ,v) planewhere one contract dominates the other,when v(µ) is constant.

Figure 3: Risk premiums when v = 1.

πCC

πAC

πDCC = πDAC

v = 1 CC moreefficient

AC moreefficient

Figure 4: Regions on the (λ,v) planewhere one contract dominates the other,when v(µ) is constant.

CC moreefficient

AC moreefficient

Figure 4: Regions on the (λ,v) planewhere one contract dominates the other,when v(µ) is constant.

From a practical point of view, this analysis suggests that contracting decisions must be made

carefully: incentive structures can be completely altered depending on equipment characteristics

(i.e., reliability) and the performance metric speci�ed in a contract. In the cases of CC and AC,

they end up creating very di¤erent supplier responses.

Optimal penalty rates in the � ! 0 limit. Both pCC and pAC approach in�nity in the

� ! 0 limit: when equipment failures are extremely unlikely the supplier has little incentive to

invest in capacity because the chance of his being penalized for poor service time realization is very

small. To convince the supplier otherwise and to induce the target capacity �m, the customer must

threaten him with a very high penalty rate.

Relative magnitudes of risk premiums. We �nd that �CC and �AC are not equal in

general, in contrast to the deterministic case where we have found that �DCC = �DAC (Proposition

1). See Figure 3 for an example with v = 1. We saw earlier that the reason behind �DCC = �DAC

was that CVs inherent in CC and AC were identical under the deterministic failures. Part (iii)

suggests, then, that the CVs di¤er from each other when failures are random, and that it is this

di¤erence that causes �CC 6= �AC . In fact, we can gain greater insight into when one contract leads

to a more e¢ cient outcome than another by considering a special case in which v(�) is constant.

Corollary 1 Suppose that v(�) does not vary in � and let v � v(�). De�ne

!(�) �

s(1� e��)=�� e���(�)� (1� e��)=�: (18)

Then �CC � �AC i¤ v � !(�).

24

According to Corollary 1, the customer (and hence the supply chain) can achieve better e¢ ciency

with AC than with CC if and only if v, the variability of service time, is less than or equal to !(�).

Although a complete analytical description of the shape of !(�) is intractable, numerical plotting

shows that it is a convex function that has a unique minimizer at � = 0:79 with !(�) = 1:4 at that

point (see Figure 4). For � > 0:79, !(�) is an increasing function. Therefore, AC always performs

better than CC whenever v < 1:4, regardless of �. Considering that the coe¢ cient of variation of

1.4 is a very large number for most well-known distributions, and that allowing v(�) to decrease

with � has a larger impact on reducing �AC than �CC (as can be veri�ed by numerical examples),

we conclude that AC is preferable unless the service time S exhibits extremely large variability.

What drives AC to be more e¢ cient when the service time S does not vary too much? Why is

CC more e¢ cient in some cases? We �nd the key insight by separately analyzing the two additive

terms that appear in each CV,pV CCm (�) and

pV ACm (�):

V CCm (�) =1

�+1

�v2 and V ACm (�) =

e��

1� e�� +�(�)

1� e�� v2: (19)

As we can see from these expressions, each V jm(�) is separated into terms that are either independent

or dependent of v2. The identities of the independent (�rst) terms are revealed by rewriting them

as1

�=�

�2=Var[N ](E[N ])2

ande��

1� e�� =e��(1� e��)(1� e��)2 =

Var[1(N > 0)]

(E[1(N > 0)])2:

In other words, they are the squares of the CVs that originate from uncertainty in the number

of equipment failures N . However, they manifest themselves in di¤erent forms for two contracts:

while it is the CV of N that enters into V CCm (�), it is the CV of 1(N > 0) that enters into V ACm (�).

The latter comes from the no-failure e¤ect of AC. Var[N ] is present in CC because uncertainty in

N is one of the two components of the total variance in cumulative downtime (see (9)), whereas

under AC, Var[N ] is eliminated through division ofPNi=1 Si by N , leaving the no-failure e¤ect as

the only residual of uncertainty from N . As intuition suggests, the variability of CC turns out to

be greater than that of AC when only the �rst terms of V CCm (�) and V ACm (�) in (19) are compared:

it can be shown that 1=�� e��=(1� e��) > 0.

