Courier Dispatch in On-Demand Delivery

Courier Dispatch in On-Demand Delivery

Mingliu ChenThe Department of Industrial Engineering and Operations Research, Columbia University, [email protected]

Ming HuRotman School of Management, University of Toronto, [email protected]

We study a courier dispatching problem in an on-demand delivery system where customers are sensitive

to delay. Specifically, we evaluate the effect of temporal pooling by comparing systems using the dedicated

strategy, where only one order is delivered per trip, vs. the pooling strategy, where a batch of consecutive

orders is delivered on each trip. We capture the courier delivery system’s spatial dimension by assuming

that following a Poisson process, demand arises at a uniformly generated point within a service region, as

a generalization of the circular city model. With the same objective of revenue maximization, we find that

the dispatching strategy depends critically on customers’ patience level, the size of the service region, and

whether the firm can endogenize the demand. We obtain concise but informative results when there is a

single courier and customers’ underlying arrival rate is large enough, meaning a crowded market such as rush

hour delivery. In particular, when the firm has a growth target and needs to achieve an exogenously given

demand rate, using the pooling strategy is optimal if the service area is large enough to fully exploit the

pooling efficiency. Otherwise, using the dedicated strategy is optimal. In contrast, if the firm can endogenize

the demand rate by varying the delivery price, using the dedicated strategy is optimal for a large service

area, and vice versa. The reason is that it is optimal for the firm to sustain a relatively low demand rate by

charging a high price for a large service radius: within this large area, the pooling strategy would lead to a

long wait time because the multiple orders required for pooling accumulate slowly. Moreover, under market

penetration with exogenous demand, customers’ patience level has no impact on the dispatch strategy, but

when the demand rate can be endogenized, the dedicated strategy is preferable if customers are impatient,

and vice versa. Furthermore, we extend our model to account for social welfare maximization, a hybrid

delivery policy, a general arrival rate that does not have to be large, a non-uniform distribution of orders in the

service region, and multiple couriers. We also conduct numerical analysis and simulations to complement our

main results and find that most observations in our base model still hold in these extensions and numerical

studies.

1. Introduction

On-demand delivery of food and groceries has gained traction nowadays. Given the prevalence of

smart devices and the existence of a flexible labor force of independent contractors, many food and

grocery stores have started on-demand delivery for relatively small orders. For example, Starbucks

plans to expand their coffee delivery services across the United States and has already established

delivery services in China in 30 cities and more than 2000 stores (Jargon 2018). Unlike traditional

package delivery services, coffee delivery involves spontaneous orders for small quantities. Typically,

customers who order consumables like coffee do not order in advance and expect the coffee to still

be hot on arrival. A customer may choose not to order if the expected delivery time is too long.

1

2 Chen, Hu: Courier Dispatch in On-Demand Delivery

In the domain of hyper-fast (or so-called instant) delivery, delivery companies offer a wait time

expectation coupled with a price tag, e.g., 10-minute grocery delivery for $2 by Gorillas and 30-

minute grocery and food delivery for $1.95 by Gopuff with additional markups on product prices.

To meet such a promise of rapid delivery, companies such as Gorillas and Gopuff employ and

staff couriers who are dedicated to multi-hourly shifts, fulfilling orders from “dark” warehouses or

micro-fulfillment centers. The Covid-19 pandemic has solidified this trend. Many more vendors are

hiring dedicated couriers to deliver their goods. According to Rana and Haddon (2021b), about a

half of 150 registered restaurant on Spread, a start-up delivery platform hires dedicated drivers to

their own deliveries. Furthermore, in order to compete with large platforms, they set much lower

delivery prices to carve out a market share. This trend is particularly significant in pizza delivery.

Melton (2021) reports that Domino’s has established market penetration by using dedicated drivers

and offering cheaper than market price pies. During the pandemic, Domino’s market share has

increased by 31%.

Since on-demand deliveries are by nature sensitive to delay, many delivery systems dispatch a

courier whenever an order arrives. Thus, the couriers can serve only one order per trip, in the hope

of reducing delivery time for each customer. The empirical analysis of Mao et al. (2019) shows

that delivery delay significantly reduces future orders. However, there are still many occasions

on which a firm can utilize batch delivery if multiple orders are placed around the same time in

the same area. That is, a courier may deliver multiple orders per trip; we refer to this as the

temporal pooling strategy. In this strategy, a courier is not necessarily dispatched as soon as an

order arrives; orders are allowed to accumulate over time and then a batch of sequential orders

is delivered in one trip. We show that this strategy achieves delivery efficiency in the form of

a shorter expected travel distance per order and lower variability in traveling distance per trip.

However, while this pooling strategy benefits the supply side, it undoubtedly affects customers’

service experiences on the demand side, which may deter customers from using the service or may

require the firm to compensate customers monetarily for the longer wait and hence reduce the

strategy’s attractiveness. Therefore, each delivery strategy has its own advantages: the dedicated

delivery system may mean a shorter wait for each customer, while batch delivery appears more

efficient from the firm’s perspective.

The on-demand courier dispatch problem differs from traditional delivery problems (such as the

celebrated Traveling Salesman Problem) where there are many stops per trip. Orders containing

on-demand supplies (such as coffee, food, and medicine) typically have short delivery windows.

According to Rana and Kang (2021), food delivery platforms such as DoorDash and Uber are

researching on bundling orders together but unlike traditional delivery services, they also plan to

deliver all orders in an hour. Thus, on-demand delivery services cannot deliver with large batch

Chen, Hu: Courier Dispatch in On-Demand Delivery 3

sizes consistently. In particular, according to an internal study conducted by one of the largest

delivery platforms in China, for food delivery their couriers carry less than two orders on average

per trip, even during peak lunch and dinner hours (see Figure 1).

(a) (b)

Figure 1 The distribution of orders per courier during peak (a) lunch and (b) dinner hours

Another important factor in the operations of delivery systems is whether the actual demand

is exogenous or can be endogenized through pricing. On one hand, an emerging delivery platform

needs to maintain growth and carve out its market share by sustaining a certain demand rate,

also known as market penetration (Rana and Haddon 2021a).1 Studies on market penetration can

be traced back to Buzzell et al. (1975), followed by empirical evidence (see, e.g., Szymanski et al.

1993), stating there is a positive correlation between the market share and (long-term) profitability.

Thus, for a vendor in its early stage of operations under market penetration, the demand can be

exogenously determined to achieve certain market share. On the other hand, a delivery platform,

who has already established a stable market base, can endogenize the demand by varying prices or

fees to further optimize its revenue.

In this paper, we take the perspective of a vendor providing delivery service and attempt to

address the following research questions: when is temporal pooling beneficial and when should

a courier be dedicated to one order per trip? More specifically, we consider scenarios where the

delivery system with dedicated couriers has exogenous and endogenous demand, respectively, and

identify the key factors affecting its operating strategy. In either scenario, we use the vendor’s

revenue as the performance measure. For simplicity, we refer to the delivery strategy with temporal

pooling as the batch or pooling strategy and the one serving a single order per trip as the dedicated

strategy. In the exogenous demand case, depending on the expected wait time associated with each

strategy, the vendor sets the price to achieve the targeted demand rate. In the endogenous demand

case, the vendor has the full freedom in varying the price to moderate the demand rate.

We build a stylized model capturing the spatial aspect of delivery systems under different dis-

patch strategies. Following a Poisson process, demand arises at a uniformly distributed point in a

1 See also https://gadallon.substack.com/p/premature-scaling-will-gorillas-go.

https://gadallon.substack.com/p/premature-scaling-will-gorillas-go


Exogenousdemand

Endogenousdemand

Small area Dedicated BatchLarge area Batch Dedicated

Patient customers — BatchImpatient customers — Dedicated

Table 1 Optimal delivery strategy according to nature of demand

service region. By using a disk-shaped service area and recognizing the similarities between delivery

and queueing systems, we obtain concise but informative analytical results. We find that whether

the demand is endogenized critically affects the vendor’s optimal dispatch strategy. In our base

model, we assume there is a single courier for dispatch (which is relaxed in an extension). We first

analyze a large market where customers’ potential arrival rate is large (which is relaxed in another

extension). We show that in such a crowded market, if the demand rate is exogenously given as

under market penetration, there is a threshold size for the service area below which it is optimal to

use the dedicated strategy and above which it is optimal to use the pooling strategy. We find that

whichever strategy that produces a shorter expected wait time for the exogenously given fraction

of customers is optimal for the vendor. Thus, customers’ patience level does not directly impact

the decision on the delivery strategy because it does not affect the length of wait time itself.

Results are very different if the firm can endogenize the demand rate as under revenue max-

imization. With endogenized demand, there is a threshold size for the service area below which

it is optimal for the firm to deliver in batches and above which it is optimal to adopt dedicated

delivery. This result is in stark contrast to the one for exogenous demand, and runs counter to

popular belief that serving in batches leads to higher delivery efficiency in a large service area than

dedicated delivery (which is likely gained under the assumption that the demand rate is exoge-

nously given). The intuition of our finding is that, in a relatively large service area, both strategies

involve substantial travel distances, leading to long wait times. By maintaining a high demand

rate, the firm needs to sacrifice a lot of profit margin to ensure customers joining the service. As a

result, the firm favors a relatively small endogenized demand rate for both strategies. The pooling

strategy loses its efficiency edge in this case since it takes a long time to accumulate multiple orders

with a low demand rate. The dedicated strategy is more efficient since its optimal demand rate is

lower than the one under the pooling strategy. Furthermore, we also find that there is a threshold

on customers’ patience level below which the pooling strategy is optimal and above which the

dedicated strategy is optimal. We summarize these results in Table 1.

We then examine a variety of extensions of the base model. First, we consider social welfare

maximization as the vendor’s objective. We find that all the insights in our base model carry over.


Second, based on the dedicated and batch strategies in the base model, we consider a contingent

hybrid policy. To be more specific, the courier uses batch strategy as long as there are more than

one outstanding orders and use dedicated strategy otherwise. We show that this hybrid policy

dominates the dedicated strategy. However, the trade-off between the dedicated and batch strategies

still persists between the contingent and batch strategies.

Third, we relax the large-market assumption by studying general customer arrival rates. We find

that all results for an exogenous demand rate still hold. When the firm can endogenize the demand,

there are still thresholds on the size of the service area and customers’ patience level above which

it is optimal to use the dedicated strategy.

Fourth, we relax the batch-of-two assumption in our base model. In our numerical calculations,

we find that our main insights in the base model still hold. However, with a larger batch size, there

are other issues that need to be addressed, which we leave for future research.

Fifth, we consider an extension in which the demand is not uniformly dispersed inside the service

area. Specifically, we consider a circular city structure in which all demands are distributed only on

the edge of the disk. Not only do we confirm that all results in our base model still hold, but also we

compare the thresholds to those in the base model. We find that the thresholds for both the service

radius and customers’ patience level, above which it is optimal to use the dedicated strategy and

vice versa, are lower in this setting, compared with those in the base model. Under this setting,

the pooling strategy is at a disadvantage since the courier has to travel a longer distance to the

edge of the service region before exploiting the efficiency created by pooling.

Finally, we generalize the base model to allow for multiple couriers. Again, we obtain analytical

results and numerical observations consistent with the base model of a single courier.

2. Literature Review

Two of the papers most closely related to ours are Cao et al. (2020a) and Yildiz and Savelsbergh

(2019). Cao et al. (2020a) study the optimal deployment strategy for vendors with high mobility,

often referred to as the stall economy. Although their main focus is on using the combination

of the analytical model and machine learning algorithms to explain the scalability of the stall

economy, the authors also empirically evaluate the benefit of demand pooling. They divide the

service area into several subregions and consider demand pooling that serves orders arriving within

the same time window in the same subregion together, before moving to the next subregion. In

their empirical study, they find that such demand pooling is more beneficial when customers are

patient, which is consistent with our analytical results under the endogenous demand rate. Yildiz

and Savelsbergh (2019) also consider a circular delivery area similar to that in our model, where

a single restaurant located at the center of the disk serves the entire disk-shaped area. They only


consider the dedicated strategy. Their focus is on the optimal service radius and compensation for

crowdsourced couriers, whereas ours is on evaluating the benefit of temporal demand pooling.

Our paper belongs to the stream of research on spatial queueing models. This literature typically

considers a logistical setting in which vehicles are modeled as servers and their traveling time to

serve customers equals the service time. Berman et al. (1985) and Berman et al. (1987) focus on

finding one or multiple service hubs in a network to minimize the expected response time to random

demand. They model the service system using queueing models incorporating the spatial features

of the network. Bertsimas and Van Ryzin (1990, 1992) consider stochastic and dynamic routing of

vehicles to serve service requests that are randomly generated over a service region. The authors

evaluate the performances of various policies and identify optimal and near-optimal policies under

light and heavy traffic. Recently, spatial queueing models are also utilized in smart city design

(see, e.g., He et al. 2017, Mak 2020). In contrast to these papers, we focus on the comparison of

the dedicated and the pooling strategy and also incorporate the pricing decision to examine the

interactions between the demand side’s pricing decision and the supply side’s dispatch decision.

Our paper is also related to the recent research that uses queueing models to study the on-demand

economy. Taylor (2018) and Bai et al. (2019) treat freelancers in the on-demand economy as servers

in queueing models. They approximate the customers’ wait time with the help of M/M/k queues.

