Dynamic Matching for Real-Time Ridesharingpublic.kenan-flagler.unc.edu/2017msom/SIGs/Service...ket asymptotic regime. The relevant objective is to maximize the number of customers

Submitted to Manufacturing & Service Operations Managementmanuscript (Please, provide the manuscript number!)

Authors are encouraged to submit new papers to INFORMS journals by means ofa style file template, which includes the journal title. However, use of a templatedoes not certify that the paper has been accepted for publication in the named jour-nal. INFORMS journal templates are for the exclusive purpose of submitting to anINFORMS journal and should not be used to distribute the papers in print or onlineor to submit the papers to another publication.

Dynamic Matching for Real-Time Ridesharing

Erhun Ozkan, Amy R. WardMarshall School of Business, University of Southern California, Los Angeles, California 90089,

[email protected], [email protected],

In a ridesharing system such as Uber or Lyft, arriving customers must be matched with available drivers.

These decisions affect the overall number of customers matched, because they impact whether or not future

available drivers will be close to the locations of arriving customers. A common policy used in practice is

the closest driver (CD) policy that offers an arriving customer the closest driver. This is an attractive policy

because it is simple and easy to implement. However, we expect that a forward-looking parameter-based

policy can achieve better performance.

We propose to base the matching decisions on the solution to a continuous linear program (CLP) that

accounts for (i) the differing arrival rates of customers and drivers in different areas of the city, (ii) how long

customers are willing to wait for driver pick-up, and (iii) the time-varying nature of all the aforementioned

parameters. We prove asymptotic optimality of a CLP-based policy in a large market regime. However,

solving the CLP is difficult, thus we also propose matching policies based on a linear program (LP). We prove

asymptotic optimality of an LP-based policy in a large market regime in which drivers are fully utilized.

We conduct simulation experiments to test the performances of the CD, the LP-based, and the CLP-based

matching policies.

Key words : ridesharing platforms, dynamic matching, asymptotic optimality.

History :

1. Introduction

Ridesharing platforms are online mobile platforms which match paying customers who

need a ride with drivers who provide transportation. Some examples of these platforms are

Uber and Lyft in America, Grab in Southeast Asia, and Didi Chuxing in China. Rideshar-

ing companies emerged in the last ten years, as the market penetration of smartphones

exploded, and have grown exponentially fast. The ridesharing companies all make use of

the relatively recent ability to track the locations of the customers and drivers continuously

1

mailto:[email protected], [email protected]

Author: Dynamic Matching for Real-Time Ridesharing2 Article submitted to Manufacturing & Service Operations Management; manuscript no. (Please, provide the manuscript number!)

(unlike the taxicab companies) which enables responsive and efficient market control (cf.

Azevedo and Weyl (2016)).

Ridesharing platforms are two-sided matching markets (cf. Rochet and Tirole (2006))

that pair customers and drivers. Typically, when a customer needs a ride, she requests the

ride via an application on a smartphone. The system controller, who is the correspond-

ing ridesharing company, matches the customer with a driver, if both sides approve the

matching. Then, the driver picks up the customer and takes her to her destination.

The ridesharing company would like to ensure there is always a nearby driver to offer an

arriving customer. This is because if a customer must wait too long for pick-up, then the

customer may refuse the ride and use another transportation option. However, maintaining

adequate driver supply is difficult. Not only do customers choose when to request a ride,

but also drivers choose when to begin work, how long to work, and where to go to search

for customers. The result can be dramatic changes in customer demand and driver supply

across different locations and over the course of a day, which sometimes results in significant

driver shortages (cf. Figures 1-3 in Hall et al. (2016)).

One common operational strategy is to match customers with the closest driver (CD),

and to use pricing to incentivize drivers to move to undersupplied locations. However,

surges in price can lead to negative publicity (cf. White (2016), The Economist (2016), and

Michallon (2016)). This leads to the question of whether better matching decisions could

reduce the need for price surges. An ideal is to solve a joint pricing and matching problem

that accounts for the impact of differing customer and driver location. However, that is

a very difficult problem. As a first step, we assume customer and driver arrival rates are

given, and focus on optimizing the matching decisions. The aforementioned rates can be

time-varying, and could potentially be thought of as the result of a pricing policy (as well

as other underlying incentives that may be given to drivers or customers, such as demand

information sharing or discount coupons). Our underlying assumption is that a decoupling

between the pricing and matching strategies is helpful. That is, the pricing strategy can be

used to balance the overall number of customers and drivers in the system, and the match-

ing strategy can be used to improve the geospatial driver supply and customer demand

balance.

The following example motivates why the matching decisions made by the CD policy

can be improved. Figure 1 represents part of a city, partitioned into ten disjoint hexagonal

Author: Dynamic Matching for Real-Time RidesharingArticle submitted to Manufacturing & Service Operations Management; manuscript no. (Please, provide the manuscript number!) 3

areas, as is done by Uber (cf. Figure 3 in Chen and Sheldon (2016)). Suppose a customer

arrives at area 1 and requests a ride. There are three drivers idle in area 4 and there is

a single idle driver in area 3. Moreover, a concert recently ended in area 8, which implies

a high potential customer arrival rate in that area. If the destination of the current area

1 customer is far away, the driver assigned to that customer will not return to area 1 for

a long time. Then, the system controller has two options: He can either offer one of the

drivers in area 4 or the driver in area 3 to the customer in area 1. Under the CD policy,

the system controller offers an area 4 driver. However, offering the driver in area 3 saves

all drivers in area 4 to match with the potential customers departing the concert in area

8. This prevents the potential future need to offer an area 8 customer a far-away area 3

driver, whom the customer will likely refuse, ending in no match being made. We conclude

there is a nontrivial tradeoff between offering the closest driver in accordance with the CD

policy and offering a farther driver in order to maximize the future number of customers

matched.

!

"

#

$

%

&

'

(

)

!*

Figure 1 An intuitive explanation of why the CD policy may not assign the right driver to a customer.

1.1. Contributions of the Paper

We propose a matching policy for ridesharing and prove its optimality in a large mar-

ket asymptotic regime. The relevant objective is to maximize the number of customers

matched with drivers. The reason we focus on this objective rather than other common

objectives such as maximizing profit or revenue is that early-stage companies (such as most

ridesharing companies) often are mainly interested in growing their market share. This

leads to the following main contributions.


Formulating an analytically tractable ridesharing system model that captures location:

The essential first step to find an optimal, or near-optimal, matching policy for ridesharing

is the development of an analytically tractable model. That model must be rich enough to

capture the salient features of driver behavior. The issue is that tracking each individual

driver’s movement makes the analysis very difficult, if not impossible. Our novel insight is

that the area partitioning shown in Figure 1 leads naturally to the use of Poisson processes

to capture driver location. In particular, our assumption is that in the aggregate, the

individual driver decisions regarding when to begin and end work, if and where to move

when not transporting a customer, and what to do after customer drop-off result in non-

homogeneous Poisson arrival and departure processes to and from each of the different

areas.

Deriving an upper bound on the performance of any (admissible) matching policies: We

consider a large matching market, in which the arrival rates of the customers and drivers

grow without bound, so as to approximate the case of a large city. For any matching

policy that does not know the future with certainty, we establish that the solution to a

continuous linear program (CLP) is an asymptotic upper bound on the cumulative number

of matchings done in a finite time horizon, under fluid scaling (cf. Theorem 1). That upper

bound is very strong in the sense that it is valid on almost every sample path.

Proposing a matching policy based on the CLP: We provide matching policies that

asymptotically mimic the performance of any feasible CLP matching process (cf. Theorem

2). When the CLP is solvable, this leads to an asymptotically optimal matching policy (cf.

Corollary 1 and Theorem 3). Otherwise, there may exist a feasible CLP matching process

that can be used to develop a forward-looking policy that ameliorates the problematic

myopic behavior of the CD policy as in the example in Figure 1.

Proposing an asymptotically optimal matching policy based on a linear program (LP):

The upper bound CLP leads to an easily solvable upper bound LP when drivers are fully

utilized. We provide intuitive sufficient conditions on an underlying network game such

that drivers are fully utilized whenever prices “surge”. This motivates an asymptotically

optimal LP-based matching policy (cf. Theorem 4).

Providing simulation experiments to illustrate the performance of our CLP-based and

LP-based policies: Our simulation experiments in Section 5 show that both the CLP-

based and the LP-based proposed matching policies can significantly outperform the CD


policy. Consistent with intuition, we see that demand spikes coupled with low nearby driver

availability, such as in Figure 1, lead to the poor performance of the CD policy. This is

exactly when we recommend spending the extra effort of estimating parameters, such as

customer and driver arrival rates, in order to be able to implement a CLP- or an LP-

based matching policy. (In comparison, the CD policy requires no network information to

implement.)

The remainder of this paper is organized as follows. We conclude this section with a

literature review (cf. Section 1.2) and a summary of our mathematical notation (cf. Section

1.3). Section 2 presents our model. We formalize our large matching market regime, an

asymptotic upper bound on the cumulative number of matchings, and prove that a CLP-

based matching policy achieves that upper bound in Section 3. We motivate a condition

under which an LP-based matching policy achieves the asymptotic upper bound in Section

4. Section 5 presents some simulation experiments. Finally, we make concluding remarks

in Section 6. The proofs of all results can be found in the E-Companion.

1.2. Literature Review

In recent years, academic research related to ridesharing platforms has grown rapidly,

alongside the use of these platforms. An overview of research problems on ridesharing

platforms can be seen in Azevedo and Weyl (2016). While the effects of pricing have been

well-studied (cf. Chen et al. (2015), Cachon et al. (2016), Chen and Sheldon (2016), Hall

et al. (2016), Riquelme et al. (2016), Bimpikis et al. (2017), Guda and Subramanian (2017),

and Taylor (2017)), we are aware of no research on other ways to maximize matchings.

Dynamic matching control has been studied in the literature in the context of kid-

ney exchanges (cf. Unver (2010)), housing markets (cf. Leshno (2016)), online matching

platforms such as Upwork or Airbnb (cf. Arnosti et al. (2016)), and assemble-to-order

manufacturing systems (cf. Plambeck and Ward (2006) and Reiman and Wang (2015)).

However, the ridesharing model is different enough that it is not clear any of the results of

these studies carry over.

A method to control dynamic matching systems is the market thickening approach,

where the system controller first waits for jobs (that is, customers and drivers in the

context of this paper) to build-up, or thicken, in the market and then makes the matching

decisions. In the queueing literature, Gurvich and Ward (2014) consider a matching system

where jobs with multiple classes arrive in the system dynamically and randomly and prove


asymptotic optimality of a discrete review matching policy as the arrival rates of the jobs

increase (i.e., a large market assumption). The discrete review policy first lets the jobs

thicken in the system between the review epochs and then makes the matching decisions by

solving an LP at the review epochs. Akbarpour et al. (2016) consider a dynamic matching

network where jobs made to wait too long to be matched may abandon, or leave the system

without being matched. They prove that if the system controller can identify which jobs

are about to abandon, and match only those jobs, then waiting for other jobs to arrive

and thicken the market is very valuable. A straightforward implementation of the market

thickening approach does not work in our case because customers expect drivers to be

assigned to them immediately, and the system controller does not know when drivers will

decide to finish working for the day. However, a large market assumption that is similar

in spirit to the ones in Plambeck and Ward (2006), Gurvich and Ward (2014), Akbarpour

et al. (2016), Arnosti et al. (2016), Leshno (2016), and Riquelme et al. (2016) is the key

methodological insight we use to derive our matching policies.

A recent study applicable to ridesharing platforms is Hu and Zhou (2016), which con-

sider dynamic matching control of a two-sided, discrete-time matching system where both

supply and demand can abandon the system and the objective is to maximize the expected

total discounted profit. By using dynamic programming, Hu and Zhou (2016) derive some

structural properties of the optimal matching policies, that are different from ours (see

Remark 8).

The spirit of our methodology is drawn from the queueing control literature, where (i)

an asymptotic regime is defined, (ii) a control policy is derived from the solution of an

optimization problem, and (iii) asymptotic optimality of the control policy is proven (cf.

Harrison (2000) and Ata and Kumar (2005)). Similar methodology is used in Braverman

et al. (2017) to propose an empty car routing policy in a ridesharing model with limited

driver supply. However, that methodology does not in general allow for the derivation of

asymptotically optimal controls when the queueing network has time-varying arrival rates.

In contrast, in the context of matching, Gurvich and Ward (2014), Hu and Zhou (2016),

and our study all allow for time-varying arrival rates. Moreover, Hu and Zhou (2016) and

our study both consider the situation of a time-varying distribution that represents the

allowed waiting time for a match.


There are some recent studies related to the stability of dynamic matching networks,

cf. Caldentey et al. (2009), Adan and Weiss (2012), Busic et al. (2013), Buke and Chen

(2015), and Moyal and Perry (2016). However, stability is not an issue in our case because

customers do not wait for pick-up from far away drivers and drivers leave their area if not

matched with a customer quickly enough.

1.3. Notation and Terminology

The set of nonnegative integers and strictly positive integers are denoted by N and N+,

respectively. The k dimensional (k ∈N+) Euclidean space is denoted by Rk and R+ denotes

[0,+∞). For x, y ∈ R, x ∨ y := maxx, y, x ∧ y := minx, y, and (x)+ := x ∨ 0. We let

B(Rk) denote the Borel σ-algebra on Rk for all k ∈N+ and L(R) denote the collection of

Lebesgue measurable subsets of R, which is a σ-algebra on R. For all T ∈ R+, B([0, T ])

and L([0, T ]) denote the Borel and Lebesgue σ-algebra on the interval [0, T ], respectively.

A function f :X→R defined in measure space (X,X ) is called X -measurable (denoted by

f ∈X ), if it is (X ,B(R))-measurable. If the measure space (X,X ) is (R,B(R)) ((R,L(R))),

we say that f is Borel (Lebesgue) measurable.

For each k ∈N+, Dk denotes the the space of all ω : R+→Rk which are right continuous

with left limits. Let 0, e∈D be such that 0(t) = 0 and e(t) = t for all t∈R+. We abbreviate

the phrase “independent and identically distributed” by “i.i.d.” and “almost surely” by

“a.s.”. The notationa.s.−−→ denotes almost sure convergence. We assume that all of the

random variables and stochastic processes are defined in the same complete probability

space (Ω,F ,P), E denotes the expectation under P, and P(A,B) := P(A ∩ B). We let

σX denote the σ-field generated by the random variable X, I denote the indicator

function, and ⊥ denotes independence.

2. A Ridesharing Model

We partition the city into N ∈ N+ disjoint areas, as illustrated in Figure 1, and assume

that in the aggregate the individual driver decisions (regarding their work schedule and

location) result in Poisson process arrivals and departures to and from each area. Specifi-

cally, type i drivers arrive at area i ∈N := 1,2, . . . ,N according to a non-homogeneous

Poisson process having instantaneous rate λi(t) at time t∈R+, and cumulative rate func-

tion Λi(t) :=∫ t

0λi(s)ds. The instantaneous rate at which type i drivers neither transporting

customers nor on their way to pick-up a customer leave area i∈N is θi(t).


Customers arrive in the system and request to be matched. The J ∈ N+ different cus-

tomer types are categorized by factors such as their origin and destination area, and

their priority status. Type j ∈ J := 1,2, . . . , J customers arrive in accordance with a

non-homogeneous Poisson process having instantaneous rate µj(t) at time t ∈ R+, and

cumulative rate function Γj(t) :=∫ t

0µj(s)ds. A matching policy π= (π1, . . . , πJ) determines

which driver type to offer an arriving customer. Each component πj tracks the sequence

of driver types offered to type j customers; i.e., when πj(k) = i, then the system controller

attempts to match the kth arriving type j customer with a type i driver, for j ∈ J , k ∈

N+, i ∈ N ∪ 0. The notation πj(k) = 0 implies no driver is offered to the customer, in

which case the customer is lost. The customer accepts the offered driver if the waiting time

required for pick-up is not too large (in a sense specified precisely below), and otherwise

the customer is lost and the offered driver stays in his current area. The implication is that

customers are classified as matched or unmatched (i.e., lost) at the time of their arrival,

even though a matched customer must still wait to be picked-up by a driver. The process

Dπij(t) tracks the cumulative number of type i drivers matched to type j customers under

policy π in [0, t].

The number of drivers in area i at time t∈R+ depends on the matching policy π. Then,

for Ai and Ri unit rate Poisson processes, the number of unmatched type i drivers at time

t is:

Qπi (t) =Qi(0) +Ai (Λi(t))−Ri

(∫ t

0

θi(s)Qπi (s)ds

)−∑j∈J

Dπij(t)≥ 0, (1)

where Qi(0), i ∈N are random variables independent of all other stochastic primitives.

The second term in the right-hand side of (1) represents the cumulative number of driver

arrivals to area i in [0, t], while the third and fourth terms represent the cumulative number

of driver departures from area i in [0, t] by unmatched and matched drivers, respectively.

We would like to match as many customers with drivers as possible. This is straightfor-

ward if there are many matching policies under which Qπi (t)> 0 for all i ∈N and t ∈R+,

because then there is always a driver near to an arriving customer. The difficulty is that in

general not every area will have an available driver – and which areas have available drivers

depends on earlier matching decisions. In this case, the arriving customer is matched only

if the time required for the driver to pick-up the customer is less than the amount of time

that customer is willing to wait for a driver.


The time a customer is willing to wait for a driver can depend on the time of day. For

example, during working hours, a customer may be more time-constrained than during

non-working hours. We represent this using a step function that depends on the customer

arrival time. First, we partition the time horizon into countably many disjoint intervals by

defining the deterministic sequence τm,m ∈ N such that τm ∈ R+ and τm ≤ τm+1 for all

m∈N, and τm→∞ as m→∞. Second, to allow for potentially changing traffic conditions,

we define the pick up time of a type i driver for a type j customer who arrived in the system

at time t ∈ [τm, τm+1) by tij(m) ∈ R+ for all i ∈ N , j ∈ J , and m ∈ N. Third, we denote

the time the kth type j customer arrival is willing to wait for pick-up given the arrival

occurred at time t∈ [τm, τm+1) by the random variable akj (m) for all k ∈N+, j ∈J , m∈N.

The sequence akj (m), k ∈N+ is i.i.d., and independent of all other stochastic primitives

for all m∈N and j ∈J . Then, the probability that the kth type j customer accepts a type

i driver, given the arrival time is t, is

Fij(t) =∞∑m=0

P(akj (m)≥ tij(m)

)I (t∈ [τm, τm+1)) (2)

for all i∈N , j ∈J , k ∈N+, and t∈R+.

We connect our model to the queueing literature by showing its visual relationship to

a stochastic processing network (cf. Harrison (2000)) in Figure 2. The main difference is

that customers that do not accept their offered “server” (i.e., driver) are lost, with the

probabilities specified by Fij(t), for i∈N , j ∈J , and t∈R+.

The closest driver (CD) policy, denoted by πCD, offers a type j customer that arrives at

time t∈R+ a driver type from the set

arg mini∈N :Q

πCDi (t−)>0

∑m∈N

tij(m)I(t∈ [τm, τm+1)). (3)

If the set in (3) is not a singleton, the offered driver is chosen randomly from the closest

drivers. The question is: Does the CD policy match as many customers as possible?

The total cumulative number of matchings under any policy π depends on the number

of customers willing to wait for their offered driver. Specifically, if Ej is a unit rate Poisson

process and νj(k) := inft ∈ R+ : Ej(Γj(t)) = k is the arrival time of the kth type j ∈ Jcustomer, k ∈N+, then

Dπij(t) :=

Ej(Γj(t))∑k=1

∑m∈N

I(νj(k)∈ [τm, τm+1), a

kj (m)≥ tij(m), πj(k) = i

), (4)

Author: Dynamic Matching for Real-Time Ridesharing10Article submitted to Manufacturing & Service Operations Management; manuscript no. (Please, provide the manuscript number!)

! " !

!" # !$ # !% #

&" #

'" #

&$ #

'$ #

&( #

'( #

)*"" #)*$" # )*(" #

Figure 2 A queueing-inspired visualization of our model formulation.

for all i ∈ N , j ∈ J , t ∈ R+, and under any policy π. Our objective is to find a policy π

that maximizes the total cumulative number of matchings made in a finite time horizon

over a specified class of admissible matching policies Π; that is, to solve

maxπ∈Π

∑i∈N ,j∈J

Dπij(T,ω), (5)

for all ω ∈Ω. In studying this problem, we obtain some results on the more general objective

maxπ∈Π

∑i∈N ,j∈J

wijDπij(T,ω) (6)

that also allows the system manager to provide weights wij, i ∈N , j ∈ J , on the possible

matchings. The objectives (5) and (6) are very strong objectives because solving either

requires specifying a policy that maximizes the number of matchings on every sample path.

