Line-Sitting Services

Abstract

Waiting in line is a common yet disagreeable experience for customers seeking service, and the desire to bypass a long line has spawned the innovative practice of line-sitting—instead of joining the queue themselves, customers can hire surrogates to stand in line on behalf of them. These surrogates are referred to as line-sitters. They enter queues upon customer requests, and unlike true customers they sit in line not for the underlying service that customers are waiting for but for monetary gains from those who are willing to pay to avoid waiting. The line-sitters can notify customers through text messaging or a mobile app when they move close to the front of the line. As a result, the hiring customer can simply show up to take the line-sitter’s spot when the service is about to start. In this chapter, we present a model of line-sitting and show that it can increase the social welfare or revenue of the service provider over a first-in-first-out (FIFO) policy. This chapter is primarily based on Cui et al. (Manag Sci 66(1):227–242, 2020) and Wang and Wang (Oper Res Lett 47(5):447–451, 2019) where interested readers can find proofs of the findings shown in this chapter.

1 Introduction

Queues are an integral part of many service systems. However, waiting in a long line is universally unpleasant. With the rise of the gig economy, line-sitting (i.e., customers hire a third-party line stander to sit in line on their behalf) has taken off and seeped into all kinds of waiting lines. Examples include queues for congressional hearings in DC, for buying Hamilton tickets and Cronuts in New York, for paying government bills in Italy, for seeing doctors in China, and for eating at trendy restaurants. Line-sitters have also appeared at passport/visa application lineups and COVID-19 testing sites all over the world.

The recent years have seen line-sitting morphing into a burgeoning business. Same Ole Line Dudes (SOLD), a New York City-based company, provides line-sitting services for customers buying Dominique Ansel cronuts or Hamilton tickets; Skip the Line, a Washington DC-based business, can save space at popular restaurants, museum exhibits, and shopping events; Washington DC Line Standing, another DC-based company, helps clients stand in line for Congressional hearings. These firms typically set an hourly rate for their line-sitting services, based on the nature of the lines. For instance, SOLD charges $10–$15 per hour; Skip the Line, a fee of $30 per hour; and Washington DC Line Standing, $40 per hour. Additionally, thanks to technological advances, various mobile-application startups have been launched to facilitate on-demand booking of line-sitters. Notable examples include LineAngel based in Los Angeles and Placer in New York City.

At first sight, one may be prone to the thought that line-sitting is a minor variant of the more traditional priority purchasing scheme which is widespread and extensively studied. After all, isn’t it all about customers being able to pay extra for reduced physical wait? Yes and no. There are two key differences between line-sitting and priority purchasing. First, priority is sold by the service provider (e.g., Universal Studios selling Front-of-Line passes), whereas line-sitting is managed by an independent third-party company (e.g., SOLD). Second, a priority customer reduces her wait by bumping customers who opt out of the priority-upgrade option, whereas a line-sitting adopter skips wait by swapping position with the hired line-sitter when the line-sitter nears the front of the line without pushing back non-adopters. These two subtle distinctions not only lead to divergent customer behavior but also have rich revenue implications for the service provider and welfare implications for the customer population. As such, a game-theoretical queueing model was studied in Cui et al. (2020) to formally analyze the impact of line-sitting on the service provider’s revenue and customer welfare.

Based on Cui et al. (2020), we employ a canonical first-in-first-out (FIFO) queueing model as a benchmark in which customers with heterogeneous waiting costs decide whether to join a service system or balk, based on the service reward, service fee, and the waiting cost. Next, we set up the model of line-sitting in which, in addition to joining themselves and balking, customers can also choose to hire a line-sitter to stand in the line at an hourly rate specified by the line-sitting firm. Furthermore, we formulate a model of priority purchasing, in which, besides joining a regular line and balking, customers can also choose to join the priority line by paying an additional priority premium set by the service provider. We fully characterize the equilibrium in each of the three schemes, namely, (1) FIFO without line-sitting or priority, (2) line-sitting, and (3) priority purchasing. We then perform pairwise comparisons of the three schemes in terms of the service provider’s revenue and customer welfare. Finally, we combine the results from the pairwise comparisons to identify the optimal scheme out of the three that either maximizes the service provider’s revenue or achieves the highest customer welfare, or both.

2 Model Preliminaries and FIFO Benchmark

Consider an $M/M/1$ service system with the first-in-first-out (FIFO) queue discipline. Customer needs for the service arise according to a Poisson process with rate $\Lambda$ per hour. The service provider has an exponentially distributed service time with rate $\mu$ per hour. Let $\rho =\Lambda /\mu$ denote the potential workload of the system. If a customer joins the service system, she earns a service reward R and pays the service provider a base service fee B. The hourly customer waiting cost in the service system, c, is uniformly distributed over the interval $(0,\overline {c})$. The uniform distribution is assumed for tractability, while our key insights continue to hold under various forms of distributions, including the normal, beta, power, and triangular distributions. The arrival rate of potential customers $\Lambda$, service rate $\mu$, service reward R, service fee B, and waiting-cost distribution are common knowledge. Customers do not observe the queue length when they make decisions but are privately informed of their individual waiting cost rate. We assume an unobservable queue because (1) customers typically decide whether to hire a line-sitter when their service needs arise, which tends to occur before they observe the queue length (we shall introduce the line-sitting model in Sect. 3), and (2) the unobservable model is more amenable to analysis. Customers do not renege if they join the service system.

2.1 FIFO Benchmark

We first set up a FIFO benchmark model without line-sitters. Each customer decides whether to join the system or balk when they experience a need for the service, in order to maximize their expected utility. Balking gives zero utility. For a customer with hourly waiting cost c, the expected utility from joining the queue $U^{FIFO}(c)$ given system throughput $\lambda _e$ (or the effective joining rate) is equal to the service reward less the service fee less the expected waiting cost:

$$\displaystyle \begin{aligned} U^{FIFO}(c) = R -B - \frac{c}{\mu-\lambda_e} , {} \end{aligned} $$

(3.1)

where $1/(\mu -\lambda _e)$ is the expected waiting time (including the time at service) in the $M/M/1$ queue. Because the expected utility function $U^{FIFO}(c)$ is decreasing in c, strategic customers will adopt a threshold joining strategy for any given throughput $\lambda _e$, i.e., there exists a cost threshold $c^{FIFO}$ such that a customer with hourly waiting cost c joins the system if $c\le c^{FIFO}$ and balks otherwise. Intuitively, this means that customers who are less sensitive to waiting would join, whereas those who are more sensitive to waiting choose to balk. In equilibrium, system throughput $\lambda _e^{FIFO}$ must satisfy the condition that $\lambda _e^{FIFO} = \Lambda c^{FIFO}/\bar {c}$.