Frazelle et al. (2020) examine different contracts between a delivery platform and a single restaurant

and compare their performance to that in a centralized setting where the restaurant controls prices.

Chen et al. (2020a) study a similar research question by examining a setting with two streams of

customers, tech-savvy and traditional. Both papers model the food-serving restaurant as a stylized

M/M/1 queue. Cui et al. (2019, 2020) model line-sitting and queue-scalping, respectively, based on

M/M/1 queues. They treat line-sitting and queue-scalping as innovative service models, as opposed

to traditional First-Come-First-Serve, and compare their performances in equilibria.

Similar to our paper, a stream of literature in operations management also uses couriers’ travel

distances to quantify the cost of delivery. These papers typically deal with a large number of orders

per delivery trip and resort to asymptotic analysis of variants of the Traveling Salesman Problem

(TSP) to quantify the expected travel distance (see, e.g., Cachon 2014, Carlsson and Song 2017,

Qi et al. 2018, Cao et al. 2020b). In contrast, we assume that a courier delivers no more than

two orders per trip, supported by empirical evidence (see Figure 1). Furthermore, with a spatial

queueing formulation, our analysis is anchored by not only the expected travel distance, but also

the variability in traveling during delivery trips. More recently, He et al. (2020) also recognize that

using TSP may not accurately depict the trip length in food delivery as couriers and the platform

may not share the same information. For example, the couriers may have additional information


on the road condition, driving pattern, etc., which are ignored in the TSP formulation. Thus, they

propose prediction models on travel time using machine learning models.

Many papers also discuss the impact of dispatch policies on operational efficiency. Klapp et al.

(2018a,b) consider the dynamic dispatch wave problem. In their setting, dispatch decisions are

made at pre-determined times of a day, and the decision maker decides on which orders to be

delivered in each wave. The major trade-off in whether to deliver an order or not is between

reducing the number of outstanding orders so they can be delivered by the end of the day versus

waiting for nearby orders to show up so the delivery efficiency can be improved. Voccia et al. (2019)

also consider a multi-vehicle dynamic pickup and delivery problem with same-day delivery as the

time constraint. Other papers such as Azi et al. (2012) and Ulmer et al. (2018) also study the

optimal order assignment along with the optimal timing for vehicle departure in a single-depot

setup. Unlike these papers, we consider the pricing decision besides the dispatching policies.

Finally, our spatial modeling approach relates to Hotelling’s circular city model in economics.

The original model has suppliers and consumers evenly dispersed on a circle, and consumers have

preferences over suppliers based on their relative locations. We extend the original circular city

model (see, e.g., Salop 1979) to have the supplier sitting at the center of the circle and customers

located inside the circle, forming a disk-shaped service area. In one of our extensions, we also

investigate the extreme case where customers only reside on the edge of the disk. Some recent papers

in operations management also use spatial models based on a circular city. Chen et al. (2020b)

consider a matching problem in ride-sharing where drivers and riders depart from the center of a

circle going to different locations on the edge of the circle. They use the circular angle induced by

the circular city to characterize the mismatch between drivers and riders. Feng et al. (2020) also

use a circular city to study ride-hailing where drivers travel clockwise or counterclockwise with

a constant speed, picking up riders on the circle. Unlike our spatial model, none of these papers

consider areas inside the circle as part of the service region.

3. Model

Consider a vendor who has a facility located at the center of a disk-shaped region with radius r > 0

and hires a single courier serving customers in the area. We relax the single-courier assumption

and consider multiple couriers in Section 6.6. The structure of our service area is a generalization

of the “Circular City Model” (see, e.g., Salop 1979), in the sense that customers also occupy areas

inside the disk, whereas the original model only considers the edge of the disk. The centrally

located facility can be a store, urban warehouse, restaurant or ghost kitchen. We assume the arrival

process of customers is Poisson with rate Λr2, which scales with the area πr2 of the service region.2

2 We can also assume that the arrival rate scales with the circumference of the circle, which is linear in r. That is,the arrival rate is Λr. Our results still hold.


Upon arrival, each customer’s location is independent and uniformly distributed on the disk. Each

customer is also subjected to a wait cost with rate c per unit of time. Furthermore, we assume that

each customer has a valuation v for the delivery service, which follows a general distribution with

the cumulative density function (c.d.f.) F . Without loss of generality, we normalize the support

of F to [0,1]. The vendor can decide the charge for each delivery service at price p. We assume

that customers are sensitive towards the “virtual” in-line delay (see, e.g., Liu and Huang 2021),

which represents the time between an order is placed and the delivery courier is en-route. That is,

a customer would be satisfied once a courier is on the way to make a delivery of her order. Thus,

given a wait time w (from the order time to the start of making the dedicated delivery, even in

a batch delivery), a customer’s utility from using the delivery service is simply v − p− cw. Note

that the above utility expression assumes that the wait cost is linear in time. This is indeed a

simplification of the reality but also a reasonable one. In practice, most instant delivery services

offer a “soft promise” in wait time (e.g., 10 minutes for Gorillas, 30 minutes for Gopuff, and 1 hour

for Instacart). This implies that the wait cost by customers tends to be an increasing function of

wait time. A linear approximation of this function may be more appropriate than a step function

in which there is no cost if the wait time is below a cutoff and a constant one otherwise. Indeed,

if a delivery service like Instacart can consistently deliver in 30 minutes, the company would alter

its sales pitch by emphasizing 30 minutes as the expected wait time rather than 1-hour delivery.

Before moving on, we want to provide an alternative interpretation of our model, which accounts

for location-based valuation and pricing. We can consider each customer has valuation V = v+ cd

for the delivery service, where v is still the valuation of the service, and d is the expected travel

time required for a courier to reach the focal customer. Thus, our setup can be interpreted as that

besides the base valuation of the service, customers are willing to pay for a higher delivery price if

customers are sensitive to the wait time until they receive the order, and their location is further

away from the expected location of a courier when she embarks on the dedicated delivery trip.

The vendor can decide the charge for each delivery service at price P = p+ cd, so that the price

is also location dependent. Given a wait time w (from the order time to the start of making the

dedicated delivery), a customer’s utility from using the delivery service is V −P − cw= v−p− cw.

Therefore, the effects of the location-dependent valuation and price cancel out in our model, which

means that customers’ ordering decision is independent of their locations on the disk. Readers can

make either interpretation of the model based on the applications.

Obviously, only customers with nonnegative utilities will use the delivery service from the vendor.

Customers with negative utilities may choose to pick up the orders by themselves, or not order at

all. Denote by λ ∈ [0,Λr2] the effective demand rate of the delivery service. Since each customer’s


valuation v follows a distribution with the c.d.f. F , the demand rate λ satisfies λ/Λr2 = 1−F (p+

cw), which implies that for all λ∈ (0,Λr2], we have

p= F−1

(1− λ

Λr2

)− cw, (1)

where function F−1 is the inverse function of the c.d.f. F . Thus, there is a one-to-one mapping

between price p and positive demand rate λ. Note that if the demand rate λ is exogenous, then

it is possible to have p < 0, as the vendor needs to subsidize customers for the service, which may

happen when the vendor wants to grow a market. This would not happen when the demand rate

is endogenized. Using the expression in (1) for positive demand rate, the vendor’s revenue function

can be written as

V (λ,w) = λp= λ

(F−1

(1− λ

Λr2

)− cw

), λ > 0. (2)

The vendor makes operational decisions based on the revenue it generates according to (2).

We emphasize that wait time for each customer, w, in a steady state also depends on the effective

demand rate λ. In later sections, when comparing the vendor’s revenue functions under different

delivery modes, we replace w by the expected wait time for each customer, which is a function of

the demand rate λ. The underlying assumption is that customers anticipate a wait time and use

it to anticipate and decide on whether to adopt the service. In equilibrium, their anticipated wait

time is consistent with their actual experiences over repeated interactions.

We consider and compare two delivery strategies, the dedicated strategy and the pooling strategy.

On one hand, with the dedicated strategy, the courier serves orders one by one in the First-Come-

First-Serve fashion (referred to as “dedicated delivery”). On the other hand, with the pooling

strategy, the courier is not en-route for delivery until exactly two3 orders are accumulated, which

can be interpreted as serving orders in batches of two (referred to as “batch delivery”). Figure

2 illustrates the differences between the two strategies, using the example of restaurant delivery.

When serving dedicated delivery, a courier leaves the restaurant immediately when an order arrives.

After delivering the food, the courier comes back to the restaurant to pick up or to wait for the

next order. When serving batch, the courier does not leave the restaurant until two orders have

arrived. Then, the courier delivers both orders in a single delivery trip before coming back to the

restaurant for the next batch. We do not specify the fulfillment sequence within a batch, as long as

the resulting order is purely random; for example, the sequence can follow the time order of arrivals

or a spatial order, such as always traveling clockwise. The fulfillment sequence within a batch does

not affect the total travel distance of a courier, but may affect the wait time of a specific order.


Figure 2 Serving dedicated v.s. serving batch

If the resulting fulfillment order is purely random, customers would still have the same expected

wait time over repeated interactions with the system.

We recognize the similarity between our delivery system and a single-server queue where a courier

acts as the server and customers’ orders queue up. Since potential customer arrivals follow a Poisson

process and a fraction of the customers choose the delivery service based on the expected wait,

the arrival process of orders is also Poisson with the rate equal to the effective demand rate λ. As

for the service process, we assume that the courier picks up the delivery goods at the centrally

located facility instantly and spends no time at each customer’s location. Thus, the service time

only consists of the courier’s traveling time between the facility and customers’ location(s). We

define a delivery trip as the process starting when the courier picks up the delivery goods at the

facility and ending when he returns. We utilize the results in the queueing literature to derive the

expected wait time of customers under each delivery strategy in the next two subsections.

3.1. Dedicated Delivery

Suppose the courier uses the dedicated delivery strategy to serve customer orders, i.e., dedicated

delivery. As mentioned, orders arrive following a Poisson process with rate λ in equilibrium. The

service time is the time that the courier spends in delivering each order. When dedicated delivery is

adopted, each delivery trip is simply the round trip between the facility and a random customer’s

location. By assuming a constant travel speed and normalizing it to 1, the service time equals the

travel distance per delivery trip.

Denote by a random variable XD the shortest Euclidean distance of a delivery trip when serving

orders under dedicated delivery. That is, XD is two times the distance between the center of the

3 According to an internal study by one of the largest delivery platforms in China, their couriers carry less than twoorders per trip on average, see Figure 1.


disk with radius r and a uniformly distributed point on the disk. According to the “Disk Point

Picking” literature (see, e.g., Solomon 1978), we have

E[XD] =1

2πr2

∫ r2

0

∫ 2π

0

2√xdθdx=

4

3r, and E[X2

D] =1

2πr2

∫ r2

0

∫ 2π

0

4xdθdx= 2r2. (3)

Note that the first moment of random variable XD represents the expected distance of the delivery

trip, which is also the expected service time under our normalization of the travel speed. Then we

can treat this delivery system as an M/G/1 queue with the service rate and load factor equal to

µD =1

E[XD]=

3

4r, and ρD =

λ

µD=

4

3λr, (4)

respectively.

We define the expected wait time by WD when serving orders under dedicated delivery as a

function of demand rate, service rate, and the coefficient of variation of the arrival and service

processes. That is, we have

WD(λ,µ,C) :=λ

µ(µ−λ)

C

2, ∀λ,C ≥ 0, µ > 0, (5)

where the term represents in-line delay of an M/G/1 queue (see, e.g., Gross et al. 2008, p. 222).

Note that the summation of coefficients of variation of our M/G/1 queue’s arrival and service

processes is

CD = 1 +E[X2

D]− (E[XD])2

(E[XD])2=

9

8. (6)

Thus, according to (5), WD(λ,µD,CD) represents the expected wait time for each customer when

the courier serves orders following dedicated delivery. Therefore, we can rewrite the revenue function

in (2) as

VD(λ,WD(λ,µD,CD)) = λ

[F−1

(1− λ

Λr2

)− cWD(λ,µD,CD)

], (7)

representing the revenue rate of the delivery service when the vendor adopts dedicated delivery.

3.2. Batch or Pooling Strategy

Instead of serving orders with dedicated delivery, the courier can also deliver orders using batch

delivery. In this paper, we assume that each batch consists of two orders and that inside each batch,

orders are delivered following a predetermined rule. That is, the courier does not leave the facility

until two orders have arrived. Thus, when comparing our delivery system to a queueing system, we

consider orders entering the queue in pairs of two. That is, an arriving order does not technically

enter the queue if all outstanding orders in the system are already in pairs of two. Instead, it waits

and enters the queue together with the next order that arrives. Therefore, when the demand rate


is λ, we can effectively treat the inter-arrival time as being Erlang distributed with order 2 and

having a mean of 2/λ (with the arrival rate being λ/2).

Next, we analyze the service process of the delivery system using batch. A delivery trip needs to

include three parts: travel between the facility and the first order’s location, between the first and

second orders’ locations, and finally, back to the facility from the second order’s location. Denote

by random variable XB the shortest distance a courier needs to travel per trip. According to the

“Disk Line Picking” literature (see, e.g., Solomon 1978), we have4

E[XB] =1

πr4

∫ r2

0

∫ r2

0

∫ π

0

(√x+ y− 2

√xy cos(θ) +

√x+√y

)dθdxdy=

(128

45π+

4

3

)r,

E[X2B

]=

1

πr4

∫ r2

0

∫ r2

0

∫ π

0

(√x+ y− 2

√xy cos(θ) +

√x+√y

)2

dθdxdy≈ 5.428r2. (8)

Since the travel speed is normalized to 1, the travel distance in each delivery trip is the service

time for the courier. Using the first moment of XB, we can derive the service rate and load factor

of this service queue as

µB =1

E[XB]=

45π

4r(32 + 15π), and ρB =

λ

2µB=

2λr(32 + 15π)

45π, (9)

respectively. With both arrival and service processes characterized, we recognize that our batch

service can be analyzed through an E2/G/1 queue.