Remark 1. In our model formulation, the driver types are formed only based on the

arrival locations of the drivers. However, we can extend our results to an arbitrary (but

finite) number of driver types I ∈ N+ by updating akj (m) to akij(m) for all k ∈ N+, j ∈ J ,

m ∈ N, and i ∈ I := 1,2, . . . , I. Then, akij(m) denotes the patience time of the kth type

j customer for a type i driver given that she arrived in the system on the time interval

[τm, τm+1), for all k ∈ N+, j ∈ J , m ∈ N, and i ∈ I. All of our results hold under this

extension.

Author: Dynamic Matching for Real-Time RidesharingArticle submitted to Manufacturing & Service Operations Management; manuscript no. (Please, provide the manuscript number!)11

2.1. Admissible Policies

Roughly speaking, an admissible matching policy cannot anticipate the future. This is

formalized mathematically by defining the filtration F := F(t), t ∈ R+ that represents

the information known by the system controller as time moves forward as follows.

F(t) := σ

Ai(Λi(s)),Ej(Γj(s)),Ri

(∫ s−

0

θi(u)Qi(u)du

),Dij(s−),Qi(s−), akj (m)

∀s∈ [0, t], i∈N , j ∈J , m∈N, k ∈ 1,2, . . . ,Ej(Γj(t−))

. (7)

In (7), we see that when a customer arrives in the system at time t, the system con-

troller does not know how long she is willing to wait for driver pick-up. This is important

because otherwise the system controller could cherry-pick certain customers to offer far-

away drivers without consequence. The technical implication is that the filtration F is not

right continuous.

The information the system controller has at the arrival epoch of the kth type j customer

is

Fj(k) := σ

Ai(Λi(s∧ νj(k))),Ej′(Γj′(s∧ νj(k))),Ri

(∫ (s∧νj(k))−

0

θi(u)Qi(u)du

),

Dij′((s∧ νj(k))−),Qi((s∧ νj(k))−), ∀i∈N , j′ ∈J , s∈R+,

arj′(m), r ∈ 1, . . . ,Ej′(Γj′(νj(k)−)),∀j′ ∈J \j, arj(m), r ∈ 1, . . . , k− 1,∀m∈N

, (8)

for all k ∈ N+ and j ∈ J . Since each of the stochastic processes that generate the σ-field

in (7) is either right or left continuous and νj(k) is a stopping time with respect to the

filtration F for all j ∈J and k ∈N+, the σ-field in (8) is well defined. Since νj(k)≤ νj(k+1)

for all k ∈N+, Fj :=Fj(k), k ∈N+

is a filtration for all j ∈J .

Definition 1. (Admissible Policies) For all j ∈J , let Πj denote the set of discrete-time

stochastic processes with domain N+×Ω and range N ∪0, such that for all πj ∈Πj, πj

is Fj-adapted (i.e., πj(k)∈Fj(k) for all k ∈N+) and if Qi(νj(k)−) = 0 for some i∈N , then

πj(k) 6= i. Let Π be the set of J-dimensional discrete-time stochastic processes such that

for all π ∈Π, we have π= (π1, π2, . . . , πJ) where πj ∈Πj for all j ∈J . Then, Π is the set of

admissible policies.

Lemma 1. The CD policy is admissible; that is, πCD ∈Π.


Remark 2. Although the system controller cannot anticipate either the arrival times of

the customers, or how long they will wait for pick-up, or how long the drivers will remain

in their current area, he can accurately forecast the arrival rates of the customer and driver

types and the driver acceptance probabilities associated with the customer types. In other

words, he knows the functions Λi, Γj, and Fij for all i∈N and j ∈J .

Remark 3. The filtration F can be augmented to include additional stochastic processes

Υm : R+×Ω→R with right or left continuous sample paths for all m ∈N, provided that

the sequence Υm,m∈N does not contain any future information related to the processes

that generate F(t) for all t∈R+. For example, if the system controller randomly chooses a

driver to offer an arriving customer, then Υm,m∈N includes a sequence of i.i.d. random

variables that reflect the outcome of an N -sided die roll.

Remark 4. Since the arrival process of each customer type is a non-homogeneous Pois-

son process, the probability that more than one customer arrives in the system simultane-

ously at some time epoch is zero. Therefore, the range of all πj ∈Πj is chosen as N ∪0

instead of the m-fold Cartesian product of N ∪0 for some m≥ 2 for all j ∈J .

Remark 5. Let F(νj(k)) denote the sigma algebra defined by the stopping time νj(k)

as in Definition 1.2.12 of Karatzas and Shreve (1988). Another way to define Πj is such

that for all πj ∈Πj, πj(k)∈F(νj(k)) for all k ∈N+. We do not choose this option because

proving akj (m)⊥F(νj(k)) for all j ∈J , m∈N, and k ∈N+ is difficult1, but akj (m)⊥Fj(k)

is by construction (cf. (8)). This result is exactly what prevents the system controller

knowing how long an arriving customer is willing to wait for driver pick-up, and so is

crucial to our model and analysis.

We end this section with the following technical assumption.

Assumption 1. (Technicalities) The functions λi : R+ → R+ and µj : R+ → R+ are

Borel measurable and the function θi is defined such that θi ∈D and θi ≥ 0 for all i∈N and

j ∈J . We assume that supt∈R+λi(t)<∞, supt∈R+

θi(t)<∞, and supt∈R+µj(t)<∞ for all

i∈N and j ∈J . The unit rate Poisson processes Ai,Ri, and Ej are mutually independent

for all i∈N and j ∈J .

1 Intuitively, one can expect that Fj(k) =F(νj(k)) for all j ∈J and k ∈N+. Although such a result is proved underspecific measure spaces (cf. Lemma 5.4.18 of Karatzas and Shreve (1988) and Lemma 1.3.3 of Stroock and Varadhan(2006)) or under some assumptions (cf. Theorem 1.6 of Shiryaev (2008)), none of them is applicable to our case.


3. An Asymptotically Optimal CLP-Based Matching Policy

It is very difficult to solve the optimization problem (6) exactly. Even if we can accomplish

this very challenging task, the optimal matching policy will most likely be sample path

dependent and will be very complicated. Hence, we consider a large market where the

arrival rates of the customers and drivers grow without bound, and solve (6) under fluid

scaling in that limiting regime. Section 3.1 specifies the large market limiting regime and

the fluid scaling. Section 3.2 establishes that the solution to a CLP serves as an asymptotic

upper bound on the objective (6) under fluid scaling. Section 3.3 provides a simple policy

that can achieve asymptotically any matching process feasible for the CLP, and, therefore,

can be used to specify an asymptotically optimal policy when the CLP is solvable.

3.1. A Large Matching Market

We consider a sequence of matching systems indexed by n, n∈N+. Each matching system

has the same primitives with the one introduced in Section 2 except that the arrival rates

of the drivers and customers, and departure rates of unmatched drivers from their current

areas depend on n. In the nth matching system, for all i∈N , j ∈J , and t∈R+, let

Λni (t) :=

∫ t

0

λni (s)ds, Γnj (t) :=

∫ t

0

µnj (s)ds, (9)

where λni : R+→ R+ and µnj : R+→ R+ are Borel measurable rate functions for all i ∈N ,

j ∈ J , and n ∈ N+. Moreover, θni ∈ D and θni ≥ 0 for all i ∈ N and n ∈ N+. A policy

π = πn, n∈N+ is a sequence that specifies a policy for each n, and π is admissible if πn

is admissible for all n ∈ N+. For a policy such as CD that does not change with n, in a

slight abuse of notation, we specify π (i.e., π = πCD) and assume πn = π for all n ∈ N+.

Our notational convention is to denote a process (or random variable) X in the nth system

under the admissible policy π by Xπ,n.

Increasing the arrival rates without a bound and keeping the departure rates of

unmatched drivers from their areas bounded results in a crowded matching system where

we can use law of large numbers type of results to obtain tractable approximations for the

processes of interest.

Assumption 2. (Large Market) We assume that Λni /n → Λi uniformly on compact

intervals (u.o.c.)2 and∫∞

0|µnj (s)/n−µj(s)|ds→ 0 as n→∞ for all i∈N and j ∈J . The

2 For f ∈ D and T1 ∈ R+, we let ‖f‖T1 := sup0≤t≤T1|f(t)|. Let Xn, n ∈ N be a sequence in D and X ∈ D. Then

Xn→X u.o.c. as n→∞, if ‖Xn−X‖T1 → 0 as n→∞ for all T1 ∈R+.


latter convergence implies that Γnj /n→ Γj u.o.c. as n→∞ for all j ∈J . We assume that

θni → θi u.o.c. for all i∈N .

Assumption 2 does not necessarily imply that the arrival rates of all customer and driver

types grow to infinity for all t ∈ R+ as n→∞. In particular, arrival rates can be zero in

some areas during some time periods (e.g., λni = µnj = λi = µj = 0 for some i∈N and j ∈Jand all n∈N+), as may be true in parts of the city during very early morning hours.

We focus on the first-order imbalances between the driver supply and customer demand

by considering fluid scaling. For all i∈N , j ∈J , t∈R+, n∈N+, and admissible policy π,

define

Ani (t) :=Ai(nt)/n, Rn

i (t) :=Ri(nt)/n, Enj (t) :=Ej(nt)/n, (10a)

Λni := Λn

i /n, Γnj := Γnj /n, Qπ,ni :=Qπ,n

i /n, Dπ,nij :=Dπ,n

ij /n. (10b)

By (1), (10a), and (10b), for all i∈N , t∈R+, n∈N+, and admissible policy π,

Qπ,ni (t) = Qn

i (0) + Ani (Λn

i (t))− Rni

(∫ t

0

θni (s)Qπ,ni (s)ds

)−∑j∈J

Dπ,nij (t)≥ 0. (11)

Our analysis requires the initial number of drivers waiting to be very small. When the

horizon length represents a working day, this corresponds to assuming the number of drivers

during the late night hours is very small. In general, the horizon length is arbitrary, and

can be smaller (i.e., correspond to so-called “rush” hour) or longer (i.e., represent multiple

days).

Assumption 3. (Initial Conditions) We assume that Qni (0)

a.s.−−→ 0 as n→∞ for all

i∈N .

Assumptions 1, 2, and 3 are in force throughout the paper.

3.2. An Asymptotic CLP Upper Bound

In this section, we derive an asymptotic upper bound on the fluid scaled objective (6) by

solving a CLP. The decision variables xij : [0, T ]→ R+ for all i ∈ N and j ∈ J are such

that xij(t) denotes the fraction of type j customers who are offered type i drivers at time

t, so that µj(t)Fij(t)xij(t) approximates the instantaneous matching rate between type i

drivers and type j customers. The relevant CLP is:

max∑

i∈N ,j∈J

wij

∫ T

0

µj(s)Fij(s)xij(s)ds, (12a)


subject to:∑j∈J

∫ t

0

µj(s)Fij(s)xij(s)ds≤Λi(t), for all i∈N and t∈ [0, T ], (12b)

∑i∈N

xij(t)≤ 1, for all j ∈J and t∈ [0, T ], (12c)

xij(t)≥ 0, for all i∈N , j ∈J , and t∈ [0, T ], (12d)

xij is Lebesgue measurable for all i∈N and j ∈J . (12e)

Constraint (12b) implies that the cumulative number of customers who are assigned to

type i drivers up to time t cannot be greater than the cumulative number of type i drivers

arrived in the system up to time t for all t ∈ [0, T ] and i ∈ N . Constraint (12c) implies

that a customer cannot be offered more than one driver. A more complicated CLP also

incorporates the fluid-scaled version of (11) (so that parameter θi, i∈N appears); however,

as discussed in the E-Companion, that more complicated CLP does not lead to more

general results.

Lemma 2. There exists an optimal solution of the CLP (12).

We denote an optimal solution of the CLP (12) by the process x := xij(t), i ∈ N , j ∈

J , t∈ [0, T ]. The following theorem establishes that the optimal objective function value

of the CLP (12) is an asymptotic upper bound on the fluid scaled objective (6) under any

admissible policy for almost all sample paths.

Theorem 1. Under any admissible policy π,

lim supn→∞

∑i∈N ,j∈J

wijDπ,nij (T )≤

∑i∈N ,j∈J

wij

∫ T

0

µj(s)Fij(s)xij(s)ds, a.s.

The key challenge is to show that there exists a feasible matching process xij(t), i∈N , j ∈

J , t∈ [0, T ] (not necessarily optimal) such that the derivative of the limit of the fluid scaled

cumulative matching process Dπ,nij (t), i ∈ N , j ∈ J , t ∈ [0, T ] corresponds to the process

µj(t)Fij(t)xij(t), i ∈N , j ∈ J , t ∈ [0, T ]. If the aforementioned feasible matching process

is x, then the upper bound in Theorem 1 is attained, meaning the policy is asymptotically

optimal.

The CLP (12) assumes that drivers will wait in their current location as long as necessary

to be matched with customers, because the CLP does not incorporate θi, i∈N . Although

that provides a clear upper bound on the matching rates, that upper bound is not always


achievable. In particular, in the case of a forward-looking CLP solution, the drivers must

be patient enough to wait to be matched with closer customers arriving later in time (as

in the example depicted in Figure 1).

3.3. A CLP-Based Randomized Policy

We would like to find an admissible matching policy that can reproduce any feasible match-

ing process x := xij(t), i∈N , j ∈J , t∈ [0, T ] for the CLP (12) (including an optimal one,

x). This is important because finding a feasible matching process that improves on any

myopic matching policy (such as CD) may be possible even when finding an optimal CLP

solution is not. The question is: How do we translate between a feasible matching process

and the decision of which driver to offer an arriving customer?

Definition 2. (Randomized Policy) If a type j customer arrives in the system at time

t, the system controller makes a random selection from the set N ∪ 0 such that the

probability that the outcome is i is xij(t) for all i∈N and the probability that the outcome

is 0 is 1−∑

i∈N xij(t). If the outcome is i for some i∈N and there is a type i driver in the

system, then the system controller offers a type i driver to the customer. If the outcome

is i for some i∈N but there is no type i driver in the system or if the outcome is 0, then

no driver is offered to the customer (and so the customer is lost).

We denote the randomized policy by πR and write πR(x) when we want to emphasize

its dependence on the particular feasible matching process x. Under a technical condition

on the associated feasible matching process for the CLP (12), πR is admissible.

Lemma 3. Suppose that the feasible matching process for the CLP (12), x, is such that

xij is a Borel measurable simple function with possibly countably infinite range for all i∈N

and j ∈J . Then, πR(x) is admissible.

The randomized policy is a simple policy, because no state information is needed for

its implementation. Still, its performance replicates any feasible matching process for the

CLP (12), when drivers are willing to wait in their current locations to be matched.

Theorem 2. Suppose x is a feasible matching process that satisfies the condition in

Lemma 3. Fix an arbitrary i∈N . If θi = 0, then

sup0≤t≤T

∣∣∣∣∣∑j∈J

(DπR(x),nij (t)−

∫ t

0

µj(s)Fij(s)xij(s)ds

)∣∣∣∣∣ a.s.−−→ 0, as n→∞.


Theorem 2 shows that when an optimal solution x to the CLP (12) is a Borel mea-

surable simple function, then πR(x) attains the upper bound in Theorem 1. However, x

may not satisfy the aforementioned condition. Then, we do not know the associated ran-

domized policy is admissible, and so we cannot use Theorem 2 to ensure the upper bound

in Theorem 1 is achieved. However, we can approximate an optimal CLP (12) solution

with a sequence of Borel measurable simple functions to ensure the upper bound is nearly

achieved. Specifically, by Theorem 2.10 and Proposition 2.12 of Folland (1999), there exists

a sequence of feasible matching processes for the CLP (12) denoted by xr, r ∈N+ such

that xr := xrij(t), i ∈ N , j ∈ J , t ∈ [0, T ], for all i ∈ N , j ∈ J , and r ∈ N+, xrij is a Borel

measurable simple function, 0≤ xrij(t)≤ xr+1ij (t)≤ xij(t) for all t∈ [0, T ], and xrij(t)→ xij(t)

as r→∞ for all t ∈ [0, T ] except on a set of zero measure. Then, by Theorem 2 and the

bounded convergence theorem used on the sequence µj(t)Fij(t)xrij(t), t ∈ [0, T ], r ∈ N+,

we have the following corollary.

Corollary 1. If θi = 0 for all i ∈ N , then for any ε > 0, there exists an r0(ε) ∈ N+

such that if r≥ r0(ε),

limn→∞

∑i∈N ,j∈J

DπR(xr),nij (T )≥

∑i∈N ,j∈J

∫ T

0

µj(s)Fij(s)xij(s)ds− ε, a.s.

Solving the CLP (12) or accurately approximating an optimal solution of it is a very

challenging task (cf. Perold (1981)). If µj(t)Fij(t) was a constant function of t for all i∈N

and j ∈ J , then the CLP (12) would become a Separated Continuous Linear Program

(SCLP). Although an SCLP is solvable (cf. Anderson et al. (1983) and Weiss (2008)), it

is an NP-hard problem (cf. Bertsimas et al. (2015)). The even more restricted assumption

that eliminates time dependent parameters implies that the CLP (12) can be solved by an

LP.

Theorem 3. If λi, µj, and Fij are constant functions of t for all i∈N and j ∈J , then

there exists an optimal solution of the CLP (12) denoted by x such that x(t) = x∗ for all

t∈ [0, T ], where x∗ := x∗ij, i∈N , j ∈J is an optimal solution of the following LP, having

the decision variables xij, i∈N , j ∈J :

max∑

i∈N ,j∈J

wijµjFijxij, (13a)


subject to:∑j∈J

µjFijxij ≤ λi, ∀i∈N ,∑i∈N

xij ≤ 1, ∀j ∈J , xij ≥ 0, ∀i∈N , j ∈J .

(13b)

Furthermore, if wij = 1 for all i∈N and j ∈J , then πR(x) achieves the asymptotic upper

bound given in Theorem 1; i.e.,

limn→∞

∑i∈N ,j∈J

DπR(x),nij (T ) =

∑i∈N ,j∈J

∫ T

0

µj(s)Fij(s)xij(s)ds, a.s.

In contrast to Theorem 2 and Corollary 1, Theorem 3 does not require θi = 0 for all

i ∈N . This is because an LP-based policy is myopic. In particular, such a policy will not

prolong driver waiting in some areas in anticipation of future demand changes.

The assumption of time homogeneous parameters in Theorem 3 is severe. We would like

to find another sufficient condition under which the CLP (12) can be easily solved.

4. An Asymptotically Optimal LP-Based Randomized Policy

An LP-based matching policy arises naturally when we consider an underlying network

game in which raising prices decreases customer demand and increases driver supply. Then,

a“smart” system controller will not raise prices beyond what is needed to match driver

supply with customer demand. The implication is that the CLP constraint (12b) should be

binding, as discussed in Section 4.1. Then, an LP-based randomized policy is asymptotically

optimal, as shown in Section 4.2.

4.1. Connection of the CLP (12) with Pricing

Suppose we view λi(t), µj(t), and θi(t) for all i∈N and j ∈J as outcomes of an underlying

network game played by the customers, the drivers, and the system controller at time t

for all t ∈R+. Then, the arrival rates of customers and drivers (λi(t) and µj(t)), and the

departure rates of unmatched drivers (θi(t)), can depend on prices, and potentially other

incentives, such as travel times and shared demand information. Provided λi, µj, and θi

for all i∈N and j ∈J do not depend on the matching policy, all of the analyses hold; in

particular, Theorem 1 provides an upper bound on the cumulative number of matchings

and Theorem 2 implies that any feasible matching process for the CLP (12) is achievable

by the randomized policy.

We would like to provide intuition for why an underlying network game can result in the

ability to specify a CLP optimal solution through LP solutions. To do that, suppose that


the CLP constraint (12b) is not binding at some times in an area. The implication is that

there are excess drivers. Then, assuming that decreasing prices encourages more customers

but less drivers to arrive, the system controller would like to decrease prices. However, the

system controller would only do that until there are no more excess drivers, that is until all

drivers are matched. Furthermore, the system controller would not decrease prices more

because that would result in more customers than drivers.

Requirement on Underlying Network Game If there are unmatched drivers in

the system, then there exists a pricing adjustment under which all drivers being matched

increases the total matchings.

This suggests that “good” pricing decisions lead to an outcome in which driver supply and

customer demand over all the areas are equal under “good” matching decisions.

Assumption 4. (Fully Utilized Drivers) There exists a feasible matching process for the

CLP (12), denoted by x, such that∑i∈N ,j∈J

∫ t

0

µj(s)Fij(s)xij(s)ds=∑i∈N

Λi(t), for all t∈ [0, T ].

In other words, a “smart” system controller lowers prices when there are excess drivers in

the system when he can.

