Distance-Based Service Priority

Abstract

Obtaining a service usually comes with both rewards and costs. For example, dining at a popular restaurant provides customers with a pleasant dining experience, but also incurs costs to them. The costs include the direct service charge that customers have to pay to receive the service (the meal cost) and indirect hassle costs such as going to the restaurant and waiting in line for the service. When the service reward is too low or the service fee is too high, customers will be less likely interested in the service. The same happens when the hassle cost of seeking service is too high, which is especially true when customers must travel a long distance to obtain a service that tends to have cumbersome wait times. It is without question that increasing the number of customers being served (i.e. system throughput) is critical for service providers, as it translates directly into service revenue in the case of a for-profit company and service provision in the case of a non-profit organization. Therefore, how may service providers incentivize customers located far away from the service location to seek service? In this chapter, we introduce an innovative but natural distance-based service priority mechanism to help service providers increase their system throughput. The idea is to assign higher service priority to customers who have to travel farther for the service, thus giving them with new incentives to consider using the service. We shall demonstrate, among other results, that such a mechanism can significantly increase system throughput by attracting more customers to use the service, and the increase can be up to 50% compared to the ordinary first-come-first-served service discipline. This chapter is primarily based on Wang et al. (Manuf Serv Oper Manag, 25(1), 353–369, 2023) where interested readers can find proofs of the findings shown in this chapter.

1 Introduction

The basic idea of the distance-based service priority policy covered in this chapter is to reduce service waiting time for customers who have to travel farther for the service by giving them higher service priority. The inspiration of the policy first occurred to us when one of us was seeking service from the Chinese Embassy in the United States. In the U.S., the Chinese Embassy is located in Washington D.C. and there are five additional Chinese Consulates-General located in New York City, Chicago, San Francisco, Los Angeles and Houston. Each of these six consulates provides service to a specific U.S. region, that is, a customer who seeks any in-person service needs to visit the consulate that holds jurisdiction over the region in which the customer resides in. For example, customers who reside in D.C., Delaware, Idaho, Kentucky, Maryland, Montana, Nebraska, North Carolina, North Dakota, South Carolina, South Dakota, Tennessee, Utah, Virginia, West Virginia and Wyoming can only seek in-person service from the Embassy in Washington D.C. To alleviate the travel hassle (driving or flying) of customers living in states far away from D.C., the Embassy has unwritten rules for those customers to receive service priorities upon their arrival over others who live closer. It helps the long-distance customers obtain service during a day trip without having to stay overnight in D.C. which would otherwise be impractical given the long wait time to obtain service at the Embassy.

We thought the policy could potentially increase the business for a congestion-prone service provider. This is because under the ordinary first-in-first-out (FIFO) service scheme also known as first-come-first-served, customers who live far away from the service location would have little or no interest in a service that requires a long wait due to their already significant travel costs. Service priority reduces the wait time to obtain the service once they arrive at the service location, thereby providing distant customers with a new incentive to seek service. Nevertheless, the policy de-prioritizes service requests of customers who live near the service location and the extra wait time could disincentivize them to continue to seek service. As a result, the system throughput can go down if additional service seekers contributed by distant customers cannot compensate for the losses caused by nearby customers. To this end, we constructed a game-theoretical queueing model in Wang et al. (2023) to study the effectiveness of the policy by carefully examining the resultant customer behavior, system throughput, and welfare.

In the model, customers live at different distances from a service location and need to decide whether to seek service. They make their decisions based on the service value, service fee, travel cost and expected wait time to obtain the service once they arrive at the service location. The most relevant prior work to the model is the work by Rajan et al. (2019) which considers travel cost and waiting cost to obtain service for patients who need to see a medical specialist. Similar to our model, customers (patients) in theirs have heterogeneous travel costs because their distances from the service (specialist) location are different, and they decide whether to seek service based on comparing costs to service value. The researchers focus on whether the specialist should offer telemedicine service in addition to in-person service, in order to accommodate patients who live far away and are not attracted to in-person service. To achieve the same goal, we explore the effectiveness of the distance-based service priority policy which gives customers who have to travel farther to the service location higher service priority. Although both the approach in Rajan et al. (2019) and our approach can improve system throughput, the advantage of our approach is that it may still be effective for services that cannot be provided remotely, e.g., dining services from restaurants or vaccination services from healthcare providers.