Since the inter-arrival time follows an Erlang-2 distribution, combining the first and second

moments of XB, we have the summation of the coefficients of variation for arrival and service

processes as

CB =1

2+

E[X2B]− (E[XB])2

(E[XB])2≈ 0.583. (10)

Unfortunately, we do not have a closed-form expression for the expected in-line delay of the E2/G/1

queues. Seeking analytical results, we use Kingman’s formula (see, e.g., Gross et al. 2008, p. 344)

to approximate the in-line delay of this E2/G/1 queue as a G/G/1 queue. That is, we have

Wq ≈1

2µB

ρB1− ρB

CB =CB2

λ

µB(2µB −λ), (11)

where CB is defined in (10). The Kingman’s formula we adopt serves as an upper bound (see,

e.g., Kingman 1962) on the in-line delay and is asymptotically exact in the heavy traffic regime.

All of our results in favor of the pooling strategy can be refined to be analytically exact, as we

use the upper bound of the in-line delay under batch delivery in comparison with the dedicated

4 Note that the only approximation in equation (8) is on the coefficient in the second moment, which is computedaccurately using numerical integration.


strategy. All of our results still hold for a numerical verification in which the expected in-line delay

is computed from a simulated system of the E2/G/1 queue. In Online Appendix D, we provide

simulation results on the accuracy of all the approximations in this paper. In summary, all the

closed-form approximations considered in this paper are fairly accurate.

Note that the batch delivery has a shorter in-line delay compared to a hypothetical M/G/1

dedicated delivery system where the arrival rate is λ/2. The reason is that the batch system has a

lower coefficient of variation, i.e., CB ≤CD, which means there is less variability in both the arrival

and service processes of the batch system. More specifically, the variability in the arrival process

is reduced from 1 in the dedicated system to 1/2 in the batch system due to temporal pooling of

orders, and the variability in the service process is reduced from 1/8 in the dedicated system to

about 0.083 in the batch system due to spatial pooling of two delivery trips into one.

Recall that when using an E2/G/1 queue to analyze our batch system, a single order does not

enter the queue until a second order arrives. In other words, the in-line delay does not include the

time to form a batch of two orders, which is on average 1/λ. We assume that the customer does

not know the exact state of the system, as is the case in practice. That is, she has no information

on her position in the queue. Thus, from a customer’s perspective, her expected wait time consists

of three parts: the expected wait time for a second order to arrive if her order does not enter the

queue immediately, the average in-line delay once her batch enters the queue, and if she is the

second in her batch to be served, the time it takes to serve the first. Define the expected wait time

WB as a function of the demand rate, service rate, and the coefficient of variation. That is, we have

WB(λ,µ,C) :=1

2λ+

λ

µ(2µ−λ)

C

2+

1

2

E[XD]

2=

1

2λ+

λ

µ(2µ−λ)

C

2+r

3, ∀λ,C ≥ 0, µ > 0, (12)

where the components correspond to the three parts in the customer’ s expected wait time, respec-

tively. In particular, the last term E[XD]/4, represents the expected extra delay if the courier serves

her order in second. So in a half of the time, she needs to wait the courier delivers the other order

first (taking E[XD]/2 time in expectation) before en-route with her order. Thus, WB(λ,µB,CB) rep-

resents a customer’s expected wait time when the courier is serving batch. Note that WB(λ,µB,CB)

approaches infinity as λ goes to 0. The reason is that the courier never leaves the facility with a

single order, so a customer may need to wait for a long time when a second order takes some time

to arrive. Thus, the revenue function in (2) becomes

VB(λ,WB(λ,µB,CB)) = λ

[F−1

(1− λ

Λr2

)− cWB(λ,µB,CB)

], λ∈ (0,2µB) . (13)

It is worth pointing out that limλ→0

VB(λ,WB(λ,µB,CB)) = − c2< 0 as the expected wait time

WB(λ,µB,CB) approaches infinity when λ approaches 0. Thus, in batch serving, if the vendor needs


to maintain a low demand rate close to 0, the vendor has a negative revenue rate. In other words,

maintaining a low demand rate in batch serving is unprofitable for the vendor, because it requires

a significant subsidy to customers. However, we only use this limit case to provide intuitions on a

disadvantage of batch serving, since to gain profitability, the vendor can simply serve dedicated,

which generates nonnegative revenue when the demand is very low.

4. Exogenous Demand Rate

In this section, we evaluate the performance of adopting the dedicated delivery and pooling (batch)

strategies when the demand is exogenous. Throughout the base model, we use the vendor’s revenue

as the performance measure. That is, although the demand rate is exogenous, the vendor can still

make the decision on which delivery mode to operate, coupled with the corresponding price to

achieve the targeted demand rate, in order to attain a higher revenue. This is the case when the

firm has an exogenously given demand segment to cover, due to the needs of growing or penetrating

a market or other goals that are not directly related to revenue creation from delivery services,

e.g., the need of matching the delivery capacity with the kitchen capacity. We observe immediately

that serving batch can sustain a higher demand rate than serving dedicated delivery, since when

comparing the load factors in (4) and (9), we have ρB < ρD, if λ > 0 is fixed. Furthermore, since

both ρD and ρB are linearly increasing in r, we also observe that serving batch allows the delivery

service to handle a larger service region than serving dedicated delivery.

When comparing the revenue functions in (7) and (13), if the demand rate λ is exogenous,

the delivery strategy that has the shorter expected wait time leads to higher revenue. That is,

the operating strategy with exogenous demand is efficiency driven. Therefore, in the following

two propositions, we compare the revenues generated via the two delivery strategies and their

corresponding expected wait times.

Proposition 1. Suppose the demand rate is exogenously given. There exists a threshold on the

demand rate below which serving dedicated leads to a shorter expected wait time and thus higher

revenue, and above which serving batch leads to a shorter expected wait time and thus higher

revenue.

Proposition 1 states that when the exogenous demand rate is low, operating dedicated delivery

is better than batch. The intuition is that when the demand rate is low, it takes a very long time

to accumulate two orders so that the courier can make a batch delivery trip. Figure 3(a) provides

a visual representation of the wait times. As an extreme case, when the demand rate goes to zero,

the expected wait time for each customer will approach infinity under batch. However, adopting

dedicated delivery leads to a much shorter expected wait time.


As the average time to accumulate two orders drastically decreases when the demand rate

increases, the overall expected wait time under batch decreases as well. When the demand rate

becomes very high, the in-line delay of customers dominates the average wait time for a pair of two

orders to accumulate. Thus, the expected wait time increases with a sufficiently high demand rate.

As mentioned, serving batch can handle a higher demand rate than serving dedicated because the

average travel distance associated with delivering an order is shorter. In Figure 3(a), we observe

that the expected wait time under dedicated delivery approaches infinity faster when λ becomes

sufficiently large than that under batch delivery does.

Not only is there a threshold on the demand rate that changes the vendor’s delivery strategy,

the next proposition states that there is such a threshold on the size of the service region as well.

Proposition 2. Suppose the demand rate is exogenously given. There exists a threshold on

the service radius, below which serving dedicated leads to a shorter expected wait time and thus

higher revenue, and above which serving batch leads to a shorter expected wait time and thus higher

revenue.

Proposition 2 states that operating dedicated delivery is better if the service radius is small and

serving batch is better when the service radius is large. This result appears to be intuitive as one

may think that when the service radius is large, serving batch can reduce the total travel distance

of the courier. However, the first moments of the lengths of delivery trips under both dedicated

delivery and batch scale with r with other parameters fixed in (3) and (8), respectively. Thus, one

can verify that for any service radius, compared with dedicated delivery, serving batch leads to

a longer average total travel distance but a shorter distance per order, i.e., E[XB]/2 ≤ E[XD] ≤

E[XB]. The main reason behind Proposition 2 is that when the service radius is small, the time to

accumulate two orders when serving batch is much longer than the actual travel time. On the other

hand, if the service radius is large, the travel time becomes longer than the time to accumulate

two orders, which is independent of the service radius when the demand rate is exogenous. Thus,

serving batch is more beneficial when the service radius is large. Figure 3(b) provides a visual

illustration of the expected wait time of a customer when the courier serves dedicated delivery and

batch, respectively.

Corollary 1. Suppose the demand rate is exogenously given.

(i) There exist thresholds in demand rate and service radius (which are the same as those in

Propositions 1 and 2, respectively) such that below which, the price is higher when using

dedicated delivery and above which, serving batch leads to a higher price.


(ii) There exist thresholds in demand rate and service radius (which are the same as those in

Propositions 1 and 2, respectively) such that below which, the expected wait time per order

is shorter when using dedicated delivery and above which, serving batch leads to a shorter

expected wait time per order.

Corollary 1 extends the results in Propositions 1 and 2 to price and delivery efficiency. Corollary

1 is straightforward since when the demand rate is exogenous, the price is non-increasing with

respect to the wait time. Furthermore, as we use the expected wait time per order as the measure

of delivery efficiency, serving batch is more efficient when either the demand rate is high enough

or the service radius is large enough. Otherwise, dedicated delivery is more efficient as it bypasses

the order accumulation time.

(a) (b)

Figure 3 Expected wait time when serving dedicated or batch. (a) r= 1, (b) λ= 1

As mentioned, the case with an exogenous demand rate can describe the market penetration

stage experienced by many start-up companies or applications in public or other business settings

with rigid demand requirements. For example, consider a newly formed ghost kitchen in a mega

city, which hires a given number of kitchen staff (so the maximum kitchen throughput is given)

at the operational level, or tries to carve our a targeted market share in the local takeaway food

market at the tactic level. Thus the kitchen needs to maintain a targeted demand rate through

methods such as offering delivery promotions, which greatly limits its pricing decision. If the

service area is fixed, using dedicated delivery outperforms serving batch, if and only if the targeted

demand rate is relatively low. Serving batch is only beneficial if a relatively high demand rate

needs to be maintained, so temporal pooling can add efficiency en route without losing too much

time accumulating orders. Furthermore, using dedicated delivery leads to a shorter expected wait

time for customers and higher revenue if the service area is relatively small. However, with a

predetermined larger service area, it is better to serve batch taking advantage of the efficiency en

route.


We conclude this section by pointing out that if the demand rate is exogenously determined,

only the effective demand rate λ and the service radius r impact the vendor’s delivery decision,

since we only need to compare the expected wait times for customers under the two strategies.

That is, the underlying arrival rate of customers Λ, wait cost parameter c, and the distribution

function F of customer valuations do not affect the delivery strategy once the targeted demand

rate is determined. In the next section, we compare and contrast the results of this section to the

case where the demand rate λ can be optimized.

5. Endogenous Demand Rate

The previous section covers the scenario with an exogenous demand rate that needs to be sustained.

In this section, the vendor aims at maximizing its revenue with an endogenized demand rate. That

is, there is no exogenous constraint on the demand rate and the vendor maximizes its revenue by

designing the optimal demand rate. Therefore, unlike Section 4 where the vendor can only choose

which delivery mode to operate in with a given demand rate, in this section, the vendor also chooses

the optimal demand rate in each mode which can be achieved via the freedom in varying the price.

Seeking for tractable analytical results, we first take advantage of a crowded market setting

where the underlying arrival rate of customers is high enough. Suppose the arrival rate scales with a

density factor n∈N. As n increases, the arrival rate nΛ increases as well, meaning that the market

gets more and more crowded. Thus, with customer valuations drawn from the c.d.f. F (with its

support normalized to [0,1]), the revenue function in (2) can be modified to

Vn(λ,w) = λ

(F−1

(1− λ

nΛr2

)− cw

), λ≥ 0. (14)

As in this section the vendor maximizes the revenue rate by choosing the demand rate, λ= 0 will

not be the optimal choice.

Define function

V∞(λ,w) := limn→∞

Vn(λ,w) = λ (1− cw) , λ≥ 0, (15)

where the equality follows from the fact that the upper bound on customer valuations has been

normalized to 1. The expression in (15) represents the limiting revenue when the density factor

n goes to infinity. According to (15), when the underlying arrival of customers goes to infinity,

the vendor only serves those who have a valuation almost equal to 1, the upper bound. Thus,

at the limit, the vendor’s revenue is independent of customer valuation distribution. In general,

under a given delivery strategy, when the targeted demand rate λ increases, two terms in (14)

change: (i) the base price F−1

(1− λ

nΛr2

)needs to be adjusted downwards to incentivize more

adoption, and (ii) the expected wait time w increases as a result of a higher joining rate and thus


more discount cw needs to be paid to compensate customers for the longer wait. The crowded

market assumption assumes away the first effect which is verified by Lemma 1. We will relax this

assumption in Subsection 6.3.

First, we present a lemma on utilizing the expression in (15), which greatly simplifies our analysis

for a crowded market.