One potential issue with using the underlying network game requirement to motivate

Assumption 4 is that there is generally some minimal price. Many ridesharing companies

use an area and time specific “surge” multiplier to attract more drivers when the customer

demand exceeds the driver supply. However, if that multiplier cannot go below 1, then the

result is that there can be times when there are too many drivers (i.e., the right-hand side

of the display in Assumption 4 strictly exceeds the left-hand side). This adds another layer

of complexity, because this is exactly the situation in which the system controller desires

a forward-looking policy that “saves” some of the excess drivers for future anticipated

increases in customer demand (meaning we require understanding the solution to the CLP

(12)). Still, even in this situation, focusing only on a time period during which prices are

above their minimal level provides a time period during which Assumption 4 should be

in force (although, admittedly, how to connect between that time period and another in

which prices are at their minimal level is not obvious, because a CLP cannot in general be

“broken up”).

Assumption 4 is valid in the rest of Section 4 (but not necessarily in Section 5).


Remark 6. Assumption 4 is parallel to Assumption 1 in Harrison (2000), which intro-

duces an optimization problem to define fully utilized resources in the queueing literature

when parameters do not vary with time.

4.2. An LP-Based Randomized Matching Policy

We begin by observing that Assumption 4 implies the CLP constraint (12b) binds for all

i ∈ N under at least one optimal CLP solution. This is because any feasible matching

process that satisfies Assumption 4 must be optimal because all of the drivers are matched

under this solution. Second, under any feasible matching process that satisfy Assumption 4,

if any one area i is such that the constraint (12b) has slack, then there must be another area

k such that the number of matched customers exceeds the cumulative driver arrivals (i.e.,

the left-hand side of the constraint strictly exceeds the right-hand side) which produces an

infeasible matching process. Now, the following LP is relevant at each fixed t∈ [0, T ]:

max∑

i∈N ,j∈J

wijµj(t)Fij(t)xij(t), (14a)

subject to:∑j∈J

µj(t)Fij(t)xij(t)≤ λi(t), ∀i∈N , (14b)

∑i∈N

xij(t)≤ 1, ∀j ∈J , (14c)

xij(t)≥ 0, ∀i∈N , j ∈J , (14d)

where the decision variables are xij(t) for all i∈N and j ∈J . The main difference between

the LP (14) and the CLP (12) is that (14a) and (14b) in the LP (14) are the derivatives of

(12a) and (12b) in the CLP (12), respectively. By Assumption 1 and the fact that xij(t) = 0

for all i∈N and j ∈J is feasible for the LP (14) for all t∈ [0, T ], there exists an optimal

solution of the LP (14) for all t∈ [0, T ].

Assumption 5. (Measurability) There exists a deterministic process x∗ := x∗ij(t), i ∈N , j ∈ J , t ∈ [0, T ] such that x∗ij(t), t ∈ [0, T ] is Lebesgue measurable for all i ∈ N and

j ∈ J , and x∗ij(t), i ∈ N , j ∈ J is an optimal solution of the LP (14) at time t for all

t∈ [0, T ].

If each of λi and µj is a step function (with possibly countably infinite range) for all i∈Nand j ∈ J , then there exists a deterministic function which satisfies Assumption 5, and

that function can be chosen as a step function. Assumption 5 is valid in the rest of Section

4.


The following result connects the CLP (12) with the LP (14).

Lemma 4. The process x∗ is an optimal solution of the CLP (12).

Lemma 4 combined with Theorem 1 imply the following asymptotic LP-based upper

bound on the fluid scaled objective (6).

Corollary 2. Under any admissible policy π,

lim supn→∞

∑i∈N ,j∈J

wijDπ,nij (T )≤

∑i∈N ,j∈J

wij

∫ T

0

µj(s)Fij(s)x∗ij(s)ds, a.s.

In order to obtain the upper bound in Corollary 2, the LP (14) must be solved infinitely

many times (for all t ∈ [0, T ]). However, in practice, the LP (14) can be solved at finitely

many time epochs, and the remaining x∗ij(t) values can be approximated by, for example,

linear interpolation or assuming that x∗ij is a step function with finite range.

The randomized policy πR(x∗) is LP-based. To be admissible (a necessary condition for

asymptotic optimality), the condition in Lemma 3 must be satisfied.

Theorem 4. Suppose x∗ satisfies the condition in Lemma 3. If wij = 1 for all i ∈ N

and j ∈J , then πR(x∗) achieves the asymptotic upper bound given in Corollary 2, i.e.,

limn→∞

∑i∈N ,j∈J

DπR(x∗),nij (T ) =

∑i∈N ,j∈J

∫ T

0

µj(s)Fij(s)x∗ij(s)ds, a.s.

The LP-based randomized policy is myopic. As a consequence, for the same reason

explained after Theorem 3, Theorem 4 does not require the condition θi = 0 for all i ∈

N . Another consequence is that the time-varying parameters can be estimated in real-

time, because the matching decisions do not depend on future estimated arrival rates of

customers and drivers.

5. Performance Evaluation

We begin by observing that the performance of the randomized policy can likely be

improved by incorporating state information. This is because the randomized policy can

match a customer with an area in which there are no drivers, leading to that customer

being lost. To correct this, in Section 5.1, we introduce state-dependent LP- and CLP-based

policies, that require knowledge of driver locations. Then, in Section 5.2, we compare the

performance of those policies and the LP- and CLP-based randomized policy against the

benchmark CD policy. We do this first when parameters do not vary with time and second


when they do. In the first case, the LP- and CLP-based policies coincide (cf. Theorem

3), whereas in the second, when the importance of considering future customer and driver

arrival rates becomes important, they do not.

5.1. Additional Proposed Matching Policies

The randomized policy is not match conserving in the sense that a customer arriving at a

time when at least one driver is available is always offered a driver. The benchmark CD

policy (in (3)) is match conserving. We propose two new policies, one deterministic and one

not, that are match conserving and can be based on either a CLP (12) or LP (14) solution,

depending on whether or not Assumption 4 holds. Assume x is a feasible matching process

for the CLP, which includes x∗ when Assumption 4 holds.

Deterministic Policy: If a type j ∈J customer arrives in the system at time t, the system

controller offers a driver type from the set

arg mini∈N :Qi(t−)>0

Dij(t−)−

∫ t

0


. (15)

If the set in (15) is not a singleton, then the system controller can use any tie breaking

rule which does not use any future information. If there is no driver in the system, then no

driver is offered. (Note that we have suppressed the dependence of Dij, µj, and Fij on n in

the notation to emphasize that the policy can be applied directly to the model in Section 2,

even though any results obtained on its asymptotic performance would require reference

to the sequence of matching systems introduced in Section 3 and used in Sections 3 and 4.)

Hybrid Policy: Suppose that a type j customer arrives in the system, and there is a

driver in the system. Then the system controller makes a random selection in the same

way explained under the randomized policy. If the outcome is i for some i∈N and there is

a type i driver in the system, the system controller offers a type i driver to the customer.

If the outcome is i for some i ∈ N but there is no type i driver in the system or if the

outcome is 0, the customer is offered the driver type specified in (3) by the CD policy. If

there is no driver in the system, then the customer is lost.

Remark 7. Let α : J → N be a function which specifies the arrival location of each

customer type. If xij = 0 for all i∈N and j ∈J such that i 6= α(j), then the Hybrid policy

is the CD policy defined in (3).

Lemma 5. The deterministic and hybrid policies are admissible under the condition on

x in Lemma 3.


The deterministic and hybrid policies can be implemented regardless of whether or not

they are admissible. However, in order to prove asymptotic optimality results parallel

to those for the randomized policy (specifically, Corollary 1 and Theorems 2, 3, and 4),

admissibility is required (and can be achieved using the approximation explained prior to

Corollary 1). The proofs of the aforementioned asymptotic optimality results are much

more difficult due to the state dependence.

5.2. Simulation Experiments

In this section, we present simulation experiments where we test the performances of the

CD policy and the LP- and CLP-based matching policies. We associate the LP- and CLP-

based policies based on an optimal solution of the LP (14) and an optimal solution of the

CLP (12), respectively. We present an experiment with time homogeneous parameters in

Section 5.2.1 and an experiment with time dependent parameters in Section 5.2.2.

The term “offered driver” refers to the driver offered to an arriving customer, that is

determined by the matching policy. The customer may or may not accept being matched

with the driver offered at the arrival time t ∈ [0, T ], depending on the driver type i ∈N ,

the customer type j ∈J , and the acceptance probability Fij(t). Any variation of the word

“match” means that the relevant customer has both been offered a driver, and has accepted

that driver, meaning that driver picks up that customer.

We assume that the larger the Fij(t) at time t∈ [0, T ], the smaller the pick up time of a

type i driver for a type j customer. Then, under the CD policy, the offered driver set for a

type j customer arriving at time t in (3) is exactly arg maxi∈N :Qi(t−)>0 Fij(t) for all j ∈Jand t∈R+ (and the match occurs if also the resulting pick-up time is lower than the time

that customer is willing to wait for pick-up).

We choose N = J = 3 in both simulation experiments (so the customer type represents

the customer’s arrival location). This is the smallest network size possible that allows for us

to illustrate the more general insight behind when and why LP-based policies outperform

CD. The reason a two area network will not work is that as long as Fii(t) ≥ Fij(t) for

all i, j ∈ 1,2,3 and t ∈ [0, T ], the LP-based policies give more priority to within-area

matchings than they give to between-area matchings, which exactly coincides with the CD

policy. In contrast, the reason CLP-based policies outperform CD (as well as any other

myopic policy such as an LP-based one) has to do with their potential to be forward-

looking, which can be seen in a two-area network.


Implementation Details. We used Omnet++ discrete-event simulation freeware. In each

experiment, we started with an empty system; i.e., Qi(0) = 0 for all i∈N . At each simula-

tion instance associated with each matching policy, we have done Rep number of replica-

tions. The maximum of the margin of errors associated with the 95% confidence interval

for the percentage of all customers matched (= t0.025,Rep−1× Sample Stdv./√Rep) among

all policies in all instances is less than 0.42.

!" # $

%" # &

!' # &

%' # &

!( # &

%( # $

)*"' # )*'" # $+,, )*'( # )*(' # $+,-!" #" $"

)*"( # )*(" # $

Figure 3 The parameters of the first simulation experiment are: Qi(0) = 0, θi = 0.1, Fii = 1, wij = 1 for all i, j ∈

1,2,3, and n∈ 1,10,100.

5.2.1. Time Homogeneous Parameters We present a simulation experiment which

shows that our LP-based policies can potentially match more customers and drivers in

a finite time horizon than the CD policy does. Figure 3 provides all input parameters.

Since all parameters are time homogeneous, the LP- and CLP-based policies coincide (cf.

Theorem 3). We omit “(t)” from the notation. Moreover, n ∈ 1,10,100 measures the

market size, as in Assumption 2.

The intuition for why we do not expect the CD policy to perform well in this example

is as follows. Under the CD policy, when a type 1 or 2 customer arrives in the system,

the offered driver is type 2 (if there is one in the system) because λ1 = 0, F21 > F31, and

F22 > F32. Since µ1 + µ2 = 2n > λ2 = n, some of the type 1 or 2 customers cannot be

offered type 2 drivers and must be offered type 3 drivers. The type 2 customers who are

not offered type 2 drivers are offered type 3 drivers and so 98% of those customers are

matched. However, the type 1 customers offered type 3 drivers are lost because F31 = 0.

In comparison, the optimal solution of the LP (13) has x∗21 = 1, x∗22 = 0.01, and x∗32 = 0.99.

In words, the LP-based policies match type 1 customers with type 2 drivers and type 2

customers with type 3 drivers, to prevent losing a non-trivial percentage of the type 1

customers.


CD Rand. Det. Hyb.50

60

70

80

90

100%

Cu

sto

me

rs M

atc

he

d LP Ub.

(a) n= 1


60

70

80

90

100

% C

usto

me

rs M

atc

he

d LP Ub.

(b) n= 10


60

70

80

90

100

% C

usto

me

rs M

atc

he

d LP Ub.

(c) n= 100

Figure 4 The percentage of all customers matched in the simulation experiment corresponding to Figure 3.

Figure 4 shows the percentage of all customers matched with drivers in the simulation

experiment (i.e., the objective (5), since wij = 1 for all i, j ∈ 1,2,3) under the CD pol-

icy and the LP-based policies. “LP Ub.” shows one hundred times the optimal objective

function value of the LP (13) divided by the total customer arrival rate to the system,

which is an approximate upper bound on the percentage of all customers matched under

any admissible policy by Theorems 1 and 3. The key observations are that

(i) the LP-based policies outperform the CD policy in all traffic intensities;

(ii) the LP-based policies are very close to the approximate upper bound based on The-

orem 1 as the arrival rates become large, which verifies Theorem 3 numerically.

Remark 8. The LP-based matching decisions contradict Corollary 1 of Hu and Zhou

(2016), which roughly states that a ridesharing company should always match a demand

with a supply if they are originated from the same geographic region. This is due to

differences in the model formulations. In particular, in Hu and Zhou (2016), matchings

between different types occur with either probability one or zero. In contrast, in our setting

whether or not a matching takes place (i.e., whether or not a customer accepts the offered

driver) is random.

!" # $

%"&'(

!) # *

%) # $

!+ # $

%+ # *

,-") # ,-)" # *.// ,-)+ # ,-+) # *./0!" #" $"

,-"+ # ,-+" # *

Figure 5 The parameters of the second simulation experiment are: Qi(0) = 0, Fii = 1, wij = 1 for all i, j ∈

1,2,3, n∈ 0.1,1,10,100, θi = θ is constant in i and θ ∈ 10−2,10−3,10−4,10−5, T = 1800, µ1(t) =

0 for all t∈ [0, T/2], and µ1(t) = 2n for all t∈ [T/2, T ].


5.2.2. Time Dependent Parameters Motivated by the example in Figure 1 in the

Introduction, our next simulation experiment is designed to show that CLP-based policies

can significantly outperform LP-based policies and the CD-policy by taking into account

the future customer and driver arrival rates. Our assumption is that the customer arrival

rate in one area will spike; in particular, we assume µ1(t) = 0 for all t ∈ [0, T/2] and

µ1(t) = 2n for all t ∈ [T/2, T ], where the length of the time horizon is T = 1800, and

n∈ 0.1,1,10,100 denotes the traffic intensity of the system. Figure 5 provides the other

input parameters, which are time homogeneous, and so “(t)” is omitted.

Table 1 shows the percentage of all customers that are matched with drivers in the

simulation experiment (i.e., the objective (5), since wij = 1 for all i, j ∈ 1,2,3) under the

CD policy, the LP-based policies, and the CLP-based policies.

Table 1 The percentage of all customers matched in the simulation experiment corresponding to Figure 5.

LP-Based Policies CLP-Based Policiesθ n CD Rand. Determ. Hybrid Rand. Determ. Hybrid

10−2 0.1 63.8 57.9 67.1 67.5 62.5 68.1 67.91 66.1 69.1 72.6 72.9 72.3 73.9 74.110 66.1 72.9 73.9 74.1 75.2 75.5 75.5100 66.1 74.0 74.2 74.3 75.9 75.9 75.9

10−3 0.1 67.0 69.4 73.5 75.4 82.0 83.4 83.71 66.4 72.7 74.1 74.8 85.8 86.1 86.310 66.3 73.9 74.2 74.5 86.8 86.8 86.9100 66.2 74.2 74.2 74.4 86.9 86.9 86.9

10−4 0.1 68.0 70.3 73.7 77.0 92.1 92.8 93.51 66.7 73.0 74.2 75.2 96.0 96.0 96.110 66.3 74.0 74.2 74.7 96.8 96.8 96.8100 66.2 74.2 74.2 74.5 96.9 96.9 96.9

10−5 0.1 67.8 70.4 73.9 77.0 93.6 93.7 94.31 66.8 73.2 74.1 75.4 97.4 97.4 97.610 66.4 74.0 74.2 74.7 98.5 98.5 98.6100 66.2 74.2 74.3 74.5 98.8 98.7 98.8

Table 1 confirms that the CLP-based policies outperform the LP-based policies, and

the LP-based policies outperform the CD policy. Furthermore, both the CLP-based and

LP-based policies achieve the expected performance derived from the optimal solution of

the CLP (12) and the LP (14) as the market size n becomes large. Most intriguing, the

performance of the CLP-based policies improves as θ becomes small, whereas the perfor-

mance of the CD and LP-based policies is independent of θ. This is because the CLP-based

policies are forward-looking but the CD and LP-based policies are not. As a result, the

CLP-based policies are the only ones that require the drivers arriving in the first half of

the simulation to be willing to wait until sometime during the second half of the simu-

lation to be offered to a customer – and that “willingness-to-wait” is determined by the


parameter θ. This observation is exactly the reason Theorem 2 and Corollary 1 (related to

the asymptotic optimality of the potentially forward-looking CLP-based policies) require

θ = 0 but Theorems 3 and 4 (related to the asymptotic optimality of myopic LP-based

policies) do not.

The next three paragraphs explain the intuition for the percentages shown in Table 1.

The CD policy: In the first half of the simulation, i.e., in the time interval [0, T/2],

the CD policy offers type 1 drivers to type 2 customers and in total 0.99n(T/2) type 2

customers are matched. In the second half of the simulation, the CD policy offers type

1 drivers to both type 1 and type 2 customers. All of the type 1 customers who are

offered type 1 drivers will accept those drivers and 99% of the type 2 customers who are

offered type 1 drivers will accept those drivers. Then, the arrival rate of the customers

who will accept a match with type 1 drivers if offered is 2n+ 0.99n= 2.99n but the arrival

rate of type 1 drivers is only n. Thus, only 1/2.99 of the type 1 customers are offered

and matched with type 1 drivers and 1.99/2.99 of them are lost. Hence, nT/2.99 type 1

customers are matched in the second half of the simulation. Similarly, among the 99% of

the type 2 customers who accept type 1 drivers if offered, 1/2.99 of them will be offered

type 1 drivers. Hence, 0.99n(T/2)/2.99 type 2 customer are matched with type 1 drivers

and 0.98× 1.99n(T/2)/2.99 type 2 customers are matched with type 3 drivers, and the

remaining type 2 customers are lost in the second half of the simulation. Dividing the total

number of matchings by the total number of arriving customers (2nT ) leads to 66.1% of

all customers being matched, which agrees with Table 1 for larger n.

The LP-based policies: The optimal solution of the LP (14) is such that x∗12(t) = 1 for

all t ∈ [0, T/2] and x∗11(t) = 0.5 and x∗32(t) = 1 for all t ∈ [T/2, T ], and all other decision

variables at all other times are 0. In the first half of the simulation (t∈ [0, T/2]), x∗12(t) = 1

implies type 2 customers are offered type 1 drivers, resulting in 0.99n(T/2) matchings.

In the second half of the simulation (t ∈ [T/2, T ]), x∗11(t) = 0.5 dictates type 1 customers

and drivers being matched; however, the disparity between the arrival rates results in only

half the customers being matched, or n(T/2) matchings. Also, x∗32(t) = 1 results in type

2 customers matching with type 3 drivers, leading to 0.98n(T/2) matchings. Dividing the

total number of matchings by the total number of arriving customers (2nT ) leads to 74.3%

of all customers being matched, which agrees with Table 1 for larger n


The CLP-based policies: An optimal matching process for the CLP (12) has: x32(t) = 1

for all t∈ [0, T ], x11(t) = 0 for all t∈ [0, T/2], x11(t) = 1 for all t∈ [T/2, T ], all other decision

variables are 0. (To see that process is optimal, note that as many drivers as possible

are matched.) This solution is forward-looking because the type 1 drivers arriving before

time T/2 are “saved” to be offered to the type 1 customers arriving after time T/2. Then,

2n(T/2) type 1 customers are matched in the second half of the simulation. Since x32(t) = 1

for the entire simulation, there are 0.98nT matchings for type 2 customers. Dividing the

total number of matchings by the total number of arriving customers (2nT ) leads to 99%

of all customers being matched, which, in contrast to the CD and LP-based policies does

not agree with Table 1 for all larger values of n.

The issue is that not all drivers arriving in the first half of the simulation will wait for

customers to arrive in the second half of the simulation. The number that wait depends on

the parameter θ. This is why we see 99% of all customers matched in Table 1 only when

both n is larger and θ is smaller.

6. Concluding Remarks

The decisions on which driver to offer to each arriving customer in a ridesharing system

impact the overall number of customers matched. This is because those decisions determine

whether or not future available drivers will be close to the locations of arriving customers.

We have formulated an optimization problem whose solution serves as an asymptotic upper

bound on the matching rates as the market becomes large. That optimization problem

accounts for (i) the differing arrival rates of drivers and customers in different areas of the

city, (ii) how long customers are willing to wait for driver pick-up, and (iii) the time-varying

nature of all the aforementioned parameters.

