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Capacity Management in Rental Businesses

with Two Customer Bases

Sergei V. Savin, Morris A. Cohen, Noah Gans, Ziv Katalan

Department of Operations and Information Management,

The Wharton School, University of Pennsylvania,

Philadelphia, PA 19104

May 27, 2003

Abstract

We consider the allocation of capacity in a system in which rental equipment is accessed

by two classes of customers. We formulate the problem as a continuous-time analogue of the

one-shot allocation problems found in the more traditional literature on revenue management,

and we analyze a queueing control model that approximates its dynamics. Our investigation

yields three sets of results.

First, we use dynamic programming to characterize properties of optimal capacity alloca-

tion policies. We identify conditions under which “complete sharing” – in which both classes

of customer have unlimited access to the rental ﬂeet – is optimal.

Next, we develop a computationally eﬃcient “aggregate threshold” heuristic that is based

on a ﬂuid approximation of the original stochastic model. We obtain closed-form expressions

for the heuristic’s control parameters and show that the heuristic performs well in numer-

ical experiments. The closed-form expressions also show that, in the context of the ﬂuid

approximation, revenues are concave and increasing in the ﬂeet size.

Finally, we consider the eﬀect of the ability to allocate capacity on optimal ﬂeet size.

We show that the optimal ﬂeet size under allocation policies may be lower, the same as,

or higher than that under complete sharing. As capacity costs increase, allocation policies

allow for larger relative ﬂeet sizes. Numerical results show that, even in cases in which dollar

proﬁts under complete sharing may be close to those under allocation policies, the capacity

reductions enabled by allocation schemes can help to lift proﬁt margins signiﬁcantly.

Keywords: Service Systems, Queueing Control, Stochastic Knapsack, Fluid Models.

1 Introduction

Rental businesses found in many sectors of the economy share some fundamental attributes.

The rental company invests in equipment for which there is a potential demand, and a stream

of customers patronizes the company, renting its equipment. After each rental, the equipment is

returned to the company, and rental durations are typically signiﬁcantly shorter than the life of

the equipment, so that each unit may be used repeatedly.

For those who manage rental businesses, important managerial decisions focus on matching

rental demand with the equipment supply. These decisions create a hierarchy of managerial

controls at the company’s disposal. Longer term, capital-investment decisions set the company’s

overall level of rental capacity and attempt to capture as much demand for rental services as

is (marginally) proﬁtable. While they provide for long-term matching between supply and de-

mand, ﬂeet sizing decisions may not be used to counterbalance short-term supply and demand

mismatches. On a tactical time scale, capacity allocation decisions may be needed to determine

which customers are served when rental capacity becomes scarce.

In this paper we consider a simple, stationary model of a rental problem in which capacity

must be rationed among two classes of arriving customers. We address both the lower-level,

allocation problem and the higher-level capacity sizing problems, with an emphasis on the former.

Our approach to the tactical allocation problem follows in the spirit of early formulations of

seat allocation problems in the airline yield-management literature. (For example, see Littlewood

(1972), Alstrup et al. (1986), and Belobaba (1989).) When should arriving customers of each of

the classes be allowed to rent equipment, and when would they be “closed out?”

Two common assumptions made in traditional revenue management models make them in-

adequate for our purposes, however: they assume that there exists a ﬁnite horizon over which

units of capacity can be sold and that each unit of capacity can be used only once. For example,

in aviation there are c seats on a ﬂight, and once they are sold or the plane takes oﬀ they are

not available for sale.

While hotel problems could (and perhaps should) in principle be formulated as rental prob-

lems, most academic literature only addresses the problem of allocating the rooms available

on a single night. (For example, see Rothstein (1974), Ladany (1977), Williams (1977), Liber-

man and Yechiali (1978), Bitran and Gilbert (1996).) An exception is the application of linear-

programming (LP) based “bid price” controls to hotel stays. (See Williamson (1992) and Weath-

erford (1995)). In this case, multiple nights are considered, but the problem is modeled as

1

deterministic.

In rental businesses, however, the problem is most naturally treated as a problem in dynamic

and stochastic control. An arriving customer rents a unit, which becomes unavailable for the

length of the person’s rental. When the rental period ends, the unit becomes available again.

Over any short period of time, the numbers of arriving and departing customers may be uncertain,

and managers must develop eﬀective policies for controlling the rental of system capacity.

We view the allocation of rental capacity as a continuous time, inﬁnite horizon problem

in which arrivals of customers and durations of rentals are both uncertain. We formulate this

problem as one of admission control to a multiple-server loss system. We assume that, if admitted

into service, a customer pays a daily rental fee which depends on the class to which she belongs.

If the rental request is rejected then a class-dependent, lump-sum penalty is incurred. We show

that this capacity allocation problem can be reduced to a special case of the stochastic knapsack

problem introduced in the telecommunications literature (Ross and Tsang (1989)), one in which

arriving “objects” (demands) are all of size one.

We note that this formulation does not capture the use of prior information on rental duration.

In some contexts, such as truck-trailer leasing (the application that originally motivated this

paper) and storage-locker rentals, this information may not be available. In others, such a hotel

systems, customer-stated projections of expected duration are readily available and can be of

great value in improving the eﬀectiveness of capacity allocation decisions. Thus, our approach

has important limits.

Nevertheless, the simplicity of our approach allows us to make a number of contributions:

1. We demonstrate that the allocation problem with lump-sum penalties can be reduced to one

with no penalties by appropriately adjusting the values of the rental fees. The adjustment

factors are proportional to the penalty values and the service rates.

2. We characterize two conditions under which the complete sharing policy that is often used

in practice is optimal: the ﬁrst is in the “oﬀ-season,” when the overall demand for service

is low relative to capacity; the second is in the “peak season” of high demand, given that

diﬀerent customer classes are suﬃciently similar.

3. We analyze a ﬂuid approximation to the original system, and we derive closed-form expres-

sions that characterize the controls and the performance obtained when allocating capacity

using an “aggregate threshold” policy. These expressions allow us to eﬃciently calculate

2

admission thresholds that appear to perform well in the original, stochastic model.

4. Closed-form expressions for the ﬂuid model also allow us to demonstrate the concavity of

the ﬂuid model’s revenues with respect to the ﬂeet size when the aggregate threshold policy

is used. This concavity is the essential property required for the eﬃcient solution of the

related, long-term problem of capacity sizing.

5. We show that, in the presence of capacity rationing, the optimal ﬂeet size can be either

higher or lower than that obtained when no rationing is employed. The relationship between

the two ﬂeet sizes varies systematically with the cost of capacity.

6. We present numerical experiments that highlight the potential beneﬁt of jointly optimizing

ﬂeet size and tactical controls. In particular, there appear to be cases in which the sub-

optimal use of complete sharing results in near-optimal dollar proﬁts. Even in these cases,

however, the return on in investment in capacity suﬀers signiﬁcantly.

More broadly, these numerical results complement our characterization of suﬃcient conditions

for the optimality of complete sharing policies. Complete sharing policies maximize physical

measures of system utilization. When complete sharing is optimal, this physical measure of

system utilization is a good proxy for economic utilization. When complete sharing is not optimal,

however, its use can degrade proﬁt margins and, by extension, economic measures of resource

eﬃciency, such as return on assets. In this case, physical and economic measures of eﬃciency do

not coincide.