This, however, does not tell the entire story because we have not taken into account the inter-

25

actions of N with S that are present in the second terms of V CCm (�) and V ACm (�). The interactions

occur because variabilities in performance measures are also impacted by how many samples are

collected, i.e., by N . It turns out that there is more variability in AC with regard to these inter-

actions because division ofPNi=1 Si by a random variable N introduces more noise than when it is

not divided by N , as is the case under CC. We con�rm this insight by showing that the di¤erence

of the second terms in V CCm (�) and V ACm (�) is negative: 1=���(�)=(1� e��) < 0, which follows

from the property (iv) of �(�) in Lemma 3.

Combined, the sign of V CCm (�) � V ACm (�) is ambiguous. However, we can infer from (19) and

the preceding arguments that V CCm (�) > V ACm (�) if v is su¢ ciently small but V CCm (�) < V ACm (�)

otherwise. This result answers the questions that we posed above, as the risk premium �j , and

hence the supply chain e¢ ciency, is completely determined by V jm(�) in the constant v(�) case

(from (11) and (17)): AC is more e¢ cient when v is relatively small but the reverse is true if v is

large. See Figure 4 that divides the (�; v) space in terms of relative e¢ ciency of the two contracts.

A similar argument can be made for the case where v(�) is allowed to vary, although it is more

complicated than what we have presented here. The basic insight, however, remains the same.

Changes in risk premiums as a function of �. Part (iv) shows that the risk premiums

�CC and �AC , and hence the e¢ ciency loss in the supply chain, decrease in �. This is a direct

consequence of risk pooling, as evidenced by the fact that both V CCm (�) and V ACm (�) are decreasing

functions: the more i.i.d. service time samples are collected, the less uncertain the supplier�s

performance is due to risk pooling through sampling. The reduction in performance uncertainty

means less revenue risk for the supplier, and, hence, a smaller risk premium. Note that this e¤ect

exists regardless of whether the failure process is deterministic or random; �DCC and �DAC also

decrease in �. Risk pooling under stochastic failures is, in fact, more pronounced, because sampling

is performed against the random variable N , rather than against the known value n.

From this result, we conclude that contracting e¢ ciency is worst when the equipment is most

reliable. The intuition is as follows. If the equipment fails only rarely, the supplier has few if any

opportunities to present signals about his capacity decision. As the advantage of risk pooling is

limited, large variability exists in performance realization, creating ine¢ ciency. Therefore, �rms

face a dilemma when they outsource restoration services: while it is best for them to have equipment

that rarely fails, it can potentially become very expensive under such condition to contract with

26

a supplier to deliver a fast restoration service. How expensive can it become? We answer this

question in the following discussion.

Risk premiums in the � ! 0 limit. Part (v) of the proposition states that risk premiums

under the two contracts converge to the same �nite number c�m2

�1+v2m1+�m

�in the � ! 0 limit. In

fact, this result is more general than what our model allows for, i.e., it continues to hold even if

the failure process is not Poisson.

Proposition 5 Let fYig be arbitrary but i.i.d. random variables representing the failure inter-

arrival time and F (� j�) be their cdf when the arrival rate is �. Suppose that F (� j�) satis�es

lim�!0 F (� j�) = 0. Let �jY , j 2 fCC;ACg be the risk premium under either CC or AC. Then

lim�!0 �CCY = lim�!0 �

ACY = c�m

2

�1+v2m1+�m

�.

In the vicinity of � = 0, that is, when it is highly unlikely that an equipment failure occurs within

the contracting period, the customer faces the following situation. Since the chance is high that

the supplier�s service is not required, even a fairly large penalty rate will not convince the supplier

to invest in capacity. Therefore, the customer has to provide a very high contractual incentive

(large p) in order to ensure that the supplier reserves the target capacity �m. In fact, optimal p

approaches in�nity in the �! 0 limit, as was shown in part (ii) for Poisson failures. At the same

time, however, uncertainty in the supplier�s performance Var[X j�; �m], where X =PNi=1 Si for

CC and X = bS1(N > 0) for AC, approaches zero since the supplier does not get a chance to reveal

his ability to perform if there is no equipment failure. Since the risk premium combines these two

e¤ect, i.e., � = �p2Var[X j�; �m], a tension exists between p that goes to in�nity and Var[X j�; �m]

that goes to zero. Remarkably, Proposition 5 states that a middle ground is chosen between these

two opposing forces in the �! 0 limit, and that this asymptote depends neither on the supplier�s

risk aversion coe¢ cient � or the equipment failure process.