The aforementioned optimization problem is in general a CLP, which can be difficult to

solve. We establish that a simple randomized matching policy can asymptotically achieve

any matching process feasible for the CLP, so that there is potential to develop “good”

CLP-based matching policies, even when an optimal CLP solution is unknown. When

an optimal CLP solution is known, then a CLP-based randomized policy asymptotically

achieves the aforementioned optimization problem upper bound (assuming drivers are

“patient enough”).

Under the assumption that drivers are fully utilized, the CLP solution can be specified

through LP solutions. Then, an LP-based randomized policy asymptotically achieves the


aforementioned optimization problem upper bound. The fully utilized driver assumption

arises when the customer and driver arrival rates arise as the outcome of an underlying

network game in which a decrease in price in a given area causes less driver arrivals but

more customer arrivals. However, the resulting customer and driver arrival rates must

be independent of the matching policy. An excellent question for future research is to

understand how to jointly optimize over the incentive decisions (such as pricing) and the

matching decisions.

Many ridesharing companies have a car pooling option in which more than one customer

can share a single driver. In this case, a “matched” driver is still regarded as potentially

available. The issue, as studied in Gopalakrishnan et al. (2016), is that “adding” a customer

to an already matched driver causes inconvenience to the customer(s) that are already

passenger(s) (i.e., already riding with that driver). More research is required to fully under-

stand how to make matching decisions that both account for this inconvenience and ensure

the highest possible overall matching rates.

Acknowledgments

We would like to thank Siddhartha Banerjee for his valuable comments that helped us to write Section 4.1.

References

Adan, I., G. Weiss. 2012. Exact FCFS matching rates for two infinite multitype sequences. Operations

Research 60 475–489.

Akbarpour, M., S. Li, S. O. Gharan. 2016. Thickness and information in dynamic matching markets. Working

paper, http://ssrn.com/abstract=2394319.

Anderson, E. J., P. Nash, A. F. Perold. 1983. Some properties of a class of continuous linear programs.

SIAM Journal on Control and Optimization 21 758–765.

Arnosti, N., R. Johari, Y. Kanoria. 2016. Managing congestion in matching markets. Working paper,

http://ssrn.com/abstract=2427960.

Ata, B., S. Kumar. 2005. Heavy traffic analysis of open processing networks with complete resource pooling:

Asymptotic optimality of discrete review policies. The Annals of Applied Probability 15 331–391.

Azevedo, E. M., E. G. Weyl. 2016. Matching markets in the digital age. Science 352 1056–1057.

Bertsimas, D., E. Nasrabadi, I. C. Paschalidis. 2015. Robust fluid processing networks. IEEE Transactions

on Automatic Control 60 715–728.

Bimpikis, K., O. Candogan, D. Saban. 2017. Spatial pricing in ride-sharing networks. Working paper,

https://ssrn.com/abstract=2868080.

http://ssrn.com/abstract=2394319


https://ssrn.com/abstract=2868080


Braverman, A., J. G. Dai, X. Liu, L. Ying. 2017. Empty-car routing in ridesharing systems. Working paper,

arXiv:1609.07219v2.

Buke, B., H. Chen. 2015. Stabilizing policies for probabilistic matching systems. Queueing Systems 80

35–69.

Busic, A., V. Gupta, J. Mairesse. 2013. Stability of the bipartite matching model. Advances in Applied

Probability 45 351–378.

Cachon, G. P., K. M. Daniels, R. Lobel. 2016. The role of surge pricing on a service platform with self-

scheduling capacity. Working paper, http://ssrn.com/abstract=2698192.

Caldentey, R., E. H. Kaplan, G. Weiss. 2009. FCFS infinite bipartite matching of servers and customers.

Advances in Applied Probability 41 695–730.

Chen, K., M. Sheldon. 2016. Dynamic pricing in a labor market: Surge pricing and flexible work on

the Uber platform. Working paper, http://www.anderson.ucla.edu/faculty_pages/keith.chen/

papers.htm.

Chen, L., A. Mislove, C. Wilson. 2015. Peeking beneath the hood of Uber. Proceedings of the 2015 ACM

Conference on Internet Measurement Conference. New York, 495–508.

Folland, G. B. 1999. Real Analysis Modern Techniques and Their Applications. 2nd ed. Wiley.

Gopalakrishnan, R., K. Mukherjee, T. Tulabandhula. 2016. The costs and benefits of ridesharing: Sequential

individual rationality and sequential fairness. Working paper, arXiv:1607.07306v1.

Guda, H., U. Subramanian. 2017. Strategic surge pricing and forecast communication on on-demand service

platforms. Working paper, https://ssrn.com/abstract=2895227.

Gurvich, I., A. Ward. 2014. On the dynamic control of matching queues. Stochastic Systems 4 479–523.

Hall, J., C. Kendrick, C. Nosko. 2016. The effects of Uber’s surge pricing: A case study. Working paper,

http://faculty.chicagobooth.edu/chris.nosko/.

Harrison, J. M. 2000. Brownian models of open processing networks: Canonical representation of workload.

The Annals of Applied Probability 10 75–103.

Hu, M., Y. Zhou. 2016. Dynamic type matching. Working paper, http://ssrn.com/abstract=2592622.

Karatzas, I., S. E. Shreve. 1988. Brownian Motion and Stochastic Calculus. 1st ed. Springer-Verlag.

Leshno, J. D. 2016. Dynamic matching in overloaded waiting lists. Working paper, http://jleshno.

weebly.com/.

Michallon, C. 2016. ‘Shame on you’: Uber comes under fire as users complain of price

surge in the aftermath of Manhattan explosion - with rides costing up to three times

their usual prices. Dailymail.com http://www.dailymail.co.uk/news/article-3795474/

Uber-comes-fire-users-complain-price-surge-aftermath-Manhattan-explosion.html.


http://www.anderson.ucla.edu/faculty_pages/keith.chen/papers.htm



http://faculty.chicagobooth.edu/chris.nosko/


http://jleshno.weebly.com/


http://www.dailymail.co.uk/news/article-3795474/Uber-comes-fire-users-complain-price-surge-aftermath-Manhattan-explosion.html



Moyal, P., O. Perry. 2016. On the instability of matching queues. Working paper, arXiv:1511.04282.

Perold, A. F. 1981. Extreme points and basic feasible solutions in continuous time linear programming.

SIAM Journal on Control and Optimization 19 52–63.

Plambeck, E. L., A. R. Ward. 2006. Optimal control of a high-volume assemble-to-order system. Mathematics

of Operations Research 31 453–477.

Reiman, M. I., Q. Wang. 2015. Asymptotically optimal inventory control for assemble-to-order systems with

identical lead times. Operations Research 63 716–732.

Riquelme, C., S. Banerjee, R. Johari. 2016. Pricing in ride-share platforms: A queueing-theoretic approach.

Working paper, http://ssrn.com/abstract=2568258.

Rochet, J. C., J. Tirole. 2006. Two-sided markets: a progress report. RAND Journal of Economics 37

645–667.

Shiryaev, A. N. 2008. Optimal Stopping Rules. Springer. Reprint of the 1978 edition.

Stroock, D. W., S. R. S. Varadhan. 2006. Multidimensional Diffusion Processes. Springer. Reprint of the

1997 edition.

Taylor, T. A. 2017. On-demand service platforms. Working paper, https://ssrn.com/abstract=2722308.

The Economist. 2016. A fare shake. http://www.economist.com/news/finance-and-economics/

21698656-jacking-up-prices-may-not-be-only-way-balance-supply-and-demand-taxis.

Unver, M. U. 2010. Dynamic kidney exchange. Review of Economic Studies 77 372–414.

Weiss, G. 2008. A simplex based algorithm to solve separated continuous linear programs. Mathematical

Programming 115 151–198.

White, D. 2016. Uber users are complaining about pricey New Year’s Eve rides. Time http://time.com/

4165410/uber-new-years-eve-price-surge-rides/.



http://www.economist.com/news/finance-and-economics/21698656-jacking-up-prices-may-not-be-only-way-balance-supply-and-demand-taxis


http://time.com/4165410/uber-new-years-eve-price-surge-rides/


e-companion to Ozkan and Ward: Dynamic Matching for Real-time Ridesharing ec1

ELECTRONIC COMPANION

In this electronic companion, we provide the proofs for results stated in the main body of the

manuscript titled “Dynamic Matching for Real-time Ridesharing”. Section EC.1 provides the proof

of Theorem 1, which provides an asymptotic upper bound on the fluid-scaled objective (6). That

proof relies on the fluid equations, and the details of their derivation are given in Section EC.2.

Section EC.3 provides a new regulator mapping result which is used in the proofs for asymptotic

optimality results in Section EC.4. The proofs of Lemmas 1, 2, 3, 4, and 5 are found in Section

EC.5. Finally, we present a relative compactness result in space D, which is used in the derivation

of fluid equations, in Section EC.6.

EC.1. Proof of Theorem 1

The proof of Theorem 1 relies on understanding the behavior of fluid limits. In particular, the proof

uses the fact that all fluid limits satisfy a set of equations that can be connected to the constraints

in the CLP (12). The implication is that any fluid limit gives a feasible matching process for the

CLP (12), which implies the optimal objective function value of the CLP (12) is an asymptotic

upper bound for the objective (6) under fluid scale.

For all i2N , j 2J , t2R+

, n2N+

, and admissible policy = n, n2N+

, let us define

G,nij (t) :=

Ej(nj (t))X

k=1

I

nj (k) = i

, (EC.1)

which is the cumulative number of times the system controller o↵ers a type i driver to a type j

customer up to time t and let G,nij :=G,n

ij /n be the fluid scaled process. Clearly, D,nij G,n

ij for

all i 2N , j 2 J , n 2N+

, and admissible policy by (4). For all n 2N+

and admissible policy ,

let X,n 2D2NJ+3N+J be defined such that

X,n :=

Ai ni , Ej n

j , Rni

Z ·

0

ni (s)Q,ni (s)ds

, Q,ni , D,n

ij , G,nij , 8i2N , j 2J

, (EC.2)

where “” denotes the composition map. Let X,n := X,n/n be the fluid scaled version of X,n.

Then, we define fluid limit(s) of X,n, n 2 N+

similar to the definition on page 299 of Dai and

Tezcan (2011) in the following way.

ec2 e-companion to Ozkan and Ward: Dynamic Matching for Real-time Ridesharing

Definition EC.1. (Fluid Limit) Let us fix an arbitrary admissible policy = n, n2N+

. Then,X is a fluid limit of X,n, n 2N

+

if there exists an ! 2 and a subsequence nk, k 2N+

such

that X,nk(·,!)! X u.o.c. as k!1.

Proposition EC.1. (Fluid Equations) For any admissible policy = n, n 2 N+

, there exists

a full set A (i.e., P(A) = 1) such that for any ! 2 A, X,n(·,!), n 2 N+

is relatively

compact (i.e., every subsequence has a convergent subsequence) in the Skorokhod space D2NJ+3N+J

endowed with the u.o.c. topology. Thus, fluid limits exist in almost all sample paths and any fluid

limit

X =

i, j,

Z ·

0

i(s)Qi (s)ds, Q

i , D

ij, G

ij, 8i2N , j 2J

and satisfies the following equations for all t2R+

:

Qi (t) =i(t)

Z t

0

i(s)Qi (s)ds

X

j2JD

ij(t) 0, 8i2N , (EC.3a)

Dij(0) = G

ij(0) = 0, 8i2N , j 2J , (EC.3b)

Dij and G

ij are nondecreasing and absolutely continuous for all i2N and j 2J , (EC.3c)

X

j2JD

ij(t)i(t), 8i2N , (EC.3d)

X

i2N

Gij(t2) G

ij(t1) j(t2)j(t1), 8j 2J and t

1

, t2

2R+

such that t2

t1

, (EC.3e)

Dij(t) D

ij(m) Fij(m)

Gij(t) G

ij(m)

I (t2 [m, m+1

)) = 0,

for all m2N, i2N , and j 2J . (EC.3f)

The proof of Proposition EC.1 is presented in Section EC.2. Let us fix an arbitrary admissible

policy and an arbitrary ! 2A. Then there exists a subsequence nk, k 2N+

such that

limsupn!1

X

i2N ,j2JwijD

,nij (T,!) = lim

k!1

X

i2N ,j2JwijD

,nkij (T,!).

Since D,nij (·,!) is relatively compact (cf. Proposition EC.1) for all i 2N and j 2 J , there exists

a subsequence of nk, k 2 N+

denoted by nl, l 2 N+

such that D,nlij (·,!) converges to a limit

Dij(·) u.o.c. as l!1 for all i2N and j 2J . Hence,

limsupn!1

X

i2N ,j2JwijD

,nij (T,!) = lim

k!1

X

i2N ,j2JwijD

,nkij (T,!),


= liml!1

X

i2N ,j2JwijD

,nlij (T,!) =

X

i2N ,j2JwijD

ij(T ).

Then, Theorem 1 follows by the following result.

Proposition EC.2. Let = n, n2N+

denote an arbitrary admissible policy. Under any fluid

limit of X,n, n2N+

, we have

X

i2N ,j2JwijD

ij(T )

X

i2N ,j2Jwij

Z T

0

µj(s)Fij(s)xij(s)ds.

Next, we will prove Proposition EC.2. Let us consider the following CLP which has the decision

variables zij : [0, T ]!R+

for all i2N and j 2J that replace µj(t)Fij(t)xij(t) in the CLP (12):

maxX

i2N ,j2Jwij

Z T

0

zij(s)ds, (EC.4a)

subject to:X

j2J

Z t

0

zij(s)dsi(t), for all i2N and t2 [0, T ], (EC.4b)

X

i2N :

¯Fij(t)>0

zij(t)

Fij(t) µj(t), for all j 2J and t2 [0, T ], (EC.4c)

zij(t)I

Fij(t) = 0

= 0, for all i2N , j 2J , and t2 [0, T ], (EC.4d)

zij(t)I (µj(t) = 0) = 0, for all i2N , j 2J , and t2 [0, T ], (EC.4e)

zij(t) 0, for all i2N , j 2J , and t2 [0, T ], (EC.4f)

zij is Lebesgue measurable for all i2N and j 2J . (EC.4g)

By an argument very similar to Lemma 2, there exists an optimal solution of the CLP (EC.4)

which is denoted by zij(t), i 2 N , j 2 J , t 2 [0, T ]. We consider the CLP (EC.4) instead of the

CLP (12) because of three main reasons. First, the feasible region of the CLP (EC.4) is smaller

than the one of the CLP (12) in the sense that there exist a surjective but not bijective mapping

from the feasible region of the CLP (12) to the one of the CLP (EC.4). Second, it is easier to create

connection between a fluid limit of D,nij , i 2N , j 2 J , n 2 N

+

and the decision variables of the

CLP (EC.4) than between the same fluid limit and the decision variables of the CLP (12). Third,

we have the following result.


Lemma EC.1. The optimal objective function values of the CLP (12) and the CLP (EC.4) are

equal.

Proof: First, consider the deterministic process µj(t)Fij(t)xij(t), i 2 N , j 2 J , t 2 [0, T ]

(remember that xij(t), i 2N , j 2 J , t 2 [0, T ] is an optimal solution of the CLP (12)). It is easy

to see that this process is feasible for the CLP (EC.4). Thus,

X

i2N ,j2Jwij

Z T

0

µj(s)Fij(s)xij(s)dsX

i2N ,j2Jwij

Z T

0

zij(s)ds. (EC.5)

Second, for all i2N , j 2J , and t2 [0, T ], let

yij(t) :=

8

>

>

>

<

>

>

>

:

zij(t)

µj(t) ¯Fij(t), if Fij(t)> 0 and µj(t)> 0,

0, otherwise.

(EC.6)

Then, it is easy to see that the process yij(t), i2N , j 2J , t2 [0, T ] is feasible for the CLP (12),

thus

X

i2N ,j2Jwij

Z T

0

µj(s)Fij(s)yij(s)ds=X

i2N ,j2Jwij

Z T

0

zij(s)I

Fij(s)> 0, µj(s)> 0

ds (EC.7a)

=X

i2N ,j2Jwij

Z T

0

zij(s)dsX

i2N ,j2Jwij

Z T

0

µj(s)Fij(s)xij(s)ds, (EC.7b)

where the equality in (EC.7b) is by (EC.4d) and (EC.4e). Hence, the optimal objective function

values of the CLP (12) and the CLP (EC.4) are equal to each other by (EC.5) and (EC.7b).

Let us consider an arbitrary fluid limit of an arbitrary admissible policy . By (EC.3c) and

Theorem 3.35 of Folland (1999), both Dij and G

ij are di↵erentiable almost everywhere on [0, T ] for

all i 2N and j 2 J . Let dij and gij be the (nonnegative) derivatives of Dij and G

ij, respectively,

on the points where they are di↵erentiable and without loss of generality be equal to 0 on the

points where they are not di↵erentiable for all i2N and j 2J on the interval [0, T ]. By (EC.3b),

(EC.3c), and the fundamental theorem of calculus for Lebesgue integrals (cf. Theorem 3.35 of

Folland (1999)), for all i2N , j 2J , and t2 [0, T ],

Dij(t) =

Z t

0

dij(s)ds, Gij(t) =

Z t

0

gij(s)ds. (EC.8)


Then, dij(t), i2N , j 2J , t2 [0, T ] satisfies (EC.4b) by (EC.3d) and (EC.8); (EC.4f) and (EC.4g)

by (EC.3c), Theorem 3.35 of Folland (1999), and construction.

Next, fix an arbitrary customer type j 2J and t2 [0, T ], and let t2 [m, m+1

) for some m2N.

By choosing t2

= t, t1

= 0, and taking the derivatives of the both sides in (EC.3e), we have

X

i2Ngij(t) µj(t). (EC.9)

By (EC.3f) and (EC.8), for all i2N ,

dij(t) =

8

>

>

>

<

>

>

>

:

Fij(m)gij(t), if Fij(t)> 0,

0, otherwise,

(EC.10)

where the equality in (EC.10) holds for all t 2 [0, T ] except on a set of zero measure and we

modify dij such that dij(t) = 0 in that set of zero measure. Then by (EC.9) and (EC.10) and the

nonnegativity of gij for all i2N ,

X

i2N :

¯Fij(t)>0

dij(t)

Fij(t) µj(t),

so that dij(t), i2N , j 2J , t2 [0, T ] satisfies (EC.4c). Moreover, it satisfies (EC.4d) by (EC.10).

Lastly, by (EC.9), we have

gij(t)I (µj(t) = 0) = 0, 8i2N . (EC.11)

Thus, if Fij(t)> 0 for some i2N , then dij(t)I (µj(t) = 0) = 0 by (EC.10) and (EC.11). Otherwise,

dij(t) = 0 by (EC.10). Hence, dij(t), i2N , j 2J , t2 [0, T ] satisfies (EC.4e) as well, so it is feasible

for the CLP (EC.4). Then,

X

i2N ,j2Jwij

Z T

0

zij(s)dsX

i2N ,j2Jwij

Z T

0

dij(s)ds=X

i2N ,j2JwijD

ij(T ), (EC.12)

where the equality in (EC.12) is by (EC.8). Therefore, (EC.12) together with Lemma EC.1 proves

Proposition EC.2.


EC.2. Proof of Proposition EC.1 (Properties of Fluid Limits)

Let us fix an arbitrary T1

2 R+

. For all k 2 N+

, let Dk[0, T1

] denote the the space of functions

with domain [0, T1

] and range Rk which are right continuous with left limits. We will first prove

Proposition EC.1 except the property (EC.3f) with respect to processes defined in D2NJ+3N+J [0, T1

],

then we will extend these results to the processes in D2NJ+3N+J . Lastly, we will prove (EC.3f). By

the functional strong law of large numbers, random time-change theorem (cf. Theorems 5.10 and

5.3 of Chen and Yao (2001), respectively), and Assumption 2,

Ani n

ia.s.!i, u.o.c., Rn

ia.s.! e, u.o.c., En

j nj

a.s.! j, u.o.c. as n!1, (EC.13)

for all i2N and j 2J . Let

A1

:=n

! 2 : Qni (0,!)! 0, An

i ni (·,!)!i u.o.c., Rn

i (·,!)! e u.o.c.,

Enj n

j (·,!)! j u.o.c. as n!1 for all i2N and j 2Jo

. (EC.14)

Then, A1

is a full set (i.e., P (A1

) = 1) by (EC.13) and Assumption 3. Let us fix an arbitrary

! 2A1

and an arbitrary admissible policy = n, n 2N+

. We omit and ! from the notation

up to Lemma EC.2 below for notational convenience.