Given $c^{FIFO}$ (and thus the equilibrium system throughput $\lambda _e^{FIFO}$), the service provider’s revenue (rate) in the FIFO benchmark is the service fee times the system throughput, i.e., $B \lambda ^{FIFO}_e$; customer welfare is equal to $\Lambda \int _{0}^{c^{FIFO}} U^{FIFO}(c) /\overline {c} \, d c$, where $U^{FIFO}(c)$ is specified in (3.1). For now, we treat the service fee B as given to obtain clean results; in Sect. 6, we shall find the revenue-maximizing B for the service provider under various schemes and demonstrate the robustness of our findings. For notational convenience, we define $\overline {R}=R\mu /\overline {c}$ and $\overline {B}=B\mu /\overline {c}$, i.e., $\overline {R}$ and $\overline {B}$ are the normalized service reward ${R}$ and normalized service fee ${B}$, respectively. Thus, the FIFO benchmark can be fully described by $(\overline {R},\overline {B}, \rho , \bar {c})$. We assume $\overline {R}-\overline {B}\geq 1$ throughout this chapter, which is equivalent to $R-B\geq {\overline {c}}/{\mu }$. It is without loss of generality because if the assumption was violated, then customers whose hourly waiting cost c falls within the interval $[\mu (R-B),\overline {c})$ would never join the service system even if there is no waiting at all, which implies that we can always scale $\overline {c}$ down to $\mu (R-B)$ to exclude those irrelevant customers. As a result, we can derive customers’ equilibrium decisions (characterized by $c^{FIFO}$), the corresponding service provider’s revenue $\Pi ^{FIFO}$, and customer welfare $CW^{FIFO}$.

3 Line-Sitting

Building on the FIFO model, we now set up the line-sitting model. There is a third-party line-sitting firm that provides customers with line-sitting services for an hourly charge r. If a customer (she) uses the line-sitting service, a line-sitter (he) joins the queue and waits on behalf of her; her total payment to the line-sitting firm is the product of the hourly rate r and the amount of time he spends in the queue. We assume there are a sufficient number of line-sitters available to work whose hourly opportunity cost is normalized to zero. Line-sitting tasks are generic and do not involve specific skills; consequently, the supply of line-sitters may not be a grave concern. In Sect. 7, we show indeed that our insights continue to hold even when only finitely many line-sitters are available. The simplifying assumption that line-sitters have negligible opportunity costs may be considered a reasonable representation of practice to the extent that line-sitters have been reported to be students, stay-at-home moms, or low-income individuals.

When a need for the service arises, each risk-neutral, expected-utility-maximizing customer decides whether to (i) join the service system by hiring a line-sitter, (ii) join themselves, or (iii) balk (which gives zero utility). It is clear that a joining customer with hourly waiting cost c should use the line-sitting service only if c exceeds the hourly line-sitting rate r. Moreover, because both the total line-sitting payment and waiting cost are linear in time, each hiring customer finds it optimal to let the line-sitter stand in line as long as possible. That is, a hiring customer would ask the line-sitter to take over the entire wait in the queue. In real life, line-sitters can notify customers through text messaging or a mobile app when they move close to the front of the line. For simplicity, we assume that the hiring customer heads to the service system when she learns that her line-sitter is close to the front of the queue, and she is able to show up to take the line-sitter’s spot when the service is about to start. Hence, for a customer with hourly waiting cost c, her expected utility for choosing to join the system by hiring a line-sitter (given system throughput $\lambda _e$), $U^{LS}(c)$, is equal to the service reward less the service fee less the expected waiting cost at service less the expected total payment for line-sitting:

$$\displaystyle \begin{aligned} U^{LS}(c) = R-B - \frac{c}{\mu} - \frac{r \lambda_e}{\mu(\mu-\lambda_e)}, {} \end{aligned} $$

(3.2)

where $\lambda _e/[\mu (\mu -\lambda _e)]$ is the expected waiting time in the queue (excluding the time at service) of an $M/M/1$ system or equivalently the line-sitter’s expected duration of standing in line. In contrast, if a customer decides to join the service system by herself, her expected utility (given system throughput $\lambda _e$) is the same as $U^{FIFO}(c)$ which is already specified in (3.1). Finally, it is useful to point out that in our model while customers decide whether to hire a line-sitter when their service need arises, the total payment is not due until line-sitting is complete because the payment is based on the realized waiting time in the queue. In Sect. 8, we shall consider the impact of pre-commitment payment when customers pay a prespecified amount of fee for service even if the line-sitter’s actual waiting time is less than the pre-committed wait.

Under the line-sitting setting, similar to $U^{FIFO}(c)$, the expected utility function $U^{LS}(c)$ is also decreasing in c; thus, strategic customers will adopt a double-threshold strategy (if $r<c^{FIFO}$), i.e., there exists a cost threshold $c^{LS}$ such that a customer with hourly waiting cost c joins the service system by herself if $c< r$, joins by hiring a line-sitter if $c\in [r,c^{LS})$, and balks if $c>c^{LS}$. Intuitively, it means that customers who are relatively insensitive to waiting join the system by themselves; those who are intermediately sensitive to waiting join but hire line-sitters; and those who are highly sensitive to waiting balk. In equilibrium, system throughput $\lambda _e^{LS}$ must satisfy the condition that $\lambda _e^{LS} = \Lambda c^{LS}/\bar {c}$. If $r\geq c^{FIFO}$, no customers will hire a line-sitter in equilibrium, and the system degenerates to the FIFO benchmark. In this case, only customers with $c>r \ge c^{FIFO}$ might potentially use line-sitting, but these customers either do not exist (if $c^{FIFO}=\bar {c}$) or would not join (if $c^{FIFO}<\bar {c}$) because their expected utility from joining would be negative even by hiring a line-sitter (since $U^{LS}(c)< U^{FIFO}(c^{FIFO})=0, \forall c>c^{FIFO}$).

The line-sitting firm sets the hourly rate r to maximize total revenue:

$$\displaystyle \begin{aligned} \max_{r\in (0,\overline{c})} \quad r\cdot \frac{\Lambda (c^{LS} - r)}{\overline{c}}\cdot \frac{\lambda^{LS}_e}{\mu(\mu-\lambda^{LS}_e)} {} \end{aligned} $$

(3.3)

where $\lambda _e^{LS} = \Lambda c^{LS}/\bar {c}$. The total line-sitting revenue (per hour) in (3.3) is the product of three terms: (i) the hourly rate r; (ii) the expected number of arriving customers who choose to hire line-sitters per hour, ${\Lambda (c^{LS} - r)}/{\overline {c}}$; and (iii) the expected waiting time each line-sitter spends in the queue ${\lambda ^{LS}_e}/{[\mu (\mu -\lambda ^{LS}_e)]}$. In Proposition 1, we solve for the line-sitting firm’s optimal hourly rate $r^*$, and give the corresponding cost threshold $c^{LS}$ and system throughput $\lambda _e^{LS}$ in equilibrium.