Lemma 1. Consider n∈N and a c.d.f. F such that F−1 is Lipschitz continuous. There is

limn→∞

maxλ∈[0,µD)

Vn(λ,WD(λ,µD,CD)) = maxλ∈[0,µD)

V∞(λ,WD(λ,µD,CD)), (16)

and

limn→∞

maxλ∈[0,2µB)

Vn(λ,WB(λ,µB,CB)) = maxλ∈[0,2µB)

V∞(λ,WB(λ,µB,CB)). (17)

Lemma 1 implies that we can simply optimize the demand rates for serving dedicated and batch

using the limiting revenue function in (15) when n approaches infinity. Therefore, the vendor’s

demand-rate decision is independent of the customer valuation distribution. Since function V∞

has a much more concise expression than the non-limiting revenue function, it is much easier

to be analyzed and used for comparing optimal solutions under different delivery strategies. In

particular, the next two propositions summarize the results for a crowded market when the vendor

can endogenize the demand rate.

Proposition 3. Assume a large market and suppose the demand rate can be endogenized.

(i) There exists a threshold c∞ on customers’ wait cost parameter c, below which serving batch

leads to higher revenue and above which serving dedicated leads to higher revenue.

(ii) As c crosses the threshold c∞ such that the optimal strategy switches from serving batch

to serving dedicated delivery, the optimal demand rate has a discontinuous drop, i.e.,

limc→c∞− λ∗(c)> limc→c∞+ λ∗(c), where λ∗(c) is the optimal demand rate as a function of the

wait cost coefficient c, and the corresponding optimal price has a discontinuous surge.

Proposition 3(i) states that if the vendor can optimize the revenue rate by endogenizing the

demand rate, serving dedicated is better if customers are impatient (i.e., c is sufficiently high).

With patient customers, it is optimal to serve batch (i.e., c is sufficiently low). This is in contrast

to the result in Section 4: when the demand rate is fixed, the wait cost parameter c has no impact

on the vendor’s delivery decision, since it does not affect the expected wait time. Proposition 3(ii)

says that there is a sudden drop in the optimal demand rate and a surge in the optimal price

when the cost of waiting crosses the threshold such that the optimal delivery strategy changes

from batch to dedicated. When customers are impatient, it is better for the vendor to have a less


crowded system with a relatively low demand rate, which gives an edge to dedicated fulfillment.

If customers are patient, it is better to sustain a higher demand rate while implementing batch

strategy, which shortens the time needed to accumulate two orders and hence the overall expected

wait time. This intuition is consistent with Proposition 1 that a low (resp., high) demand rate

favors dedicated (resp., batch).

Proposition 4. Assume a large market and suppose the demand rate can be endogenized.

(i) There exists a threshold r∞ on the service radius r, below which serving batch leads to higher

revenue and above which serving dedicated leads to higher revenue.

(ii) As r crosses the threshold r∞ such that the optimal strategy switches from serving batch

to dedicated, the optimal demand rate has a discontinuous drop, i.e., limr→r∞− λ(r) >

limr→r∞+ λ(r), where λ(r) is the optimal demand rate as a function of the service radius r,

and the corresponding optimal price has a discontinuous surge.

Proposition 4 states that when the market is crowded, the vendor should serve dedicated if the

service radius r is large enough. Instead, serving batch is optimal if the service radius is sufficiently

small. This result contrasts with Proposition 2, in which the demand rate is exogenous. With a

large service radius, the courier’s travel time is long under either dedicated or batch, which leads

to a relatively long expected wait time for customers. Thus, it is better for the vendor to sustain

a relatively small demand rate, otherwise the compensation for the long wait would be significant.

Again, there is a sudden drop in the optimal demand rate and a surge in the optimal price when

the service radius crosses the threshold such that the optimal delivery strategy changes from batch

to dedicated. Recall that serving batch is less profitable than serving dedicated when the demand

rate is low, since serving batch has a much longer expected wait time. That is, an order may have

to wait for a long time for another order to arrive and form a batch before it is en route for delivery.

When the service radius is small, it is beneficial to operate under a relatively high demand rate, as

the average travel distance is shorter under either delivery strategy than it is with a large service

radius. As mentioned, serving batch is more profitable for a relatively high demand rate.

Next, we discuss the practical implications of our results by discussing a few examples. During

rush hour for a delivery system, the vendor may have far more potential customers than it has the

capacity to serve. Customers ordering a cup of coffee may be impatient because hot coffee will be

cold if not delivered in time, whereas a grocery vendor or restaurant that only serves cold dishes

like sushi may have more patient customers. Thus, as implied by Proposition 3, even though the

two businesses have the same service area, the coffee shop may prefer the dedicated strategy, and

the grocery vendor or sushi restaurant, the pooling strategy. As an implication of Proposition 4,

even if their customers have the same patience level, a restaurant serving only a 10-block radius in


Midtown Manhattan may prefer the batch strategy but a restaurant with similar characteristics

delivering throughout Midtown Manhattan may want to use the dedicated strategy, since the

latter has a much bigger service area. This implication may seem counterintuitive at first glance,

as a larger service area may require more emphasis on delivery efficiency that may be achieved

by the pooling strategy (as conveyed in Proposition 2). The key to understanding this seemingly

counterintuitive insight is that for a large service area, the dedicated strategy is coupled with a

high delivery price, while the pooling strategy needs to keep the delivery price relatively low to

compensate customers for the wait. With the profit margin being taken into account as the demand

rate is endogenized, the dedicated strategy becomes optimal for a large service area.

We conclude this section by summarizing the results and contrasting them with those when the

demand rate is exogenously given. First, we observe that with an endogenous demand rate, it is

optimal to serve dedicated if the service area is large. This result directly contrasts with the one

for an exogenous demand rate, where it is optimal to serve batch for a large service area. Second,

customers’ patience level, which has no impact if the demand rate is exogenous, greatly affects the

vendor’s delivery strategy for the endogenized demand rate. With the demand rate endogenously

determined, if customers are patient, the vendor should serve batch. However, if customers are

impatient, serving dedicated generates higher revenue. Finally, for a crowded market, we are able

to identify the optimal delivery strategy analytically for the entire spectrum of customers’ patience

level and the service area’s size, respectively.

6. Extensions

In this section, we consider three extensions of our base model. We investigate each one and examine

the robustness of our results and intuitions obtained from Sections 4 and 5.

6.1. Social Welfare

Another objective of interest is the social welfare generated by the delivery system. We define the

social welfare generated per order as the summation of the vendor’s revenue and the customer’s

profit, i.e., v − cw, where w is the expected wait time, in view of that the price is an internal

transfer between the vendor and a customer. Thus, the social welfare generated per order is

SW (λ,w) = Λr2P(v≥ p+ cw)E[v− cw |v≥ p+ cw] = Λr2

∫ 1

F−1(1− λΛr2

)(v− cw)dF (v). (18)

The next proposition characterizes the impacts on the social welfare when the vendor focuses on

market penetration or maximizing revenue, respectively.

Proposition 5. (i) Suppose the demand rate is exogenous, there exist thresholds on the

demand rate and service radius, below which serving dedicated leads to higher social welfare

and above which serving batch leads to higher social welfare.


(ii) Suppose the demand rate is endogenous and the market is crowded. There exist thresholds

on the service radius and customers’ patience level, below which serving batch leads to higher

social welfare and above which serving dedicated leads to higher social welfare.

Essentially, we recover the results in Sections 4 and 5 in Proposition 5. Thus, even when the

performance measure changes from the vendor’s revenue to social welfare, our major insights in

the previous sections still hold. When the demand rate is exogenous, the key factor in operations is

the delivery efficiency. On the other hand, when the demand can be endogenized, the vendor needs

to consider the optimal demand rate to sustain, which has a tremendous impact on the system

efficiency.

6.2. Contingent Policy

Another natural extension to our base model is to consider a contingent policy alternating between

serving dedicated and batch depends on the size of the queue. Suppose the courier serves the orders

in batch if and only if there are more than one outstanding orders in the queue and serves dedicated

otherwise (i.e., when there is a single unfilled order). At the first glance, it seems this contingent

policy takes advantage of both delivery methods considered in this paper. In the next proposition,

we show its relationship with dedicated and batch delivery.

Proposition 6. For any demand rate λ > 0, the contingent policy leads to a shorter expected

wait time for customers, compared to that of dedicated delivery. But there still exists a trade-off

between this contingent policy and batch delivery.

Proposition 6 states that in terms of the expected wait time, the contingent policy always

dominates dedicated delivery. Thus, we can conclude that the contingent policy indeed outperforms

dedicated delivery. However, the major trade-off between dedicated and batch delivery still persists

between this contingent policy and batch delivery. As batch serving always waits to accumulate

two orders before dispatch, it can take advantage of a large demand rate setting where the expected

wait time to accumulate another order is shorter than a delivery trip with a single order. On the

other hand, the contingent policy is better suited when the demand rate is relatively low, providing

the flexibility to avoid long wait time for order accumulation.

To better analyze the performance of the contingent policy considered here or any other state-

dependent delivery policy, we believe a dynamic program model is needed and this is beyond the

scope of this paper. We hope our discussion on the contingent policy can stimulate future research

in this direction.


6.3. General Arrival Rate

First, we relax the large-market assumption in Section 5. In this subsection, we consider the general

arrival rate of customers. We investigate whether observations such as Propositions 3 and 4 still

hold without the arrival rate being at the limit. To keep our results concise and informative, we also

assume that customer valuations are uniformly distributed on [0,1]. That is, F (v) = v for v ∈ [0,1]

and F (v) = 0 otherwise. Note that our result does not anchor on the uniform distribution assump-

tion. Statements in this subsection can be generalized to more general valuation distributions as

well. We leave the detailed discussion to Online Appendix C.2.

When the courier serves dedicated, the revenue maximization problem for the vendor is

maxλ∈[0,µD)

VD(λ,WD(λ,µD,CD)), (19)

where the constraint on the demand rate λ reflects the load factor ρD < 1 so that the system is

stable. Similarly, when the courier serves batch, the maximization problem is

maxλ∈[0,2µB)

VB(λ,WB(λ,µB,CB)), (20)

where functions VD and VB are defined in (7) and (13), respectively; the constraint on λ reflects

ρB < 1. Note that we do not include constraint λ≤Λr2 in either (19) or (20). The reason is that for

any demand rate that is greater than Λr2 (which is still mathematically possible), the corresponding

revenue function has a negative value, so it cannot be optimal. The next two propositions summarize

the results when the vendor optimizes its revenue according to (19) and (20).

Proposition 7. Fix r,Λ> 0. Consider F (v) = v for v ∈ [0,1] and F (v) = 0 otherwise. With the

demand rate endogenized, there exists a threshold cen on the customers’ wait cost parameter c, such

that for all c≥ cen, it is optimal to serve dedicated.

Proposition 7 complements the results in Proposition 3 while assuming that each customer’s

valuation follows an independent standard uniform distribution. Even with general arrival rates

of customers, it is still optimal to serve dedicated when customers are impatient (i.e., c is large

enough). Unfortunately, it is challenging to demonstrate analytically that it is optimal with general

arrival rates to serve batch when customers are very patient, unlike the case in the limiting regime.

With general arrival rates, both the distribution of customers’ valuations and the expected wait

time affect the overall revenue as mentioned in Section 5. The distribution of valuations determines

the optimal base price, which, unlike the crowded market, is not independent of the demand

anymore. Furthermore, finite arrival rates may prevent the delivery system from achieving the

optimal demand rate when customers are patient. This hurts serving batch specifically since the

pooling strategy shines under a high demand rate and its efficiency may not be fully exploited


in this case. Moreover, the price compensation would have to be significant in order to sustain a

large demand rate with finite arrivals. However, we can still numerically verify that there exists a

threshold on wait cost parameter c below which it is optimal to serve batch. Figure 4(a) provides

such a visual illustration: the optimal revenue functions of serving dedicated and batch only cross

once.

Proposition 8. Fix c,Λ> 0 and constant L such thatΛ

c3>L (with the exact expression of con-

stant L provided in the online appendix). Consider F (v) = v for v ∈ [0,1] and F (v) = 0 otherwise.

With the demand rate endogenized, there exists a threshold ren on the service radius r, such that

for all r≥ ren, it is optimal to serve dedicated.

Proposition 8 extends the result in Proposition 4 when each customer’s valuation follows an

independent standard uniform distribution. We show that with general customer arrival rates, it

is still optimal to serve dedicated when the service radius is large enough. We only require an

extra minor condition that either the arrival rate of customers is high enough or their wait cost

parameter is low enough. Similar to Proposition 7, it is very difficult to establish optimal conditions

for serving batch. In fact, in our numerical experiments, we find counterexamples where it may

not be optimal to serve batch when the radius is small. Instead, as in the counterexample shown

in Figure 4(b), it is only optimal to serve batch when the service radius is medium. For sufficiently

small or large service radii, it is always better to serve dedicated. As mentioned, serving batch

has the edge over dedicated when the demand rate is relatively high. When the service radius

is sufficiently small, it is beneficial to sustain a high demand rate for both dedicated and batch.

However, due to the finite arrival rate of customers, the demand rate cannot reach the magnitude

at which serving batch outperforms serving dedicated; otherwise, the price discount to sustain a

high demand rate for batch delivery would be too great. This also explains why we only observe a

single threshold on the service radius in Proposition 4 in the large market limiting regime.