Clearly, for all i2N and j 2J , Ani n

i , n2N+

and Enj n

j , n2N+

are relatively compact

in the u.o.c. topology by (EC.14). Moreover, i and j are Lipschitz continuous by Assumption 1;

and for all i2N , j 2J , and t1

, t2

2R+

such that t2

t1

, we have

X

i2N

Gnij(t2) Gn

ij(t1) En

j nj (t2) En

j nj (t1), (EC.15)

Dnij(t2) Dn

ij(t1) Gnij(t2) Gn

ij(t1), (EC.16)

where (EC.15) is by the fact that the system controller can o↵er a driver to a customer only at the

customer arrival epochs, and (EC.16) is by (4) and (EC.1). Then by Lemma EC.14 (cf. Section

EC.6), Gnij, n 2N

+

and Dnij, n 2N

+

are relatively compact in D[0, T1

] endowed with the u.o.c.

topology for all i 2N and j 2 J such that all of their subsequential limits are Lipschitz and so


absolutely continuous. Since Gnij and Dn

ij are nondecreasing and Gnij(0) = Dn

ij(0) = 0 for all i 2N ,

j 2J , n 2N+

, any fluid limits of these processes satisfy (EC.3b) and (EC.3c). Moreover, (EC.3e)

follows by (EC.14) and (EC.15).

By Assumption 1, we define i := supt2R+i(t) <1 and µj := supt2R+

µj(t) <1 for all i 2N

and j 2 J . Moreover, there exists a constant 2 R+

and n0

2 N+

such that if n n0

, ni (t)

for all t 2 R+

and i 2N by Assumptions 1 and 2. Let us fix an arbitrary i 2N and consider the

processR ·0

ni (s)Qni (s)ds. By (EC.14), An

i ni (T1

)!i(T1

) iT1

<1 and Qni (0)! 0 as n!1.

Hence, there exists an n1

2N+

and a constant C <1 such that if n n1

, Qni (0)+ An

i ni (T1

)C.

Then

Z t2

0

ni (s)Qni (s)ds

Z t1

0

ni (s)Qni (s)ds=

Z t2

t1

ni (s)Qni (s)ds

Z t2

t1

Qni (s)ds (EC.17a)

Z t2

t1

Qni (0)+ An

i ni (s)

ds

Qni (0)+ An

i ni (T1

)

(t2

t1

) C(t2

t1

) (EC.17b)

for all n n0

_ n1

and t1

, t2

2 [0, T1

] such that t2

t1

, and the first inequality in (EC.17b) is by

(11). Then,R ·

0

ni (s)Qni (s)ds,n 2N

+

is relatively compact in the u.o.c. topology with Lipschitz

continuous subsequential limits by Lemma EC.14 (cf. Section EC.6). By the fact that Rni ! e u.o.c.

(cf. (EC.14)) and Lemma 11 of Ata and Kumar (2005),

Rni

R ·0

ni (s)Qni (s)ds

, n2N+

is relatively

compact in the u.o.c. topology and all of its limits are Lipschitz continuous. Hence, Qni , n 2N

+

is also relatively compact in the u.o.c. topology with Lipschitz continuous and nonnegative limits

by (11) and (EC.14).

Let us fix an arbitrary subsequence nl, l 2N+

such thatR ·

0

ni (s)Qnli (s)ds, l 2N

+

converges

to a (continuous) limit under the uniform norm. Then, there exists a subsequence of nl, l 2N+

,

denoted by nk, k 2N+

, such that

Qnki Qi

T1! 0 as k!1 for all i 2N , where Qi 2D[0, T

1

]

is nonnegative and continuous for all i2N . Then,

supt2[0,T1]

Z t

0

nki (s)Qnk

i (s)dsZ t

0

i(s)Qi(s)ds

supt2[0,T1]

Z t

0

nki (s)Qnk

i (s)dsZ t

0

i(s)Qnki (s)ds

+ supt2[0,T1]

Z t

0

i(s)Qnki (s)ds

Z t

0

i(s)Qi(s)ds

T1

Qnki

T1knk

i ikT1+T

1

Qnki Qi

T1! 0, as k!1,


by Assumption 2. Therefore, each convergent subsequence ofR ·

0

ni (s)Qni (s)ds,n2N

+

converges

toR ·0

i(s)Qi(s)ds u.o.c. where Qi is a fluid limit of Qni , n2N

+

for all i2N . Then, by Lemma 11 of

Ata and Kumar (2005), each convergent subsequence of

Rni

R ·0

ni (s)Qni (s)ds

, n2N+

converges

toR ·0

i(s)Qi(s)ds u.o.c. where Qi is a fluid limit of Qni , n2N

+

for all i2N . This proves (EC.3a).

Next, (EC.3d) follows by (EC.3a) and the fact that Qi is nonnegative for all i2N .

By Theorems 16.2 and 16.4 of Billingsley (1999) and the fact that T1

2R+

is arbitrarily chosen,

we can extend the results above to the processes in D2NJ+3N+J , which proves Proposition EC.1

except (EC.3f). Lastly, in order to prove (EC.3f), it is enough to prove the following result. From

this point forward, we keep and ! in the notation.

Lemma EC.2. For all i2N , j 2J , m2N, and admissible policy = n, n2N+

,

supt2[m,m+1)

D,nij (t) D,n

ij (m) Fij(m)

G,nij (t) G,n

ij (m)

a.s.! 0, as n!1. (EC.18)

Proof: Let us fix arbitrary i 2 N , j 2 J , m 2 N, and an admissible policy = n, n 2 N+

.

Then the converging number in (EC.18) is equal to

supt2[m,m+1)

1

n

Ejnj (t)

X

k=Ejnj (m)+1

I

akj (m) tij(m)

Fij(m)

I

nj (k) = i

, (EC.19)

by (4), (EC.1), and (10b). For notational completeness, for any n,m 2 N and sequence of real

numbers xk, k 2N, if n>m, thenPm

k=n xk := 0. Let

Yk := I

akj (m) tij(m)

Fij(m), Znk := I

nj (k) = i

, Xnk := Yk Zn

k .

Then, (EC.19) is equal to

supt2[m,m+1)

1

n

Ejnj (t)

X

k=Ejnj (m)+1

Xnk

. (EC.20)

By the definition of the admissible policies (cf. Definition 1), nj (k) is Fn

j (k)-measurable for all

k,n2N+

, so does Znk . Moreover, ak

j (m) is independent of Fnj (k) by construction (cf. (8)), so does

Yk. Then, E

Xnk |Fn

j (k)

= 0 for all k,n 2 N+

because E [Yk] = 0. Therefore, we expect to have a

Martingale strong law of large numbers result for triangular arrays:

1

n

Ejnj (m)+nX

k=Ejnj (m)+1

Xnk

a.s.! 0, as n!1. (EC.21)


We present the formal proof of (EC.21) done by the technique introduced by de Jong (1996) in

Section EC.2.1.

Let us first choose an arbitrary > 0 and then choose arbitrary 1

> 0 and 2

> 0 such that

1

(2

+ µjm+1

)< . Let the set of ! 2 which satisfy (EC.21) be denoted by A2

. Then P(A2

) = 1.

Let us choose an arbitrary ! 2 A1

\A2

, where A1

is defined in (EC.14). Then, there exists an

n2

(2

,!) 2N such that |Enj n

j (m+1

,!)j(m+1

)|< 2

for all n n2

(2

,!) by (EC.14). By the

definition of j, (10a), (10b), and Assumption 1,

Enj n

j (m+1

,!)/n< 2

+ µjm+1

, 8n n2

(2

,!). (EC.22)

By (EC.21), there exists an n1

(1

,!)2N such that

1

n

Ejnj (m,!)+nX

k=Ejnj (m,!)+1

Xnk (!)

< 1

, 8n n1

(1

,!). (EC.23)

Let us fix an arbitrary t2 [m, m+1

). On the one hand, if Ej nj (t,!)Ej n

j (m,!) n1

(1

,!),

then

1

n

Ejnj (t,!)

X

k=Ejnj (m,!)+1

Xnk (!)

< 1

Ej nj (t,!)Ej n

j (m,!)

n

1

Ej nj (m+1

,!)

n, (EC.24)

by (EC.23). On the other hand, if Ej nj (t,!)Ej n

j (m,!)<n1

(1

,!), then

1

n

Ejnj (t,!)

X

k=Ejnj (m,!)+1

Xnk (!)

1

n

Ej nj (t,!)Ej n

j (m,!)

<n1

(1

,!)

n. (EC.25)

Hence, for all t2 [m, m+1

), if n n2

(2

,!),

1

n

Ejnj (t,!)

X

k=Ejnj (m,!)+1

Xnk (!)

< (1

(2

+ µjm+1

))_ n1

(1

,!)

n< _ n

1

(1

,!)

n, (EC.26)

by (EC.22), (EC.24), (EC.25), and the fact that 1

(2

+ µjm+1

)< . Let

n0

(,!) :=maxn2

(2

,!), n1

(1

,!)/. (EC.27)

Then, for all > 0, there exists an n0

(,!) such that if n n0

(,!),

supt2[m,m+1)

1

n

Ejnj (t,!)

X

k=Ejnj (m,!)+1

Xnk (!)

< , (EC.28)

by (EC.26) and (EC.27). Since (EC.28) is true for all ! 2A1

\A2

, the proof is complete.

We complete the proof of Proposition EC.1 by defining the full set A :=A1

\A2

.


EC.2.1. Proof of (EC.21)

Since the indices in the sum in (EC.21) are random, we first prove that YEjnj (m)+k ?Zn

Ejnj (m)+k

and E[XnEjn

j (m)+k] =E[YEjnj (m)+k] = 0 for all n2N

+

and k 2N+

, so that we can use a Martin-

gale strong law of large numbers result for triangular arrays in order to prove (EC.21).

For notational convenience, we omit the superscript from the notation. Remember that we

fix arbitrary i 2N , j 2J , m2N in Lemma EC.2. Let Knk :=Kn

Ejnj (m)+k for all K 2 X,Z and

k 2N+

, and Yk := YEjnj (m)+k for all k 2N

+

. Then proving (EC.21) is equivalent to proving

1

n

nX

k=1

Xnk

a.s.! 0, as n!1. (EC.29)

Let Fnj (k) := Fn

j (Ej nj (m) + k) for all k 2 N

+

(cf. (8)) and Fnj (k) := ;, for all k 2 Z\N

+

,

where Z denotes the set of integers. Since nj (Ej nj (m) + k) is a stopping time with respect to

the filtration Fn for all k 2N+

and is increasing in k, Fnj (k) is well defined and

Fnj (k), k 2 Z

is

a filtration.

Let Snk :R+

!RN be defined such that

Snk(t,!) :=

Ai ni (t^ nj (k,!),!),Ej0 n

j0(t^ nj (k,!),!),Qni ((t^ nj (k,!)),!),

Ri

Z

(t^nj (k,!))

0

ni (u)Qni (u,!)du, !

!

, Dnij0((t^ nj (k,!)),!), 8i2N , j0 2J ,

ar^Ej0n

j0 (nj (k,!),!)

j0 (m,!), ar^(k1)

j (m,!), 8r,m2N!

, (EC.30)

for all k,n2N+

and ! 2. By (8),

Fnj (k) = Sn

k(t), t2R+

. (EC.31)

Then we have the following result.

Lemma EC.3. Fix arbitrary k,n2N+

and j 2J . Let g be an arbitrary Fnj (k)-measurable function.

Then, there exist a B(RN)-measurable function f and a sequence of nonnegative real numbers tl, l 2

N such that g(!) = f(Snk(tl,!), l 2N) for all ! 2.


Proof: The proof follows by the fact that RN is a Polish space (cf. Corollary 3.39 of Aliprantis

and Border (2006)) and Exercise 1.5.6 of Stroock and Varadhan (2006).

Lemma EC.4. We have nj

Ej nj (m)+k

2 Fnj (k) and a

Ejnj (m)+k

j (m)2 Fnj (k+ l) for all j 2J ,

k, l, n2N+

, and m2N.

Proof: We have aEjn

j (m)+k

j (m) 2 Fnj (k + l) by definition of Fn

j (k + l) (cf. (8)). Let us fix

arbitrary j 2J , k,n2N+

, and m2N. For notational convenience, let E :=Ej nj (m). Then

nj

Ej nj (m)+ k

=1X

r=0

nj

r+ k

I

E = r

=1X

r=0

fr

Snr+k(t

rl ), l 2N

I

E = r

(EC.32)

=1X

r=0

fr

Sn˜E+k(t

rl ), l 2N

I

E = r

(EC.33)

where the second equality in (EC.32) is by Lemma EC.3. First, I

E = r 2 Fn

j (k) for all r 2 N

by definition of Fnj (k). Second, let y : ! RN be such that y(!) :=

Sn˜E(!)+k

(trl ,!), l 2 N

. Since

B(RN) = B(R)B(R) . . . (cf. Lemma 1.2 of Kallenberg (1997)), B(RN) is generated by the sets

of type B :=

Q

l2NBl : Bl 2 B(R) (cf. Proposition 1.3 of Folland (1999)). Then, since Fnj (k) =

n

Sn˜E+k

(t), t2R+

o

(cf. (EC.31)), we have

! 2 : y(!)2B=n

! 2 : Sn˜E(!)+k(t

rl ,!)2Bl, l 2N

o

=\

l2N

n

! 2 : Sn˜E(!)+k(t

rl ,!)2Bl

o

2 Fnj (k).

Since fr 2B(RN) by definition, the mapping in (EC.33) is Fnj (k)-mesurable.

Then, we have the following result.

Lemma EC.5. We have aEjn

j (m)+k

j (m)? Fnj (k) for all j 2J , k,n2N

+

, and m2N.

Proof: For notational convenience, again let E :=Ej nj (m). Let us fix arbitrary j 2J , k,n2

N+

, m2N, and c2R and an arbitrary set B 2 Fnj (k). Then

P

a˜E+kj (m)< c, B

=1X

r=0

P

ar+kj (m)< c, B, E = r

=1X

r=0

Eh

I

ar+kj (m)< c

I(B)I

E = ri


=1X

r=0

Eh

I

ar+kj (m)< c

f

Sn˜E+k(tl), l 2N

I

E = ri

(EC.34)

=1X

r=0

Eh

I

ar+kj (m)< c

f

Snr+k(tl), l 2N

I

E = ri

=1X

r=0

P

ar+kj (m)< c

Eh

f

Snr+k(tl), l 2N

I

E = ri

(EC.35)

=P

a1

j(m)< c

1X

r=0

Eh

f

Snr+k(tl), l 2N

I

E = ri

(EC.36)

=P

a1

j(m)< c

1X

r=0

P

B, E = r

=P

a1

j(m)< c

P (B)

=1X

r=0

P

a1

j(m)< c

P

E = r

P (B) =1X

r=0

P

ar+kj (m)< c

P

E = r

P (B)

=1X

r=0

P

ar+kj (m)< c, E = r

P (B) =1X

r=0

P

a˜E+kj (m)< c, E = r

P (B) (EC.37)

=P

a˜E+kj (m)< c

P (B) , (EC.38)

where (EC.34) is by Lemma EC.3. (EC.35) is by (EC.31) and the fact that ar+kj (m)? Fn

j (r+ k)

(cf. (8)), I

E = r2Fn

j (r+k) for all r 2N, and f

Snr+k(tl), l 2N

2Fnj (r+k) (see proof of Lemma

EC.4 for a similar argument explained in detail). (EC.36) is by the i.i.d. property of the sequence

arj(m), r 2 N, and the first equality in (EC.37) is by the fact that ar+k

j (m) ? Fnj (r + k) and

I

E = r 2 Fn

j (r+ k) for all r 2N. By definition of independence (cf. page 41 of Durrett (2010)),

the equality in (EC.38) proves the desired result.

Remark EC.1. If the filtration defined in (7) is generated by the sequence of processes m,m2

N defined in Remark 3, we can still prove Lemmas EC.4 and EC.5 by updating the definition in

(EC.30) such that the infinite dimensional process Snk includes the sequence m,m2N.

Now we are ready to prove (EC.21) by the technique introduced by de Jong (1996). For any

x2R, let bxc denote the greatest integer which is smaller than or equal to x. Then,

1

n

nX

k=1

Xnk =

1

n

nX

k=1

Xnk E

h

Xnk

Fnj (k+ bn0.25c 1)

i

+1

n

nX

k=1

Eh

Xnk

Fnj (kbn0.25c)

i

+1

n

nX

k=1

Eh

Xnk

Fnj (k+ bn0.25c 1)

i

Eh

Xnk

Fnj (kbn0.25c)

i

=:An1

+An2

+An3

. (EC.39)


We will consider each of An1

, An2

, and An3

separately. Let us choose an arbitrary > 0. First, let

us consider

1X

n=1

P (|An1

|> ) =1X

n=1

P

nX

k=1

Xnk E

h

Xnk

Fnj (k+ bn0.25c 1)

i

>n

!

. (EC.40)

Note that Znk is Fn

j (k)-measurable by Lemma EC.4 and Yk ? Fnj (k) by Lemma EC.5. Moreover,

Eh

Yk

i

=Eh

I

aEjn

j (m)+k

j (m) tij(m)

Fij(m)i

=E

" 1X

r=0

I

ar+kj (m) tij(m)

I

Ej nj (m) = r

#

Fij(m)

=1X

r=0

E

I

ar+kj (m) tij(m)

I

Ej nj (m) = r

Fij(m)

=1X

r=0

E

I

ar+kj (m) tij(m)

E

I

Ej nj (m) = r

Fij(m) (EC.41)

=1X

r=0

Fij(m)E

I

Ej nj (m) = r

Fij(m) (EC.42)

= Fij(m)

1X

r=0

E

I

Ej nj (m) = r

1

!

= 0, (EC.43)

where (EC.41) is by the fact that ar+kj (m)?Fn

j (r+k) (cf. (8)) and I

Ej nj (m) = r

2Fnj (r+k)

for all k 2N+

and r 2N, and (EC.42) is by definition of Fij(m) (cf. (2)). Hence, on the one hand,

Eh

Xnk

Fnj (k)

i

=Eh

YkZnk

Fnj (k)

i

= ZnkEh

Yk

i

= 0, for all k,n2N+

, (EC.44a)

Eh

Xnk

Fnj (k l)

i

=Eh

Eh

Xnk

Fnj (k)

i

Fnj (k l)

i

= 0, for all k, l, n2N+

, (EC.44b)

by (EC.43). On the other hand, Xnk is Fn

j (k+ l)-measurable for all k 2N+

and l 2N+

by Lemma

EC.4. Hence, for all k,n2N+

,

Eh

Xnk

Fnj (k+ bn0.25c 1)

i

=

8

>

>

>

<

>

>

>

:

0, if n< 16,

Xnk , n 16.

(EC.45)

Since

Xnk

1 for all k 2 N+

and n 2 N+

, the sum in the the right-hand side in (EC.40) is less

than or equal to 15<1 by (EC.45).


Second, by (EC.44), we have

1X

n=1

P (|An2

|> ) =1X

n=1

P

nX

k=1

Eh

Xnk

Fnj (kbn0.25c)

i

>n

!

= 0. (EC.46)

Third,

1X

n=1

P (|An3

|> ) =1X

n=1

P

nX

k=1

Eh

Xnk

Fnj (k+ bn0.25c 1)

i

Eh

Xnk

Fnj (kbn0.25c)

i

>n

!

=1X

n=1

P

0

@

nX

k=1

bn0.25c1

X

l=bn0.25c+1

Eh

Xnk

Fnj (k+ l)

i

Eh

Xnk

Fnj (k+ l 1)

i

>n

1

A

1X

n=1

P

0

@

bn0.25c1

X

l=bn0.25c+1

nX

k=1

Eh

Xnk

Fnj (k+ l)

i

Eh

Xnk

Fnj (k+ l 1)

i

>n

1

A

1X

n=1

bn0.25c1

X

l=bn0.25c+1

P

nX

k=1

Eh

Xnk

Fnj (k+ l)

i

Eh

Xnk

Fnj (k+ l 1)

i

>n

2bn0.25c

!

. (EC.47)

For all r 2N+

, let

Mnl (r) :=

rX

k=1

Eh

Xnk

Fnj (k+ l)

i

Eh

Xnk

Fnj (k+ l 1)

i

.

First, E

Mnl (r)

2r <1 and Mnl (r) 2 Fn

j (l+ r) (even when l+ r 0 in which Mnl (r) = 0 by

definition) for all r 2N+

. Second,

Eh

Mnl (r+1)

Fnj (l+ r)

i

=Eh

Eh

Xnr+1

Fnj (l+ r+1)

i

Eh

Xnr+1

Fnj (l+ r)

i

+Mnl (r)

Fnj (l+ r)

i

= 0+Mnl (r).