To put this last result into a perspective, we compare it to the analysis in Abreu et al. (1991).

One of the implications of the analysis in Abreu et al. (1991) is that it becomes in�nitely expensive

in the �! 0 limit to implement an incentive-compatible �xed-price contract in a repeated setting,

which is known to allow for achieving the �rst-best solution if the discount rate is close to one.9

9Rather than showing that supply chain cost approaches in�nity, Abreu et al. (1991) show that an incentive-compatible �xed-price contract that satis�es a budget constraint does not exist if the agent�s action is evaluated too

27

Although a side-by-side comparison between our model and that of Abreu et al. (1991) is not

possible because our model assumes a single interaction between the customer and the supplier,

we �nd evidence from Proposition 5 that PBC o¤ers a unique advantage of containing cost even in

extreme situations, as an upper bound exists. Given that repeated interactions can only improve

the e¢ cacy of a contract, as is well known in the contracting literature, this advantage over a

�xed-price contract would become even more pronounced in a setting with repeated interactions.

Thus, this result advocates the use of a performance-based contract over a �xed-price contract (i.e.,

the contract that is independent of the performance outcome) in high-reliability environments, if

outsourcing is required.

6 Extension: The Pro�t-Maximizing Customer

We based our analysis in Section 5 on the assumption that the customer is subject to the ser-

vice constraint (1). As we have mentioned in Section 3.2, this constraint is appropriate when the

customer is concerned with per-incident service outcomes that follow each equipment failure event

(e.g., when �rms o¤er service time guarantees to consumers). However, many �rms also operate in

settings where they are more concerned with an aggregate service measure, such as total equipment

downtime. The Samsung example that we cited in the introduction �ts into this scenario, since the

main cost that a¤ects Samsung�s bottom line is its lost opportunity cost, which is probably pro-

portional to the total equipment downtime. To model such cases, we consider a pro�t-maximizing

customer who does not impose a constraint of the type (1). The main question that we ask is:

under this new environment, will there be fundamental changes to the insights that we have gained

from Section 5?

To this end, let r be the revenue per unit time that the customer receives while the equipment is

functioning. The pro�t-maximizing customer minimizes her total expected cost rE[PNi=1 Si j�; �]+

E[T j�; �], which is composed of her expected opportunity cost due to the total downtimePNi=1 Si

and the amount of expected contractual payment to the supplier. Other aspects of the problem,

frequently. Note that frequent action evaluation (i.e., short period length) in their model is equivalent to infrequentproduct failures in our model, in that they assume that the signal frequency is �xed, whereas we assume that it isthe period length that is �xed.

28

r/c = 5 x 103

r/c = 104

r/c = 5 x 103

r/c = 104

η = 0.01 η = 0.001

(a) η = 0.01 (b) η = 0.001

Figure 5: πCC and πAC as a function of λ. Thick lines represent πCC. πj = 0 at the lower endof λ range since no contracting is established there. Note that the unit of measure for r/c isdollar per period length. For example, assume that the unit capacity cost is $10,000 andthere are 100 days in the contracting period. Then r/c = 104 means that the customer loses$1M per day while the equipment is down. µ = 100 is chosen for this example.

πCC

πAC

πCC

πAC

πCC

πAC

πCC

πACr/c = 5 x 103

r/c = 104

r/c = 5 x 103

r/c = 104

η = 0.01 η = 0.001

(a) η = 0.01 (b) η = 0.001

Figure 5: πCC and πAC as a function of λ. Thick lines represent πCC. πj = 0 at the lower endof λ range since no contracting is established there. Note that the unit of measure for r/c isdollar per period length. For example, assume that the unit capacity cost is $10,000 andthere are 100 days in the contracting period. Then r/c = 104 means that the customer loses$1M per day while the equipment is down. µ = 100 is chosen for this example.

πCC

πAC

πCC

πAC

πCC

πAC

πCC

πAC

including the supplier�s capacity choice, remain the same. One striking di¤erence that emerges

from this problem setup is that a pro�t-maximizing customer could be content with the minimum

capacity � whenever equipment failures occur only rarely, unlike in the previous case with the

service constraint (1) where the assumption �m > � ensured that the optimal capacity level should

always be greater than �. This happens because the pro�t-maximizing customer discounts rare

failures, as evidenced by the fact that the expected total downtime E[PNi=1 Si j�; �] = �=� that

appears in her objective function is proportional to the failure rate �: the lower the failure rate, the

less likely she is to lose revenue due to downtime. However, when the equipment actually fails, the

customer has to deal with a potentially lengthy downtime because she did not o¤er strong enough

incentive to the supplier that would ensure a high capacity level. For many organizations that

value consistent mission readiness, such scenario is deemed unacceptable. For them, employing the

service constraint (1) is more sensible because it expresses the desire to provide consistent service

standard regardless of the frequency of failures.