Thus, within the context of the stationary problem developed in this paper, we are able to

characterize how the use of tactical controls aﬀects longer-term decisions regarding ﬂeet size, as

well as longer-term and economic eﬃciency. While a complete analysis of the problem, which

should account for seasonal changes in demand patterns, is beyond the scope of this paper, our

current results represent a promising ﬁrst step.

Finally, we note that our analysis and results complement that of two recent papers that

have independently considered the stochastic knapsack. Our analysis parallels that of Altman

et al. (2001), which uses dynamic programming techniques to study optimal capacity allocation

rules and develops and solves (numerically) a ﬂuid approximation to the problem. Our special

problem structure, however, allows us to more fully characterize properties of optimal and heuris-

tic admission controls. We are able to develop a number of additional useful structural results

concerning optimal policies and to develop precise, closed-form characterizations in the context

3

of ﬂuid control.

¨

Ormeci et al. (2001) also uses dynamic programming techniques to develop

similar characterizations of structural properties of the optimal policy. It does not, however,

consider heuristic controls. Neither of these papers considers how the use of tactical controls

aﬀects longer-term ﬂeet-sizing decisions.

The remainder of the paper is organized as follows. In the next section we formulate and ana-

lyze the capacity allocation problem and demonstrate how the problem with lump-sum penalties

can be reduced to one without penalties. We also discuss properties of optimal capacity allo-

cation policies and establish conditions for the optimality of the complete sharing policy. In

Section 3 we introduce a heuristic aggregate threshold policy based on a ﬂuid-model version of

our system, and we compare the performance of this heuristic to that of the optimal policy. In

Section 4, we investigate the interaction between capacity sizing and capacity allocation problems

and establish how optimal ﬂeet capacity changes in the presence of capacity rationing. We then

conclude with a discussion of the results and describe open issues and worthwhile extensions. All

proofs may be found in the Appendix.

2 The Capacity Allocation Problem

In this section we analyze the capacity allocation decision. We formulate it as a problem in the

control of queues, and we use dynamic programming techniques to investigate properties of the

optimal control policies.

2.1 Model Description

Consider a ﬂeet of c identical vehicles or pieces of rental equipment accessed by 2 customer

classes whose arrival processes are independent and Poisson with intensities λ

1

and λ

2

.Letthe

durations of their rentals be independent, exponentially distributed random variables of mean

µ

−1

1

and µ

−1

2

. Suppose, further, that each arrival wishes to rent exactly one unit of capacity.

At each arrival epoch a system controller, such as the manager of the rental location, can

decide whether or not to admit an arriving customer for service – if one of the c units of capacity

is free – or to reject the arrival. Arrivals that are admitted to service are permitted to complete

the duration of their (randomly distributed) rental periods uninterrupted. Rejected customers

do not queue; they exit the system. Similarly, customers that arrive when all c units of capacity

are rented are lost.

4

Rewards and penalties associated with the system state and action are as follows. Arrivals

that are admitted to service pay respective rental fees of $a

1

and $a

2

per unit of time. When

a customer’s rental request is denied – either due to the absence of available rental capacity

or because of the particular capacity allocation policy used – a lump-sum penalty of $π

1

or

$π

2

is incurred, depending on the customer’s class. (For more on rejection penalties and their

relationship to service-level constraints, please see Appendix A.)

The assumption that interarrival and service times are exponentially distributed implies that,

at times between these event epochs, the system evolves as a continuous time Markov chain. At

these times, the system state can be completely described by the numbers of class-1 and class-2

customers renting units. Furthermore, system control – in the form of acceptance or rejection

of an arriving customer – is exercised only at arrival epochs, and it is suﬃcient to consider only

the discrete-time process embedded at arrival and departure epochs when determining the form

of eﬀective system controls (see Chapter 11 in Puterman (1994)). That is, the system can be

modeled as a discrete time Markov Decision Process (MDP).

In Appendix B we formally deﬁne discounted and average-cost versions of this MDP. For

both cases, we also indicate why there exist stationary, deterministic policies that are optimal.

Therefore, we will only consider policies of this class. Furthermore, rather than directly analyze

the MDPs’ objective functions, we use well-known results concerning the convergence of the

value-iteration procedure to analyze the problems.

2.2 Value Iteration Formulation

We begin our deﬁnition of the value iteration procedure by “uniformizing” the system. (See

Lippman (1975) and Serfozo (1979).) Formally, we let Γ = λ

1

+ λ

2

+ cµ

1

+ cµ

2

and, for the

discounted problem with a continuous-time discount rate of α, we uniformize the system at rate

α +Γ.

Without loss of generality, we can deﬁne the time unit so that α +Γ=1. Thus, λ

i

≡

λ

i

α+Γ

and µ

i

≡

µ

i

α+Γ

become, respectively, the probability that the next uniformized transition is a

type-i arrival or service completion. Similarly, a

i

≡

a

i

α+Γ

is the expected discounted revenue per

type-i rental until the time of the next uniformized transition.

Note that the uniformization rate includes the discount factor, α. In fact, it is well known

that discounting at rate α is equivalent to including a constant intensity at which the process

terminates, after which no more proﬁts will be earned. Thus, one may think of α as the per-

5

period probability that the next transition is a terminating one. (For example, see Section 5.3

in Puterman [12].)

The rate also includes rental completions of “phantom” customers. For example, if the current

system state is (k

1

,k

2

), then the probability that one of (c − k

1

) phantom type-1 customers or

(c − k

2

) phantom type-2 customers completes a rental is (c − k

1

)µ

1

+(c − k

2

)µ

2

.Attheendof

such a phantom rental, the observed state remains the same, (k

1

,k

2

).

Given these uniformized system parameters, we deﬁne the value-iteration operator T as

Tf(k

1

,k

2

)=a

1

k

1

+ a

2

k

2

+ λ

1

H

1

[f(k

1

,k

2

)] + λ

2

H

2

[f(k

1

,k

2

)]

+ µ

1

k

1

f(k

1

− 1,k

2

)+µ

2

k

2

f(k

1

,k

2

− 1)

+((µ

1

+ µ

2

)c − µ

1

k

1

− µ

2

k

2

)f(k

1

,k

2

). (1)

The heart of the procedure is carried out via the maximizations

H

1

[f(k

1

,k

2

)] =

max[f(k

1

,k

2

) − π

1

,f(k

1

+1,k

2

)] when k

1

+ k

2

<c,

f(k

1

,k

2

) − π

1

when k

1

+ k

2

= c,

(2)

and

H

2

[f(k

1

,k

2

)] =

max[f(k

1

,k

2

) − π

2

,f(k

1

,k

2

+1)] whenk

1

+ k

2

<c,

f(k

1

,k

2

) − π

2

when k

1

+ k

2

= c,

(3)

which are speciﬁed for any function f deﬁned on the state space S = {(k

1

,k

2

) ∈ Z

2

| k

1

≥ 0,k

2

≥

0,k

1

+ k

2

≤ c}.