(a) (b)

Figure 4 Optimal revenue under dedicated and batch delivery. (a) r= 1, Λ = 25, (b) c= 0.03, Λ = 25


6.4. Batch Size Greater Than Two

In our base model, we consider batches with size of two, in view of applications in food delivery,

to better illustrate the main trade-offs in our delivery policies. Here we extend the model to a

batch size greater than two and conduct numerical studies. Namely, we consider each batch has

size of three or more. In Figure 5, we provide the empirical cumulative distribution functions on

the courier’s travel time (service time) per order when using batch with different sizes. As we can

see, increasing the batch size can reduce the chance of inducing long service times. However, this

margin of improvement gets smaller as the batch size increases. In addition, smaller batch sizes

have higher chances to induce very short service time. In addition to these observations, we choose

not to consider batch sizes greater than three for the following two reasons. First, for any batch

greater than two, we need to consider proper routing policies in delivery, which is not the focus of

our paper. When the batch size equals to three, in the following, we consider the courier delivers

orders with a purely random ordering. But it will be observed that the gap between a random

fulfillment policy versus delivery policy based on the shortest path gets larger when the batch

size increases. Second, as the batch size increases, it may be in the best interest of the vendor to

consider contingent policies as in Section 6.2, which we leave as a future research direction as they

should be analyzed using non-stationary models.

Figure 5 Empirical cumulative distribution functions with various batch sizes. r= 1

Now, we analyze this batch system using an Erlang-3 arrival process. That is, an arriving order

does not enter the dispatch queue until a batch of three is formed. Then, batches have the arrival

rate of λ/3, with the inter-batch time following the Erlang-3 distribution. Following similar deriva-

tion to (8), we have

E[X3B] =

∫x,y,z∈[0,r2], θ,φ∈[0,π]

(√x+ y− 2

√xy cos(θ) +

√y+ z− 2

√yz cos(φ)

+√x+√z

)dUx dUy dUz dUθ dUφ,


E[X2

3B

]=

∫x,y,z∈[0,r2], θ,φ∈[0,π]

(√x+ y− 2

√xy cos(θ) +

√y+ z− 2

√yz cos(φ)

+√x+√z

)2

dUx dUy dUz dUθ dUφ, (21)

where Ux, Uy, Uz are independent uniform distributions on interval [0, r2], and Uθ, Uφ are indepen-

dent uniform distributions on interval [0, π]. Using the above first and second moments, we can get

the service rate and load factor as

µ3B =1

E[X3B]and ρ3B =

λ

3µ3B

. (22)

Furthermore, the coefficient of variation is

C3B =1

3+

E [X23B]− (E [X3B]

2)

(E [X3B])2. (23)

As a result, the expected wait time can be calculated using

W3B(λ,µ,C) =1

λ+

λ

µ(3µ−λ)

C

2+

1

3

(4

3+

128

45π

)r, (24)

where the first term is the average wait time an order has to wait to form a batch (an order needs

to wait for 0, 1, or 2 more orders with equal probability to form a batch), the second term is the

in-line delay, and the last term is the extra delay if other order(s) in the batch needs to be delivered

firstly. Then, W3B(λ,µ3B,C3B) is the expected wait time.

(a) (b)

Figure 6 Revenue functions when serving dedicated versus batch of size 3 under a large market.

(a) c= 0.1, (b) r= 1

In our numerical calculations, we always observe that there are single thresholds in the service

radius r and wait cost c, respectively, such that below which, the vendor should serve with batches,

and above which, dedicated delivery generates more revenue. This is consistent with our findings

in the case with a batch size of two. Thus, our main insights are not limited by the simplification

of considering batches with the size of two.


Consistent with Section 6.3, for general arrival rates, when the radius is large enough, the vendor

should use dedicated delivery. We can again find numerical examples such that there are two

thresholds on the service radius such that only between these thresholds, serving batch outperforms

dedicated delivery in terms of revenue maximization. However, we can also find extreme parameters

such that batch delivery is completely dominated by dedicated delivery for all service radii. We

believe that the possible inferior performance of batch delivery with a size greater than two is

contributed to the random routing policy and lack of contingent policies as mentioned earlier,

which are beyond the scope of this paper and left for future research.

6.5. Circular Service Area

So far we have compared serving dedicated and batch on a disk-shaped service area where customers

are uniformly located inside the disk. In this subsection, we consider a service area that only

constitutes the edge of the disk, i.e., the circumference of the circle. That is, we still have the facility

located at the center of the disk but orders are only coming from locations that are uniformly

distributed on the edge of the disk with radius r. This type of city structure has been examined

by many researchers before, most notably by Salop (1979). The so-called “circular city” model

has a lot of practical implications because many major cities have this kind of circular or ring

structure (e.g., Beijing and Moscow). These cities have massive business areas in the inner rings,

with residential areas surrounding the city center in an outer ring. The circular city model also

captures scenarios where the storage warehouse is in a relatively remote area and couriers have

to travel long distances in each direction to reach the nearest residential area. Furthermore, it

also serves as an extreme case where customers’ locations are not uniformly distributed inside the

service area.

As we do in the previous sections, we propose an appropriate queueing system and analyze the

delivery strategies for it. When serving dedicated, the service time for each order is deterministic

since the time travel from the center to any point on the edge of the circle is fixed. Thus, this

delivery system can be treated as an M/D/1 system under dedicated delivery. We still denote the

effective demand rate for this system by λ, under the arrival rate Λr2. Then this M/D/1 queue has

the service rate and load factor as

µD,C =1

2r, and ρD,C =

λ

µD,C= 2λr, (25)

respectively, where the subscript represents dedicated in a circular service area. The wait time for

a customer in this system is simply characterized by function WD(λ,µD,C ,CD,C) in (5) with

CD,C = 1, (26)


since the arrival process is Poisson and the service process is deterministic. Furthermore, the revenue

function can be written as

VD(λ,WD(λ,µD,C ,CD,C)) = λ

[F−1

(1− λ

Λr2

)− cWD(λ,µD,C ,CD,C)

].

Next, we consider serving batch in a circular service area with radius r. Again, the arrival process

has the inter-arrival time following an Erlang-2 distribution. The service time needs to include

three parts: first, the travel time from the center of the circle to a random point on its edge; second,

the travel time between two uniformly distributed points on the edge of the circle; and finally

the travel time from the edge of the circle back to the center. Denote by random variable Y the

distance a courier needs to travel per trip. Then we have random variable Y following a uniform

distribution on [2r,2r+πr], with

E[Y ] = 2r+πr

2, and σ2

Y =(πr)2

12.

Thus, this queueing system has the service rate and load factor as

µB,C =1

E[Y ]=

1

2r+ πr2

, and ρB,C =λ

2µB,C=λr(4 +π)

4, (27)

respectively, where the subscript represents batch in the circular service area.

Next, again, we use Kingman’s formula to approximate the average wait time for each customer.

The average wait time for each order follows from WB(λ,µB,C ,CB,C), with

CB,C =1

2+

σ2Y

(E[Y ])2=

1

2+

π2

3(4 +π)2. (28)

Furthermore, the revenue function in (13) incorporates the adjusted wait time as

VB(λ,WB(λ,µB,C ,CB,C)) = λ

[F−1

(1− λ

Λr2

)− cWB(λ,µB,C ,CB,C)

], λ∈ (0,2µB,C) .

We find that all the major results in Sections 4 and 5 still hold even if we change the service

area from a disk to a circle. We relegate the formal statements and detailed derivations to Online

Appendix C.3 to avoid repetition.

Due to the change in the city’s geometry, there are no orders coming from areas inside the disk.

Thus, the courier always needs to travel a fair distance before reaching the delivery area (the outer

ring). As a result, the thresholds for switching delivery strategy also change, though the threshold

structure remains. The next proposition provides the relationship between thresholds in a circular

city and those in the base model.

Proposition 9. Assume a crowded market and suppose that the demand rate can be endoge-

nized.


(i) In a circular service area, there exist thresholds c∞ and r∞ such that the vendor should serve

batch if the wait cost and radius parameters c and r fall below the thresholds, respectively.

Otherwise, the vendor should serve dedicated.

(ii) We have

c∞ > c∞, and r∞ > r∞, (29)

where c∞ and r∞ are thresholds in Propositions 3 and 4, respectively, for the counterpart of

the base model serving the entire disk.

Proposition 9 shows that both thresholds on the wait cost coefficient and service radius are

lower if orders only come from the edge of the disk. The reason is that the courier needs to travel

to the edge of the disk before benefiting from the pooling effect of batch delivery. Thus, serving

dedicated has more advantage in this setting. This implies that dedicated delivery will more likely

be beneficial when the orders tend to be distributed on the outskirts of a service region than when

they have a more uniform distribution inside the region.

6.6. Multiple Couriers

So far we have focused on cases with a single courier. Suppose the vendor hires k couriers to serve

the disk-shaped delivering area at the same time. We reassess the performance of dedicated vs.

batch delivery.

First, note that having k couriers does not change the service process for each courier individually.

Thus, when serving dedicated, the arrival process is still Poisson and the service rate remains µD

for each courier. However, the load factor is different since we have k couriers instead of one. That

is, we have

ρD,k =λ

kµD=

4λr

3k,

where the subscript of the load factor represents serving dedicated with k couriers. Thus, this

service system can be analyzed through an M/G/k queue. In order to obtain a tractable expected

wait time, we utilize two approximations together. Recall that the summation of coefficients of

variation of the arrival and service processes is CD = 9/8. We approximate the in-line delay of an

M/G/k queue as

Wq {M/G/k} ≈ CD2Wq {M/M/k} ≈ CD

2

ρ

√2(k+1)

D,k

λ(1− ρD,k), (30)

where we first use an M/M/k queue with the same input to approximate the in-line delay of the

M/G/k counterpart (see, e.g., Gross et al. 2008, p. 345), and then use a well-studied approximation


for the M/M/k queue itself (see, e.g., Sakasegawa 1977). This approximation is consistent with

recent studies on on-demand economy, see, e.g., Bai et al. (2019), Taylor (2018), and Benjaafar

et al. (2020). As the result of such an approximation, the expected wait time for each customer is

WF,k(λ)≈ CD2

ρ

√2(k+1)

D,k

λ(1− ρD,k)=

CD2(kµD−λ)

(λ

kµD

)√2(k+1)−1

. (31)

After these setups, the revenue function is simply V (λ,WF,k(λ)).

Next, we consider serving batch. Similar to serving dedicated, each courier’s service rate µB =45π

4r(32 + 15π)remains the same but the load factor needs to take k couriers into consideration.

That is, we have

ρB,k =λ

2kµB=

2λr(32 + 15π)

45kπ.

Using the same approximation method as in (30), which can be applied to G/G/k systems as well,

the expected wait time for each customer is

WB,k(λ)≈ 1

2λ+

CB2(2kµB −λ)

(λ

2kµB

)√2(k+1)−1

+r

3. (32)

It is worth pointing out that under this approximation scheme, the queueing systems reduce to

those in Section 3 where k= 1, both for dedicated and for batch.

With multiple couriers, we can still show analytically that there is a threshold on the exogenous

demand rate below which serving dedicated generates higher revenue and above which serving

batch is more profitable. This result is consistent with Proposition 1. If the vendor can endogenize

the demand, there is still a threshold on customers’ wait cost parameter above which it is optimal to

serve dedicated, consistent with Proposition 7. We leave the formal statements of these analytical

results in Online Appendix C.4. Other results in Sections 4 and 5, such as Propositions 2 and 8, are

very difficult to prove analytically with multiple couriers. However, we still observe these results

in our numerical experiments.

Figure 7 provides two examples. Figure 7(a) shows the relationship between the expected wait

time when serving dedicated and that when serving batch. Similar to Proposition 2(i) and Figure

3(a), there appears to be a threshold on the service radius r below which serving dedicated leads

to a shorter wait time than serving batch when the demand rate λ is fixed (and vice versa).

Furthermore, Figure 7(b) gives an example of the revenue function with respect to the service

radius. As we can see, serving dedicated still outperforms batch when the service radius is large

enough, just as in Proposition 8 in Section 5.


(a) (b)

Figure 7 Expected wait time and revenue functions with multiple couriers.

(a) k= 5, and λ= 0.5, (b) k= 3, c= 0.1, and Λ = 10

7. Conclusion

In this paper, we compare and contrast the fundamentals of using dedicated vs. pooling delivery

strategy. We model the two strategies as queueing systems serving dedicated and batch, respec-

tively. In addition, we incorporate a spatial feature in these systems using a generalized circular

city model. This spatial feature makes our service system relevant to the daily operations of the

on-demand delivery industry. We highlight the scenarios in which dedicated or pooling delivery

strategy is optimal, and our results remain robust in a variety of extensions.

Our research contributes to the literature of innovative operations and smart cities. One of the

major managerial insights is that, contrary to the common belief, temporal pooling, such as serving

batch, may not always increase delivery efficiency in a large service area and lead to higher revenue

for the vendor. When the vendor can endogenize the demand, pooling should only be used when

the vendor can profitably sustain a relatively large demand rate. With impatient customers or a

large service area, it is better for the vendor to use the dedicated strategy, but charge a relatively

high delivery price. This paper also contributes to the spatial queueing literature by providing an

analytically tractable framework using a generalized circular city model, which is relevant to many

practical applications. In particular, our model accurately depicts delivery systems with a small

number of orders per trip.

This paper can shed light on operational strategies for on-demand delivery services in the emerg-

ing markets such as food or grocery delivery where dedicated couriers or robots/drones are deployed

to make deliveries. Although our focus is mainly on investigating the benefit of temporal and spa-

tial pooling in delivery, there are many other interesting research questions in the delivery business.