2 Model Preliminaries and FIFO Benchmark

We consider a service provider offering a physical service to customers living in a specific area. The service requires traveling and waiting and is modeled as an unobservable queueing system. More specifically, the travel cost of a customer is determined by the distance between the customer’s residence location and the service location, and the expected wait time to obtain service is determined by the service discipline chosen by the service provider. As a benchmark, we first consider the scenario where the service provider uses the ordinary FIFO service discipline. It specifies that anyone who arrives at the service location earlier will be served before others who arrive later. Customers’ service needs arise according to a Poisson process with rate $\Lambda$, that is, $\Lambda$ indicates the arrival rate of potential customers to the system. A single server serves the queue, and the service time is exponentially distributed with rate $\mu$ and independent among customers. We use $\rho \equiv \Lambda /\mu$ to denote the system’s potential workload.

When a service need arises, customers decide whether to seek service based on the service value, service fee, travel cost and expected wait time to obtain service once they arrive at the service location, and they do not renege if they decide to join the service system. We denote the service reward by V  and service fee by B. We assume the distances between customers’ locations and the service location form a uniform distribution with support $[0,\bar {x}]$ where $\bar {x}$ is the farthest distance where service demand arises. In other words, customers who have the service need live at heterogeneous distances from the service location. We denote customers’ unit travel cost as d. Finally, we denote the expected wait time to obtain service once customers arrive at the service location (which includes the delay in the queue and service time) as W and customers’ unit waiting cost as c. Based on these notations, the expected utility of seeking service for a customer with “distance” x, denoted by $U(x)$, can be expressed as

$$\displaystyle \begin{aligned} U(x)=V-B-d\cdot x-c\cdot W \mbox{ for } x\in[0,\bar{x}],{} \end{aligned} $$

(6.1)

and we normalize the utility of customers who decide not to seek service to 0. Because whether or not customers decide to seek service determines the expected wait time to obtain service (W), which in turn determines the utility of customers who seek service ($U(x)$), an equilibrium analysis is necessary.

To simplify the notation for subsequent analysis, we use $S\equiv \mu (V-B)/c=(V-B)/(c/\mu )$ to denote the net service value (not considering travel and waiting costs), $V-B$, normalized by the expected service cost, $c/\mu$. We will refer to S as the normalized service value in the rest of the paper. We assume $S\geq 1$ because otherwise no customers will choose to join the system even if no other person is in the system (i.e., even when $W=1/\mu$). We also use $T\equiv \mu d\bar {x}/c=(d\bar {x})/(c/\mu )$ to denote the maximum travel cost, $d\bar {x}$, normalized by the expected service cost, $c/\mu$. We will refer to T as the normalized travel cost in the rest of the chapter. In particular, when customers’ unit travel cost increases (i.e., $d\uparrow$) or when they are distributed over longer distances (i.e., $\bar {x}\uparrow$), the normalized travel cost T will also increase.

Because customers are heterogeneous in terms of their distance to the service location, we consider an asymmetric equilibrium strategy parameterized by their distance $x\in [0,\bar {x}]$. In particular, we consider a general, mixed strategy $q(x)\in [0,1]$ which specifies that a potential customer with distance $x\in [0,\bar {x}]$ will choose to seek service with probability $q(x)$ and not to seek service with probability $1-q(x)$. A pure strategy of either joining or not joining (i.e., $q(x)\in \{0,1\}$) is naturally a special case. An equilibrium, denoted by subscript e, is such that no customer can improve his or her expected utility by a unilateral change of strategy, that is, any customer cannot benefit from changing his or her probability of seeking service while all other customers stick to the equilibrium strategy.