Proposition 1

The optimal hourly rate for the line-sitting firm,$r^*$, and the corresponding cost threshold$c^{LS}$, system throughput$\lambda _e^{LS}$, and fraction of joining customers who end up using the line-sitting service$(c^{LS}-r^*)/c^{LS}$are given in Table3.1.

Table 3.1 Results of Proposition 1

The cases presented in Proposition 1 are mutually exclusive and collectively exhaustive of the parameter space. In Cases (1) and (2), the potential workload of the system $\rho$ is relatively small, and all customers join the service system in equilibrium. In the FIFO benchmark, the all-joining scenario occurs only when $\rho \leq {(\overline {R}-\overline {B}-1)}/{(\overline {R}-\overline {B})}$. In the presence of the line-sitting firm, all customers would still join when this condition holds; even when it is violated, the all-joining scenario may still arise (note that, e.g., Case 1 subsumes $\rho \leq {(\overline {R}-\overline {B}-1)}/{(\overline {R}-\overline {B})}$). This shows the value line-sitting brings to high-waiting-cost customers—a customer not patient enough to wait in a FIFO queue may now choose to join the system by hiring a line-sitter because doing so eliminates her waiting time in the queue at a price lower than her personal waiting cost.

Although Cases (1) and (2) have the same all-joining customer behavior, they differ in the line-sitting firm’s pricing pattern. In Case (1), the optimal hourly rate $r^\ast$ is flat in the potential workload $\rho$, whereas in Case (2), it is decreasing in $\rho$. The line-sitting firm trades off the hourly rate with the number of customers who use the line-sitting service and each customer’s expected in-line waiting time in equilibrium. When the potential workload is small (Case (1)), the firm only needs to balance the hourly rate and the number of line-sitting users without worrying about the expected waiting time since all customers join regardless of the line-sitting rate. However, when the potential workload gets relatively large (Case (2)), the third effect kicks in, i.e., an increase in the hourly rate would decrease not only the number of customers who use the line-sitting service but also the expected in-line waiting time (due to customer balking). Therefore, a larger potential workload would compel a lower hourly rate in order to prevent customers from balking.

When the potential workload $\rho$ becomes even larger, heavy congestion makes customer balking inevitable, as shown in Cases (3) and (4) of Proposition 1. Similar to Case (2), we can show that the optimal hourly rate is decreasing in $\rho$ also in Case (3). Notably, in Case (4) when the potential workload is sufficiently large, customer demand for line-sitting becomes so price-elastic that the line-sitting firm finds it optimal to keep lowering the hourly rate for more line-sitting users and longer in-line waiting times. As a result, the optimal hourly rate is set as low as possible (zero in this case but in general a value equal to the opportunity cost of the line-sitters if we had not normalized it to zero). The resulting expected line-sitting time tends to be very large (infinity in the zero-opportunity-cost case), making the total expected payment of each transaction a positive and finite amount, equal to $R-B-{\overline {c}}/{\Lambda }$.

Given the hourly rate r of the line-sitting firm, and the cost threshold $c^{LS}$ (and thus throughput $\lambda _e^{LS}$), the service provider’s revenue (rate) in the presence of the line-sitting firm is equal to $B \lambda _e^{LS}$; and customer welfare is given by

$$\displaystyle \begin{aligned} \Lambda \left[\int_{0}^{r} U^{FIFO}(c) /\overline{c} \, d c + \int_{r}^{c^{LS}} U^{LS}(c) /\overline{c} \, d c\right] , \end{aligned}$$

where $U^{FIFO}(c)$ and $U^{LS}(c)$ are specified in (3.1) and (3.2), respectively. We can then derive closed-form expressions for the service provider’s revenue $\Pi ^{LS}$ and customer welfare $CW^{LS}$ when the line-sitting firm charges the optimal hourly rate.

3.1 Comparison Between Line-Sitting and FIFO

We now investigate the impact of line-sitting on the service provider’s revenue and customer welfare (at the optimal hourly line-sitting rate $r^*$ set by the line-sitting firm).

Theorem 1

The service provider receives a higher revenue in the presence of the line-sitting firm than in the FIFO benchmark, i.e.,$\Pi ^{LS}\geq \Pi ^{FIFO}$.

Theorem 1 is intuitive—the service provider increases revenue by accommodating line-sitting because some high-waiting-cost customers who would balk in the FIFO benchmark are now willing to join the system by hiring line-sitters (to stand in line on behalf of them). The line-sitting firm strictly increases the service provider’s throughput when the potential workload of the system is large enough that not all potential customers join in the FIFO benchmark, i.e., when $\rho > {(\overline {R}-\overline {B}-1)}/{(\overline {R}-\overline {B})}$. Because service fee B is fixed, a higher throughput translates into a higher revenue for the service provider. We refer to line-sitting’s ability to increase system throughput as the demand expansion effect, which is at the heart of line-sitting.

If the service provider can adjust the service fee to the line-sitting firm’s entry, its revenue will still be higher with line-sitting than without because adjusting the service fee will only further improve revenue, i.e., $\Pi ^{LS}(B^{LS})\geq \Pi ^{LS}(B^{FIFO})\geq \Pi ^{FIFO}(B^{FIFO})$ if we use $\Pi ^X(B^Y)$ to denote the service provider’s revenue under setting X when it charges the optimal service fee for setting Y  in $B^Y$. Thus, it would still be wise for a revenue-oriented service provider to accommodate line-sitting. In practice, however, it is not unusual for a service provider to prohibit line-sitting, possibly due to non-revenue-related factors which will be discussed in more detail in Sect. 9. Now that we have investigated the impact of line-sitting on the service provider’s revenue, what is the effect on customer welfare? We provide the answer in the following Theorem 2.

Theorem 2

Customer welfare in the presence of line-sitting, $CW^{LS}$ , and customer welfare in the FIFO benchmark, $CW^{FIFO}$ , compare as follows:

• (1) if$\rho < 1-\sqrt {3}/{3}$, $CW^{LS}>CW^{FIFO}$;

• (2) if$\rho \in [1-\sqrt {3}/{3},1)$, there exist two thresholds$\overline {R}_{a}$, $\overline {R}_{b}$such that$CW^{LS}>CW^{FIFO}$if and only if$\overline {R}<\overline {R}_{a}$or$\overline {R}>\overline {R}_{b}$, where$\overline {R}_{a}=\overline {B}+\frac {2\rho -1+\sqrt {4\rho ^2+1}}{2 \rho }$and$\overline {R}_{b}=\overline {B}+\frac {\rho ^2-12\rho +8-\sqrt {\rho ^4+8\rho (\rho ^2-2\rho +2)}}{8(\rho -1)(2\rho -1)}$;

• (3) if$\rho \in [ 1, (\sqrt {5}+1)/2]$, there exists a unique threshold$\overline {R}_{c}=\overline {B}+\frac {\sqrt {5}+1}{2\rho }$such that$CW^{LS}>CW^{FIFO}$if and only if$\overline {R}<\overline {R}_{c}$.