Our modeling framework can potentially serve as a building block for future research in areas such

as, but not limited to, contracting and compensation for couriers and incentive management with

freelancers, e.g., by endogenizing the number of dedicated couriers or freelancers in a shift through

a wage decision or a payout contract, which is currently missing in the model.


Our work is not without limitations. First, given a general arrival rate, for some parameters, e.g.,

when the wait cost parameter or the service radius has a sufficiently low value, we cannot obtain

an unambiguous preference for the dedicated or pooling delivery strategy. For these parameters

and a general valuation distribution, one needs to resort to a numerical comparison. Second, the

empirical demand distribution is most likely not a uniform distribution. A data-driven approach

needs to be adopted to prescribe the best strategy for a specific practical setting. Third, we assume

that the couriers are employees of the vendor and their delivery speed is independent of their

workload. As mentioned, the compensation and behavioral issues for couriers may also need to be

examined. Finally, we assume the firm commits to either the dedicated or pooling strategy, as the

resulting pricing and response time could be easily conveyed to consumers. In practice, the firm

can further improve its performance by making optimal contingent decisions about dispatching

and routing depending on the realized locations of outstanding orders, which are outside the scope

of our stylized model.

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Online Appendix to“Courier Dispatch in On-Demand Delivery”:

Supplementary Derivations and Proofs

A. Major Proofs

Proof of Propositions 1 and 2. We first show the result of Proposition 1. First, note that

we have limλ→0WB(λ,µB,CB) =∞ and limλ→µDWD(λ,µD,CD) =∞, according to (4) and (12).

Thus, we only need to show that WD(λ,µD,CD)−WB(λ,µB,CB) = 0 has a unique solution in λ to

established the desired threshold result in λ.

Consider function

f(a) :=1

λ

[a2C1

2T1(T1− a)− 1

2− a2C2

T2(T2− a)− a

3

], ∀a> 0, (A.1)

where

T1 =3

4, T2 =

45π

2(32 + 15π), C1 =

9

8, C2 = 0.583. (A.2)

Then, according to (4) and (12), we have f(λr) =WD(λ,µD,CD)−WB(λ,µB,CB). Note that f(a) =

0 is a single variable cubic equation, which can be solved using standard methods. In particular,

equation f(a) = 0 only has two positive solutions: a1 ≈ 0.5689 and a2 ≈ 1.150 (the exact symbolic

solutions with T1, T2, C1, C2 are cumbersome, thus omitted). Thus, we have 0� a1� T1 = µDr and

T1� a2, which implies that 0 =WD(λ,µD,CD)−WB(λ,µB,CB) has a unique solution λex = a1/r

on (0, µD).

We use the same technique to show Proposition 2. According to (4) and (12), with slight abuse

of notation we write WB(λ,µB,CB) and WD(λ,µD,CD) as WB(r,λ,µB,CB) and WD(r,λ,µD,CD),

respectively, to emphasize their dependence on r. have

limr→0

WB(r,λ,µB,CB) =1

2λ> 0 = lim

r→0WD(r,λ,µD,CD),

and limr→T1WD(r,λ,µD,CD) =∞. Next, since a1 is the unique solution to f(a) = 0 on a∈ (0, T1),

there is a unique rex = a1/λ such that WD(r,λ,µD,CD) = WB(r,λ,µB,CB). This completes the

proof.

Proof of Corollary 1. (i) This part follows Proposition 1 and 2 directly, where we have shown

that serving dedicated leads to shorter wait time comparing to serving batch when either the

demand is low or the radius is small, and vice versa. Thus, using the expression for the price in

(1), we reach the desired result.


(ii) We use the exactly same proof technique as in the proof of Proposition 1 and 2. We only

need to modify the definition of function f(·) is in (A.1) to

f(a) :=1

λ

[a2C1

2T1(T1− a)− 1

4− a2C2

2T2(T2− a)− a6

], ∀a> 0,

where parameters T1, T2, C1, and C2 are defined in (A.2). Then, we have f(λr) =WD(λ,µD,CD)−

WB(λ,µB,CB)/2. Again, function f(a) = 0 is a simple cubic equation, having a unique solution on

a≈ 0.5087∈ (0, T1]. We omit the details to avoid repetition.

Before proving Propositions 3 and 4, we first provide some properties of the optimal demand

rate. For notational convenience, denote

V ∞D (λ) := V∞(λ,WD(λ,µD,CD)) and V ∞B (λ) := V∞(λ,WB(λ,µB,CB)),

and the optimal solutions to the optimization problems

maxλ∈[0,µD)

V ∞D (λ), and maxλ∈[0,2µB)

V ∞B (λ), (A.3)

as λ∞F and λ∞B , respectively. Further, denote TB = 2µBr=45π

2(32 + 15π)and TD = µDr=

3

4.

Next, we present the optimal solutions and objective values to the optimization problems in

(A.3). Note that the objective functions in (A.3) are strictly concave in λ, since

d2V ∞D (λ)

dλ2=

cCDTD

r(λ− TDr

)3< 0, and

d2V ∞B (λ)

dλ2=

2cCBTB

r(λ− TBr

)3< 0,

respectively. Furthermore, by solving the first order conditions when cr < TB < 2TD, we have

dV ∞D (λ)

dλ= 1 +

crCD

(1− T2

D(TD−λr)2

)2TD

= 0, anddV ∞B (λ)

dλ= 1 +

crCB

(1− T2

B(TB−λr)2

)TB

− cr3

= 0,

respectively, which imply

λ∞F =TDr

(1− crCD√

crCD[2TD + crCD]

), λ∞B =

TBr

(1−

√3crCB√

crCB[3TB + cr(3CB −TB)]

),

(A.4)

respectively, and

V ∞D (λ∞F ) = cCD +TDr− 1

r

√crCD[2TD + crCD], (A.5)

V ∞B (λ∞B ) = c

(2CB −

1

2− 1

3TB

)+TBr− 4

r

√3crCB[3TB + cr(3B −TB)], (A.6)

where we have omitted the solutions that are outside the feasible regions.


Proof of Proposition 3. Fix r > 0 and define function

g(α) := r (V ∞B (λ∞B )−V ∞D (λ∞F )) = α

(2CB −

1

2− TB

3

)+TB − 4

√3αCB[3TB +α(3CB −TB)]

−[αCD +TD−

√αCD[2TD +αCD]

], (A.7)

where α := cr. By plugging in CD = 9/8, CB = 0.583, TD = 3/4, TB = 45π/(2(32+15π)) and solving

g(α) = 0, we obtain a unique solution α∗ ≈ 0.1355, which implies that function g(·) only “crosses” 0

once. Finally, one can easily verify that limα→0 g(α)> 0. Thus, we have g(α)> 0 for all α∈ (0, α∗)

and g(α)≤ 0 when α≥ α∗. Therefore, it is better for the vendor to serve dedicated if c≥ α∗/r and

to operate batch otherwise, which implies the first statements in Proposition 3.

Next, we show the second statements in Proposition 3. Recall that the threshold on the wait

cost c can be expressed as α= cr and the vendor switches from serving batch to dedicated when

α= α∗. Thus, we only need to show that λ∞B >λ∞F when cr= α= α∗, where the optimal non-zero

demands λ∞F and λ∞B are defined in (A.4).

Note that for fixed r and cr= α∗ ≈ 0.1355, we have

λ∞B −λ∞F =1

r

(TB −TD +

αCDTD√αCD[2TD +αCD]

−√

3αCB√αCBTB[3TB +α(3CB −TB)]

)> 0,

where we plug in the value of α∗, CD, CB, TD, and TB to reach the inequity. Thus, we conclude

that when switching from serving batch to dedicated, the optimal demand rate decreases. Finally,

recall the revenue function in (2) equals to demand times price, i.e., λp. When the optimal demand

switches from λB to λF < λB at cr = α∗, the corresponding optimal price surges upwards accord-

ingly. This completes the proof.

Proof of Proposition 4. This proof follows from the same steps as the proof of Proposition 3.

Fix c > 0 and define function

h(α) :=1

c(V ∞B (λ∞B )−V ∞D (λ∞F )) =

(2CB −

1

2− TB

3

)+TBα−

4√

3αCB[3TB +α(3CB −TB)]

α

−

[CD +

TDα−√αCD[2TD +αCD]

α

], (A.8)

where α= cr. By solving h(α) = 0, we obtain the same unique solution α∗ ≈ 0.1355. We omit the

details for the rest of the proof to avoid repetition.

Proof of Proposition 5. To show the first statement, we show that the social welfare function

in (18) is an non-increasing function w.r.t. w by taking the first order derivative:

∂SW (w)

∂w=−c

∫ 1

F−1(1− λΛr2

)dF (v) =−c

(1− λ

Λr2

)≤ 0.


Therefore, when fixing λ, the smaller expected wait time translates to higher social welfare. There-

fore, the first statement in Proposition 5 simply follows Propositions 1 and (2).

In order to show the second statement, we consider the large market regime and redefine the

social welfare rate as

SWn(λ,w) = Λnr2

∫ 1

F−1(1− λΛnr2

)(v− cw)dF (v). (A.9)

First, note that we have

limn→∞

SWn(λ,w) = limn→∞

Λnr2

∫ 1

F−1(1− λΛnr2

)(v− cw)dF (v) (A.10)

= λ

∫ 1

F−1(1− λΛnr2

)(v− cw)dF (v)∫ 1

F−1(1− λΛnr2

) dF (v)(A.11)

= λ limn→∞

−dF−1(1− λ

Λcr2)

dn(F−1

(1− λ

Λcr2

)− cw)f

(F−1

(1− λ

Λcr2

))−dF−1(1− λ

Λcr2)

dnf(F−1

(1− λ

Λcr2

)) (A.12)

= λ(1− cw) := SWinf(λ,w), (A.13)

where the first equality follows (1); the second equality follows L’Hopital rule and Leibniz rule.

Finally, note that we have both max{SWn(λ,WD(λ,µD,CD)),0} and

max{SWn(λ,WB(λ,µB,CB)),0} are Lipschitz continuous. To see this, take

max{SWn(λ,WD(λ,µD,CD)),0}= max

{−Λnr2

∫ 1

F−1(1− λΛnr2

)v dF (v) +λcWD(λ,µD,CD),0

},

as the example. It is easy to verify there exists a λ such that max{SWn(λ,WD(λ,µD,CD)),0}= 0

for all λ ∈ [λ, µD) since WD(λ,µD,CD) is convex, increasing and approaching infinity as λ→ µD.

Thus, we only need to focus on λ ∈ [0, λ]. Note that the derivative of −Λnr2

∫ 1

F−1(1− λΛnr2

)v dF (v)

is

Λnr2∂F−1

(1− λ

Λnr2

)∂λ

F−1

(1− λ

Λnr2

)f

(F−1

(1− λ

Λnr2

))<∞,

since F−1 (·) is a Lipschitz continuous function and thus∂F−1

(1− λ

Λnr2

)∂λ

is finite. Therefore,

the term −Λnr2

∫ 1

F−1(1− λΛnr2

)v dF (v) is also Lipschitz continuous. It is also straight forward

to verify that λcWD(λ,µD,CD) is Lipschitz continuous (see the proof of Lemma 1). Thus,

max{SWn(λ,WD(λ,µD,CD)),0} is a Lipschitz continuous function. Then one can show that

limn→0

maxλ∈[0,µD)

SWn(λ,WD(λ,µD,CD)) = maxλ∈[0,µD)

limn→0

SWn(λ,WD(λ,µD,CD)) = maxλ∈[0,µD)

λ (1−WD(λ,µD,CD))


limn→0

maxλ∈[0,2µB)

SWn(λ,WB(λ,µB,CB)) = maxλ∈[0,2µB)

limn→0

SWn(λ,WB(λ,µB,CB)) = maxλ∈[0,2µB)

λ (1−WB(λ,µB,CB)) ,

following the exactly same proof techniques in the one of Lemma 1. We omit the details to avoid

repetition.

Thus, under a crowded market, social welfare maximization over the demand rate is equivalent

to revenue maximization in Section 5, thus, producing the same results.

Proof of Proposition 6. We prove this result using a coupling argument. First, we recognize

that the underlying stochasticity is the Poisson arrival of orders with rate λ and their locations,

which are independent and uniformly distributed on a disk. Furthermore, in all dispatch policies,

since we normalized courier’s speed to 1, the service time is simply the travel distance.

Denote LD, and LC as the number of unfilled orders under dedicated and contingent policies,

respectively. We show that there is path-wise dominance: LD ≥ LC when the underlying random

events are coupled.

Consider an hypothetical alternative policy (short for “alternative contingent policy”), which

mimics the contingent policy and following the delivery decision for each order. To be more specific,

if two arrivals under the contingent policy are served in a batch, in the alternative policy, they

are also served in the same batch. However, unlike the original contingent policy, the courier does

not travel the shortest distance between locations when serving batch. Instead, the courier travels

to the first location but then travels back to the hub before heading to the second location. That

is, although two orders are leaving the hub together, the delivery routing is actually the same as

that of dedicated. We first argue that the alternative policy is feasible at any time t, only requiring

information up to time t of the contingent policy, and does not require clairvoyant information.