Therefore, for all fixed n2N+

and l 2Z, Mnl (r), Fn

j (l+ r), r 2N+

is a martingale sequence such

that E[Mnl (r)] = 0. Moreover,

Mnl (r)Mn

l (r1)

2 for all r 2N+

, i.e., the martingale di↵erences

are bounded by 2. Then, the sum in (EC.47) is equal to

1X

n=1

bn0.25c1

X

l=bn0.25c+1

P

|Mnl (n)|>

n

2bn0.25c

1X

n=1

bn0.25c1

X

l=bn0.25c+1

2exp

0

@2

n

2bn0.25c

2

nX

k=1

42!1

1

A

=1X

n=1

2

2bn0.25c 1

exp

n2

32bnc0.5

<1, (EC.48)

where the inequality is by Azuma’s inequality (cf. Theorem 6.3.3 of Ross (1996)).


Finally, (EC.29) follows by the fact that

P

limsup

1

n

nX

k=1

Xnk

>

!

=P (limsup |An1

+An2

+An3

|> ) = 0,

where the second equality is by the fact that the sum in (EC.40) is finite, (EC.46), (EC.48), and

Borel-Cantelli Lemma (cf. Theorem 2.3.1 of Durrett (2010)).

EC.3. A Regulator Mapping Result

We need regulator mapping results in order to prove Theorems 2, 3, and 4 in Section EC.4. Since

the generalized one-sided regulator mappings defined in the literature (cf. Reed and Ward (2004,

2008) and Ward and Kumar (2008)) are not applicable to our case, we introduce a new one-sided

and nonlinear regulator mapping in this section. Let , : D! D be such that for all x 2 D and

t2R+

,

(x)(t) := sup0st

(x(s))+, (x)(t) := x(t)+ (x)(t). (EC.49)

Then, is the conventional one-sided and one-dimensional regulator mapping (cf. Chapter 13.5 of

Whitt (2002)). We define the following regulator mapping.

Definition EC.2. (A time-dependent, one-sided, and nonlinear regulator mapping): Let x, y 2

D be such that x(0) 0 and supt2R+|y(t)| < 1. The time-dependent, one-sided, and nonlinear

regulator mapping (M, M) :D2 !D2 is defined by (M, M) (x, y) = (z, `) where

CONDITION 1 (C1). z(t) = x(t) R t

0

y(s)z(s)ds+ `(t) 0 for all t2R+

.

CONDITION 2 (C2). `(0) = 0, ` is nondecreasing, andR10

z(t)d`(t) = 0.

If y= 0 in Definition EC.2, then the time-dependent, one-sided, and nonlinear regulator mapping

becomes the conventional one-sided and one-dimensional regulator mapping defined in (EC.49) (cf.

Theorem 6.1 of Chen and Yao (2001)); if y 2D is a constant function, then it becomes the linearly

generalized one-sided mapping described in Ward and Kumar (2008).

In order to write the time-dependent, one-sided, and nonlinear regulator mapping in terms of

the conventional one-sided and one-dimensional regulator mapping defined in (EC.49), we make

the following definition.


Definition EC.3. (Integral equation): Let x, y 2D be such that supt2R+|y(t)|<1. Let M :D2 !

D is a mapping such that u :=M(x, y) solves the integral equation

u(t) = x(t)Z t

0

y(s)(u)(s)ds, 8t2R+

. (EC.50)

A mapping g :D!D is Lipschitz continuous with respect to the uniform norm, if for all T1

2R+

,

there exists a constant 2R+

which may depend on T1

such that kg(x) g(y)kT1 kx ykT1 for

all x, y 2D. Then, the first main result of this section is the following.

Proposition EC.3. For any given x, y 2D such that x(0) 0 and supt2R+|y(t)|<1, there exists

a unique pair of functions (M, M) (x, y) which satisfies conditions C1 and C2 in Definition EC.2.

Moreover,

1. M(x, y) = (M(x, y)) and M(x, y) = (M(x, y)),

2. both M(·, y) :D!D and M(·, y) :D!D are Lipschitz continuous with respect to the uniform

norm.

In order to prove Proposition EC.3, we need the following result.

Lemma EC.6. For any given x, y 2 D such that supt2R+|y(t)| < 1, there exists a unique u 2 D

which solves the integral equation (EC.50). Moreover, the mapping M(·, y) : D ! D is Lipschitz

continuous with respect to the uniform norm.

Proof: Fix an arbitrary pair x, y 2D such that y := supt2R+|y(t)|<1. Let y :D!D be such

that for all u2D and t2R+

, y(u)(t) := y(t)(u)(t). Then for all u1

, u2

2D and t2R+

,

ky(u1

) y(u2

)kt y k(u1

)(u2

)kt 2y ku1

u2

kt ,

where the last inequality is by the fact that the mapping is Lipschitz continuous with respect

to the uniform norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)). Thus, y is

Lipschitz continuous with respect to the uniform norm. Since (EC.50) is equivalent to

u(t) = x(t)Z t

0

y(u)(s)ds, 8t2R+

,


there exists a unique u2D which solves (EC.50) by Lemma 1 of Reed and Ward (2004). Moreover,

the mapping M(·, y) :D!D is Lipschitz continuous with respect to the uniform norm by the same

Lemma.

Proof of Proposition EC.3: Let us fix an arbitrary pair x, y 2 D such that x(0) 0 and

supt2R+|y(t)| < 1. Let u := M(x, y) (note that such a u 2 D uniquely exists by Lemma EC.6).

Moreover, let (z, `) := (, )(u). Then, by (EC.49), (EC.50), and the fact that x(0) = u(0) 0,

z(t) = u(t)+ `(t) = x(t)Z t

0

y(s)(u)(s)ds+ `(t) = x(t)Z t

0

y(s)z(s)ds+ `(t) 0, 8t2R+

,

thus condition C1 in Definition EC.2 is satisfied by (z, `). Since u(0) = x(0) 0, then `(0) =

(u)(0) = 0 and `(·) is nondecreasing by the definition of the mapping (cf. (EC.49)). Lastly,

Z 1

0

z(t)d`(t) =

Z 1

0

(u)(t)d (u)(t) = 0

by definition of the conventional one-sided, one-dimensional regulator mapping (cf. Theorem 6.1

of Chen and Yao (2001)). Therefore, the pair (z, `) = (, )(u) satisfies the conditions C1 and C2

in Definition EC.2.

Next, we will prove uniqueness. Let (z1

, `1

) be another pair which satisfies the conditions C1 and

C2 and g 2D be such that

g(t) := x(t)Z t

0

y(s)z1

(s)ds, 8t2R+

.

Then, z1

(t) = g(t) + `1

(t) for all t 2 R+

. By condition C2 and the uniqueness of the Skorokhod

mapping (cf. Theorem 6.1 of Chen and Yao (2001)), (z1

, `1

) = (, )(g), so

g(t) = x(t)Z t

0

y(s)(g)(s)ds, 8t2R+

. (EC.51)

By Lemma EC.6, there exists a unique solution of (EC.51) which is M(x, y) = u = g. There-

fore, (z1

, `1

) = (, )(g) = (, )(u) = (z, `), which proves uniqueness. Moreover, (M, M)(x, y) =

(z, `) = (, )(u) = (, )(M(x, y)).

Next, let us consider arbitrary x1

, x2

2D such that x1

(0) 0 and x2

(0) 0. Let u1

:=M(x1

, y)

and u2

:=M(x2

, y). By Lemma EC.6, the mapping M(·, y) : D! D is Lipschitz continuous with


respect to the uniform norm. Let y(t) be the corresponding Lipschitz constant for t 2R+

. Then,

for all t2R+

,

M(x1

, y)M(x2

, y)

t= k(u

1

)(u2

)kt 2ku1

u2

kt 2y(t)kx1

x2

kt ,

where the first inequality is by the fact that the mapping is Lipschitz continuous with respect to

the uniform norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)). Hence, M(·, y)

is Lipschitz continuous with respect to the uniform norm. Lastly, for all t2R+

,

M(x1

, y) M(x2

, y)

t= k (u

1

) (u2

)kt ku1

u2

kt y(t)kx1

x2

kt ,

where the first inequality is by the fact that the mapping is Lipschitz continuous with respect to

the uniform norm with Lipschitz constant 1 (cf. Lemma 13.4.1 of Whitt (2002)). Hence, M(·, y)

is also Lipschitz continuous with respect to the uniform norm.

Next, we present the following preliminary result.

Lemma EC.7. Let T1

2R+

be an arbitrary constant, x, y 2D be such that y 0 and there exists a

constant K 2R+

such that y(t)K for all t2R+

. Then, there exists a constant C =C(K,T1

)2R+

such that kM(x, y)kT1 CkxkT1.

Proof of Lemma EC.7: Let u := M(x, y) be the unique solution of the integral equation

(EC.50). Note that (u) 0 by (EC.49). Then by (EC.50) and the fact that y 0 and (u) 0,

u(t) kxkT1 , 8t2 [0, T1

]. (EC.52)

Let dae denote the smallest integer which is greater than or equal to a for all a 2 R. Let us

partition the interval [0, T1

] into subintervals with length 1/(4K) except the last interval which is

[(d4KT1

e 1)/(4K), T1

]. Then there are d4KT1

e subintervals. Let,

fn := sup

|u(t)| : t2

n 1

4K,n

4K

, 8n2 1, . . . , d4KT1

e 1 , (EC.53)

fd4KT1e := sup

|u(t)| : t2d4KT

1

e 1

4K,T

1

.


Then,

kM(x, y)kT1 = kukT1 = maxn21,...,d4KT1e

fn.

By (EC.50),

u(t) =x(t)+

Z t

0

y(s)(u)(s)ds, 8t2R+

. (EC.54)

By (EC.54) and the fact that the mapping is Lipschitz continuous with respect to the uniform

norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)),

u(t) kxkT1 +1

4K2Kkuk

1/(4K)

, 8t2

0,1

4K

) f1

2kxkT1 (EC.55)

where the second inequality above is by (EC.52) and (EC.53). By (EC.54),

u(t) =x(t)+x

1

4K

u

1

4K

+

Z t

1/(4K)

y(s)(u)(s)ds, 8t2

1

4K,1

2K

)u(t) 2kxkT1 + f1

+1

4K2Kkuk

1/(2K)

, 8t2

1

4K,1

2K

)u(t) 2kxkT1 + f1

+1

2(f

1

+ f2

) , 8t2

1

4K,1

2K

) f2

4kxkT1 +3f1

,

where the last inequality follows by (EC.52) and (EC.53). By induction, we can show that

fn 4kxkT1 +2fn1

+n1

X

k=1

fk, 8n2 2, . . . , d4KT1

e . (EC.56)

By (EC.55) and (EC.56), one can show that

kukT1 = maxn21,...,d4KT1e

fn (d4KT1

e+4)! kxkT1 .

We complete the proof by defining C(K,T1

) := (d4KT1

e+4)!.

The second main result of this section is the following.

Lemma EC.8. Let xn, n 2N and yn, n 2N be sequences in D such that xn(0) 0 and yn 0

for all n 2 N, there exists a constant K 2 R+

such that yn(t) K for all n 2 N and t 2 R+

,

and there exist x 2 D and y 2 D such that xn ! x u.o.c. and yn ! y u.o.c. as n ! 1. Then,

(M, M) (xn, yn)! (M, M) (x, y) u.o.c. as n!1.


Proof of Lemma EC.8: For each n2N, let un 2D be the unique solution of the equation

un(t) = xn(t)Z t

0

yn(s)(un)(s)ds, 8t2R+

, (EC.57)

which exists by Lemma EC.6. For each n2N, let us define

fn(t) :=

Z t

0

yn(s)(un)(s)ds, 8t2R+

. (EC.58)

Note that fn is continuous for all n 2 N and un = xn fn for all n 2 N by (EC.57) and (EC.58).

Let T1

be an arbitrary constant in R+

. We will show that the sequence fn, n 2 N restricted to

the compact domain [0, T1

] is relatively compact by the Arzela-Ascoli Theorem (cf. Theorem 7.2

of Billingsley (1999)). First, supn2N |fn(0)|= 0<1. Second,

lim!0

supn2N

sup|t1t2|

|fn(t2) fn(t1)|= lim!0

supn2N

sup|t1t2|

Z t2

t1

yn(s)(un)(s)ds

lim!0

2 supn2N

kynkT1supn2N

kunkT1

lim!0

2K supn2N

kunkT1

, (EC.59)

lim!0

2KC(K,T1

) supn2N

kxnkT1

= 0, (EC.60)

where the first inequality in (EC.59) is by the fact that the mapping is Lipschitz continuous with

respect to the uniform norm with Lipschitz constant 2 (cf. Lemma 13.5.1 of Whitt (2002)). The

inequality in (EC.60) is by Lemma EC.7. Note that xn ! x u.o.c where x is a bounded function in

[0, T1

], thus there exists a suciently large n0

2N such that supnn0kxnkT1

<1, and without loss

of generality we assume supn2N kxnkT1<1. Hence, we obtain the convergence result in (EC.60).

Therefore, fn, n2N restricted to the compact domain [0, T1

] is relatively compact.

Since both xn, n2N and fn, n2N are relatively compact in D[0, T1

] endowed with the u.o.c.

topology, so does un, n2N by (EC.57) and (EC.58). Let us consider an arbitrary subsequence of

un, n2N, denoted by unk, k 2N, such that unk

! u u.o.c. as k!1 where u2D[0, T1

]. Then,

supt2[0,T1]

u(t)x(t)+

Z t

0

y(s)(u)(s)ds

(EC.61)

= supt2[0,T1]

u(t)unk(t)+unk

(t)x(t)+

Z t

0

y(s)(u)(s)ds


uunk

T1+

xnkx

T1+ sup

t2[0,T1]

Z t

0

y(s)(u)(s) ynk(s)(unk

)(s)

ds

uunk

T1+

xnkx

T1+ sup

t2[0,T1]

Z t

0

y(s)(u)(s) ynk(s)(u)(s)

ds

+ supt2[0,T1]

Z t

0

ynk(s)(u)(s) ynk

(s)(unk)(s)

ds

uunk

T1+

xnkx

T1+2T

1

kukT1

ynk y

T1+2T

1

K

unku

T1. (EC.62)

As k!1, all of the terms in (EC.62) converges to 0, so (EC.61) is equal to 0. Thus, u=M(x, y)

and u is the unique solution of (EC.50) by Lemma EC.6. Therefore, each subsequence of un, n2N

has convergent subsequence which converges to the same limit, which implies that un ! u u.o.c.

as n!1 where u=M(x, y). Note that the process (M, M) (xn, yn) 2D2 is well defined for all

n 2N by Proposition EC.3. Since the mappings and are Lipschitz continuous with respect to

the uniform norm,

M, M (xn, yn) = (, ) (M(xn, yn)) = (, ) (un)

! (, ) (u) = (, ) (M(x, y)) =

M, M (x, y), u.o.c. as n!1,

which completes the proof. EC.4. Proofs of Theorems 2, 3, and 4

We prove Theorems 2, 3, and 4 by representing the queue length process of each driver type with

the time-dependent, one-sided, and nonlinear regulator mapping defined in Definition EC.2. We

first present some preliminary results that will be used in the proofs of all of the three theorems in

Section EC.4.1. Then, we will prove Theorems 2, 4, and 3 in Sections EC.4.2, EC.4.3, and EC.4.4,

respectively. Lastly, we make a discussion about whether introducing a di↵erent CLP can extend

the results of this section and the main results of the paper in Section EC.4.5.

EC.4.1. Preliminary Results

In this section, we present some preliminary results that will be used in the proofs of Theorems

2, 3, and 4. We consider R(x) where x = xij(t), i 2 N , j 2 J , t 2 [0, T ] is a feasible matching


process for the CLP (12) solution such that xij is a Borel measurable simple function with possibly

countably infinite range for all i2N and j 2J in this section. We start with the formal definition

of this randomized policy based on Definition 1.

Let xij(m), i 2N , j 2 J ,m 2 N be a sequence of real numbers in R+

and Bm,m 2 N be a

disjoint partition of the interval [0, T ] such that Bm is a Borel measurable set for all m 2 N. We

assume that

xij(t) =1X

m=0

xij(m)I (t2Bm) , 8t2 [0, T ], i2N , j 2J . (EC.63)

Without loss of generality, we choose the partition Bm,m 2 N such that Bm [m, m+1

) for

all m 2 N. By (12c),P

i2N xij(m) 1 for all j 2 J and m 2 N. Let us define the sequence of

independent random variables pkj (m), k 2 N+

, j 2 J ,m 2 N, which is independent of all other

stochastic primitives and F(0)-measurable such that, P(pkj (m) = i) = xij(m) and P(pkj (m) = 0) =

1Pi2N xij(m) for all i2N , j 2J , k 2N+

, and m2N. Note that the sequence pkj (m), k 2N+

, j 2

J ,m2N corresponds to m,m2N defined in Remark 3 associated with the randomized policy.

With a slight abuse of notation, let R(x) = (n1

,n2

, . . . ,nJ ) in the nth system. Then,

nj (k) =

X

i2N

X

m2N

i I

nj (k)2Bm, pkj (m) = i, Q

R(x),ni (nj (k))> 0

, (EC.64)

for all j 2 J and k,n 2N+

. Then, clearly, nj (k) is Fn

j (k)-measurable for all j 2 J and k,n 2N+

,

thus R(x) is admissible by Definition 1.

Let : N ! N be a function such that (m) := l 2 N : Bm [l, l+1

). Under R(x), for all

i2N , j 2J , t2 [0, T ], and n2N+

, we have

DR(x),nij (t) =

Ejnj (t)

X

k=1

X

m2N

I

nj (k)2Bm, akj ((m)) tij((m)), pkj (m) = i, Q

R(x),ni (nj (k))> 0

.

Let us define the following stochastic process such that for all i2N , j 2J , n2N+

, and t2 [0, T ],

HR(x),nij (t) :=

Ejnj (t)

X

k=1

X

m2N

Fij((m)

)xij(m)I

nj (k)2Bm, QR(x),ni (nj (k))> 0

,

=

Z t

0

X

m2N

Fij((m)

)xij(m)I

s2Bm, QR(x),ni (s)> 0

dEj nj (s),

=

Z t

0

Fij(s)xij(s)I

QR(x),ni (s)> 0

dEj nj (s),


where the second equality is the Lebesgue-Stieltjes integral (cf. page 107 in Folland (1999)), and

the third equality is by (2) and (EC.63). Let HR(x),nij := (1/n)HR(x),n

ij . Then we have the following

result whose proof is presented in Section EC.4.1.1.

Lemma EC.9. For all i2N and j 2J , we have

DR(x),nij H

R(x),nij

T

a.s.! 0, as n!1.

We omit the superscript R(x) from the notation for convenience in presentation in the rest

of Section EC.4.1. Let us define the following stochastic processes such that for all i 2N , j 2 J ,

n2N+

, and t2 [0, T ],

Mnij(t) :=

Z t

0

Fij(s)xij(s)I (Qni (s)> 0)

dEj nj (s) dn

j (s)

,

Lnij(t) :=

Z t

0

Fij(s)xij(s) (I (Qni (s)> 0) I (Qn

i (s)> 0))dnj (s),

Inij(t) :=

Z t

0

Fij(s)xij(s)I (Qni (s) = 0)µn

j (s)ds. (EC.65)

Let Mnij := (1/n)Mn

ij, Lnij := (1/n)Ln

ij, and Inij := (1/n)Inij for all i2N , j 2J , and n2N+

. Then, by

(9), (11), and after some algebra,

Qni (t) = Xn

i (t)Z t

0

ni (s)Qni (s)ds+

X

j2JInij(t), (EC.66)

for all i2N , n2N+

, and t2 [0, T ], where

Xni (t) := Qn

i (0)+ Ani n

i (t)

Rni

Z t

0

ni (s)Qni (s)ds

Z t

0

ni (s)Qni (s)ds

X

j2J

Dnij(t) Hn

ij(t)X

j2JMn

ij(t)+X

j2JLn

ij(t)1

n

X

j2J

Z t

0

Fij(s)xij(s)µnj (s)ds

for all i2N , n2N+

, and t2 [0, T ]. Without loss of generality, we define

Qni (t) := Qn

i (T ), Xni (t) := Xn

i (T ),X

j2JInij(t) :=

X

j2JInij(T ), ni (t) := 0 (EC.67)

for all i 2N , n 2N+

, and t T for mathematical completeness, so (EC.66) is well defined for all

t2R+

.

The following result allows us to approximate Xni by a deterministic process.


Lemma EC.10. For all i2N and j 2J , we have

Lnij

T= 0 a.s. 8n2N

+

, and

Mnij

T

a.s.! 0, sup0t<1

Xni (t) Xi(t)

a.s.! 0, as n!1,

where

Xi(t) :=i(t)X

j2J

Z t

0

Fij(s)xij(s)µj(s)ds,

for all t 2 [0, T ] and Xi(t) := Xi(T ) for all t T . Moreover, Xi is absolutely continuous and non-

negative for all i2N .