Although restating the problem does not generate di¢ culties, unfortunately it is analytically

intractable to specify the optimal solutions for the pro�t-maximization case similarly as those in

Propositions 3-4. In particular, we cannot show the convexity of the optimization problem or

sensitivity results with respect to �. Our numerical examples show some interesting deviations

29

from previous results. First, in most cases we have observed that the optimal penalty rates pCC

and pAC increase monotonically in �. This happens because the customer�s objective function

contains expected total downtime E[PNi=1 Si j�; �] = �=�, which is scaled by the failure rate �.

Intuitively, the customer has to o¤er increasingly large incentive p to the supplier because she

loses more revenue with increased total downtime as failures become more frequent. We observe

a similar deviation for the risk premium �CC under CC. On the other hand, we �nd that the

behavior of �AC under AC is more irregular; see Figures 5(a) and 5(b) that show examples of

�AC increasing, non-monotonic, or decreasing in �. Although there are clear departures from our

analysis in Section 5, we also �nd that the key result remains unchanged: �AC < �CC continues

to hold for a reasonable range of v, as was the case in Corollary 1. The reason is that, despite

the new customer objective that we have considered in this section, the underlying mechanism by

which the performance uncertainties are dealt with under CC and AC, and, hence, the contracting

e¢ ciencies, remains the same. In sum, while some results from Proposition 4 need to be restated

under the pro�t-maximization criterion, important results need not be: we continue to �nd that

AC is superior to CC for most reasonable cases where v, the CV of service time, is not too large.

7 Conclusion

In this paper we study issues arising from performance-based contracting for restoration services,

which are essential in minimizing the impact of disruptions on mission-critical operations. Despite

a large volume of literature on service contracting, surprisingly little attention has been directed to

outsourced services in an environment characterized by low-frequency, high-impact events such as

equipment failures. With the increasing use of PBC for service outsourcing in both the commercial

and government sectors, analyzing the merits and pitfalls of PBC in such an environment provides

important managerial guidelines to practitioners who face unique issues that arise in those arenas.

We �nd that the main source of ine¢ ciency when contracting in this environment is the low rate

of equipment failures. Since equipment failures are relatively rare, the customer has few opportu-

nities to observe signals about a supplier�s choice of service capacity through repeated realizations

of the supplier�s performance, namely service completion times. In an extreme scenario, the equip-

ment may not fail at all within a contracting period, revealing no signals about the supplier�s

30

capacity choice. With limited information about the supplier�s decision, it is di¢ cult to provide a

high-powered incentive via PBC. We show that, if the customer sets a standard for expected service

time for each equipment failure incident, PBC is in fact most costly to implement when the equip-

ment is most reliable. This creates a dilemma for �rms that value both fast restorations and high

reliability. While it is obviously best to avoid disruptive events in the �rst place, if one happens, the

customer has to resolve it as quickly as possible; however, it may become very expensive to contract

with a supplier to achieve that objective. This analysis shows that PBC should be implemented

with discretion. Currently, there is a major policy shift in the government sector, especially at

the DoD, which advocates a complete switch to PBC for all logistics and service acquisitions. Our

study reveals, however, that for managing highly reliable equipment systems PBC may create great

ine¢ ciency. In such cases it might be prudent to consider in-sourcing or expend signi�cant e¤ort

to continuously monitor a supplier�s capacity investments. Otherwise, contracting may result in

prohibitively high agency costs. In-sourcing can be achieved, for example, by hiring highly trained

maintenance personnel or by acquiring the spare parts inventory that may be needed for repairs.