• (4) if$\rho > (\sqrt {5}+1)/2$, $CW^{LS} < CW^{FIFO}$.

Theorem 2 presents precise conditions under which line-sitting has a positive (or negative) impact on customer welfare. The results are illustrated in Fig. 3.1. On the high level, line-sitting exerts three forces on customer welfare. First, line-sitting users can retain more surplus because they pay at an hourly rate lower than their own waiting cost rate; this puts an upward pressure on customer welfare. Second, the demand expansion effect distilled from Theorem 1 implies that line-sitting brings value to new customers who would not otherwise join the system, which also puts an upward pressure on customer welfare. Third, these additional customers nonetheless generate negative congestion externalities, which put downward pressure on customer welfare. Note that for customers who join by themselves, an increase in congestion can hurt them directly by exacerbating their waiting costs, whereas for line-sitting users, an increase in congestion can hurt them indirectly because line-sitters can pass it on to their users through a larger line-sitting bill.

Fig. 3.1

Customer welfare comparison between line-sitting and FIFO

When the potential workload $\rho$ is small ($\rho < 1-\sqrt {3}/{3}$), most customers, if not all, would join the service system under both FIFO and line-sitting due to light system congestion. The line-sitting option would reduce the cost of high-waiting-cost customers (those with $c\ge r^*$) without raising the congestion level dramatically. Hence, overall customer welfare improves.

Now consider intermediate values of the potential workload ($\rho \in [1-\sqrt {3}/{3},1)$). When the service reward is sufficiently small, i.e., $\overline {R}<\overline {R}_a$, customers do not have strong joining incentives; hence, there is light congestion under FIFO. The line-sitting option would prompt more customers to join the system, and since the system load is not too high, the extra service utility customers receive outweighs the extra negative congestion externalities, thereby improving customer welfare. When the service reward is sufficiently large, i.e., $\overline {R}>\overline {R}_b$, almost all potential customers join under both FIFO and line-sitting. Thus, line-sitting again only increases congestion slightly. However, when the service reward is intermediate, i.e., $R\in [\overline {R}_a,\overline {R}_b]$, the throughput increase creates more negative congestion externalities than positive service utilities, thus lowering customer welfare. Observe from Fig. 3.1 that welfare deterioration becomes more predominant (i.e., the interval $[\overline {R}_a,\overline {R}_b]$ expands) as $\rho$ increases. This is because a higher workload amplifies the effect of congestion externalities.

Finally, consider a large potential workload ($\rho \ge 1$). When the service reward is small, i.e., $\overline {R}<\overline {R}_c$, customer welfare improves for similar reasons as discussed in the earlier $\overline {R}<\overline {R}_a$ case. In contrast, when the service reward is large, i.e., $\overline {R}>\overline {R}_c$, the line-sitting option would incentivize joining by a sizable amount, which considerably prolongs the system wait (in this case, recall from Proposition 1 that the line-sitting firm would set a low optimal hourly rate but charge each user for a long line-sitting time). As such, the line-sitting option creates too much congestion to compensate for the gains in service utilities, and hence customer welfare suffers. Observe from Fig. 3.1 that welfare deterioration becomes more predominant (i.e., the interval $(\overline {R}_c,\infty )$ expands) as $\rho$ increases, similar to our earlier observation with an intermediate $\rho$. In particular, when $\rho >(\sqrt {5}+1)/2$, line-sitting would harm customer welfare for any service reward.

We conclude with two remarks from the comparison between line-sitting and FIFO. First, because the service provider’s revenue is always weakly higher with line-sitting ($\Pi ^{LS}\ge \Pi ^{FIFO}$; see Theorem 1), the shaded area in Fig. 3.1, which corresponds to the region for $CW^{LS}>CW^{FIFO}$, also represents the region in which both the service provider’s revenue and customer welfare are (weakly) higher in the presence of line-sitting. This implies that allowing customers to use line-sitting can be a win-win proposition for both the service provider and customers, relative to FIFO. Second, we would also like to emphasize that giving customers an extra option to hire line-sitters does not necessarily guarantee improvement in customer welfare, because the decisions of self-interested customers can generate an overwhelming amount of negative congestion externalities in the system.

4 Accommodating Line-Sitting or Selling Priority?

Thus far, we have introduced the model of line-sitting based on a classical FIFO model, and showed that line-sitting improves the service provider’s revenue because the high-waiting-cost customers, who would balk in the FIFO case, are now willing to join the system by paying a line-sitter to stand the line. The idea of paying a premium to skip the wait bears a resemblance to the well-established practice of priority queues (e.g., visitors to the London Eye in the UK can pay an extra £9, in addition to the base admission fee for an online Fast Track ticket. While a priority queue is typically managed by the same service provider, line-sitting is run by a third-party firm. The priority premiums go directly to the service provider, whereas the line-sitting payments do not. As such, wouldn’t a service provider always favor implementing a priority purchasing scheme over accommodating a third-party line-sitting firm? And what about customer welfare? In this section, we seek to answer these questions by first setting up the model for priority purchasing.

4.1 Priority Purchasing

The model of priority purchasing is based on the FIFO benchmark introduced in Sect. 2. The service provider sets and collects a premium P for priority service. Each customer selects one of the three available options when their need for the service arises: (i) join the system as a regular customer, or (ii) join it as a priority customer, or (iii) balk (which gives zero utility). A priority customer obtains non-preemptive priority for service over regular customers (i.e., a regular customer who has started service will not be bumped by a priority customer), and the service disciplines within the regular line and within the priority line are both FIFO.

Given the expected waiting time in the regular line (denoted by $w_1$) and in the priority line ($w_2$) both including the time at service, for a potential customer with hourly waiting cost c, the expected utility from joining as a regular customer, $U^{REG}(c)$, and that as a priority customer, $U^{PRI}(c)$, are given by

$$\displaystyle \begin{aligned} U^{REG}(c) = R- B - c w_1\quad \text{and} \quad U^{PRI}(c) = R - (B+P) - c w_2, {} \end{aligned} $$

(3.4)

respectively, where a regular customer pays the (base) service fee B and expects a waiting time of $w_1$, whereas a priority customer pays a total of $B+P$ and expects a waiting time of $w_2$. It is rational for a joining customer with hourly waiting cost c to purchase priority if and only if her expected waiting-cost savings exceed the priority premium, i.e., $c (w_1- w_2) \ge P$.