Let v

0

(k

1

,k

2

) ≡ 0 represent an initial estimate of the optimal expected discounted proﬁt, and

v

n

represent the estimated value after n iterations of the value-iteration algorithm:

v

n

(k

1

,k

2

)=a

1

k

1

+ a

2

k

2

+ λ

1

H

1

[v

n−1

(k

1

,k

2

)] + λ

2

H

2

[v

n−1

(k

1

,k

2

)]

+ µ

1

k

1

v

n−1

(k

1

− 1,k

2

)+µ

2

k

2

v

n−1

(k

1

,k

2

− 1)

+((µ

1

+ µ

2

)c − µ

1

k

1

− µ

2

k

2

)v

n−1

(k

1

,k

2

). (4)

Then the fact that

λ

1

+ λ

2

+(µ

1

+ µ

2

)c<1(5)

for α>0 ensures that T is a contraction operator and that {v

n

} converges to the optimal “value

function”

v(k

1

,k

2

)=a

1

k

1

+ a

2

k

2

+ λ

1

H

1

[v(k

1

,k

2

)] + λ

2

H

2

[v(k

1

,k

2

)]

+ µ

1

k

1

v(k

1

− 1,k

2

)+µ

2

k

2

v(k

1

,k

2

− 1)

+((µ

1

+ µ

2

)c − µ

1

k

1

− µ

2

k

2

)v(k

1

,k

2

) , (6)

6

whose value equals that of the MDP’s optimal objective function (see Porteus (1982)).

The ﬁrst two terms on the right-hand side of (6) represent the expected discounted revenue

earned until the next uniformized transition. The following four represent the probabilities and

associated proﬁts-to-go associated with system arrivals and service completions. The last term

represents the probability and proﬁt-to-go of a “phantom” rental completion. (Without loss of

generality, we omit the probability, α, and value, 0, associated with a terminating transition.)

If no rejection penalties are used (π

1

= π

2

= 0), then (6) directly reduces to the stochastic

knapsack problem, well known from the telecommunications literature (Ross and Tsang (1989)).

Furthermore, for any given rental fees and penalty values (a

1

,a

2

,π

1

,π

2

), there exists an equivalent

stochastic knapsack formulation with adjusted rental fees: (a

1

, a

2

, 0, 0).

Theorem 1

For any problem with rewards and penalties (a

1

,a

2

,π

1

,π

2

), and optimal value function v(k

1

,k

2

),

there exists an alternative formulation with rewards

a

i

= a

i

+ π

i

(µ

i

+ α) i =1, 2 , (7)

zero penalties, and optimal value function v(k

1

,k

2

) for which

v(k

1

,k

2

)=v(k

1

,k

2

)+

λ

1

π

1

α

+ λ

2

π

2

α

+ π

1

k

1

+ π

2

k

2

. (8)

Furthermore, a policy is optimal for the original problem if and only if it is optimal for the

transformed problem with adjusted revenues and zero penalties.

Therefore, in the analysis that follows we will consider only the transformed problem v

n

(k

1

,k

2

)

with adjusted fees (a

1

, a

2

). Observe that the adjustment factors are linear in the penalty values

and the service rates.

We note that the paper’s numerical results are performed using an average-cost MDP formu-

lation. (Because they do not depend on the starting state, “average-cost” results are easier than

discounted results to interpret.) In this case, a similar result holds, with

a

i

= a

i

+ µ

i

π

i

,i=1, 2 . (9)

For a formal development of the value iteration procedure and the analogue of Theorem 1 for

the average cost problem, please see Appendix C.

7

2.3 Optimality of switching-curve policies

To establish structural properties of the optimal control policy, it is suﬃcient to show that

certain properties of the functions deﬁned on S are preserved under the action of the value

iteration operator, T (see Porteus (1982)). In particular, we are interested in submodularity. We

say that f(k

1

,k

2

) is submodular in k

1

and k

2

if

f(k

1

+1,k

2

+1)− f (k

1

,k

2

+1)≤ f (k

1

+1,k

2

) − f (k

1

,k

2

),k

1

+ k

2

+2≤ c. (10)

Let F be the set of all f deﬁned on S that are submodular in k

1

and k

2

.

The following Theorem states that F is closed under T, so that the value iteration operator

preserves submodularity of the value function. This, in turn, implies that the optimal capacity

allocation policy is of a special form; it is a “switching curve” policy.

Theorem 2 (Altman et al. (2001);

¨

Ormeci et al. (2001); Savin (2001))

a) f ∈ F ⇒ Tf ∈ F , and therefore v(k

1

,k

2

) ∈ F

b) In turn, for each k

1

it is optimal to admit customers of class 1 when in state (k

1

,k

2

) if

and only if k

2

<k

min

2

(k

1

),where

k

min

2

(k

1

)=

c − k

1

, if v(k

1

+1,c− k

1

− 1) > v(k

1

,c− k

1

− 1)

min(k

2

:0≤ k

2

≤ c − k

1

− 1, v(k

1

+1,k

2

) ≤ v(k

1

,k

2

)), otherwise.

Similarly, for each k

2

it is optimal to admit customers of class 2 when in state (k

1

,k

2

) if and

only if k

1

<k

min

1

(k

2

),where

k

min

1

(k

2

)=

c − k

2

, if v(c − k

2

− 1,k

2

+1)> v(c − k

2

− 1,k

2

)

min (k

1

:0≤ k

1

≤ c − k

2

− 1, v(k

1

,k

2

+1)≤ v(k

1

,k

2

)) , otherwise.

Part b) of the Theorem can be interpreted as follows: when a given number of customers of a

particular class is already renting equipment, the “next” customer of the same class is admitted

if and only if the number of customers of the other class present in the system does not exceed

some critical value. This is switching curve policy, characterized by c critical indices for each of

the customer classes.

For the average cost case we can develop analogous results. At every pass of the value iteration

procedure, the operator preserves the submodularity of the estimate of the value function. This

ensures that the results of Theorem 2 apply to the optimal control policy for this case as well.

As an illustration of the optimal capacity allocation policies we consider an example with

a

1

= 10, a

2

=5,λ

1

= 25, λ

2

= 10, µ

1

=5,µ

2

=1,c = 10 for the case when average revenue per

8

period is maximized. Figure 1 describes the capacity allocation decisions for class 2 customers

and illustrates the notion of the “switching curve.”

Figure 1: The optimal capacity allocation policy for class 2 customers when the average adjusted

revenue per period is maximized (a

1

=10, a

2

=5,λ

1

=25,λ

2

=10,µ

1

=5,µ

2

=1,c= 10).

One feature of this example worth noting is the following: class 1 customers are always

allowed to rent equipment, i.e., k

min

2

(k

1

)=c − k

1

for all feasible k

1

(and so we did not include

the graph of optimal allocation for class 1). In this case, we say that class 1 customers are a

preferred class. While in every numerical example we tested there existed a preferred class, we

have not been able to prove that such a class exists universally. Nevertheless, we have been able

to characterize a great deal about preferred customer classes.

2.4 Preferred classes and the optimality of the complete sharing policy

In this section we investigate the conditions which make a particular customer class a preferred

one. Closely connected to the question about the nature of preferred classes is the issue of

the optimality of the complete sharing policy: complete sharing is optimal when both customer

classes are preferred. The following theorem provides suﬃcient conditions under which one – or

both – classes may be preferred.

Theorem 3

a) Deﬁne λ = λ

1

+ λ

2

, µ =min(µ

1

,µ

2

) , a =max(a

1

, a

2

) and

c

∗

i

=2+

λ

µ

a

a

i

µ

i

+ α

µ

i

6+4

λ +2µ

µ

i

+

µ

i

µ + α

2+

λ +2

µ

µ

i

+ α

− 1

,i=1, 2. (11)

Then for systems with capacity c>c

∗

i

,is always optimal to admit class i customers, i =1, 2.

9

b) In turn, for c ≥ max (c

∗

1

,c

∗

2

) the policy of complete sharing of the service ﬂeet is optimal.