Note that the two contingent systems behave the same until the first time two orders needs to

be batched. For any two orders with location x1 and x2, the travel distance under the contingent

policy is always no greater than that of the alternative policy due to triangular inequality (i.e.

|x1|+ |x2| ≥ |x1−x2|). So whenever a batch delivery occurs, the alternative policy induces longer

service time. Thus, by an induction argument, the next delivery decision under the contingent

policy (and thus the alternative policy) is always made no later than the time either the courier

becomes idle or the arrival of the next order in the alternative policy, requiring no clairvoyant

information. Next, denote LA as the number of unfilled orders. Since the decisions of which orders

are served in batches are the same between the two systems, and the alternative policy leads to no

shorter service time, we have LA ≥LC path-wise.

Next, note that the alternative policy is identical to the dedicated policy in terms of the total

service time when coupled together, since the two policies induces the exactly same routing policy.


The only difference between the two policies is that whenever a batch decision is made in the

alternative policy, two orders leave the queue, but in the corresponding dedicated system, only one

order leave the system at a time. Therefore, we have LD ≥LA and the (strict) inequality only holds

during periods when the courier serves batch and is completing the first order in a batch of two.

Thus, we have reached LD ≥LA ≥LC . Finally, the results in expected wait time holds by Little’s

Law.

We prove the second statement by considering extreme cases. When the demand rate is close

to zero, the contingent policy leads to shorter expected wait time since the time to accumulate

orders in batch goes to infinity. On the other hand, when the demand rate is very large, which

leads to shorter order accumulation time than the travel time for an single order (1/λ� 2r), then

it is beneficial for the courier to serve batch. This completes the proof.

Before proving the rest of Propositions in Section 6, we first present a lemma on the properties

of the revenue function of serving batch, when customer valuations follow a standard uniform

distribution without the large market assumption.

Lemma 2. Consider function

g(λ, r, c,Λ) := λ

{1− λ

Λr2− c[

1

2λ+

λr2C

T (T −λr)+r

3

]}, (A.14)

with C,T > 0.

(i) Fix Λ, r > 0. Function g is submodular in (λ; c) for c > 0 and λ∈ (0, T/r).

(ii) Fix c,Λ> 0 and C < 1. Function g is submodular in (λ; r) for r >

(T 2

cCΛ

) 14

and λ∈ (0, T/r).

(iii) Fix c, r > 0. Function g is supermodular in (λ;Λ).

For the proofs of Propositions 7 and 8, denote by λ∗D and λ∗B the optimal solutions to (19) and

(20), respectively.

Proof of Proposition 7. We show that, when fixing r,Λ> 0, there exist some cen such that we

have VB(λ∗B,WB(λ∗B, µB,CB))≤ VD(λ∗D,WD(λ∗D, µD,CD)) if c≥ cen.

With slight abuse notation, denote the optimal solution to

maxλ∈[0,2µB)

λ

[1− λ

Λr2− cWB(λ,µB,CB)

], (A.15)

by λ∗B(c) for every c > 0. Also recall Proposition 1 and denote by λex the unique solution to

f(λ, r) = 0 for r > 0, which is independent of c.

Consider T =45π

2(32 + 15π)and C = CB for function g defined in (A.14). Recall CB ≈ 0.583< 1

and we recognize function g is the revenue function in (A.15) with T = TB and C =CB, so we can


apply the result of Lemma 2 (i) directly. Thus, we have λ∗B(c) is decreasing when c is increasing

using Topkis’s Theorem (see, e.g., Topkis 1978) since the objective function is submodular in (λ; c).

It is obvious that when c is large enough, we must have λ∗B(c) = 0. Thus, as λex > 0 is independent

w.r.t. c where λex is the threshold on which the wait times are equal under dedicated and batch in

Proposition 1, there exists some cen ∈ (0,2µB) such that for all c > cen, we have λex >λ∗B(c) since

λ∗B(c) is decreasing w.r.t. c. Therefore, when c≥ cen, we have

VB(λ∗B(c),WB(λ∗B(c), µB,CB)) ≤ λ∗B(c)

[1− λ

∗B(c)

Λr2− cWD(λ∗B(c), µD,CD)

]≤ VD(λ∗D,WD(λ∗D, µD,CD)),

where the first inequality follows from Proposition 1 since λex > λ∗B(c) and the second inequality

follows from the definition of λ∗D as the optimal solution.

Proof of Proposition 8. We show that, when considering Λ, c > 0 such thatΛ

c3> L :=

1 + 8(√

2CB + 6CB + 8√

2C3B + 8C2

B)

16CB

(2(32 + 16π)

45π

)2

≈ 13.39, there exists a threshold, ren, on the

service radius r, such that we have VB(λ∗B,WB(λ∗B, µB,CB))≤ VD(λ∗D,WD(λ∗D, µD,CD)) if r≥ ren.

Denote by λ∗B(r) the optimal solution to

maxλ∈[0,1/(2µB))

λ

[1− λ


], (A.16)

when fixing r > r := 3

√5π

32 + 15π

1

(0.583Λc)14

. Note that we only need to consider the case where

λ∗B(r) solves the first order condition since any λ∗B(r) that approaches the boundaries leads

to a negative objective value, which implies VD(λ∗D,WD(λ∗D, µD,CD)) ≥ VB(λ∗B,WB(λ∗B, µB,CB))

immediately as VD(λ∗D,WD(λ∗D, µD,CD)) ≥ 0. Furthermore, note that by plugging in T =2µBr

=

45π

2(32 + 15π)and C = CB, function g in (A.14) is the revenue function of serving batch in (20).

Since we have

CB ≈ 0.583< 1, and r > r= 3

√5π

32 + 15π

1

(0.583Λc)14

=

(T 2

ΛcCB

) 14

,

we can apply Lemma 2(ii) directly and conclude that λ∗B(r) is decreasing w.r.t r when r > r.

Denote r= 45π/(c(32 + 15π)(1 + 2√

1.166)). When r > r have

λ

[1− λ


]= λ

{1− λ

Λr2− c[

1

2λ+

λr2CBTB(TB −λr)

]}≤ λ

{1− c

[1

2λ+

λr2CBT (T −λr)

]}≤ λ

{1− cr1 + 2

√2CB

2T

}≤ 0,


where the second inequality follows the fact that the term1

2λ+

λr2CBT (T −λr)

is convex in λ and

attains its minimum r1 + 2

√2CB

2Twhen λ=

T (√

2CB − 1)

r(2CB − 1), and the second inequality follows r > r.

Note that there is

r=45π

c(32 + 15π)(1 + 2√

1.166)=

2T

c(1 + 2√

2CB)>

(T 2

ΛcCB

) 14

= r,

where the inequality follows fromΛ

c3>L. As λ∗B(r) is decreasing w.r.t. r, we have limr→r λ

∗B(r) = 0.

Recall that for a finite r, we have λex, the solution to WD(λ,µD,CD) =WB(λ,µB,CB), never equals

to 0. Thus, denote ` = minr∈(r,r) λex, which is strictly greater than 0. Then by definition of the

limit, there exists some ren ∈ (r, r) such that λ∗B(r)≤ `≤ λex for all r≥ ren.

Finally, consider r≥ ren. We have

VB(λ∗B,WB(λ∗B(r), µB,CB)) ≤ λ∗B(r)

[1− λ

∗B(r)

Λr− cWD(λ∗B(r), µD,CD)

]≤ VD(λ∗D,WD(λ∗D, µD,CD)),

where the first inequality follows from Proposition 1 since we have λ∗B(r)≤ λex for all r≥ ren and

the second inequality follows from that λ∗D is the optimal solution. This completes the proof.

Proof of Proposition 9. The proof of the first statement follows from the exactly same steps

as the proof of Propositions 3 and 4. We only need to substitute the values of constants by TB =

2µB,Cr=4

4 +π, TD = 2µD,Cr=

1

2, CB,C =

1

2+

π2

3(4 +π)2, and CD,C = 1. We still denote α= cr and

let α∗ be the new threshold in cr when serving a circular region. We have α∗ ≈ 0.057� 0.1809≈ α∗,

where α∗ is the optimal value of alpha when the service region is a disk. That is, when cr ≥ α∗,

serving dedicated is better than batch, and vice versa. We omit the details to avoid repetition.

B. Supplementary Proofs in Section 5

Proof of Lemma 1. We only show the argument for function Vn(λ,WD) since the same steps can

be applied to function Vn(λ,WB).

Denote function Jn := min{−Vn(λ,WD),0}. Thus, maximizing Vn(λ,WD) is equivalent to mini-

mizing Jn. Furthermore, by equation (15), denote function

J := limn→∞

Jn = min{−λ(1− cWD(λ,µD,CD)),0}.

The rest of this proof is broken into three steps:

1. We show that for each λ∈ [0, µD), there exists a sequence {λ′n} converging to λ such that

limn→∞

Jn(λ′n) = J(λ). (B.1)


2. We show that for every λ∈ [0, µD) and for every sequence {λ′n} converging to λ, there is

lim infn→∞

Jn(λ′n) = J(λ). (B.2)

3. Once conditions in Steps 1 and 2 are satisfied, we have that function Jn Γ−converges (see,

e.g., Dal Maso 1993) to function J , which implies that

limn→∞

minJn(λ) = min limn→∞

Jn(λ).

Thus, the desired result can be obtained.

Step 1: We show a stronger result here: for every λ ∈ [0, µD) and for every sequence {λn} con-

verging to λ, the equation in (B.1) holds. This stronger result also helps us to show the statement

in Step 2.

We begin by showing that function Jn is Lipschitz continuous for every n ∈N. First, note that

0≤ F−1

(1− λ

Λnr2

)≤ 1, since w.l.o.g., the bounded support of the valuation distribution function

F is normalized to [0,1]. From the proof of Proposition 1, we know that the wait time function

WD(λ,µD,CD) is strictly convex and increasing in λ with limλ→µDWD(λ,µD,CD) =∞. Therefore,

there exists some λ < µD, such that

F−1

(1− λ

Λnr2

)− cWD(λ,µD,CD)≤ 0, so that Jn(λ) = 0, ∀λ > λ.

Furthermore, denote KF as the Lipschitz constant for function F−1 and

KW =∂WD(λ,µD,CD)

∂λ

∣∣∣λ=λ

. (B.3)

Then we have that function WD is Lipschitz continuous when λ ∈ (0, λ) with constant KW since

function WD is strictly convex from the proof of Proposition 1. Furthermore, we have function

gn(λ) :=−λ[F−1

(1− λ

Λnr2

)− cWD(λ,µD,CD)

],

is also Lipschitz continuous with constant

Kn = µD max

{KF

Λnr2, cKW

}, (B.4)

since function gn(λ)/λ is the difference between two Lipschitz continuous functions and λ < µD.

Since there is Jn = min{0, gn}, we conclude that function Jn is Lipschitz with factor Kn.

Next, we have

|Jn(λ)−J(λ)| ≤ |gn(λ) +λ(1− cWD(λ,µD,CD))|

= λ

(1−F−1

(1− λ

Λnr2

))


< µD

(1−F−1

(1− λ

Λnr2

)), (B.5)

according to the definitions of functions Jn and J , and λ< µD.

Consider any λ∈ [0, µD) and any sequence {λ′n} converging to λ. By the definition of convergence,

consider ε > 0 and we can find N1 such that there is

|λ′n−λ|<ε

2K, ∀n≥N1, (B.6)

where K = max{Kn|n ≥ N1}. Furthermore, since F−1 is Lipschitz continuous and monotone

increasing, fixing ε > 0, we can find N2 such that

1−F−1

(1− λ

Λnr2

)≤ ε

2µD, ∀n≥N2. (B.7)

Fix n>Nε := max{N1,N2}, we have

|Jn(λn)−J(λ)| ≤ |Jn(λn)−Jn(λ)|+ |Jn(λ)−J(λ)|

≤ K|λn−λ|+µD

(1−F−1

(1− λ

Λnr2

))≤ K|λn−λ|+

ε

2

≤ ε

2+ε

2= ε,

where the first inequality follows from the triangular inequality; the second inequality follows from

(B.5) and that function Jn is Lipschitz with constant Kn ≤ K; the third inequity follows from

(B.7); and the last inequality follows from (B.6). Therefore, we conclude that (B.1) holds for every

λ∈ [0, µD) and for every sequence {λn} converging to λ.

Step 2: The stronger statement we have shown in step 1 implies the desired result in this step.

Consider any λ ∈ [0, µD) and any sequence {λn} converging to λ. Fix n >Nε = max{N1,N2} and

denote m∗ ∈ arg infm≥n Jm(xm). We have

|Jm∗(λ∗m)−J(λ)| ≤ |Jm∗(λ∗m)−Jm∗(λ)|+ |Jm∗(λ)−J(λ)|

≤ K|λ∗m−λ|+µD

(1−F−1

(1− λ

Λnr2

))≤ ε.

Step 3: Now we can conclude that function Jn Γ-Converges to function J as n approaches

infinity, since we have: 1) for each λ ∈ [0, µD), there exists a sequence {λn} converging to λ such

that equation (B.1) holds from step 1; 2) equation (B.2) holds for every λ ∈ [0, µD) and for every

sequence {λn} converging to λ from step 2. Thus, by the property of Γ-Convergence, we have

limn→∞maxλ∈[0,µD) VnD (λ) = maxλ∈[0,µD) limn→∞ V

nD (λ), and this completes the proof.