We also highlighted other nontrivial issues that arise in the settings characterized by rare outages

by analyzing two widely used contracts that function identically but which actually yield quite

di¤erent incentive e¤ects. One contract is based on cumulative downtime, and the other is based

on sample-average downtime. Although both provide the supplier with incentives to invest in

service capacity, they also create very di¤erent risk pro�les, which in turn a¤ect the way optimal

contracts are structured. In the case of the contract based on average downtime, we show that

the solution behavior that emerges is counterintuitive. For example, when equipment reliability is

very high, the optimal penalty rate decreases in � because of the no-failure e¤ect, whereas for large

enough �, the opposite happens because the variance reduction through sample averaging becomes

a dominant force. To the best of our knowledge, such dynamics have not been analyzed in the

contracting literature. We also compare the relative e¢ ciencies of the two contracts, and �nd that

the contract based on average downtime is superior in the most practical situations.

The model we propose in this paper is not without shortcomings, as we have made a few

simplifying assumptions that permit tractable analysis. One potential criticism lies in the speci�c

functional forms we have chosen to represent contracts and the supplier�s utility function. Because

of these assumptions, it is possible that some of the insights that this model generates may not

31

apply fully to more general situations, e.g., when nonlinear contracts are used. However, we believe

that our model captures the most important aspects of our problem setting, i.e., using PBC for

rarely requested restoration services, and they serve as useful guidelines to practitioners.

As to future directions for research, we envision multiple ways in which our model can be

extended. One promising idea is to consider heterogeneous failure processes within the equipment.

Although analysis becomes more complicated, considering heterogenous failure processes might be

more realistic since many equipment systems consist of multiple components that have di¤erent

reliability characteristics and may be serviced by di¤erent suppliers. Since the end-user is usually

less concerned with how fast individual components are restored than with how fast the composite

product is restored to working condition, she faces a resource allocation problem. For example, she

may opt to allocate more service capacity to a component that fails more often, ceteris paribus. The

question is: how should the customer structure a contract that induces the supplier to make such

a decision? Moreover, is it better to o¤er a contract based on separate service times for individual

components, or a contract based on a composite service time (i.e., equipment downtime) for the

end product? Finally, given that there can be multiple suppliers, each responsible for restoring

di¤erent components (as in the airline industry, where the engine service provider is distinct from

the avionics component service provider), it would be interesting to investigate contracting in such

situations as well. Yet another fruitful direction would be to extend this model to a setting in

which the contracting decisions are made repeatedly, so that a direct comparison with Abreu et

al. (1991) can be made. In the defense industry, for example, PBC is sometimes augmented with

contract renewals (called �award-term contracts�), which provide an added incentive to the supplier

to invest in capacity because future revenue streams can be secured when the supplier establishes

a reputation for solid service. We believe that our model paves the way for analyzing this practice

as well.

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Appendix: Proofs

Proof of Lemma 1. Note that

E [Pni=1 Si j�; �] = b�c =�, Var [

Pni=1 Si j�; �] = b�c v(�)

2=�2,

E[(Pni=1 Si)=n j�; �] = 1=�; and Var[(

Pni=1 Si)=n j�; �] = v(�)

2=(b�c�2):

The corresponding supplier utilities under each contract are, respectively, udc(�) � w � p b�c =��

�p2 b�c v(�)2=�2� c(���) and uda(�) � w�p=���p2v(�)2=(b�c�2)� c(���), which are derived

from (2). Di¤erentiating udc(�) we obtain

u0dc(�) = p b�c =�2 + 2�p2 b�c �(�)=�3 � c;

u00dc(�) = �2p b�c =�3 � 6�p2 b�c �(�)=�4 + 2�p2 b�c �0(�)=�3:

Observe that

�0(�) = v(�)v0(�)� �[v0(�)]2 � �v(�)v00(�) � 0:

Hence, u00dc(�) < 0, i.e., the supplier�s utility maximization problem with CC is concave. The

sensitivity analysis results can be found from implicit di¤erentiations, which we omit. The same

results for uda(�) can be shown analogously.

35

Proof of Proposition 1. For each contract, the customer�s contract design problem

minw;p

E[Tj j�; ��j ] subject to ��j � �m and u(��j ) � 0;

j 2 fdc; dag, can be reduced to

minp

dc(p) � c(��dc � �) + �p2 b�c v(��dc)2=(��dc)2 subject to ��dc � �m and

minp

da(p) � c(��da � �) + �p2v(��da)2=(b�c (��da)2) subject to ��da � �m,

as the IR constraint u(��) � 0 binds at the optimal solution by selecting appropriate values for w.