Theorem 3 provides a lower bound on the amount of capacity suﬃcient to ensure that a

particular customer class (or both classes) has unrestricted access to the available equipment.

Of course, for proﬁt-maximizing ﬁrms, capacity costs may prevent c from becoming large enough

to optimally operate in the complete-sharing regime. In Section 4 we investigate the interaction

among capacity cost, ﬂeet size, and tactical control in more detail.

We note that for each customer class this lower bound is, as expected, a non-increasing

function of the penalty-adjusted fee paid by customers of this class. We observe that in the

simple case of µ

1

= µ

2

α, (11) implies that c

∗

1

,c

∗

2

λ/µ. Thus, in the presence of seasonal

demand patterns, these results describe the “oﬀ-peak” season when the demand for rentals may

be signiﬁcantly lower than the available capacity.

Note that Theorem 3 is stronger than a limiting statement. In general, it is not hard to

imagine that as c → +∞, a complete sharing policy will be asymptotically optimal. Theorem

3, however, says that there is a ﬁxed, ﬁnite c above which complete sharing is optimal. This is

because, as more and more pieces of equipment are rented, the probability that the next event

is a service completion, rather than an arrival, grows. Thus, the busier the system, the stronger

its drift toward emptying out. For large enough c the expected loss of revenue due to blocking

becomes small when compared to the immediate gain of taking the next customer, no matter

which class she belongs to.

Theorem 3 states that for suﬃciently high service capacity the complete sharing policy is

optimal. It is also possible to show that the complete sharing is optimal even in the “peak

season”, when capacity is tight, provided that the customer classes are similar in terms of their

penalty-adjusted rental fees:

Theorem 4

For either class i ∈{1, 2},andj = i,if

a

i

max[µ

i

,µ

j

]

≥

λ

j

λ

j

+ µ

i

a

j

µ

j

, (12)

then it is always optimal to admit type i customers.

The statement of Theorem 4 is intuitively appealing: all other parameters of the problem

being ﬁxed, there exists a minimum value of the adjusted rental fee a

i

which ensures that cus-

tomers of this class should be freely admitted into the system. Complete sharing of service ﬂeet

is optimal when (12) is satisﬁed for both classes, i.e. when a

1

and a

2

are “close”.

10

Furthermore, recall that a

i

= a

i

+ π

i

(µ

i

+ α) depends on both the revenue earned when

accepting a class i customer and the penalty paid when rejecting class i demand. That is, a

preferred customer may be proﬁtable to serve, unproﬁtable not to serve, or some combination

of the two. For example, a high-volume customer, such as a national account, may receive a

favorable rental rate in return for a large stream of rentals. At the same time, contractual

service-level requirements or the customer’s market power may imply a large rejection penalty,

so that class i arrivals become VIP. (For more on the relationship between service-level constraints

and rejection penalties, see Appendix A.)

The suﬃcient conditions of Theorem 4 are direct analogues to expressions for protection levels

in airline seat allocation models. (For example, see Belobaba (1989).) Both sets of inequalities

can be interpreted in terms of simple marginal analysis. For instance, for i =1, the right hand

side of (12) describes (a bound on) the expected cost of admitting an arriving class 1 customer.

It is the expected revenue lost from a blocked class 2 customer that might have been served.

Here

λ

2

λ

2

+µ

1

is the probability that a class 2 arrives before the admitted class 1 ﬁnishes service,

and

a

2

µ

2

is the expected revenue lost, given the blocking occurs.

In fact,

¨

Ormeci et al. (2001) develops a characterization of preferred classes that mirrors

this “marginal analysis” result. The left hand side of (12) is more complex – and more stringent

– than simply

a

1

µ

1

, however. This diﬀerence better reﬂects the more complex dynamics of our

system.

Observe that there exists a broad range of circumstances under which a class of customers may

be preferred. First, note that if a

i

> a

j

and a

i

/µ

i

≥ a

j

/µ

j

,thentype-i customers have higher

penalty-adjusted rental rates and higher expected rental durations – and they are preferred.

Second, even though a

j

/µ

j

≤ a

i

/µ

i

,type-j customers may also be preferred, as long as a

j

is not

too far below a

i

.

Conversely, it is possible to construct examples in which neither of the suﬃcient conditions of

Theorem 4 is satisﬁed. This occurs when a

i

> a

j

, µ

i

>µ

j

,anda

i

/µ

i

< a

j

/µ

j

. Of course, failure

to satisfy the suﬃcient conditions does not demonstrate that there exists no preferred class.

Finally, we note that the conditions of Theorem 4 are broadly applicable in that they do

not depend on the service capacity, c, or on the intensity of arrivals of the customer class being

considered for admission. The required parameters are simple to estimate from observable data,

and the results are simple to interpret.

11

3 Heuristic Capacity Allocation Policies

In general, it is optimal to base the control of admissions into the service on the numbers of

customers of both classes 1 and 2 that are in the system at the time each control decision is

made. In practice, however, these “vector” policies may be diﬃcult to implement, especially for

rental systems with large capacities.

Admission control decisions that are based on the value of a particular scalar metric derived

from this vector state, rather than the detailed state of the system, may also provide eﬀective (if

suboptimal) controls. One of the most widely used heuristics is the aggregate threshold (trunk

reservation) policy.

The aggregate threshold (AT) policy assumes that there exists a preferred customer class,

and it is the class that oﬀers higher revenue per unit of time. The AT policy admits second-class

customers as long as the total number of customers already in the system does not exceed some

critical threshold value.

Besides being intuitively appealing, aggregate threshold policies have been proven to be

optimal whenever µ

1

= µ

2

(see Miller (1969)). More generally, we expect them to perform well

in cases when the expected service times for diﬀerent customer classes are similar.

Figure 2 illustrates the best AT policy, as well as the optimal control policy, for the same

example shown in Figure 1. While the control exercised by the AT policy diﬀers from that of the

optimal policy, the revenues it generates are nearly optimal, falling below optimality by about

0.15%.

Figure 2: The optimal and the best AT policies for class 2 customers (a

1

=10, a

2

=5,λ

1

=

25,λ

2

=10,µ

1

=5,µ

2

=1,c= 10).

AT policies, however, do not yield closed-form expressions for system performance measures.

12

In general, the task of computing the value of the best aggregate threshold level can be compa-

rable in its complexity to the task of computing the optimal control policy.

Ideally, we would like to have a policy that combines ease of calculation with the robust

performance of AT controls. In the following section we develop such a heuristic. It uses a ﬂuid-

model approximation of the stochastic model to derive closed-form expressions for the aggregate

threshold values.

3.1 Fluid models and scaling

In many practical situations, both the size of rental ﬂeet c and the oﬀered rental intensities

ρ

1

=

λ

1

µ

1

and ρ

2

=

λ

2

µ

2

are large. Under these conditions a deterministic ﬂuid model may oﬀer

a good approximation to the original control problem. Indeed, Altman et al. (2001) oﬀer a

heuristic derivation of such a ﬂuid model as the limit of a linearly scaled sequence of MDPs, and

they numerically evaluate the resulting Hamilton-Bellman-Jacobi equations.

We follow the approach of Altman et al. (2001), but given the underlying structure of our

problem, in which there are two classes of customers, we can directly analyze the trajectory of

the ﬂuid system. This allows us to develop an aggregate threshold heuristic whose performance is

robust and whose closed-form expressions allow for immediate calculation of policy parameters.