Because the expected utility functions $U^{REG}(c)$ and $U^{PRI}(c)$ are both linearly decreasing in c and $U^{REG}(c)$ is decreasing at a larger rate than $U^{PRI}(c)$, strategic customers will follow a double-threshold strategy, i.e., there exist two cost thresholds $(c_{a}^{PRI},c_b^{PRI})$ such that customers with waiting cost $c\leq c_{a}^{PRI}$ choose to join the system as regular customers; those with waiting cost $c\in (c_{a}^{PRI},c_{b}^{PRI}]$ join as priority customers, and those with waiting cost $c>c_{b}^{PRI}$ balk. rst in, rst out rm’s revenue in Gavirneni and Kulkarni’s (2016) case, Let $q=(c_{b}^{PRI}-c_{a}^{PRI})/{c_{b}^{PRI}}$ denote the fraction of joining customers who purchase priority. Given system throughput $\lambda _e$ and priority purchase fraction q, it is well-known that $w_1$ and $w_2$ can be derived as follows (see, e.g., Chapter 33 of Harchol-Balter 2013):

$$\displaystyle \begin{aligned} \begin{array}{rcl} w_{1} (\lambda_e,q) = \frac{\lambda_e}{(\mu-\lambda_{e})(\mu-\lambda_e q)}+\frac{1}{\mu}, \quad w_{2} (\lambda_e,q)= \frac{\lambda_e}{\mu(\mu-\lambda_e q)}+ \frac{1}{\mu}. \end{array} \end{aligned} $$

In equilibrium, the effective joining rate (system throughput) must satisfy the condition $\lambda _e^{PRI}=\Lambda c_{b}^{PRI}/\overline {c}$, and $w_1(\lambda _e^{PRI}, q)$ and $w_2(\lambda _e^{PRI}, q)$ must be consistent with $(c_{a}^{PRI},c_b^{PRI})$ in that $q=(c_{b}^{PRI}-c_{a}^{PRI})/{c_{b}^{PRI}}$, $c_{a}^{PRI}= P/(w_1-w_2)$ and $c_b^{PRI}=(R-B-P)/w_2$ (if $c_{a}^{PRI}$ and $c_b^{PRI}$ are less than $\overline {c}$). For given priority premium P, we can characterize the customer equilibrium $(c_{a}^{PRI},c_b^{PRI})$ in closed form. The service provider sets priority premium P to maximize total revenue:

$$\displaystyle \begin{aligned} \max_{P\ge 0}\lambda^{PRI}_{e}(B+q P). {} \end{aligned} $$

(3.5)

The service provider’s revenue (rate) in (3.5) consists of the base service fee collected per hour, $\lambda ^{PRI}_{e} B$ and the priority premium collected per hour $\lambda ^{PRI}_e q P$. As before, we continue to treat the base service fee B as given, and leave the joint optimization of the base service fee and the priority premium to Sect. 6. Customer welfare in this setting can be written as

$$\displaystyle \begin{aligned} \Lambda \left[\int_{0}^{c_a^{PRI}} U^{REG}(c) / \overline{c} dc + \int_{c_a^{PRI}}^{c_b^{PRI}} U^{PRI}(c) / \overline{c} dc \right], \end{aligned}$$

where $U^{REG}(c)$ and $U^{PRI}(c)$ are specified in (3.4). We can derive closed-form expressions for the service provider’s optimal priority premium $P^*$, the corresponding revenue of the service provider $\Pi ^{PRI}$, and customer welfare $CW^{PRI}$ under $P^*$.

4.2 Comparison Between Priority and FIFO

We benchmark the priority model against FIFO in the following Theorem 3, similar to what was done for the line-sitting case in Sect. 3.1. Doing so will facilitate the formal comparison between priority purchasing and line-sitting in Sect. 4.3.

Theorem 3

The service provider obtains a higher revenue in the priority purchasing scheme than in the FIFO benchmark, i.e.,$\Pi ^{PRI}>\Pi ^{FIFO}$, but it results in lower customer welfare, i.e.,$CW^{PRI}<CW^{FIFO}$.

The service provider can boost its revenue by selling priority to customers because doing so is essentially a practice of price discrimination that takes advantage of customers’ heterogeneous waiting costs. If the priority premium is zero, all joining customers will choose to upgrade, and no one would receive any actual priority. On the other hand, if the priority premium is prohibitively high, none of the customers will upgrade and the priority model again degenerates into the FIFO case. The optimal priority premium is such that only a fraction of joining customers opt into priority, making the service provider strictly better off. Customer welfare, however, is compromised, in part, because the priority purchasing scheme allows the service provider to appropriate more surplus value from customers.

4.3 Comparison Between Line-Sitting and Priority

This subsection compares line-sitting and priority purchasing in terms of the service provider’s revenue and customer welfare. Recall from Theorems 1 and 3 that either accommodating line-sitting or implementing priority would benefit a FIFO service provider. One might expect the service provider to benefit more from introducing a priority purchasing scheme than permitting line-sitting, because the priority premium contributes directly to the service provider’s revenue, whereas the line-sitting business is run by a third-party company. Somewhat surprisingly, Theorem 4 reveals that this is not always the case.

Theorem 4

If$\overline {B}\leq {(\sqrt {1+\rho }-1)}/{(1+\rho )}$, the service provider’s revenue in the priority purchasing scheme,$\Pi ^{PRI}$, is higher than that in the line-sitting scheme,$\Pi ^{LS}$, i.e.,$\Pi ^{PRI}>\Pi ^{LS}$. Otherwise, there exists an$\overline {R}'$such that$\Pi ^{LS}>\Pi ^{PRI}$if and only if$\overline {R}<\overline {R}'$, where$\overline {R}'$is unimodal in$\rho$.

The results of Theorem 4 are illustrated in Fig. 3.2. The two practices help improve the service provider’s revenue (relative to FIFO) by different means: the priority system improves revenue by charging high-waiting-cost customers a priority premium, i.e., price discrimination, whereas line-sitting does so by increasing system throughput, i.e., demand expansion (see Theorem 1). When the service fee B is too low, i.e., when $\overline {B}\leq {(\sqrt {1+\rho }-1)}/{(1+\rho )}$, the additional throughput brought by the line-sitting firm, compared to FIFO, only translates into limited extra revenue for the service provider, and therefore, implementing the priority purchasing scheme is always superior.

Fig. 3.2

The service provider’s revenue comparison between line-sitting and priority

However, when the base service fee is not too low, i.e., $\overline {B}>{(\sqrt {1+\rho }-1)}/ {(1+\rho )}$, the priority system is better than line-sitting for the service provider only when the service reward R is sufficiently large, i.e., when the condition $\overline {R}>\overline {R}'$ is satisfied. A large service reward would justify joining for those high-waiting-cost customers that have a stronger willingness to pay for priority than their low-waiting-cost counterparts. On the other hand, when the service reward is not large enough, only customers with relatively low waiting costs find it worthwhile to join the service system (in the priority model), which circumscribes the service provider’s ability to profit from implementing priority; in the meantime, the demand expansion effect from line-sitting becomes particularly effective in luring high-waiting-cost customers who would not otherwise join, and consequently, accommodating line-sitting becomes more revenue-improving for the service provider.