C. Supplementary Results and Proofs in Section 6

C.1. General Arrival Rate

Proof of Lemma 2. Noting that the function g is continuous and twice differentiable w.r.t. each

variable, we can verify the statements by taking the mixed second derivatives.

(i) Fix Λ, r > 0 and we have

∂2g(λ, r, c,Λ)

∂λ∂c=Cr

(1

T− T

(T −λr)2

)− r

3=Cλr2(λr− 2T )

T (T −λr)2− r

3< 0,

where the inequality follows from λr < T .

(ii) Fix c,Λ> 0 and there is

∂2g(λ, r, c,Λ)

∂λ∂r=

4λ

Λr3+cC

T

(1− T

2(T +λr)

(T −λr)3

)− c

3(C.1)

We verify the right-hand-side of (C.1) is decreasing in λ by taking its derivative w.r.t. λ:

4

Λr3− 2crCT (2T +λr)

(T −λr)4<

4

Λr3− 4crC

T 2< 0,

where the first inequality follows that the term2crCT (2T +λr)

(T −λr)4is increasing in λ, thus, letting

λ= 0, and the second inequality follows r >

(T 2

cCΛ

) 14

. Therefore, the expression in (C.1) reaches

its maximum −c/3 when λ= 0, suggesting submodularity.

(iii) Fix c, r > 0 and we have

∂2g(λ, r, c,Λ)

∂λ∂Λ=

2λ

Λ2r2> 0,

and this completes the proof.

C.2. Discussion of the Uniform Distribution Assumption on Customers’ Valuation

As mentioned in Section 6, the assumption that function F (v) = v for v ∈ [0,1] and F (v) = 0 is

not restrictive. According to the proof of Proposition 7 and 8, all we need is that the revenue

function when serving batch VB is submodular in (λ; c) and (λ; r), respectively. Thus, as long

as the distribution function F of customer valuations induces a inverse function F−1 leading to

submodularity, similar to Lemma 2 (i) and (ii), we can still find thresholds in wait cost c and radius

r above which, serving dedicated is optimal.

In particular, for any continuous and twice differentiable function F−1, the result in Proposition

7 still holds. To see this, note that the base price F−1

(1− λ

Λr2

)is not a function of c, so it does

not affect submodularity of the revenue function. Thus, the result in lemma 2 still holds.


Similarly, we can extend Proposition 8 under any continuous and twice differentiable function

F−1 such that

λ∂2F−1

(1− λ

Λr2

)∂λ∂r

+∂F−1

(1− λ

Λr2

)∂r

+cC

T

(2− T

2(T +λr)

(T −λr)3

)− c

3< 0,

where T = 2µB/r and C =CB.

C.3. Circular City

Proposition C.1. Consider a circular service area with radius r. When the demand rate λ> 0

is exogenous:

(i) There exists a threshold on the demand rate, below which serving dedicated leads to a shorter

wait time and thus a higher revenue, and above which serving batch is optimal.

(ii) There exists a threshold on the service radius, below which serving dedicated leads to a shorter

wait time and thus a higher revenue, and above which serving batch is optimal.

Without the large market assumption, when the demand rate λ is endogenously determined by

the vendor and customer valuations follow a standard uniform distribution:

(iii) There exists a threshold on the customers’ wait cost parameter, above which it is optimal to

serve dedicated.

(iv) There exists a threshold on the service radius, above which it is optimal to serve dedicated.

Proposition C.1 confirms that all the major results in Sections 4 and 5 still hold even if we

change the service area from a disk to a circle.

Proof of Proposition C.1. We only present a proof sketch for each statement as the details

greatly resemble the previous proofs by substituting in CD,C , and CB,C for CD, and CB:

Statements in (i) and (ii) follow from the proof of Propositions 1 and 2, with T1 = µD,Cr, T2 =

2µB,Cr, C1 = 1 and C2 =1

2+

π2

3(4 +π)2. Then we verify that the cubic equation f(a) = 0 has a

unique solution on a∈ (0, T1), which completes the proof.

Statement (iii) follows from the proof of Proposition 7, and statement (iv) follows from the proof

of Proposition 8. We omit the details.

C.4. Multiple Couriers

Proposition C.2. Consider the vendor hires k≥ 2 couriers covering the service area.

(i) When the demand rate λ> 0 is exogenous, there exists a threshold on the demand rate, below

which serving dedicated leads to a shorter wait time and thus a higher revenue, and above which

serving batch is optimal.

(ii) When the demand rate λ is endogenously determined by the vendor and customer valua-

tions follow a standard uniform distribution, there exists a threshold on the customers’ wait cost

parameter, above which it is optimal to serve dedicated.


As Proposition C.2 suggests, there is still a threshold on the exogenous demand rate, below

which serving dedicated leads to a smaller expected wait time and above which serving batch has

the edge when there are k couriers. Furthermore, when the wait cost parameter is relatively large,

we still have that serving dedicated dominates serving batch.

Proof of Proposition C.2. (i) For notational convenience, we denote µ1 := 2µB,k, µ2 := µF,k and

a :=√

2(k+ 1). By definition, we have µ2 >µ1. Furthermore, define function

f(λ) := WF,k−WB,k

=CD

2(kµ1−λ)

(λ

kµ1

)a−1

−

(1

2λ+

CB2(kµ2−λ)

(λ

kµ2

)a−1

+r

3

), λ∈ (0, µ1). (C.2)

Note that function f represents the difference in wait times when serving dedicated and batch.

First, we show that function f is strictly increasing w.r.t. λ by taking the first order derivative.

We have

∂f(λ)

∂λ=

1

2λ2(1 +CDh(µ1)−CBh(µ2)) ,

where

h(µ,λ) =kµ(λkµ

)a(kµ(a− 1)−λ(a− 2))

(kµ−λ)2, µ∈ (λ/k,∞). (C.3)

Note that function h is non-increasing w.r.t. µ since by taking the first order derivative, we have

∂h(µ,λ)

∂µ=

k(λkµ

)a(kµ−λ)3

[(a− 1)ak2µ2− 2(a− 2)akλµ+ (a− 1)(a− 2)λ2

]=

k(λkµ

)a(kµ−λ)3

{a(kµ−λ)[(a− 1)kµ− (a− 2)λ] +λ[akµ− (a− 2)λ]} ≥ 0,

where the inequality follows from that kµ> λ. Thus, we have

∂f(λ)

∂λ≥ 1

2λ2(1 +CDh(µ1, λ)−CBh(µ2, λ))> 0,

where the first inequality follows from that function h in non-increasing in µ together with µ2 >µ1

and the second inequality follows from CD >CB.

Next, by acknowledging function f goes to negative infinity and positive infinity when λ

approaches 0 and µ1, respectively, we can reach the first statement in Proposition C.2, since f(λ) = 0

has a unique solution.

(ii) We only need to verify that the revenue function of serving batch,

U(λ, c) := λ

{1− λ

Λr2− c

[1

2λ+

CB2(2kµ1−λ)

(λ

2kµ1

)a−1

+r

3

]}, (C.4)


is submodular in (λ, c). By taking its derivatives w.r.t. λ and c, we have

∂2U(λ, c)

∂λ∂c=−

CB

(λkµ1

)a−1

2(kµ1−λ)2(akµ− (a− 1)λ)− r

3< 0,

where the inequality follows from kµ−λ> 0.

The rest of the proof follows from the exactly same steps as in the proof of Proposition 7. Thus,

we omit the details.

D. Simulation Results

In this section, we present the simulation results on the accuracy of several approximations on

expected wait times in this paper.

First, we briefly state our simulation methods. In each sample, consider a unit disk (with radius

r= 1), and generate 1,000 arrivals uniformly distributed inside the disk using a fixed arrival rate λ.

We calculate the average wait time (sample mean) for each sample. When calculating the sample

mean, we exclude the first 200 arrivals to ensure the system has reached a steady state. In total, we

simulate 100 samples for each arrival rate and take the average of sample means as the simulated

expected wait time.

Next, we present the results by comparing the simulated expected wait times versus the approx-

imated wait times. Table D.1 demonstrates that our approximation for the expected wait time in

an E2/G/1 queue using Kingman’s formula in (12) is reasonably good. The percentage difference

is calculated as

Difference (%) :=|Simulated Time−Approx. Time|

Approx. Time.

Lastly, we choose parameters λ≤ 2µB so that the approximated system is finite and the comparisons

are meaningful.

Table D.1 Performance of approximated expected wait time with Kingman’s formula on E2/G/1 queue

λ(≤ 2µB) Simulated Time Approx. Time Difference (%)0.2 2.92 3.02 3.47%0.3 2.20 2.33 5.49%0.4 1.95 2.11 7.71%0.5 1.97 2.16 8.80%0.6 2.30 2.50 8.23%0.7 3.21 3.41 5.90%0.8 6.13 6.55 6.45%

In Table D.3, we present the expected wait time with the approximation in (31) and (32), which

are compared to their simulation counterparts, respectively. We choose courier number k= 3.


Table D.2 Performance of approximated expected wait time with dedicated delivery and k= 3 couriers

λ(≤ kµD) Simulated Time Approx. Time Difference (%)0.3 0.003 0.007 57.48%0.6 0.02 0.03 32.91%0.9 0.07 0.08 10.18%1.2 0.33 0.36 5.38%1.5 0.43 0.42 8.35%1.8 0.84 0.83 0.48%2.1 3.10 3.31 6.12%

Table D.3 Performance of approximated expected wait time with batch delivery and k= 3 couriers

λ(≤ 2kµB) Simulated Time Approx. Time Difference (%)0.3 2.00 2.00 0.47%0.6 1.17 1.18 1.06%0.9 0.91 0.91 0.39%1.2 0.79 0.80 0.70%1.5 0.77 0.75 1.76%1.8 0.84 0.77 9.51%2.1 1.14 0.89 27.3%

As we can see from Tables D.2 and D.3, our approximation is relatively accurate except for light

traffic dedicated systems or nearly overloaded batch systems. In the next two figures, we show that

even with approximation errors, our insights on the thresholds in wait cost c and service radius r

still hold with simulation results.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

DedicatedBatch

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

DedicatedBatch

(b)

Figure D.1 Revenue functions when serving dedicated versus batch under a large market.

(a) r= 1 k= 3, (b) c= 0.5, k= 3

E. Other Delivery Delay

In our base model, we consider that the courier does not stay for any positive amount of time at

each order location nor has any delay caused by other factors. In this section, we provide a brief

discussion on how to extend our base model to incorporate these features to the model.


Take delays occurred at the delivery locations as the example. Suppose each order incurs an

independently and identically distributed extra delay denoted as d at each order location, where

d has mean µd and variance σ2d, following a known distribution. Further assume that this extra

delay is also independent from the courier’s travel time (the time to cover the distance between

two locations). In reality, indeed there is at most a negligible correlation between the actual travel

time and the time finding a parking space at the delivery location.

Using this setup, we can rewrite the service rate and coefficient of variation as

µD =1

E[XD] +µd=

143r+µd

, CD = 1 +E[X2

D]− (E[XD])2 +σ2d

(E[XD] +µd)2= 1 +

29r2 +σ2

d(43r+µd

)2 , (E.1)

when the courier serves dedicated delivery, and

µB =1

E[XB] +µd=

14(32+15π)

45r+µd

, CB =1

2+

E[X2B]− (E[XB])2 +σ2

d

(E[XB] +µd)2=

1

2+

5.428r2−(

4(32+15π)

45r)2

+σ2d(

4(32+15π)

45r+µd

)2 ,

(E.2)

when serving batch. Then the expected wait time of the two delivery modes can be written as

WD(λ, µD, CD) and WB(λ, µB, CB), respectively.

Using this alternative setup, we can recreate the analysis in our base model. Take Section 4 with

exogenous demand for example. We can still show that there are thresholds on demand rate λ and

service radius r, respectively, such that below which serving dedicated is optimal and above which,

the courier should serve batch. We only provide a sketch of proof here, using a similar method as

the proofs for Propositions 1 and 2.

Consider function

f(λ, r) := WD(λ, µD, CD)−WB(λ, µB, CB)

=1

180πλ(−3 + 4rλ+ 3λµd)(64rλ+ 15π(−3 + 2rλ+ 3λµd))[8192r2λ2(−3 + 4rλ+ 3λµd)− 1920πrλ(−9 +λ(2r2λ− 9µd(−3 +λµd)− 12r(−2 +λµd) + 9λσ2

d))

+225π2(4(−10 + 9a)r3λ3− 54(−1 +λµd)2 + 3r2λ2(12− 9a− 28λµd + 9aλµd)

−36rλ(−3 +λ(µd +λµ2d−λσ2

d)))

]− r

3, (E.3)

where a=E[X2B]− (E[XB])2 ≈ 0.416. One can easily verify that f(λ, r) = 0 is a cubic equation with

respect to either λ or r. Furthermore, it has a unique real solution for λ or r, just as in the proof

of Proposition 1 and 2. Lastly, we only need to verify that WB(λ, µB, CB)<WD(λ, µD, CD) when

λ (or r) approaches zero, but WB(λ, µB, CB)>WD(λ, µD, CD) when λ (or r) is sufficiently large.


As we can see, with the addition of extra delays independent of the delivery trip travel time,

the analysis becomes a lot messier due to complicated algebra operations. Thus, we decide not to

include these features into our base model to preserve a clean yet informative analysis.

References

Dal Maso, G. (1993). An Introduction to Γ−Convergence. Birkhauser, Basel.

Courier Dispatch in On-Demand Delivery

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