Furthermore, our analysis also allows us to demonstrate the concavity of discounted revenues

(of a “µ-scaled” version of our model), with respect to the ﬂeet size, c, a property that becomes

important in the capacity-sizing analysis of Section 4.

We start by deﬁning the state space and dynamics for ﬂuid approximations (in general).

Time t is continuous, and the state parameters k

1

(t)andk

2

(t) of the original model become

continuous state variables, restricted to set S

f

=(k

1

(t) ≥ 0,k

2

(t) ≥ 0,k

1

(t)+k

2

(t) ≤ c). Poisson

customer arrivals are replaced by the deterministic continuous “ﬂow” arrivals with intensities λ

1

and λ

2

. The departure process becomes deterministic as well: for the state (k

1

(t),k

2

(t)) it is

represented by an outﬂow at rate µ

1

k

1

(t)+µ

2

k

2

(t).

Arrivals are controlled as follows: at time t,acontrolpolicy(u

1

(t),u

2

(t)) results in the total

customer inﬂow of u

1

(t)λ

1

+ u

2

(t)λ

2

. Thus, for the control trajectories (u

1

(t),u

2

(t)) (0 ≤ u

i

(t) ≤

1,i =1, 2) the Kolmogorov evolution equations for the original system are replaced by

dk

1

(t)

dt

= u

1

(t)λ

1

− µ

1

k

1

(t)and

dk

2

(t)

dt

= u

2

(t)λ

2

− µ

2

k

2

(t), (13)

13

with a constraint that reﬂects the ﬁnite size of the service ﬂeet

λ

1

u

1

(t)+λ

2

u

2

(t) ≤ µ

1

k

1

(t)+µ

2

k

2

(t),i=1, 2, whenever k

1

(t)+k

2

(t)=c. (14)

The total discounted revenue is then the objective to be maximized. If at t = 0 the system is

in the state (k

1

,k

2

), then – for a feasible (under (14)) control policy ∆ which uses (u

1

(t),u

2

(t))

– the total discounted revenue is

ˆ

R

α

(k

1

,k

2

, ∆) =

∞

0

(a

1

k

1

(t)+a

2

k

2

(t)) e

−αt

dt =

a

1

k

1

µ

1

+ α

+

a

2

k

2

µ

2

+ α

+ R

α

(k

1

,k

2

, ∆), (15)

where

R

α

(k

1

,k

2

, ∆) =

∞

0

a

1

λ

1

u

1

(t)

µ

1

+ α

+

a

2

λ

2

u

2

(t)

µ

2

+ α

e

−αt

dt, (16)

is the part of the revenue that actually depends on the control policy chosen. In what follows,

the term “revenue” is used to designate R

α

(k

1

,k

2

, ∆).

Our aggregate threshold heuristic is based on a “scaled” version of the ﬂuid model:

Deﬁnition 1

A µ-scaled version of the ﬂuid model with parameters λ

1

, λ

2

, µ

1

,andµ

2

is the problem with

parameters λ

s

1

=

λ

1

µ

µ

1

, λ

s

2

=

λ

2

µ

µ

2

, µ

s

1

= µ

s

2

= µ for µ ∈ [µ

1

,µ

2

].

Note that in every µ-scaled version of the ﬂuid model, the departure rates of both customer classes

are equal and

λ

s

1

µ

s

1

=

λ

1

µ

1

,

λ

s

2

µ

s

2

=

λ

2

µ

2

. Since the departure rates of both classes are the same, one

can use arguments similar to those in Miller (1969) to show that the optimal admission control

decisions only depend on the total number of customers k(t)=k

1

(t)+k

2

(t) in the system. Thus,

given a

1

> a

2

, a control policy which admits as many class 1 customers as possible and limits

the admissions of class 2 customers is optimal for any µ-scaled problem.

3.2 Fluid aggregate threshold heuristic

In the µ-scaled model, system dynamics simplify to

dk(t)

dt

= u

1

(t) λ

s

1

+ u

2

(t) λ

s

2

− µk(t) . (17)

In turn, a ﬂuid analog of the original, stochastic system’s AT policy admits class-2 customers

if and only if the total system occupancy, k(t), does not exceed a “ﬂuid aggregate threshold”

(FAT), k

FAT

.Whenρ

1

≥ c or ρ

1

+ ρ

2

≤ c such a FAT policy is a direct analog of AT policies

in the original, stochastic system. For ρ

1

<c<ρ

1

+ ρ

2

, however, there does not exist a neat

correspondence. Therefore, in the following sections we deﬁne and analyze the FAT policy within

each subset of the relevant parameter range.

14

3.2.1 The FAT policy when ρ

1

≥ c.

For systems with ρ

1

≥ c class-1 traﬃc alone is suﬃcient to ensure complete utilization of the

rental ﬂeet, and a threshold policy can be deﬁned and analyzed in a straightforward fashion. In

this case, the control (u

1

(t),u

2

(t)) is deﬁned as follows:

(u

1

(t),u

2

(t)) =

(1, 1), for k(t) <k

FAT

,

(1, 0), for k

FAT

≤ k(t) <c,

cµ

λ

s

1

, 0

, for k(t)=c.

(18)

Note that, once the system hits the boundary and k(t)=c, customers continue to be admitted

at the maximum feasible rate, and the system state remains at the boundary thereafter.

Control (18) then implies that, at time t, the revenue generation rate r(t)=

a

1

λ

s

1

u

1

(t)+a

2

λ

s

2

u

2

(t)

µ+α

for ρ

1

≥ c is given by

r(t | ρ

1

≥ c)=

a

1

λ

s

1

+a

2

λ

s

2

µ+α

, for k(t) <k

FAT

,

a

1

λ

s

1

µ+α

, for k

FAT

≤ k(t) <c,

a

1

µc

µ+α

, for k(t)=c.

(19)

To compute the total discount revenues for a given k

FAT

, we must also account for the starting

state k ≡ k(0).

When k<k

FAT

≤ c, there are three elements to the discounted revenues: those earned as

k(t) approaches k

FAT

; those earned when k

FAT

≤ k(t) ≤ c; and those earned after the boundary

has been hit. We calculate each in turn. Let t

FAT

=

1

µ

ln

ρ

1

+ρ

2

−k

ρ

1

+ρ

2

−k

FAT

bethetimethatsystem

state hits k

FAT

,sothatk(t

FAT

)=k

FAT

. Then from (19) we have

t

FAT

0

a

1

λ

s

1

u

1

(t)+a

2

λ

s

2

u

2

(t)

µ + α

e

−αt

dt =

a

1

λ

s

1

+ a

2

λ

s

2

µ + α

1 − exp (−αt

FAT

)

α

. (20)

Similarly, let t

c

= t

FAT

+

1

µ

ln

ρ

1

+ρ

2

−k

FAT

ρ

1

+ρ

2

−c

be the time at which the system state hits c,sothat

k(t

c

)=c. Then using (19) we have

t

c

t

FAT

a

1

λ

s

1

u

1

(t)+a

2

λ

s

2

u

2

(t)

µ + α

e

−αt

dt =

a

1

λ

s

1

µ + α

exp (−αt

FAT

) − exp (−αt

c

)

α

. (21)

Finally, from (19) the revenues earned after reaching the boundary are given by

+∞

t

c

a

1

λ

s

1

u

1

(t)+a

2

λ

s

2

u

2

(t)

µ + α

e

−αt

dt =

a

1

µc

µ + α

exp (−αt

c

)

α

. (22)