Let us consider CC and suppress the subscript dc for notational convenience. We can invert ��

found from the �rst-order condition in Lemma 1 with respect to p, using the monotonicity relation

@��=@p > 0. Thus,

p(�) =�1 +

p1 + 8�c��(�)= b�c4��(�)=�

=2c�2

b�c�1 +

p1 + 8�c��(�)= b�c

� :Let us suppress the argument � in p(�), �(�), and v(�). Observe that

@�v2=�2

�=@� = (2=�3)

�vv0� � v2�0

�= (2=�3)

�vv0�v2 � �vv0

�� v2

�vv0 � �(v0)2 � �vv00

��= (2=�3)�v3v00 � 0:

Combining this result with (4), i.e., � + ��0 � 0 for � � �m, we �nd that the risk premium

� = �p2 b�c v2=�2 (from the expression of dc(p) above) is increasing in � for � � �m:

16�

b�c@�

@�= 16�2

@

@�

�(p=�)2 v2

�=@

@�

���1 +

p1 + 8�c��= b�c

�2 �v2=�2

��=

8�c

b�c�� + ��0

� 1� 1p

1 + 8�c��= b�c

!v2

�2+��1 +

p1 + 8�c��= b�c

�2 @@�

�v2

�2

�� 0:

Thus, the customer cost dc = c(���)+� increases in � in the feasible region. Since our goal is to

�nd the minimum of dc, which is increasing in �, in the feasible region [�m;1) on the �-domain,

the cost minimizer is found at the left-most boundary, � = �m. Hence, the service constraint

36

binds at the optimal solution. The equilibrium solutions pDCC and �DCC are found by substituting

� = �m in p(�) and �(�). The results for AC are obtained similarly.

Proof of Lemma 3. For notational convenience, let us suppress the conditional arguments (�; �).

First, we prove the following auxiliary lemma.

Lemma 5

E[bS] = E[S] = 1=�; (20)

Var[bS] = �(�)Var[S] = �(�)v(�)2=�2; (21)

Proof. Let M(t) � E[etbS ] be the moment generating function for bS. ThenM(t) = E[et(

PNi=1 Si)=N jN > 0] =

1

Pr(N > 0)

1Xn=1

E[et(PNi=1 Si)=N jN = n] Pr(N = n)

=1

1� e��1Xn=1

Ehet(Pni=1 Si)=n

i �ne��n!

=1

1� e��1Xn=1

�E[etSi=n]

�n �ne��n!

;

where the last equality follows from independence of fSig. Di¤erentiating,

M 0(t) =1

1� e��1Xn=1

n�E[etSi=n]

�n�1E

�SinetSi=n

��ne��

n!;

M 00(t) =1

1� e��1Xn=1

n(n� 1)�E[etSi=n]

�n�2�E

�SinetSi=n

��2 �ne��n!

+1

1� e��1Xn=1

n�E[etSi=n]

�n�1E

�S2in2etSi=n

��ne��

n!

The �rst and second moments are

E[bS] = M 0(0) =E[S]

1� e��1Xn=1

�ne��

n!= E[S];

E[bS2] = M 00(0) =(E[S])2

e� � 1

1Xn=1

�1� 1

n

��n

n!+E[S2]

e� � 1

1Xn=1

�n

n!

1

n= (E[S])2 +

Var[S]e� � 1

1Xn=1

�n

n!

1

n;

which together yield Var[bS] = E[bS2]� (E[bS])2 = � 1e��1

P1n=1

�n

n!1n

�Var[S]:

Next, we prove (13) and (14). Let I � 1(N > 0). Note that, using (20) and (21), E[bSI j I = 0] =37

0, E[bSI j I = 1] = E[bS] = E[S], Var[bSI j I = 0] = 0, and Var[bSI j I = 1] =Var[bS] = �(�)Var[S].

The mean is

E[bSI] = E[E[bSI j I]] = Pr(I = 1)E[S]:To compute the variance, �rst observe that

Var[E[bSI j I]] = E[E[bSI j I]2]� (E[E[bSI j I]])2 = E[E[bSI j I]2]� (E[bSI])2= Pr(I = 1)(E[bSI j I = 1])2 + Pr(I = 0)(E[bSI j I = 0])2 � (Pr(I = 1)E[S])2= Pr(I = 1)(E[S])2 � (Pr(I = 1))2(E[S])2 = Pr(I = 0)Pr(I = 1)(E[S])2;

where we have used the results obtained above. Therefore,

Var[bSI] = Var[E[bSI j I]] + E[Var[bSI j I]]= Pr(I = 0)Pr(I = 1)(E[S])2 + Pr(I = 1)�(�)Var[S]

= Pr(I = 1)�Pr(I = 0)(E[S])2 +�(�)Var[S]

�:

Finally, we show the properties of �(�). The following facts are useful:

�0(�) = � e�

(e� � 1)21Xn=1

��n

n!