Interestingly, Theorem 4 further reveals the range of the service reward for which line-sitting generates more revenue than priority first expands and then shrinks with the potential workload $\rho$, as illustrated by Fig. 3.2. This implies that for a fixed service reward, accommodating line-sitting is more favorable than selling priority when $\rho$ is intermediate. When the potential workload is small enough, all customers join under either scheme, so the demand expansion effect that line-sitting relies on for increasing revenue is mute, which makes selling priority more desirable. On the other hand, when the potential workload is large enough, the demand expansion effect realizes its full potential by bringing the throughput to its upper limit—system capacity $\mu$; an even higher load will not further expand demand in the line-sitting scheme but can still boost the revenue from selling priority; hence, selling priority also becomes superior. Consequently, for the service provider, accommodating line-sitting works best when $\rho$ is intermediate.

We should mention one key driver of line-sitting’s potential revenue dominance is that it introduces an extra source of cost reduction by shifting a portion of customers from waiting in the physical queue to an “offline” channel. The priority purchasing scheme does not enjoy this cost advantage since all customers, regardless of their priority status, must wait in the physical queue themselves. Such a cost advantage renders the possibility of line-sitting outperforming priority in its revenue contribution to the service provider. Next, we examine in Theorem 5 how customer welfare in the priority purchasing scheme compares with that of the line-sitting scheme.

Theorem 5

Customer welfare in the line-sitting scheme, $CW^{LS}$ , and customer welfare in the priority purchasing scheme, $CW^{PRI}$ , compare as follows: there exists ${\rho }'> 1- \sqrt {3}/3$ such that

• (1) if$\rho < {\rho }'$, $CW^{LS}>CW^{PRI}$;

• (2) if$\rho \in [{\rho }',1)$, there exist two thresholds$\overline {R}_{a}^{\prime }\leq \overline {R}_{b}^{\prime }$such that$CW^{LS}\ge CW^{PRI}$if and only if$\overline {R}\leq \overline {R}_{a}^{\prime }$or$\overline {R}\ge \overline {R}_{b}^{\prime }$;

• (3) if$\rho \geq 1$, there exists$\overline {R}_{c}^{\prime }$such that$CW^{LS}>CW^{PRI}$if and only if$\overline {R}<\overline {R}_{c}^{\prime }$.

The results of Theorem 5 are illustrated in Fig. 3.3: the shaded area below the solid curve represents circumstances under which customer welfare is higher in line-sitting than in priority, i.e., $CW^{LS}>CW^{PRI}$, whereas the area under the dotted curve corresponds to circumstances under which line-sitting achieves higher customer welfare in priority than in FIFO, i.e., $CW^{LS}>CW^{FIFO}$ (a reproduction of Fig. 3.1 from Sect. 3.1). Figure 3.3 suggests that the region corresponding to $CW^{LS}>CW^{PRI}$ actually contains that corresponding to $CW^{LS}>CW^{FIFO}$ and this is consistent with the finding that $CW^{PRI}<CW^{FIFO}$ (Theorem 3).

Fig. 3.3

Customer welfare comparison between line-sitting and priority

Given our detailed explanation of customer-welfare comparison between line-sitting and FIFO in Theorem 2, and given that customer-welfare comparison between line-sitting and priority yields qualitatively similar results, we do not discuss Theorem 5 at length besides highlighting the white region in Fig. 3.3, which shows $CW^{LS}<CW^{PRI}$. In this region, the potential workload $\rho$ is high and service reward R is large. In this case, the line-sitting firm charges a low hourly rate (see Proposition 1) to induce excessive line-sitting time (and effectively, large customer payment), making line-sitting even worse than priority from a customer-welfare standpoint.

5 Three-Way Comparison

We focus on studying the new business practice of line-sitting in this chapter. Thus far, we have set up a line-sitting model based on FIFO in Sect. 2, and also compared it with a priority purchasing scheme in Sect. 4 as a more traditional approach for customers to pay and skip wait. Building on the pairwise comparisons of the three schemes (FIFO, line-sitting, priority) in Sects. 3.1, 4.2, and 4.3, we now proceed to conduct a three-way comparison to identify the service scheme that delivers the maximum revenue for the service provider and/or maximum customer welfare, i.e., finding $\mathop{\text{argmax}} \limits _{s}\Pi ^s$ and $\mathop{\text{argmax}} \limits _{s}CW^s$ for $s\in \{FIFO, LS, PRI\}$. Figure 3.4a illustrates the outcome. Note that it is created simply by combining Fig. 3.1 from Sect. 3.1 and Fig. 3.2 from Sect. 4.3.

Fig. 3.4

Optimal schemes in terms of the service provider’s revenue and customer welfare

From Fig. 3.4a, it is evident that the FIFO system is never revenue-maximizing for the service provider (Theorems 1 and 3), whereas the priority purchasing scheme is not welfare-maximizing for customers (Theorem 3). However, line-sitting can be optimal at the same time for both the service provider and the customers (see Region I in Fig. 3.4a), and it is the only scheme out of the three that can be simultaneously optimal for both. In addition, line-sitting is by construction the only model that also benefits the line-sitting company. As a result, it can create a win-win-win situation for the service provider, customers, and the line-sitting company. Therefore, albeit a contentious issue, line-sitting should be viewed as a value-generating business under appropriate circumstances.

6 Endogenizing Service Fee B

The base service fee B and the corresponding $\overline {B}$ are given in our base models and do not vary across the three schemes under comparison. In the short run, the service provider may not be able to adjust its price to various schemes, e.g., due to menu costs. Our preceding results are best applicable to those settings in which the base service fee is not scheme-specific. In the long run, the service provider may be able to change the service fee in response to the scheme being used. This section investigates such endogenous service fees.

Let $B^{FIFO}$, $B^{LS}$, and $B^{PRI}$ denote the optimal service fees the service provider charges under FIFO, line-sitting, and priority, respectively, i.e., $B^s=\mathop{\text{argmax}} \limits _{B}\Pi ^s$ for $s\in \{FIFO, LS, PRI\}$. We can explicitly characterize these optimal service fees, based on which, we conduct numerical experiments to find the schemes that achieve the highest server revenue and greatest customer welfare, respectively. The results are illustrated in Fig. 3.4b. We observe that consistent with Fig. 3.4a which fixes the service fee, the line-sitting scheme still dominates the left corner of the $(\overline {R},\rho )$ space and remains the only scheme that can be simultaneously optimal for both the service provider and customers. We also observe that when the base service fee is endogenized, accommodating line-sitting dominates selling priority for the service provider.