Collecting the revenue terms (20)-(22), substituting for t

FAT

and t

c

, and simplifying, we then

obtain the discounted revenues for the FAT policy when ρ

1

≥ c and k ≤ k

FAT

<c:

R

FAT

α

(k, k

FAT

| ρ

1

≥ c, k ≤ k

FAT

<c)=

µ

α(α + µ)

a

1

ρ

1

+ a

2

ρ

2

−

ρ

1

+ ρ

2

− k

FAT

ρ

1

+ ρ

2

− k

α

µ

a

2

ρ

2

+ a

1

(ρ

1

− c)

α+µ

µ

(ρ

1

− k

FAT

)

α

µ

. (23)

15

When k

FAT

≤ k<c, only type-1 customers are admitted to the system. In this case, in the

above analysis we replace t

FAT

by 0 and k

FAT

by k. Then analogous calculations yield

R

FAT

α

(k, k

FAT

| ρ

1

≥ c, k

FAT

≤ k<c)=

µa

1

α(α + µ)

ρ

1

−

(ρ

1

− c)

α+µ

µ

(ρ

1

− k)

α

µ

. (24)

3.2.2 FAT policy when ρ

1

+ ρ

2

< c

When ρ

1

+ ρ

2

<c, a threshold policy with k

FAT

<cleads to incomplete utilization of the rental

ﬂeet and may trivially be improved by setting k

FAT

= c so that all customers are admitted for

service, no matter what the initial state of the system, k(0). Here, the policy is, again, a direct

analog of AT policies in the original, stochastic system. Speciﬁcally, the optimal ﬂuid-threshold

of c corresponds to complete sharing, an AT policy with a threshold of c.

Because ρ

1

+ ρ

2

<c, even with no control the boundary k(t)=c is never hit (for t>0). In

this case, the optimal control is

(u

1

(t),u

2

(t)) = (1, 1) ,

for any system state, k(t), and the rate at which revenue is earned is

r(t | ρ

1

+ ρ

2

<c)=

a

1

λ

s

1

+ a

2

λ

s

2

µ + α

.

In turn, the revenue calculation is

R

FAT

α

(k, c | ρ

1

+ ρ

2

<c)

=

∞

0

a

1

λ

s

1

u

1

(t)+a

2

λ

s

2

u

2

(t)

µ + α

e

−αt

dt =

µ

α(µ + α)

(a

1

ρ

1

+ a

2

ρ

2

) . (25)

3.2.3 FAT policy when ρ

1

< c ≤ ρ

1

+ ρ

2

.

Finally, when ρ

1

<c≤ ρ

1

+ ρ

2

there does not appear to exist a ﬂuid analog of a threshold

policy that is both eﬀective and straightforward to implement. On the one hand, a threshold

of k

FAT

<cresults in incomplete utilization of the rental ﬂeet and can be improved upon by

admitting some class-2 customers. On the other, setting k

FAT

= c and admitting all class-2

customers is infeasible, since the maximum rate at which the system can be cleared is strictly

less than the rate at which customers are arriving: cµ < λ

s

1

+ λ

s

2

.

In this case, a natural interpretation of the threshold rule deﬁnes a “soft” threshold when

k(t)=c, one that limits, but does not eliminate, the ﬂow of class-2 customers into the system:

(u

1

(t),u

2

(t)) =

(1, 1), for k(t) <c,

1,

µc−λ

s

1

λ

s

2

, for k(t)=c,

(26)

16

so that

r(t | ρ

1

<c≤ ρ

1

+ ρ

2

)=

a

1

λ

s

1

+a

2

λ

s

2

µ+α

, for k(t) <c,

a

1

λ

s

1

+a

2

(µc−λ

s

1

)

µ+α

, for k(t)=c.

(27)

Thus for ρ

1

<c<ρ

1

+ ρ + 2, the control generates system behavior and revenue that diﬀer from

those when k

FAT

<cor k

FAT

= c, and we denote this soft threshold as k

FAT

= c

−

.

Given k

FAT

= c

−

and any k ≡ k(0) ∈ [0,c], the system’s revenues can be split into two

components: those earned before reaching c, and those earned after. If t

c

=

1

µ

ln

ρ

1

+ρ

2

−k

ρ

1

+ρ

2

−c

is

the time required for the system to reach the boundary, than the ﬁrst revenue component in (27)

gives us

t

c

0

a

1

λ

s

1

u

1

(t)+a

2

λ

s

2

u

2

(t)

µ + α

e

−αt

dt =

a

1

λ

s

1

+ a

2

λ

s

2

µ + α

1 − exp (−αt

c

)

α

. (28)

After the full capacity is reached, we use the bottom revenue generation rate within (27) to

obtain

+∞

t

c

a

1

λ

s

1

u

1

(t)+a

2

λ

s

2

u

2

(t)

µ + α

e

−αt

dt =

a

1

λ

s

1

+ a

2

(µc − λ

s

1

)

µ + α

exp (−αt

c

)

α

. (29)

Adding (28) and (29), and using the expression for t

c

,wethenhave

R

FAT

α

(k | ρ

1

<c≤ ρ

1

+ ρ

2

)=

µ

α (α + µ)

(a

1

ρ

1

+ a

2

ρ

2

) − a

2

(ρ

1

+ ρ

2

− c)

α

µ

+1

(ρ

1

+ ρ

2

− k)

α

µ

. (30)

3.2.4 Optimal Thresholds and Revenues for the FAT Policy

We can use the expressions we have derived for discounted revenues to determine both opti-

mal thresholds and optimal discounted revenues. In both cases, we obtain simple, closed-form

expressions.

First we address the optimal threshold, k

∗

FAT

.Forρ

1

≥ c, its determination follows from

diﬀerentiation of (23) with respect to k

FAT

:

Theorem 5

The optimal value of the aggregate threshold, k

∗

FAT

, is independent of the starting state, k,and

is given by

k

∗

FAT

(c)=

0, for c<ρ

1

1 −

a

2

a

1

µ

µ+α

,

c − (ρ

1

− c)

a

1

a

2

µ

µ+α

− 1

, for ρ

1

1 −

a

2

a

1

µ

µ+α

≤ c ≤ ρ

1

,

c

−

for ρ

1

<c≤ ρ

1

+ ρ

2

,

c, for ρ

1

+ ρ

1

<c.

(31)

17

We observe that, all other problem parameters being ﬁxed, the optimal aggregate threshold

value described by (31) is a non-decreasing function of the ﬂeet size c. In particular, if the

available rental capacity falls below the critical value c

min

= ρ

1

1 −

a

2

a

1

µ

µ+α

, then it is optimal

not to admit any of class 2 customers into service. Conversely, if the rental capacity is suﬃciently

large, exceeding the oﬀered load from class 1, then the control on admissions of class 2 customers

should be postponed until the entire rental ﬂeet is utilized. For the rental ﬂeet values in between

these two critical quantities, some form of admission control on class 2 customers is optimal,

even in states in which some rental capacity is available. We observe that the critical index c

min

is a decreasing function of the ratio of penalty-adjusted rental fees a

2

/a

1

.

When the time discounting factor α is much smaller than µ, the optimal aggregate threshold

level, described in Theorem 5, is not particularly sensitive to the choice of µ. Even for rental

durations of several months, the service rates (inverse of the expected service time) are about

µ 10

−3

per day and are at least order of magnitude higher than any realistic values for α (for

example, 30% − 40% annual discounting rate results in α 10

−4

per day). The same argument

suggests that k

∗

FAT

is not sensitive to the choice of α. Thus, it is straightforward to use k

∗

FAT

as

a threshold for both discounted and “average-cost” versions of the problem.