Given an equilibrium strategy $q_e(x)$ for $x\in [0,\bar {x}]$, (i) the service provider’s system throughput (or the effective arrival rate of customers to the system) is

$$\displaystyle \begin{aligned} \lambda_{e}=\Lambda\int_{0}^{\bar{x}}q_e(x)/\bar{x}dx, \end{aligned} $$

(6.2)

(ii) customer welfare, denoted by CW, is the total utility of all customers given by

$$\displaystyle \begin{aligned} CW=\Lambda \int_{0}^{\bar{x}}q_e(x)U(x)/{\bar{x}}dx, \end{aligned} $$

(6.3)

and (iii) social welfare, denoted by SW, is the summation of customer welfare and service revenue, given by

$$\displaystyle \begin{aligned} SW=CW+B\lambda_{e}=\Lambda \int_{0}^{\bar{x}}q_e(x)[U(x)+B]/{\bar{x}}dx. \end{aligned} $$

(6.4)

2.1 FIFO Service Discipline

As a benchmark, we first consider the ordinary FIFO service discipline when all customers who choose to obtain the service will be served according to the order of arrival to the system. We use the superscript F to denote the FIFO case. The expected wait time to obtain service ($W^F$) is the same for all customers under FIFO, however, travel cost increases in the distance customers must travel to get to the service location. This implies that

$$\displaystyle \begin{aligned} U^{F}(x)=V-B-d\cdot x-c\cdot W^F \mbox{ for } x\in[0,\bar{x}],{} \end{aligned} $$

(6.5)

where the waiting time expression is standard (for an $M/M/1$ system). Because $U^{F}(x)$ is strictly decreasing in x, there exists some threshold $x^{F}\in [0,\bar {x}]$ such that in equilibrium potential customers will choose to join the system if and only if their distance satisfies $x<x^{F}$ where $x^{F}$ can be uniquely determined by

$$\displaystyle \begin{aligned} x^{F}=\max\{x\vert U^{F}(x)\geq 0, x\in[0,\bar{x}]\}. \end{aligned} $$

(6.6)

This implies the (unique) equilibrium strategy is a pure strategy, which can be characterized as

$$\displaystyle \begin{aligned} q_{e}^{F}(x)=\left\{ \begin{array}{ll} 1 & \mbox{if}\ x\in[0,x^{F}]; \\ 0 & \mbox{if}\ x\in(x^{F},\bar{x}]. \end{array} \right. \end{aligned}$$

We can derive the system throughput, customer welfare and social welfare under the FIFO service discipline based on (6.2)–(6.4) accordingly.

3 Distance-Based Service Priority Policy

We now study customer behavior under the distance-based service priority policy which will be simply referred to as the “priority policy” when there is no ambiguity and denoted by the superscript P for the remainder of the chapter. Under such a policy, a customer who arrives at the system with distance x will receive (preemptive) service priority over any customers in the system with a distance $x'<x$.¹ That is, the priority of customers increases with the distance they must travel to get to the service location. It is worth noting that there is a continuum of priority levels because the distances between the customers’ locations and the service provider’s location are continuously distributed. We assume that the service provider can verify customers’ distance information. This can be done at the facility check-in where customers present their IDs.

We define the distance of the farthest potential customer who decides to seek service under an equilibrium strategy as $\tilde {x}$. Based on (6.1), we must have $\tilde {x}=\max \{x\vert V-B-d\cdot x-c/\mu \geq 0, x\in [0,\bar {x}]\}$ because this customer will receive service priority over any other customers who have decided to seek service and his or her expected waiting time to obtain service is simply the expected service time $1/\mu$. It can be derived that

$$\displaystyle \begin{aligned} \tilde{x}=\min\left\{\bar{x}(S-1)/T,\bar{x}\right\} {} \end{aligned} $$

(6.7)

and any customers with distance $x>\tilde {x}$ will not join the service system under the equilibrium.