The proof of Lemma EC.10 is presented in Section EC.4.1.2. Note that for all i2N and n2N+

,

X

j2JInij(0) = 0,

X

j2JInij is nondecreasing,

Z 1

0

Qni (t)d

X

j2JInij(t)

= 0, (EC.68)

by (EC.65) and (EC.67). By Assumption 1, the fact that Xni (0) 0, (EC.66), (EC.67), (EC.68),

Definition EC.2, and Proposition EC.3, we have

Qni ,X

j2JInij

=

M, M Xni ,

ni

for all i2N and n2N+

.

There exists a constant 2R+

and n0

2N+

such that ni (t) for all t2R+

, i2N , and n n0

by Assumptions 1 and 2. Then, by Assumption 2, Lemma EC.8, and Lemma EC.10, we have

Qni M Xi,i

T

a.s.! 0,

X

j2JInij M Xi,i

T

a.s.! 0, (EC.69)

as n!1 for all i2N . Lastly, we have the following result.

Lemma EC.11. Fix an arbitrary i2N . If M Xi,i

= 0, then

sup0tT

X

j2J

Dnij(t)

Z t

0


a.s.! 0, as n!1.

Proof: By (EC.69) and the fact that M Xi,i

= 0,

X

j2JInij

T

a.s.! 0, as n!1. (EC.70)


Then,

sup0tT

X

j2J

Dnij(t)

Z t

0


= sup0tT

X

j2J

Dnij(t) Hn

ij(t)+ Hnij(t) Mn

ij(t)+ Lnij(t)+ Mn

ij(t) Lnij(t)

+ Inij(t) Inij(t)Z t

0


X

j2J

Dnij Hn

ij

T+X

j2J

Mnij

T+X

j2J

Lnij

T+

X

j2JInij

T

+ sup0tT

X

j2J

Hnij(t) Mn

ij(t)+ Lnij(t)+ Inij(t)

Z t

0


=X

j2J

Dnij Hn

ij

T+X

j2J

Mnij

T+X

j2J

Lnij

T+

X

j2JInij

T

+ sup0tT

X

j2J

Z t

0

µnj (s)

nµj(s)

Fij(s)xij(s)ds

(EC.71)

a.s.! 0, as n!1, (EC.72)

where the first inequality above is by triangular inequality and (EC.71) is by algebra. In (EC.71),

as n!1, the first term converges to 0 a.s. by Lemma EC.9, the second and third terms converge

to 0 a.s. by Lemma EC.10. The fourth term in (EC.71) converges to 0 a.s. by (EC.70). Finally, the

fifth term in (EC.71) converges to 0 by Assumption 2 and the fact that 0 Fij(t)xij(t) 1 for all

i2N , j 2J , and t2 [0, T ]. Hence, we obtain the convergence in (EC.72).

EC.4.1.1. Proof of Lemma EC.9 The proof of Lemma EC.9 is very similar to the one of

Lemma EC.2. For notational convenience, we omit the superscript R(x) from the notation in this

section. Let us fix arbitrary i2N and j 2J . Then, by definition, we have

Dnij Hn

ij

T= sup

0tT

1

n

Ejnj (t)

X

k=1

X

m2N

I

akj ((m)) tij((m)), pkj (m) = i

Fij((m)

)xij(m)

I

nj (k)2Bm, Qni (

nj (k))> 0

. (EC.73)

For all n2N+

and m2N, let

Yk(m) := I

akj ((m)) tij((m)), pkj (m) = i

Fij((m)

)xij(m),


Znk (m) := I

nj (k)2Bm, Qni (

nj (k))> 0

,

W nk (m) := Yk(m)Zn

k (m),

W nk :=

X

m2N

W nk (m)

Then, the right-hand side of (EC.73) is equal to

sup0tT

1

n

Ejnj (t)

X

k=1

W nk

. (EC.74)

Note that, if we can prove that

1

n

nX

k=1

W nk

a.s.! 0, as n!1, (EC.75)

then we can prove that the term in (EC.74) converges to 0 as n!1 by the same way we prove

Lemma EC.2. Hence, in the remaining part of this section, we will prove (EC.75), whose proof is

very similar to the one of (EC.21).

For all n2N+

, if k 2Z\N+

, we let Gnk := ;,; and if k 2N

+

, we let

Gnk :=

(

Ai ni (s^ nj (k)),Ej0 n

j0(s^ nj (k)),Ri

Z

(s^nj (k))

0

ni (u)Qni (u)du

!

,

Dnij0((s^ nj (k))),Qn

i ((s^ nj (k))), 8i2N , j0 2J , s2R+

,

arj0((m)), prj0(m), r 2 1, . . . ,Ej0 n

j0(nj (k)), 8j0 2J \j,m2N

arj((m)), prj(m), r 2 1, . . . , k 1, 8m2N

)

. (EC.76)

Then, Gnk , k 2 Z is a filtration for all n 2 N

+

. Moreover, Yk(m) ? Gnk and Zn

k (m) 2 Gnk for all

k,n2N+

and m2N; and W nk (m)2 Gn

k+l for all k,n, l 2N+

by construction. First, for all k,n2N+

,

Eh

W nk

Gnk

i

=E

"

X

m2N

W nk (m)

Gnk

#

=X

m2N

E

W nk (m)

Gnk

=X

m2N

E

Yk(m)Znk (m)

Gnk

, (EC.77a)

=X

m2N

Znk (m)E

Yk(m)

Gnk

=X

m2N

(Znk (m)E [Yk(m)]) = 0, (EC.77b)

where the second equality in (EC.77a) is by the fact that |W nk | 1 for all k,n 2N

+

and bounded

convergence theorem for conditional expectations. Second, for all k,n, l 2N+

,

Eh

W nk

Gnkl

i

=Eh

Eh

W nk

Gnk

i

Gnkl

i

= 0, (EC.78)


by (EC.77b). Third, for all k,n2N+

,

Eh

W nk

Gnk+bn0.25c1

i

=

8

>

>

>

<

>

>

>

:

0, if n< 16,

W nk , n 16.

(EC.79)

Therefore, by (EC.77b), (EC.78), and (EC.79), we can prove (EC.75) by the same method that we

use in Section EC.2.1 (starting from (EC.39)) in order to prove (EC.21).

EC.4.1.2. Proof of Lemma EC.10 By triangular inequality, for all i2N ,

Xni Xi

T Qn

i (0)+

Ani n

i i

T+X

j2J

Dnij Hn

ij

T+X

j2J

Mnij

T(EC.80a)

+X

j2J

Lnij

T+ sup

0tT

Rni

Z t

0

ni (s)Qni (s)ds

Z t

0

ni (s)Qni (s)ds

(EC.80b)

+X

j2Jsup

0tT

1

n

Z t

0

Fij(s)xij(s)µnj (s)ds

Z t

0

Fij(s)xij(s)µj(s)ds

. (EC.80c)

We will consider each term in the right-hand side (RHS) of (EC.80) separately. The first term

in the RHS of (EC.80a) converges to 0 a.s. by Assumption 3. The second term in the RHS of

(EC.80a) converges to 0 a.s. by (EC.13). The third term in the RHS of (EC.80a) converges to 0

a.s. by Lemma EC.9. Next let us consider the fourth term in the RHS of (EC.80a). For all i 2N

and j 2J ,

Mnij

T= sup

0tT

1

n

Z t

0

Fij(s)xij(s)I (Qni (s)> 0)d

Ej nj (s)n

j (s)

sup0tT

1

n

Z t

0

Fij(s)xij(s)I (Qni (s)> 0)d

Ej nj (s)_n

j (s)Ej nj (s)^n

j (s)

sup0tT

1

n

Z t

0

d

Ej nj (s)_n

j (s)Ej nj (s)^n

j (s)

= sup0tT

1

n

Ej nj (t)_n

j (t)Ej nj (t)^n

j (t)

sup0tT

1

n

Ej nj (t)_n

j (t)j(t)

+ sup0tT

1

n

Ej nj (t)^n

j (t)j(t)

a.s.! 0, as n!1, (EC.81)

where (EC.81) is by Assumption 2, (EC.13), and continuous mapping theorem (cf. Theorem 5.2 of

Chen and Yao (2001)). Therefore, the fourth term in the RHS of (EC.80a) also converges to 0 a.s.


Next, let us consider the first term in (EC.80b). For all i2N , j 2J , and n2N+

,

Lnij

T sup

0tT

1

n

Z t

0

Fij(s)xij(s) |I (Qni (s)> 0) I (Qn

i (s)> 0)|µnj (s)d(s)

1

n

Z T

0

I (Qni (s) 6=Qn

i (s))µnj (s)d(s)

= 0 a.s., (EC.82)

where (EC.82) is by the fact that Qni has finitely many jumps on a compact time interval (cf.

Lemma 12.1 of Billingsley (1999)), because Qni 2D and its jump magnitudes are exactly 1, so the

Lebesgue measure of the set t2 [0, T ] :Qni (t,!) 6=Qn

i (t,!) is 0 for almost all ! 2. Therefore,

the first term in (EC.80b) is 0 a.s. for all i2N and j 2J .

Next, let us consider the second term in (EC.80b). Let us fix an arbitrary ! 2A1

(cf. (EC.14)).

From the proof of Proposition EC.1 (cf. page ec8 in Section EC.2), we know that each subsequence

of the second term in (EC.80b) has a subsequence which converges to 0 on the sample path !,

which implies that the second term in (EC.80b) itself converges to 0 as n!1 on the sample path

!. Since P(A1

) = 1, the second term in (EC.80b) converges to 0 a.s as n!1.

Next, the term in (EC.80c) converges to 0 as n!1 by Assumption 2 and the fact that 0

Fij(t)xij(t) 1 for all i2N , j 2J , and t2 [0, T ]. Since Xni (t) = Xn

i (T ) for all t T (cf. (EC.67)),

sup0t<1

Xni (t) Xi(t)

a.s.! 0, as n!1 for all i2N .

Lastly, Xi is absolutely continuous because it is a di↵erence of two absolutely continuous func-

tions. It is nonnegative by (12b) and the fact that x is a feasible matching process for the CLP

(12).

EC.4.2. Proof of Theorem 2

Since i = 0 by assumption, M

Xi,i

= Xi by Definition EC.3. By (EC.49) and the fact that

M Xi,i

= M

Xi,i

(cf. Proposition EC.3) and Xi is nonnegative (cf. Lemma EC.10), we

have M Xi,i

=

Xi

= 0 for all i2N . Then, the proof follows by Lemma EC.11.



The process x

is a feasible matching process for the CLP (12) by Lemma 4 and it is a Borel

measurable simple function with possibly countably infinite range for all i 2 N and j 2 J by

assumption. Then, the results of Section EC.4.1 apply to this process. Moreover, we can extend

the results of Lemma EC.10 such that Xi is also nondecreasing for all i 2N by (14b), (12b), and

the fact that x is a feasible matching process for both the CLP (12) and the LP (14). Then, we

have the following result.

Lemma EC.12. M

Xi,i 0 for all i2N .

Proof: Recall that Xi is an absolutely continuous, nonnegative, and nondecreasing function for

all i2N by Lemma EC.10 and the argument mentioned above. Let us fix an arbitrary i2N . Let

ui :=M(Xi,i). Then, by Definition EC.3,

ui(t) = Xi(t)Z t

0

i(s)(ui)(s)ds, 8t2R+

. (EC.83)

Since ui is a di↵erence of two absolutely continuos functions, it is also absolutely continuous.

We will use proof by contradiction technique. Assume that ui(t)< 0 for some t2R+

. Then, there

exists an interval [t1

, t2

]R+

such that t1

:= inft2R+

: ui(t)< 0, ui(·) is nonincreasing on [t1

, t2

],

and ui(t2)< 0. Since ui(0) = 0 and ui is continuous, such an interval exists. Then, ui(t1) = 0 and

(ui)(t) = 0 for all t2 [t1

, t2

] by (EC.49) and construction, and

ui(t2) = Xi(t2)Z t2

0

i(s)(ui)(s)ds= Xi(t2)

ui(t1)+

Z t1

0

i(s)(ui)(s)ds

Z t2

t1

i(s)(ui)(s)ds

= Xi(t2) Xi(t1)Z t2

t1

i(s)(ui)(s)ds= Xi(t2) Xi(t1) 0,

which is a contradiction because ui(t2) < 0 by construction. Therefore, ui(t) 0 for all t 2 R+

,

which completes the proof.

By (EC.49), the fact that M Xi,i

= M

Xi,i

(cf. Proposition EC.3), and Lemma

EC.12, we have M Xi,i

= 0 for all i2N . Then, by Lemma EC.11, for all i2N ,

sup0tT

X

j2J

DR(x

),n

ij (t)Z t

0


a.s.! 0, as n!1,

which gives us the desired result.



By Assumption 1 and the fact that xij = 0 for all i 2N and j 2J are feasible matching fractions

for the LP (13), there exists an optimal solution of the LP (13), denoted by x = xij, i2N , j 2J .

Let x := xij(t), i2N , j 2J , t2 [0, T ] be such that xij(t) := xij for all i2N , j 2J , and t2 [0, T ].

It is easy to see that the process x is a feasible matching process for the CLP (12). Thus,

X

i2N ,j2JwijµjFij

Z T

0

xij(s)ds=X

i2N ,j2JwijµjFijx

ijT

X

i2N ,j2JwijµjFij

Z T

0

xij(s)ds, (EC.84)

where x := xij(t), i 2 N , j 2 J , t 2 [0, T ] is an optimal solution of the CLP (12), which exists

by Lemma 2. Let xij := (1/T )R T

0

xij(s)ds for all i 2 N and j 2 J . Then it is easy to see that

xij, i2N , j 2J is feasible for the LP (13), thus

X

i2N ,j2JwijµjFijxij =

X

i2N ,j2JwijµjFij

1

T

Z T

0

xij(s)ds

X

i2N ,j2JwijµjFijx

ij. (EC.85)

By (EC.84) and (EC.85), we see that x is an optimal solution of the CLP (12).

The rest of the proof of Theorem 3 is very similar to the one of Theorem 4. Since the process x is

an optimal solution of the CLP (12) and xij is a Borel measurable simple function for all i2N and

j 2 J , the results of Section EC.4.1 apply to this process. Moreover, we can extend the results of

Lemma EC.10 such that Xi is also nondecreasing for all i2N by (13b). Therefore, Lemma EC.12

holds and M Xi,i

= 0 for all i2N . Then, by Lemma EC.11, for all i2N ,

sup0tT

X

j2J

DR(˜x),nij (t)

Z t

0


a.s.! 0, as n!1,

which gives us the desired result.

EC.4.5. A CLP with departure rates of unmatched drivers

By the properties of the fluid limits of admissible policies (cf. Proposition EC.1), it is possible to

derive a CLP which considers with i, i2N in the following way:

maxX

i2N ,j2Jwij

Z T

0

µj(s)Fij(s)xij(s)ds, (EC.86a)


X

j2J

Z t

0

µj(s)Fij(s)xij(s)ds+ qi(t)+

Z t

0

i(s)qi(s)ds=i(t), 8i2N and t2 [0, T ], (EC.86b)

X

i2Nxij(t) 1, for all j 2J and t2 [0, T ], (EC.86c)

xij(t) 0, qi(t) 0, for all i2N , j 2J , and t2 [0, T ], (EC.86d)

xij and qi are Lebesgue measurable for all i2N and j 2J , (EC.86e)

where the decision variables are xij : [0, T ]!R+

and qi : [0, T ]!R+

for all i 2N and j 2 J . The

only di↵erence between the CLP (EC.86) and the CLP (12) is that the former one contains the

additional parameters i, i2N and the additional decision variables qi, i2N.

Although we can associate the matching policies defined in Definition 2 and Section 5.1 with

feasible matching processes for the CLP (EC.86), these matching policies do not provide more

general asymptotic optimality results than the ones that we have in Corollary 1 and Theorems

2, 3, and 4. Recall that we need the condition i = 0 in Theorem 2 and i = 0 for all i 2 N in

Corollary 1 because we use Theorem 2 to prove Corollary 1. In order to prove Theorem 2, we need

Lemma EC.11. However, Lemma EC.11 holds, if M Xi,i

= 0; and a sucient condition for that

result is i = 0. In order to prove Theorem 2 for the randomized policy associated with a feasible

matching process for the CLP (EC.86), we still need to prove Lemma EC.11, and we still need the

condition M Xi,i

= 0. However, the CLP (EC.86) does not facilitate a di↵erent condition than

the one i = 0 to prove M Xi,i

= 0.

Recall that we derive the LP (14) from the CLP (12) by taking the derivative of (12a) and (12b)

with respect to t. Then we prove Theorem 4 by Assumption 4. However, it is not clear how to

derive an LP from CLP (EC.86) because taking the derivative of (EC.86b) with respect to t results

in an ordinary di↵erential equation. Hence, it is not clear how to relax Assumption 4 by using the

CLP (EC.86).

In summary, considering the CLP (EC.86), which is more complicated than the CLP (12), does

not facilitate more general results. Hence, we consider the CLP (12) in this study.


EC.5. Lemma Proofs

We first prove Lemma 1, second prove Lemmas 3 and 5 (because they are related to Lemma 1),

third prove Lemma 2, and lastly prove Lemma 4.

EC.5.1. Proof of Lemma 1

Let CD = (1

,2

, . . . ,J). For notational convenience, we omit the superscript CD from the nota-

tion in this proof. Let us define the sequence of independent random variables qkj (l), k 2 N+

, j 2

J , l 2NN, which is independent of all other stochastic primitives and F(0)-measurable, such that

qkj (0,0, . . . ,0) = 0, and ifPN

i=1

li > 0, then P(qkj (l1, l2, . . . , lN) = i) = li/PN

i=1

li for all k 2N+

, j 2J ,

and l 2NN such that l= (l1

, l2

, . . . , lN). For notational convenience, let us define the set

Sj(k) := argmini2N :Qi(j(k))>0

X

m2N

tij(m)I(j(k)2 [m, m+1

) (EC.87)

for all j 2 J and k 2N+

. Let |Sj(k)| denote the cardinality of the set Sj(k). Then, under the CD

policy, for all j 2J and k 2N+

, let

j(k) := I (|Sj(k)|= 1)X

i2N

i I (i2 Sj(k))

+ I (|Sj(k)|> 1) qkj

Q1

(j(k))I(12 Sj(k)), . . . ,QN(j(k))I(N 2 Sj(k))

.

Note that Qi(j(k)) = limt!1Qi((t^ j(k))) 2Fj(k) by Proposition 2.7 of Folland (1999) and

definition of Fj(k) (cf. (8)) for all i 2N , j 2 J , and k 2 N+

. By (8), (EC.87), and the fact that

j(k) 2 Fj(k), the random variables I (|Sj(k)|= 1), I (|Sj(k)|> 1), I (i2 Sj(k)), and I (Sj(k) =N1

)

are Fj(k)-measurable for all i2N , j 2J , k 2N+

, and N1

N . Lastly,

qkj

Q1

(j(k))I(12 Sj(k)), . . . ,QN(j(k))I(N 2 Sj(k))

=X

N1NI (Sj(k) =N

1

)

X

l2NN:li=0,8i/2N1

qkj (l1, l2, . . . , lN)I

Qi(j(k)) = li,8i2N1

!

2Fj(k).

Therefore, j 2 Fj for all j 2 J under the CD policy, so it is an admissible policy. Note that, the

sequence qkj (l), k 2N+

, j 2J , l 2NN corresponds to m,m2N defined in Remark 3 associated

with the CD policy.


EC.5.2. Proofs of Lemmas 3 and 5

First, the randomized policy given in Definition 2 is admissible by the fact that (EC.64) is Fnj (k)-

measurable for all j 2J and k,n2N+

. Second, the admissibility proof of the hybrid policy follows

by the fact that it is a hybrid of the randomized policy given in Definition 2 and the CD policy

defined in (3), and both of the latter two policies are admissible. Third, the deterministic policy is

admissible because both Qni (

nj (k)) and Dn

ij(nj (k)) are Fn

j (k)-measurable for all i 2N , j 2J ,

and k,n2N+

(cf. (8)) and the tie breaking rule does not use any future information.


We use the proof technique introduced by Levinson (1966). We first present a preliminary result

from Levinson (1966), then prove Lemma 2. Let L2([0, T ]) denote the space of Lebesgue measurable

functions with domain [0, T ] and range R such that if f 2 L2([0, T ]), thenR T

0

f(t)2dt <1. We let

w! denote weak convergence in L2([0, T ]) as defined in Proposition 6 in Section 8.2 of Royden and

Fitzpatrick (2010).

Lemma EC.13. (Lemma 2.1 of Levinson (1966)) Let fr, r 2N be a uniformly bounded sequence

of functions in L2([0, T ]) such that frw! f for some f 2L2([0, T ]). Let fu, fl : [0, T ]!R be defined

as fu(t) := limsupr!1 fr(t) and fl(t) := lim infr!1 fr(t) for all t 2 [0, T ]. Then, f(t) fu(t) and

f(t) fl(t) for all t2 [0, T ] except on a set of zero measure.