1

n

�+

1

e� � 1

1Xn=1

��n�1

n!

�

= � e�

(e� � 1)21Xn=1

��n

n!

1

n

�+

1

e� � 11

�

1Xn=1

��n

n!

�= � e�

(e� � 1)21Xn=1

��n

n!

1

n

�+1

�

= � �(�)

1� e�� +1

�; (22)

1Xn=1

�n

n!

1

n= �+

1Xn=2

�n

n!

1

n

> �+

1Xn=2

�n

(n+ 1)!= �+

1

�

1Xn=3

�n

n!

= �+1

�

�e� � 1� �� �

2

2

�=1

�

�e� � 1� �+ �

2

2

�; (23)

38

and

e�� � 1� �+ �2=2; (24)

which can be shown as follows. Let �(�) � 1 � � + �2=2 � e��. Then �0(�) = �1 + � + e�� and

�00(�) = 1� e��. Since �0(0) = 0 and �00(�) � 0, we have �0(�) � 0. But this implies �(�) � 0 since

�0(0) = 0.

(i) Di¤erentiating �(�), we obtain

�0(�) = � e�

(e� � 1)21Xn=1

��n

n!

1

n

�+1

�

< �e2� � e� � �e� + �2e�=2

�(e� � 1)2 +1

�= �e

2� � e� � �e� + �2e�=2� e2� + 2e� � 1�(e� � 1)2

= �e� � �e� + �2e�=2� 1

�(e� � 1)2 = �e�(1� �+ �2=2� e��)

�(e� � 1)2 � 0;

where the �rst and second inequalities follow from (23) and (24), respectively.

(ii) By l�Hopital�s rule,

lim�!0

�(�) = lim�!0

P1n=1

�n

n!1n

e� � 1 = lim�!0

P1n=1

�n�1

n!

e�= lim�!0

1 +P1n=2

�n�1

n!

e�= 1:

(iii) Applying l�Hopital�s rule n times,

lim�!1

�(�) =1Xn=1

1

n!

1

n

�lim�!1

�n

e� � 1

�=

1Xn=1

1

n!

1

n

�lim�!1

n!

e�

�=

1Xn=1

1

n

�lim�!1

1

e�

�= 0:

(iv) The lower bound of �(�) follows from (22) and part (i).

(v) Since�(�)

e��=

1

1� e��1Xn=1

�n

n!

1

n;

39

we see that

@

@�

��(�)

e��

�=

1

(1� e��)2

(1� e��)

1Xn=1

�n�1

n!� e��

1Xn=1

�n

n!

1

n

!

� 1

(1� e��)2

(1� e��)

1Xn=1

�n�1

n!� e��

1Xn=1

�n

n!

!

=1

(1� e��)

1Xn=1

�n�1

n!� 1!=

�e� � 1

�=�� 1

1� e�� > 0;

where the last inequality comes from e� � 1 > � for � > 0.

Proof of Lemma 4. The proofs for all results in the lemma are analogous to those of Lemma 1,

except for the last result concerning the sign of @��a=@�. Suppose that � is close to zero such that

terms of order �2 and above can be dropped. Then (16) is approximated as

c =p

�2(1� e��) + 2�p

2

�3

�e�� +�(�)�(�)

�(1� e��) � p

�2�+

2�p2

�3(1 + �(�))�;

where we have used (ii) of Lemma 3. This result identical to (10). Hence, @��a=@� > 0 for small �.

On the other hand, if � is su¢ ciently large so that e��=�(�) � 0 (see Table 1), 1 � e�� � 1 and

(16) becomes

c =p

�2(1� e��) + 2�p

2

�3

�e�� +�(�)�(�)

�(1� e��) � p

�2+2�p2

�3�(�)�(�):

Since �0(�) < 0 by (i) of Lemma 3, it is clear from this expression that @��a=@� < 0.

Proof of Proposition 4.