When a strategy $q(x)$ is adopted by all customers with distance $x<\tilde {x}$, the expected utility for a customer with a particular distance x ($x\leq \tilde {x}$) is given by

$$\displaystyle \begin{aligned} U^{P}(x)=V-B-d\cdot x-{c}\cdot W^P(x) \end{aligned} $$

(6.8)

where $W^P(x)$ is the expected wait time to obtain service upon arrival and can be derived as

$$\displaystyle \begin{aligned} W^P(x)={\mu^{-1}\left[1-\rho\left(\int_{x}^{\tilde{x}}q(t)dt\right)/\bar{x}\right]^{-2}},{} \end{aligned} $$

(6.9)

where the customer with distance x will not be affected by customers with a distance smaller than x because of service priority, and the fraction of customers who own priority over this customer is $\left (\int _{x}^{\tilde {x}}q(t)dt\right )/\bar {x}$, see, e.g., Eqn (9) in Haviv and Oz (2018). We derive customers’ equilibrium strategy for the distance-based service priority policy in the following propositions.

Proposition 1

When$T\leq S-1$, we have$\tilde {x}=\bar {x}$and customers’ equilibrium strategy under the priority policy is characterized by$q_{e}^{P}(x)=\frac {T}{2\rho }\left (S-{T x}/{\bar {x}}\right )^{-\frac {3}{2}}$for$x\in [0,\max \{\hat {x},0\})$and by$q_{e}^{P}(x)=1$for$x\in [\max \{\hat {x},0\},\bar {x}]$, where$\hat {x}\in [\bar {x}-\bar {x}/\rho ,\bar {x}]$uniquely solves$S=T\hat {x}/\bar {x}+[1-\rho +\rho \hat {x}/\bar {x}]^{-2}$.

Proposition 2

When $T>S-1$ , we have $\tilde {x}=(S-1)\bar {x}/T<\bar {x}$ and customers’ equilibrium strategy under the priority policy is given as follows.

• (1) If$T\in (0, 2\rho ]$, we have$q_{e}^{P}(x)=\frac {T}{2\rho }\left (S-{T x}/{\bar {x}}\right )^{-\frac {3}{2}}$for$x\in [0,\tilde {x}]$and$q_{e}^{P}(x)=0$for$x\in (\tilde {x},\bar {x}]$.

• (2) If$T\in (2\rho ,(\sqrt {S}+ S)\rho ]$, we have$q_{e}^{P}(x)=\frac {T}{2\rho }\left (S-{T x}/{\bar {x}}\right )^{-\frac {3}{2}}$for$x\in [0,\check {x}]$, $q_{e}^{P}(x)=1$for$x\in [\check {x},\tilde {x}]$and$q_{e}^{P}(x)=0$for$x\in (\tilde {x},\bar {x}]$, where$\check {x}={[2(S-T/\rho )-1+\sqrt {4T/\rho +1}]\bar {x}}/{(2T)}$.

• (3) If$T\in (\rho (\sqrt {S}+ S),\infty )$, we have$q_{e}^{P}(x)=1$for$x\in [0,\tilde {x}]$and$q_{e}^{P}(x)=0$for$x\in (\tilde {x},\bar {x}]$.

Propositions 1–2 fully characterize customers’ equilibrium strategies under the priority policy for the entire parameter space. Note that for given system parameters ($\Lambda , \mu , V, B, c, d, \bar {x}$), customers’ equilibrium strategy is unique. We illustrate the four cases described in Proposition 1 and Proposition 2/(1),(2),(3) in Fig. 6.1a–d, respectively. In particular, the solid lines in Fig. 6.1 represent customers’ equilibrium strategies under the priority policy while the dash lines correspond to customers’ equilibrium strategies under the FIFO service discipline as a benchmark.

Fig. 6.1

Illustration of equilibrium strategies under the FIFO and distance-based service priority policies. (Note that $S=2.5$ and $\rho =\mu =c=\bar {x}=1$ for all panels while the value of T varies)