7 Finitely Many Line-Sitters

In the base model, we have assumed a sufficient supply of line-sitters such that there is always a line-sitter available to work upon customer request. In this subsection, we consider a line-sitting firm that organizes N line-sitters. The main complication is that when a customer wishes to hire a line-sitter, none would be available if all of the N line-sitters are currently standing in line on behalf of other customers. If this occurs, then a joining customer must wait herself regardless of her line-sitting preference. Hence, the line-sitters’ availability creates an additional level of heterogeneity among customers. Let the probability that all line-sitters are busy be $\pi _b$; the joining threshold when all line-sitters are busy, $c^{FIFO}$ (similar to the FIFO benchmark); and the joining threshold when at least one line-sitter is available, $c^{LS}$ (similar to the base model of line-sitting). Thus, the system throughput $\lambda _e$ is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \lambda_e= \Lambda \left[(1-\pi_b) c^{LS}/\overline{c} + \pi_b {c}^{FIFO}/\overline{c}\right]. {} \end{array} \end{aligned} $$

(3.6)

To obtain $\pi _b$, recognize that given system throughput $\lambda _e$, joining threshold $c^{LS}$, and hourly rate r, the probability that none of the N line-sitters is available is equal to the blocking probability of an Erlang loss system with N servers, Poisson arrival rate $\lambda '= \Lambda (c^{LS}-r)/\bar {c}$, and mean service time $\tau =\lambda _e[\mu (\mu -\lambda _e)]^{-1}$. Hence,

$$\displaystyle \begin{aligned} \pi_b= \frac{(\lambda' \tau)^N/N!}{\sum_{i=0}^{N} (\lambda' \tau)^i/i!}. {} \end{aligned} $$

(3.7)

For any hourly rate r, on the one hand, given conjectured joining thresholds $c^{FIFO}$ and $c^{LS}$, the throughput and blocking probability, $(\lambda _e, \pi _b)$, are jointly determined by (3.6) and (3.7); on the other hand, given conjectured throughput $\lambda _e$, joining thresholds $c^{FIFO}$ and $c^{LS}$ can be found by the usual best-response argument as in our FIFO benchmark and the base model of line-sitting. In equilibrium, $(\lambda _e, \pi _b, c^{FIFO}, c^{LS})$ must be self-consistent.

Given N, the line-sitting firm sets the hourly rate r to maximize total revenue:

$$\displaystyle \begin{aligned} \max_{r\in (0,\overline{c})} \quad r\cdot \frac{\Lambda (c^{LS}-r)}{\overline{c}} \cdot (1-\pi_b) \cdot \frac{\lambda_e}{\mu(\mu-\lambda_e)}. {} \end{aligned} $$

We run numerical experiments and find that our qualitative insights are preserved even when only a single line-sitter is available ($N=1$) and that with five available line-sitters, the directional impact of line-sitting on the service provider’s revenue and customer welfare is already reasonably similar (in terms of the parameter space partitioning) to the infinite supply case in the base model.

8 Pre-commitment Payment

In our base model, the line-sitting fee is paid at the completion of service, which can be referred to as the post-payment scheme. Nevertheless, could the line-sitting firm benefit from the pre-commitment payment scheme, in which the customer still pays a prespecified fee for service even if the line-sitter’s actual waiting time is less than the pre-committed wait? In this section, we compare the two payment schemes in Wang and Wang (2019). For tractability, we assume that balking or reneging is not allowed, and the effective arrival rate satisfies $\lambda <\mu$ to ensure system stability.

Under the pre-commitment payment scheme, a customer decides how long she wants a line-sitter to wait in line on behalf of her, denoted by t, a quantity that she commits before the waiting time in line is realized. Define w as the actual waiting time in line. If $w\leq t$, the customer still needs to pay the line-sitting the pre-specified amount payment among $r\cdot t$. The hiring customer shows up to take the line-sitter spot when the service is about to start. Otherwise, if $w>t$, the customer swaps with the line-sitter after time t and waits in line herself for the rest of the wait in line (i.e., $w-t$). As such, customers can be regarded as newsvendors who determine the optimal time t to maximize their utilities. Anticipating that, the line-sitting firm needs to adjust the optimal service rate r to maximize its revenue.

In an $M/M/1$ queue with arrival rate $\lambda$ and service rate $\mu$, the probability that the server is busy upon the arrival of a customer is $\rho =\lambda /\mu$. Conditioned on the server being busy, the waiting time in the queue ($W_{b}$) is exponentially distributed with rate $\mu -\lambda$. Thus, a joining customer with delay sensitivity c chooses t to maximize her expected utility:

$$\displaystyle \begin{aligned} U^{PC}(c) &= \max_{t} \left[- rt - \frac{c}{\mu}- c \rho E[\max\{W_{b}-t,0\}]-c(1-\rho)\cdot 0\right], \\ &= \max_{t} \left[- rt -\frac{c}{\mu}- c\rho \int_{0}^{\infty} \max\{T-t,0\} (\mu-\lambda){e}^{-(\mu-\lambda)T}dT\right], \end{aligned} $$

where $r t$ is the customer’s up-front line-sitting payment; $c/\mu$ is the expected waiting cost in service; and $c\rho E[\max \{W_{b}-t,0\}]$ is the expected waiting cost for the customer spent in the queue. Waiting time reduction is positive if and only if the server is busy upon arrival (otherwise there is no wait), which occurs with probability $\rho$. Conditioned on a positive waiting time reduction, the amount of waiting time for the customer is $\max \{W_{b}-t,0\}$, where $W_{b}$ is exponentially distributed with rate $\mu -\lambda$. The customer’s optimal line-sitting time, $t^\ast (c)$, follows the newsvendor solution:

$$\displaystyle \begin{aligned} e^{-(\mu-\lambda)t^\ast(c)} = \frac{r}{c\rho},\end{aligned}$$

which leads to $t^\ast (c) = \frac {\ln (c\rho /r)}{\mu -\lambda }$ for $c > r/\rho$. The expression of $t^\ast (c)$ shows that the customers would purchase a positive amount of line-sitting time if and only if their delay sensitivity satisfies $c> r/\rho$. Additionally, customers with a higher c would purchase a longer time because $t^\ast (c)$ is increasing in c. This is different from the post-payment case where the expected line-sitting time customers pay for would be $\frac {\lambda }{\mu (\mu -\lambda )}$ if $c>r$, and customers with delay sensitivity $c\leq r$ would not purchase the line-sitting service. That is, the threshold of purchasing the line-sitting service under the post-payment scheme (i.e., r) is lower than that under the pre-commitment payment scheme (i.e., $r/\rho$).

When the optimal quantity $t^\ast (c)$ is adopted, the expected utility of a customer with delay sensitivity c under the pre-commitment payment scheme is given by

$$\displaystyle \begin{aligned} U^{PC}(c)=\left\{ \begin{array}{ll} -\frac{c}{\mu-\lambda}, & \mbox{if}\ 0\leq c\leq r/\rho; \\ - r \frac{\ln (c\rho/r)}{\mu-\lambda} -\frac{r}{\mu-\lambda} - \frac{c}{\mu}, & \mbox{if}\ r/\rho<c\leq \overline{c}, \end{array} \right. \end{aligned}$$

and we compare it with the expected utility of the same customer under the post-payment scheme, denoted by $U^{LS}(c)$, in the following proposition.