Using expression for the optimal aggregate threshold (31), we obtain

Theorem 6

Given ﬁxed λ

s

1

, λ

s

2

, µ, a

1

, a

2

and α, deﬁne c

min

= ρ

1

1 −

a

2

a

1

µ

µ+α

.

a) If the rental system starts in state k, then the optimal total discounted revenue is

R

FAT

α

(k, k

∗

FAT

(c)) =

µ

α(α+µ)

a

1

ρ

1

− a

1

(ρ

1

−c)

α+µ

µ

(ρ

1

−k)

α

µ

, for c ≤ c

min

,

µ

α(α+µ)

a

1

ρ

1

+ a

2

ρ

2

− a

2

ρ

2

+(ρ

1

−c)

a

1

a

2

µ

µ+α

α+µ

µ

(ρ

1

+ρ

2

−k)

α

µ

, for c

min

≤ c<ρ

1

,k<k

∗

FAT

(c),

µ

α(α+µ)

a

1

ρ

1

− a

1

(ρ

1

−c)

α+µ

µ

(ρ

1

−k)

α

µ

, for c

min

≤ c<ρ

1

,k≥ k

∗

FAT

(c),

µ

α(α+µ)

a

1

ρ

1

+ a

2

ρ

2

− a

2

(ρ

1

+ρ

2

−c)

α+µ

µ

(ρ

1

+ρ

2

−k)

α

µ

, for ρ

1

≤ c ≤ ρ

1

+ ρ

2

µ

α(α+µ)

(a

1

ρ

1

+ a

2

ρ

2

) , for ρ

1

+ ρ

2

<c.

(32)

b) For ﬁxed values of rental fees, demand and service parameters, R

FAT

α

(k, k

∗

FAT

(c)) is an

non-decreasing concave function of the rental ﬂeet size c for every k ≤ c.

18

Inspection of (32) shows that R

FAT

α

(k, k

∗

FAT

(c)), like k

∗

FAT

(c), is insensitive to the choice of µ

for α µ. (Of course, this insensitivity follows from the µ-scaled problem, not necessarily from

the two-class problem in which µ

1

= µ

2

.) Part b) of Theorem 6 also states that, for any starting

state, FAT revenues are concave in c. Thus, although the concavity of revenue with respect to

ﬂeet size is diﬃcult to demonstrate in the context of the original MDP, it emerges naturally from

the µ-scaled ﬂuid approximation. This concavity property becomes important in the context of

ﬂeet sizing decisions, which we discuss in Section 4.

3.3 Numerical study of the performance of the FAT heuristic

Our motivation for developing the FAT heuristic was that it should perform well and be easy to

implement. Therefore, to test the policy’s performance we have undertaken a series of numerical

studies which compare its average revenues to those obtained using the optimal control and the

complete sharing policy.

In two of the three cases analyzed above, translation of the FAT policy (31) to the context of

a discrete, stochastic system is straightforward. For ρ

1

≥ c we assume

µ

µ+α

≈ 1, when necessary,

and then round the resulting k

∗

FAT

down to the nearest integer. For ρ

1

+ ρ

2

<c,wesetthe

aggregate system threshold equal to c, eﬀectively implementing a complete sharing policy.

When ρ

1

<c<ρ

1

+ ρ

2

, however, k

∗

FAT

= c

−

, and the inﬂow of class-2 rentals is partially

controlled. In this case, there is not a clear correspondence in a discrete system: setting the

aggregate threshold to c implements complete sharing, which does not control class-2 customers

at all; conversely, setting the threshold to c − 1 completely stops the ﬂow of class-2 customers at

the boundary.

Because both alternatives of the FAT policy are trivial to compute, we include them both in

our numerical tests. In total, in each numerical experiment, we test four polices: the optimal

policy; FAT with c

−

set to c (“c

−

= c”); FAT with c

−

set to c − 1(“c

−

= c − 1”); and complete

sharing (CS). For each set of system parameters, we evaluate the Markov chains induced by the

four policies (in the case of the optimal policy, via value iteration) to calculate long-run average

revenues.

In our numerical tests, we ﬁx the expected rental duration of class-1 rentals at 1/µ

1

=1,

and we run sets of tests in which systematically vary the oﬀered load, θ =

ρ

1

+ρ

2

c

,aswellasthe

relative processing rate of class-2 customers, µ

2

. Within each test set, θ and µ

2

also remain ﬁxed,

and we run (10 × θ + 1) experiments in which we systematically vary λ

1

and λ

2

.

19

CS Policy FAT w it h c

−

= c − 1 FAT w it h c

−

= c

θ

µ

2

µ

1

=0.1

µ

2

µ

1

=1

µ

2

µ

1

=10

µ

2

µ

1

=0.1

µ

2

µ

1

=1

µ

2

µ

1

=10

µ

2

µ

1

=0.1

µ

2

µ

1

=1

µ

2

µ

1

=10

0.5 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0)

0.8 0.0 (0.0) 0.2 (0.4) 1.5 (2.8) 0.0 (0.0) 0.2 (0.4) 1.5 (2.8) 0.0 (0.0) 0.2 (0.4) 1.5 (2.8)

1.0 0.3 (1.8) 0.9 (1.9) 1.0 (1.9) 0.3 (1.8) 0.9 (1.9) 1.0 (1.9) 0.3 (1.8) 0.9 (1.9) 1.0 (1.9)

1.5 3.1 (6.2) 4.1 (6.5) 6.7 (11.7) 1.2 (4.0) 0.7 (3.2) 0.2 (2.8) 1.9 (5.8) 2.8 (7.3) 4.3 (11.6)

2.0 7.0 (13.1) 6.3 (12.7) 9.6 (16.1) 1.5 (5.9) 0.6 (2.7) 0.3 (2.3) 2.7 (10.9) 3.3 (12.5) 4.7 (16.1)

2.5 9.2 (16.9) 9.6 (16.5) 11.6 (19.6) 1.6 (7.9) 0.7 (4.6) 0.2 (1.9) 3.1 (14.6) 3.5 (16.3) 4.6 (19.6)

3.0 10.9 (20.5) 11.3 (20.1) 12.3 (22.3) 1.6 (9.4) 0.7 (5.3) 0.2 (1.2) 3.5 (17.4) 3.6 (19.1) 4.2 (22.3)

Table 1: Numerical results. Average and maximum (in parentheses) percent shortfall from opti-

mal revenue for 10 θ +1 test cases. Penalty-adjusted service fees are a

1

=10and a

2

=5for

class 1 and 2 customers, re spectively. Service rate for class 1 customers is µ

1

=1. Rental ﬂeet

size is c =10.

More speciﬁcally, in each test set we begin with (10×θ+1) equally spaced λ

1

’s – from λ

1

=0to

λ

1

= µ

1

c –andthenchooseλ

2

in each case so that

λ

1

µ

1

c

+

λ

2

µ

2

c

= θ. We then modify the endpoints –

where either λ

1

or λ

2

equals zero – so that the arrival rate that would be zero actually equals 0.01.

For example, the set in which θ =1andµ

2

/µ

1

=1,thereare10×θ+1 = 11 test points, and their

(λ

1

,λ

2

)valuesare{(0.01, 9.99), (1, 9), (2, 8), (3, 7), (4, 6), (5, 5), (6, 4), (7, 3), (8, 2), (9, 1), (9.99, 0.01)}.