(i) Rewriting pCC and pAC ,

pCC = 2c�2m

��+

q�2 + 8�c�m (1 + �m)�

��1and (25)

pAC = 2c�2m

�(1� e��) +

q(1� e��)2 + 8�c�m (e�� +�(�)�m) (1� e��)

��1: (26)

40

@pCC=@� < 0 is clear from (25). Since pAC ! pCC as � ! 0, lim�!0 @pAC=@� < 0. On

the other hand, if � is su¢ ciently large so that e��=�(�) � 0 (see Table 1), (26) is reduced

pAC � 2c�2m

�1 +

p1 + 8�c�m�(�)�(�m)

��1, from which we �nd that @pAC=@� > 0 since

�0(�) < 0 by (i) of Lemma 3.

(ii) Term-by-term comparison of the denominators of (25) and (26) reveals that pCC < pAC ,

since 1� e�� < �, e�� < 1, and �(�) < 1: Retaining only the terms up to O(�), we see that

pAC ! pCC in the �! 0 limit. Moreover, lim�!0 pCC =1 is clear from (25).

(iii) @�CC=@� < 0 is clear from the expression (11). To show @�AC=@� < 0, let 'a(�) �e��+�(�)v2me��+�(�)�m

be the multiplicative factor that appears in �AC from (17). De�ne �m � ��mv(�m)v0(�m).

Note that

'0a(�) =d

d�

�1 +

�me��=�(�) + v2m

��1< 0;

by the property (v) of Lemma 3. Using this and �0(�) < 0, @�AC=@� < 0 follows.

(iv) Notice that lim�!0 'a(�) = 'c, where 'c ��1 + v2m

�= (1 + �m) is the multiplicative factor

that appears in �CC . Together with '0a(�) < 0, this implies 'a(�) < 'c. With the latter,

the stated condition�e�� +�(�)�m

�=(1 � e��) � (1 + �m) =� implies �CC � �AC , as can

be veri�ed from the expressions (11) and (17). The �! 0 limit is immediate from the same

expressions.

Proof of Corollary 1. When v0(�) = 0, (11) and (17) are reduced to

�CC =c�m2

�1� 2

�1 +

p1 + 8�c�m (1 + v

2) =���1�

and

�AC =c�m2

1� 2

�1 +

q1 + 8�c�m (e

�� +�(�)v2) = (1� e��)��1!

:

Then the condition v � !(�) is equivalent to�e�� +�(�)v2

�=�1� e��

���1 + v2

�=�, which

in turn implies �CC � �AC by part (iii) of Proposition 4. Conversely, �CC � �AC implies�e�� +�(�)v2

�=�1� e��

���1 + v2

�=�, v � !(�).

41

Proof of Proposition 5. The CC and AC converge to one another when � � 0 since their

performance measures become indistinguishable, asPNi=1 Si � S11(N = 1) and bS1(N > 1) �

S11(N = 1). Let us consider AC with � � 0, for which Ta � w � pS11(N = 1). Since the period

length is normalized to one, 1(N = 1) = 1(Y1 < 1). Thus, T � w � pS11(Y1 < 1). By the law of

total variance (proof is similar to that of Lemma 3), it can be shown that

E[T j�; �] � w � pF (1 j�)E[S j�],

Var[T j�; �] � p2F (1 j�)�[1� F (1 j�)](E[S j�])2 +Var[S j�]

�� p2F (1 j�)

�(E[S j�])2 +Var[S j�]

�;

where the last approximation is valid since [F (1 j�)]2 is negligible when � � 0. Compare these

expressions to their Poisson counterparts for � around zero (where the terms of order only up to

O(�) in (13) and (14) are retained):

E[T j�; �] = w � p(1� e��)E[S j�] � w � p�E[S j�],

Var[T j�; �] = p2(1� e��)�e��(E[S j�])2 +�(�)Var[S j�]

�� p2�

�(E[S j�])2 +Var[S j�]

�:

We see that they have the same forms, the only di¤erence being that F (1 j�) is substituted by �.

Hence, the analysis of AC with Y is equivalent to that with the Poisson failures. Therefore, when

� � 0,

�ACY � c�m2

�1 + v2m1 + �m

�0@1� 2 1 +s1 + 8�c�m 1 + �mF (1 j�)

!�11A ;which is obtained from (17) with 1 � e�� � � replaced by F (1 j�) and e�� ! 1 and �(�) ! 1.

lim�!0 �ACY = c�m

2

�1+v2m1+�m

�follows after letting �! 0, since lim�!0 F (� j�) = 0 by assumption.

42