Proposition 2

For any given line-sitting rate r, we have$U^{PC}(c)\leq U^{LS}(c)$.

Proposition 2 reveals that when the same line-sitting rate r is adopted, customers under the pre-commitment payment scheme receive a lower utility. This is because when the pre-commitment time is higher than the actual waiting time in line, customers are overpaying the line-sitters. On the other hand, if the pre-commitment time is lower than the actual waiting time in line, customers have to stand in line for a portion of the time by themselves, incurring a larger waiting cost. Therefore, customers’ utility is reduced either way under the pre-commitment payment scheme.

8.1 Revenue of the Line-Sitting Firm

Under the optimal strategy of the customers, the line-sitting firm’s revenue is

$$\displaystyle \begin{aligned} \pi^{PC}(r)=\frac{\lambda r}{\overline{c}} \int_{r/\rho}^{\overline{c}} \frac{\ln (c\rho /r)}{\mu-\lambda} d c. \end{aligned}$$

After some algebraic manipulation, we have:

$$\displaystyle \begin{aligned} \pi^{PC}(r)=\frac{\lambda r}{\overline{c}(\mu-\lambda)} \int_{r/\rho}^{\overline{c}} \ln (c\rho/r) dc = \frac{\rho r [r/\rho -\overline{c} + \overline{c}\ln (\overline{c}\rho/r)] }{\overline{c}(1-\rho)}, \end{aligned} $$

and we compare it with the line-sitting firm’s revenue under the post-payment scheme, denoted by $\pi ^{LS}(r)$ in the following result.

Proposition 3

There exists a unique$\bar {r}\in (0,\overline {c}\rho )$such that$\Pi ^{PC}(r)\geq \Pi ^{LS}(r)$if and only if$r\leq \overline {r}$.

Under the same line-sitting rate, Proposition 3 shows that the line-sitting firm’s revenue is higher under the pre-commitment payment scheme if and only if r is small. This is because when r is small (resp., large), the total line-sitting time purchased by customers under the pre-commitment payment scheme exceeds (resp., falls short of) that under the post-payment scheme. Next, denote by $r^{PC}$ and $r^{LS}$ the optimal line-sitting rates under the two payment schemes, and $\Pi ^{PC}$ and $\Pi ^{LS}$ the line-sitting firm’s (optimal) revenue when $r^{PC}$ and $r^{LS}$ are adopted, respectively. We have the following result.

Proposition 4

$r^{PC}<r^{LS}$and$\Pi ^{PC}< \Pi ^{LS}$.

Somewhat surprisingly, Proposition 4 shows that the line-sitting firm’s optimal revenue is actually lower under the pre-commitment payment scheme than under the post-payment scheme. This is because the post-payment scheme saves customers’ money, so they hire the line-sitters more often, leading to higher revenue for the line-sitting firm.

8.2 Welfare Implications

Finally, we investigate how the pre-commitment payment scheme affects customer welfare as well as social welfare. We have the following result.

Proposition 5

$CW^{PC}> CW^{LS}$but$SW^{PC}< SW^{LS}$.

Proposition 5 shows that the pre-commitment payment scheme can improve customer welfare. This is counterintuitive at the first sight because we have shown earlier that customers are worse off under the pre-commitment payment scheme compared to the post-payment scheme (Proposition 2). However, Proposition 2 holds only for a fixed line-sitting rate. Because customers are more reluctant to use the line-sitting service under the pre-commitment payment scheme, it is optimal for the line-sitting firm to lower the line-sitting rate (Proposition 4). As a result, customers are actually better off under the pre-commitment payment scheme compared to the post-payment scheme under the optimal line-sitting rates for both schemes. Finally, from social welfare’s perspective, the line-sitting fee is an internal transfer between the customers and the line-sitting firm. The reason why the post-payment scheme achieves higher social welfare is that customers (with high delay sensitivity) who hire line-sitters do not have to wait in line at all, reducing customers’ total waiting costs.

9 Concluding Remarks

This chapter covers the booming business of line-sitting in congestion-prone service systems and explores its economic impact. In the presence of a line-sitting firm, customers with high waiting costs are willing to pay to get a line-sitter in line for them. We first examine how line-sitting impacts the service provider’s revenue and customer welfare (relative to a FIFO queue without line-sitting). In the light of the similarity between line sitting and priority purchasing—both enable customers to skip physical wait with an extra payment—we then investigate whether the service provider should accommodate line-sitting or sell priority, and how these two schemes differ in customer welfare.

We find that, like selling priority, allowing line-sitting always improves the service provider’s revenue, relative to FIFO. Yet, the revenue improvement is made possible by different means. Selling priority increases revenue through price discrimination, whereas allowing line-sitting increases revenue through demand expansion. This distinction leads to a divergence of the two schemes in customer welfare: higher revenue from implementing priority tends to come at the expense of customer welfare, whereas line-sitting can lead to a win-win situation for both the service provider and the customers. However, demand expansion can be a double-edged sword: while customer welfare improves with line-sitting in some cases, it can also deteriorate in others, particularly when both the service reward and the potential system workload are high, due to the negative congestion externalities introduced by the extra demand.

One major difference between line-sitting and priority purchasing is that they are managed by different parties—the line-sitting business is run by a third-party company, whereas priority is sold directly by the service provider. It is thus tempting to believe that selling priority is more lucrative than accommodating line-sitting for the service provider who acquires the priority premium but not the line-sitting payment. On the other hand, line-sitting has a cost advantage in diverting its clients to an offline channel. Hence, it is unclear a priori which scheme performs better. Interestingly, we find that the latter force (line-sitting being cost advantageous) can outweigh the former (selling priority being a direct revenue stream). Not only can line-sitting be more favorable for the service provider, it is also the only scheme among the three (FIFO, line-sitting, and priority) that can potentially be optimal for the service provider and customers alike.

Despite its potential contributions to the service provider’s revenue and customer welfare, line-sitting may raise fairness concerns among customers. One phenomenon that may trigger strong customer reaction is the instance of a single line-sitter holding a place for a group of many. In fact, this is the primary reason some service providers cite when banning line-sitting. However, if one-to-one-substitution is followed (the case in our model), we suspect that the associated fairness concerns may be less protruding than those with priority purchasing because; in the former, line-sitting users only swap positions with line-sitters who are already in the system without affecting other customers’ queue positions, whereas, in the latter, priority customers can cut the line to bump regular customers. Moreover, customer complaints, if any, are less likely to be directed to service providers because the line-sitting proceeds do not accrue to them. We refer interested readers to Althenayyan et al. (2022) which provide an experimental investigation on the fairness concerns of line-sitting. Another follow-up work is Zhao and Wang (2023), who study a setting where customers waiting for service in one line can hire a line-sitter to stand in the line of a second server who offers identical service. Any hiring customer can receive service from the first available server.