Table 1 shows results for the 21 sets of experiments. In each experiment within a set we

record average penalty-adjusted revenue per period using the optimal policy (R

∗

), as well as

that obtained from the FAT and CS policies (R

FAT

and R

CS

). For each experiment we then

calculate the percentage revenue lost when using the heuristic controls ((1 − R

FAT

/R

∗

) × 100%

and (1 − R

CS

/R

∗

) × 100%). Finally, within each cell of Table 1 we report two statistics that

summarize the results across all 10 × θ + 1 experiments: the average of the percentage shortfalls,

as well as the maximum shortfall recorded over all cases (in parentheses).

Table 1’s results show that all three policies perform well at low oﬀered loads. For θ ≤ 1,

none of the three policies controls the inﬂow of class-2 requests, and all three perform consistently

close to optimality. It is also worth noting that, in these examples, the CS policy is consistently

optimal at θ =0.5. While the suﬃcient c

∗

i

’s of Theorem 3 can be very large – in the thousands

in many of these examples – the oﬀered loads at which the CS policy is actually optimal appear

to be much less extreme.

As θ climbs above 1, the three policies diverge, and the FAT heuristics outperform CS. At

θ = 2 – when the oﬀered load is twice that of the system’s capacity – the FAT with c

−

= c − 1

still performs quite well, with a worst optimality gap of less than 6% and an average gap in each

table cell that is consistently below 1%. Here, the performance of FAT with c

−

= c is noticeably

worse, with the maximum gap of 16.1% and an average gap ranging from 2.7% to 4.7%. The

20

CS policy’s worst-case performance is also 16.1% below optimal, and its average performance in

each cell trails that of FAT with c

−

= c, falling 7.0% to 9.6% below optimality.

At very high θs, the FAT heuristic with c

−

= c − 1 consistently outperforms the other

heuristics. For example, when θ = 3, the average revenue generated by the FAT policy with

c

−

= c − 1 ranged from 0.2% to 1.9% below optimal, and the worst-case examples of each of the

10θ+1 test sets ranged from 1.2% to 9.4% below optimal. In contrast, average and worst-case

performance of the FAT with c

−

= c and CS policies were 3 to 4 times worse.

Thus, as the oﬀered load increases, the performance of all three heuristics deteriorates with

respect to optimality. In general, the heuristics are exercising insuﬃcient control of class-2

customers. The relatively strong performance of the FAT heuristic with “c

−

”settoc − 1 reﬂects

the beneﬁt of reserving the last unit of rental capacity for “preferred” class-1 customers when

the traﬃc intensity is high.

Figure 3 provides additional detail on the how the setting of c

−

aﬀects the performance of

FAT heuristic. In the ﬁgure, rental capacity is c = 10, service rates are µ

1

=1.0andµ

2

=0.1,

and penalty-adjusted revenues are a

1

=10anda

2

= 5. The aggregate oﬀered load is ﬁxed at

θ =

ρ

1

+ρ

2

c

=2,andthex-axis of the ﬁgure’s parametric analysis tracks the fraction of the oﬀered

load due to class-1 customers as it is systematically increased from 0% to 100% of the total: from

ρ

1

/c =0,toρ

1

/c =2. They-axis reports the two FAT policies’ resulting percentage shortfall

from long-run average optimal revenue.

Figure 3: Performance of alternative FAT heuristics with c

−

interpreted as c (dashed line) and

as c − 1 (solid line). System has c =10, θ =

ρ

1

+ρ

2

c

=2, µ

1

=1, µ

2

=0.1, a

1

=10,anda

2

=5.

21

As Fig. 3 indicates, whenever ρ

1

/c ≥ 1.0, the two policies are identical – with the same

threshold, k

FAT

≤ c−1, and the same long-run average revenues. When ρ

1

/c < 1.0, however, the

two heuristics’ recommendations diﬀer – k

FAT

= c − 1versusk

FAT

= c – and average revenues

diﬀer as well. For moderate ρ

1

’s, the “c

−

= c − 1” policy outperforms the “c

−

= c” one, and for

ρ

1

c, the reverse is true.

It is worth noting that numerical experiments using other θs yield plots whose gross features

are directly analogous to those of Figure 3. Larger values of θ lead to more extreme performance

diﬀerences between the c

−

= c− 1andc

−

= c variants of the FAT at moderate to very low values

of ρ

1

.

4 The Eﬀect of Capacity Allocation on Optimal Fleet Size

The allocation policies investigated in Sections 2 and 3 are tactical controls intended to address

instances in which the number of rental units available falls short of the anticipated near-term

demand. The total ﬂeet size c clearly aﬀects the nature of the control. In particular, Theorem 3

shows that, given ample capacity, the optimal control is to give free access to all customers.

It is also natural to ask the converse question. How does the use of tactical control aﬀect the

ﬂeet size the rental company should use? When is the optimal ﬂeet size large enough so that, as

in Theorem 3, complete sharing is (nearly) optimal? More generally, given the ability to change

ﬂeet size, what is the economic value to a ﬁrm of exercising tactical controls? In this section we

address both of these questions.

In fact, the eﬀect of capacity allocation on optimal ﬂeet size is not immediately clear. One

might argue that, given any ﬁxed ﬂeet size, optimal rationing increases revenue per unit of time.

This revenue increase, in turn, allows the ﬁrm to more proﬁtably sustain higher overall capacity

levels. Alternatively, one might argue that rationing reduces the aggregate arrival rate to the

rental ﬂeet and that, in turn, fewer units of capacity are required to process the arrivals that are

actually served.

We can provide some insight into these trade-oﬀs by directly comparing the optimal ﬂeet

size under active allocation policies to that under complete sharing, which passively allows all

customers access to rental capacity whenever it is available. We formulate the problem of ﬁnding

the optimal ﬂeet size as

Π(∆) = max

c

(R (c, ∆(c)) − hc) , (33)

22

where R(c, ∆(c)) is the average revenue per period when operating c units under allocation policy

∆(c) and the capacity cost of $h per unit per period is ﬁxed for all c. Note that, given a ﬁxed

oﬀered load, ρ

1

+ ρ

2

, the allocation policy, ∆(c), may vary with c.

We then compare the maximizer of (33) under two regimes. In one we use ∆(c)=CS(c), the

complete sharing policy, for all c. In the other ∆(c)=∆

∗

(c), which we deﬁne as any family of

allocation policies for which the following attributes hold:

1. For any ﬁxed c, R (c, ∆

∗

(c)) ≥ R (c, CS(c)).

2. There exists a c<∞ such that for all c ≥ c, R (c, ∆

∗

(c)) = R (c, CS(c)).

3. R (c, ∆

∗

(c)) − R (c − 1, ∆

∗

(c − 1)) ≤ R (c − 1, ∆

∗

(c − 1)) − R (c − 2, ∆

∗

(c − 2)).

Condition 1 states that, for any c,∆

∗

(c) performs at least as well as complete sharing.

Condition 2 states that there exists a ﬁnite ﬂeet size above which complete sharing performs

as well as ∆

∗

(c). Note that Theorem 3 demonstrates that such a c exists in the context of the

discounted problem.

Condition 3 requires that average revenues per period under ∆

∗

are concave in c.Theorem

6 proves that this type of concavity exists for the FAT policy in the context of the discounted

ﬂuid model, and the result also suggests that the condi