First, note that xij = 0 for all i 2N and j 2J is a feasible matching process for the CLP (12).

Second, xij is nonnegative and bounded for all i2N and j 2J by (12c) and (12d). Third, both µj

and i are bounded processes on [0, T ] by Assumption 1, and wij and Fij are bounded by definition

for all i 2N and j 2 J . Moreover, µj, i, and Fij are all Borel measurable (so does the process

µj(·)Fij(·)) by construction for all i2N and j 2J .

Let x := xij, i 2N , j 2 J denote the decision variables of the CLP (12), H denote the set of

feasible decision variables of the CLP (12), i.e., H= x :x satisfies (12b)-(12e), and

M := supx2H

X

i2N ,j2Jwij

Z T

0

µj(s)Fij(s)xij(s)ds <1.


Then, there exists a sequence of feasible decision variables denoted by xr, r 2N such that xr =

xrij, i2N , j 2J for all r 2N and

limr!1

X

i2N ,j2Jwij

Z T

0

µj(s)Fij(s)xrij(s)ds=M.

By Theorem 14 in Section 8.3 of Royden and Fitzpatrick (2010), there exists a subsequence of

xr, r 2N which is again denoted by xr, r 2N for notational convenience such that xrij weakly

converges to some xij 2L2([0, T ]) for all i2N and j 2J . Let x := xij, i2N , j 2J . SinceX

i2N ,j2Jwij

Z T

0

µj(s)Fij(s)xij(s)ds=X

i2N ,j2Jwij lim

r!1

Z T

0

µj(s)Fij(s)xrij(s)ds=M, (EC.88)

it is enough to prove that a modification of x is in H in order to prove Lemma 2.

First, let us consider (12b). Let us fix an arbitrary i2N and t2 [0, T ]. Then,

X

j2J

Z t

0

µj(s)Fij(s)xij(s)ds=X

j2J

Z T

0

µj(s)Fij(s)I(s2 [0, t])xij(s)ds

=X

j2Jlimr!1

Z T

0

µj(s)Fij(s)I(s2 [0, t])xrij(s)dsi(t), (EC.89)

where the equality in (EC.89) is by the definition of weak convergence and the inequality in (EC.89)

is by (12b). Hence, x satisfies (12b).

Next, let us consider (12c) and (12d). Note that, since xrij

w! xij for all i 2N and j 2 J , it is

easy to see thatP

i2N xrij

w!P

i2N xij for all j 2J . Then, by Lemma EC.13, for all t2 [0, T ] except

on a set of zero measure,

X

i2Nxij(t) limsup

r!1

X

i2Nxrij(t) 1, xij(t) lim inf

r!1xrij(t) 0, 8i2N , j 2J . (EC.90)

Hence, x satisfies (12c) and (12d) for all t2 [0, T ] except on a set of zero measure, and we denote this

set of zero measure by H and its complement by Hc, i.e., Hc := [0, T ]\H. Let x= xij, i2N , j 2J

be such that xij(t) := xij(t) for all t2Hc, i2N , and j 2J , and xij(t) := 0 for all t2H, i2N , and

j 2J . Then, x satisfies (12c) and (12d) by construction, and satisfies (12e) by Proposition 2.11.a

of Folland (1999). Moreover, for all i2N and t2 [0, T ],

X

j2J

Z t

0



=X

j2J

Z T

0

µj(s)Fij(s)I(s2 [0, t]\H)xij(s)ds+X

j2J

Z T

0

µj(s)Fij(s)I(s2 [0, t]\Hc)xij(s)ds

=X

j2J

Z T

0

µj(s)Fij(s)I(s2 [0, t]\Hc)xij(s)ds (EC.91)

i(t)X

j2J

Z T

0

µj(s)Fij(s)I(s2 [0, t]\H)xij(s)ds=i(t), (EC.92)

where (EC.91) is by the fact that H is a set of zero measure and xij(t) = xij(t) for all t 2Hc, the

inequality in (EC.92) is by (EC.89), and the equality in (EC.92) is by the fact that H is a set of

zero measure. Therefore, x satisfies (12b), so it is a feasible matching process for the CLP (12),

and it is an optimal solution by (EC.88) and the fact that it is equal to x almost everywhere.


By Assumption 4 and the argument in the first paragraph of Section 4.2, there exists an optimal

solution of the CLP (12), denoted by x, under which constraint (12b) is binding for all driver types

at all times. Note that x satisfies (14c) and (14d) because these constraints are exactly the same

of (12c) and (12d), respectively. Since constraint (12b) is binding for all driver types at all times

under x, then by taking the derivative of both right- and left-hand sides of (12b) with respect to

t, we can see that xij(t), i 2N , j 2 J (or a modification of it) satisfies (14b) for all t 2 [0, T ], so

it is feasible for the LP (14) for all t2 [0, T ]. Then,

X

i2N ,j2Jwijµj(t)Fij(t)xij(t)

X

i2N ,j2Jwijµj(t)Fij(t)x

ij(t), 8t2 [0, T ],

)X

i2N ,j2Jwij

Z T

0

µj(s)Fij(s)xij(s)dsX

i2N ,j2Jwij

Z T

0

µj(s)Fij(s)xij(s)ds. (EC.93)

Second, x satisfies (12e) by Assumption 5 and satisfies (12c) and (12d) by satisfying (14c) and

(14d), respectively. By taking the integral of both the right- and left-hand sides in (14b) with

respect to t, we can see that x satisfies (12b) for all t2 [0, T ], so it is a feasible matching process

for the CLP (12). Then,

X

i2N ,j2Jwij

Z T

0


X

i2N ,j2Jwij

Z T

0

µj(s)Fij(s)xij(s)ds. (EC.94)

Therefore, x is an optimal solution of the CLP (12) by (EC.93) and (EC.94).


EC.6. Relative Compactness in space D

In this section, we present a relative compactness result in space D that we use in the proof of

Proposition EC.1 (cf. Section EC.2). Although this result is known in folklore, we could not find a

specific theorem to refer to, thus we provide one.

Let T1

2R+

be an arbitrary constant. We consider D[0, T1

] endowed with the usual Skorokhod

J1

topology (cf. Chapter 3 of Billingsley (1999)). For some x, y 2 D[0, T1

], let d(x, y) denote the

Skorokhod J1

distance between these processes (cf. (12.13) in Billingsley (1999)).

Lemma EC.14. Let Y n, n 2 N+

be a relatively compact sequence in D[0, T1

] endowed with the

u.o.c. topology such that all of its subsequential limits are uniformly continuous. Let Xn, n2N+

be a sequence in D[0, T1

] such that

supn2N+

|Xn(0)|<1, (EC.95a)

|Xn(t2

)Xn(t1

)|K |Y n(t2

)Y n(t1

)| , for all t1

, t2

2 [0, T1

] and n2N+

, (EC.95b)

where K 2R+

is a constant. Then, Xn, n2N+

is relatively compact in D[0, T1

] endowed with the

u.o.c. topology and all of its subsequential limits are uniformly continuous.

Moreover, if all of the subsequential limits of Y n, n2N+

are absolutely (Lipschitz) continuous,

then all of the subsequential limits of Xn, n2N+

are also absolutely (Lipschitz) continuous.

Proof: Since Y n, n 2 N+

is relatively compact with respect to the u.o.c. topology, it is also

relatively compact with respect to the Skorokhod J1

topology. Then, by Theorem 12.3 of Billingsley

(1999),

supn2N+

kY nkT1<1, lim

!0

supn2N+

w0(Y n, ) = 0, (EC.96)

where w0 is defined in equation (12.6) of Billingsley (1999). Then,

supn2N+

kXnkT1= sup

n2N+

sup0tT1

|Xn(t)| supn2N+

|Xn(0)|+ supn2N+

sup0tT1

|Xn(t)Xn(0)|

supn2N+

|Xn(0)|+K supn2N+

sup0tT1

|Y n(t)Y n(0)| supn2N+

|Xn(0)|+2K supn2N+

kY nkT1<1, (EC.97)


where the first inequality in (EC.97) is by (EC.95b) and the strict inequality in (EC.97) is by

(EC.95a) and (EC.96). Next,

0 lim!0

supn2N+

w0(Xn, )K lim!0

supn2N+

w0(Y n, ) = 0, (EC.98)

where the second inequality is by the definition of w0 (cf. equation (12.6) of Billingsley (1999))

and (EC.95b) and the last equality is by (EC.96). Therefore, Xn, n2N+

is a relatively compact

sequence in D[0, T1

] endowed with the Skorokhod J1

topology by (EC.97), (EC.98), and Theorem

12.3 of Billingsley (1999).

Let Xnl , l 2N+

be an arbitrary convergent subsequence of Xn, n2N+

such that d(Xnl ,X)!

0 for some X 2 D[0, T1

] as l ! 1. Then, there exists a subsequence of nl, l 2 N+

, denoted by

nk, k 2N+

, such that d(Y nk , Y )! 0 as k!1 where Y 2D[0, T1

] and Y is uniformly continuous.

Let us fix an arbitrary > 0. There exits a := (,K)> 0 such that if |t2

t1

|< , |Y (t2

)Y (t1

)|<

/(6K). Since convergence in Skorokhod J1

metric implies u.o.c. convergence when the limit is

continuous (cf. page 124 in Billingsley (1999)), we also have kY nk Y kT1 ! 0 as k !1. Let

denote the set of continuous, strictly increasing, and bijective mappings from the domain [0, T1

]

onto itself. Then, there exists a k0

2 N+

and a sequence ↵k, k 2 N+

in the set such that if

k k0

,

↵k e

T1_Xnk ↵k X

T1<

8^ , kY nk Y kT1 <

8K, (EC.99)

where the first inequality is by the definition of Skorokhod J1

metric (cf. (12.13) in Billingsley

(1999)) and the fact that d(Xnk ,X)! 0 as k !1. Let us fix an arbitrary k k0

. Then, for all

t1

, t2

2 [0, T1

] such that |t2

t1

| ,

|X(t2

)X(t1

)| X(t2

)Xnk(↵k(t2

))

+

Xnk(↵k(t2

))Xnk(↵k(t1

))

+

Xnk(↵k(t1

))X(t1

)

2

X

l=1

X(tl)Xnk(↵k(tl))

+K

Y nk(↵k(t2

))Y nk(↵k(t1

))

2

X

l=1

X(tl)Xnk(↵k(tl))

+K

Y nk(↵k(tl))Y (↵k(tl))

+K

Y (↵k(tl))Y (tl)

+K |Y (t2

)Y (t1

)|


< 2

Xnk ↵k X

T1+2KkY nk Y kT1 +

3+

6

<

4+

4+

2= ,

where the second inequality is by (EC.95b), the fourth inequality is by the definition of and

(EC.99), the last inequality is by (EC.99). Therefore, X is uniformly continuous, thus Xnl !X

u.o.c. as l!1 (cf. page 124 of Billingsley (1999)). This implies that Xn, n2N+

is also relatively

compact in the u.o.c. topology and all of its subsequential limits are uniformly continuous.

Next, suppose that Y is absolutely continuous. We will prove thatX is also absolutely continuous.

Let us fix an arbitrary 1

> 0. Let (al, bl)ml=1

be an arbitrary finite set of disjoint intervals such

that (al, bl) [0, T1

] for all l 2 1,2, . . . ,m. Since Y is absolutely continuous, for any 1

> 0, there

exists a 1

= 1

(1

)> 0 such that ifPm

l=1

(bl al)< 1

, thenPm

l=1

|Y (bl)Y (al)|< 1

/K. Then

mX

l=1

|X(bl)X(al)|mX

l=1

(|X(bl)Xnk(bl)|+ |Xnk(bl)Xnk(al)|+ |Xnk(al)X(al)|)

mX

l=1

|X(bl)Xnk(bl)|+ |Xnk(al)X(al)|+K |Y nk(bl)Y (bl)|

+K |Y nk(al)Y (al)|+K |Y (bl)Y (al)|

mX

l=1

2kX XnkkT1+2K kY nk Y kT1

+K |Y (bl)Y (al)|

= 2m

kX XnkkT1+K kY nk Y kT1

+KmX

l=1

|Y (bl)Y (al)|, (EC.100)

where the second inequality is by (EC.95b). By letting k!1, the first term in (EC.100) converges

to 0, thus the sum of the terms in (EC.100) becomes less than 1

. Hence,X is absolutely continuous.

Lastly, suppose that Y is Lipschitz continuous with Lipschitz constant 2 R+

. We will prove

that X is also Lipschitz continuous. For all t1

, t2

2R+

,

|X(t2

)X(t1

)| limsupk!1

(|X(t2

)Xnk(t2

)|+ |Xnk(t2

)Xnk(t1

)|+ |Xnk(t1

)X(t1

)|)

limsupk!1

|X(t2

)Xnk(t2

)|+ |Xnk(t1

)X(t1

)|+K |Y nk(t2

)Y (t2

)|

+K |Y nk(t1

)Y (t1

)| +K |Y (t2

)Y (t1

)|


limsupk!1

2kX XnkkT1+2K kY nk Y kT1

+K |Y (t2

)Y (t1

)| (EC.101)

K|t2

t1

|,

where the second inequality is by (EC.95b). Since the first term in (EC.101) converges to 0 and Y

is Lipschitz continuous, we obtain the last inequality, so X is Lipschitz continuous. References

Adan I, Weiss G (2012) Exact FCFS matching rates for two infinite multitype sequences. Operations Research

60:475–489.

Akbarpour M, Li S, Gharan SO (2016) Thickness and information in dynamic matching markets, Working

paper, http://ssrn.com/abstract=2394319.

Aliprantis CD, Border KC (2006) Infinite Dimensional Analysis: A Hitchhiker’s Guide (Springer), third

edition.

Anderson EJ, Nash P, Perold AF (1983) Some properties of a class of continuous linear programs. SIAM

Journal on Control and Optimization 21:758–765.

Arnosti N, Johari R, Kanoria Y (2016) Managing congestion in matching markets, Working paper, http:

//ssrn.com/abstract=2427960.

Ata B, Kumar S (2005) Heavy trac analysis of open processing networks with complete resource pooling:

Asymptotic optimality of discrete review policies. The Annals of Applied Probability 15:331–391.

Azevedo EM, Weyl EG (2016) Matching markets in the digital age. Science 352:1056–1057.

Bertsimas D, Nasrabadi E, Paschalidis IC (2015) Robust fluid processing networks. IEEE Transactions on

Automatic Control 60:715–728.

Billingsley P (1999) Convergence of Probability Measures (Wiley), second edition.

Bimpikis K, Candogan O, Saban D (2017) Spatial pricing in ride-sharing networks, Working paper, https:

//ssrn.com/abstract=2868080.

Braverman A, Dai JG, Liu X, Ying L (2017) Empty-car routing in ridesharing systems, Working paper,

arXiv:1609.07219v2.







Buke B, Chen H (2015) Stabilizing policies for probabilistic matching systems. Queueing Systems 80:35–69.

Busic A, Gupta V, Mairesse J (2013) Stability of the bipartite matching model. Advances in Applied Prob-

ability 45:351–378.

Cachon GP, Daniels KM, Lobel R (2016) The role of surge pricing on a service platform with self-scheduling

capacity, Working paper, http://ssrn.com/abstract=2698192.

Caldentey R, Kaplan EH, Weiss G (2009) FCFS infinite bipartite matching of servers and customers.

Advances in Applied Probability 41:695–730.

Chen H, Yao DD (2001) Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization

(Springer).

Chen K, Sheldon M (2016) Dynamic pricing in a labor market: Surge pricing and flexible work on

the Uber platform, working paper, http://www.anderson.ucla.edu/faculty_pages/keith.chen/

papers.htm.

Chen L, Mislove A, Wilson C (2015) Peeking beneath the hood of Uber. Proceedings of the 2015 ACM

Conference on Internet Measurement Conference, 495–508 (New York).

Dai JG, Tezcan T (2011) State space collapse in many-server di↵usion limits of parallel server systems.

Mathematics of Operations Research 36:271–320.

de Jong RM (1996) A strong law of large numbers for triangular mixingale arrays. Statistics & Probability

Letters 27:1–9.

Durrett R (2010) Probability: Theory and Examples (Cambridge), fourth edition.

Folland GB (1999) Real Analysis Modern Techniques and Their Applications (Wiley), second edition.

Gopalakrishnan R, Mukherjee K, Tulabandhula T (2016) The costs and benefits of ridesharing: Sequential

individual rationality and sequential fairness, Working paper, arXiv:1607.07306v1.

Guda H, Subramanian U (2017) Strategic surge pricing and forecast communication on on-demand service

platforms, Working paper, https://ssrn.com/abstract=2895227.

Gurvich I, Ward A (2014) On the dynamic control of matching queues. Stochastic Systems 4:479–523.






Hall J, Kendrick C, Nosko C (2016) The e↵ects of Uber’s surge pricing: A case study, Working paper,

http://faculty.chicagobooth.edu/chris.nosko/.

Harrison JM (2000) Brownian models of open processing networks: Canonical representation of workload.

The Annals of Applied Probability 10:75–103.

Hu M, Zhou Y (2016) Dynamic type matching, Working paper, http://ssrn.com/abstract=2592622.

Kallenberg O (1997) Foundations of Modern Probability (Springer).

Karatzas I, Shreve SE (1988) Brownian Motion and Stochastic Calculus (Springer-Verlag), first edition.

Leshno JD (2016) Dynamic matching in overloaded waiting lists, Working paper, http://jleshno.weebly.

com/.

Levinson N (1966) A class of continuous linear programming problems. Journal of Mathematical Analysis

and Applications 16:7–83.

Michallon C (2016) ‘Shame on you’: Uber comes under fire as users complain of price

surge in the aftermath of Manhattan explosion - with rides costing up to three times

their usual prices. Dailymail.com http://www.dailymail.co.uk/news/article-3795474/

Uber-comes-fire-users-complain-price-surge-aftermath-Manhattan-explosion.html.

Moyal P, Perry O (2016) On the instability of matching queues, Working paper, arXiv:1511.04282.

Perold AF (1981) Extreme points and basic feasible solutions in continuous time linear programming. SIAM

Journal on Control and Optimization 19:52–63.

Plambeck EL, Ward AR (2006) Optimal control of a high-volume assemble-to-order system. Mathematics of

Operations Research 31:453–477.

Reed JE, Ward AR (2004) A di↵usion approximation for a generalized Jackson network with reneging.

Proceedings of the 42nd Annual Allerton Conference on Communication, Control, and Computing

(Monticello, IL.).

Reed JE, Ward AR (2008) Approximating the GI/GI/1+GI queue with a nonlinear drift di↵usion: Hazard

rate scaling in heavy trac. Mathematics of Operations Research 33:606–644.

Reiman MI, Wang Q (2015) Asymptotically optimal inventory control for assemble-to-order systems with

identical lead times. Operations Research 63:716–732.

http://faculty.chicagobooth.edu/chris.nosko/







Riquelme C, Banerjee S, Johari R (2016) Pricing in ride-share platforms: A queueing-theoretic approach,

Working paper, http://ssrn.com/abstract=2568258.

Rochet JC, Tirole J (2006) Two-sided markets: a progress report. RAND Journal of Economics 37:645–667.

Ross SM (1996) Stochastic Processes (Wiley), second edition.

Royden HL, Fitzpatrick PM (2010) Real Analysis (Pearson), fourth edition.

Shiryaev AN (2008) Optimal Stopping Rules (Springer), reprint of the 1978 edition.

Stroock DW, Varadhan SRS (2006) Multidimensional Di↵usion Processes (Springer), reprint of the 1997

edition.

Taylor TA (2017) On-demand service platforms, Working paper, https://ssrn.com/abstract=2722308.

The Economist (2016) A fare shake. http://www.economist.com/news/finance-and-economics/

21698656-jacking-up-prices-may-not-be-only-way-balance-supply-and-demand-taxis.

Unver MU (2010) Dynamic kidney exchange. Review of Economic Studies 77:372–414.

Ward AR, Kumar S (2008) Asymptotically optimal admission control of a queue with impatient customers.

Mathematics of Operations Research 33:167–202.

Weiss G (2008) A simplex based algorithm to solve separated continuous linear programs. Mathematical

Programming 115:151–198.

White D (2016) Uber users are complaining about pricey New Year’s Eve rides. Time http://time.com/

4165410/uber-new-years-eve-price-surge-rides/.

Whitt W (2002) Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Appli-

cation to Queues (Springer).







Documents

Dynamic Matching for Real-Time Ridesharingpublic.kenan-flagler.unc.edu/2017msom/SIGs/Service...ket asymptotic regime. The relevant objective is to maximize the number of customers