Some interesting observations emerge from Fig. 6.1. First, while under the FIFO service discipline customers’ equilibrium strategy in terms of the probability of seeking service is always monotone decreasing in their distance to the service location, it is no longer the case under the priority policy. Under the priority policy, customers’ equilibrium strategy in terms of the probability of seeking service can be monotone increasing (Fig. 6.1a), monotone decreasing (Fig. 6.1d) or even non-monotonic (Fig. 6.1b–c). This is because customers’ hassle cost to obtain service is no longer monotone increasing with distance like in the FIFO case. In particular, customers’ hassle cost consists of travel cost to get to the service location and expected wait time/cost to obtain service once they arrive. Under the priority policy, although the travel cost continues to increase in customers’ distance from the service location, the expected wait time/cost to obtain service is decreasing. As a result, customers who live at a medium distance from the service location may end up with the highest motivation to seek service as they can avoid paying high waiting costs incurred by the short-distance customers and avoid paying high travel costs incurred by the long-distance customers (see, e.g., Fig. 6.1c). This leads to the non-monotonic equilibrium strategy among customers under the priority policy where the probability of seeking service first increases and then decreases in customers’ distance from the service location. The cases of monotone increase and monotone decrease can be regarded as special cases of the general first-increase-then-decrease structure and will be discussed in greater detail below.

We observe from Fig. 6.1 (by comparing the solid lines on the four subfigures) that as T increases progressively from 1 to 2, then to 3, and then to 6, fewer and fewer long-distance customers choose to seek service in equilibrium under the priority policy. In other words, a greater value of the normalized travel cost discourages long-distance customers from seeking service compared to a smaller value under the priority policy. Recall that an increase in the normalized travel cost can be the result of an increased unit travel cost (d) or a longer distance over which the customers are distributed ($\bar {x}$). Nevertheless, either change will reduce the priority policy’s effectiveness in attracting long-distance customers to join the service system because an increased travel cost ($d\uparrow$ and/or $\bar {x} \uparrow$) makes the waiting-cost savings for long-distance customers due to distance-based service priorities less significant relative to travel costs.

4 Comparison between Priority and FIFO Policies

To compare the (distance-based service) priority policy to the FIFO benchmark, we assume that system parameters (i.e., $\Lambda , \mu , V, B, c, d, \bar {x}$) are the same under the two. We first compare the equilibrium system throughput in the following result.

Proposition 3

Comparing $\lambda _{e}^{P}$ and $\lambda _{e}^{F}$ , we have the following.

• (1) If$T\leq S-\sqrt {S}$, then$\lambda _{e}^{P}= \lambda _{e}^{F}=\Lambda$for$\rho \leq \bar {\rho }_{P}$and$\lambda _{e}^{P}< \lambda _{e}^{F}$for$\rho > \bar {\rho }_{P}$.

• (2) If$S-\sqrt {S}<T\leq S-1$, then$\lambda _{e}^{P}= \lambda _{e}^{F}=\Lambda$for$\rho \leq \bar {\rho }_{F}$, $\lambda _{e}^{P}> \lambda _{e}^{F}$for$\bar {\rho }_{F}< \rho <T/S$, and$\lambda _{e}^{P}< \lambda _{e}^{F}$for$\rho > T/S$.

• (3) If$T>S-1$, then$\lambda _{e}^{P}> \lambda _{e}^{F}$for$\rho < T/S$, and$\lambda _{e}^{P}< \lambda _{e}^{F}$for$\rho > T/S$.

Proposition 3 provides a full comparison between $\lambda _{e}^{P}$ and $\lambda _{e}^{F}$. On the one hand, when the system load is sufficiently low and the normalized travel cost is small, both the FIFO and the priority policies achieve full market coverage (i.e., $\lambda _{e}^{P}= \lambda _{e}^{F}=\Lambda$), see Proposition 3/(1). This is because the travel and service waiting costs are both low for all customers who will enjoy joining the service under either policy. On the other hand, when the system load is sufficiently high, the FIFO policy induces a strictly higher system throughput than the priority policy, see Proposition 3/(2)–(3)—recall we commented earlier that customers are more reluctant to join the service under the priority policy with an increase in the system workload. Therefore, the priority mechanism is efficient in improving the system throughput compared to the FIFO policy only when the system load is intermediate and the travel cost is large. We show in the following result that for a given system load, the priority policy results in higher system throughput than the FIFO policy as long as the travel cost is sufficiently large.

Corollary 1

For any given$\rho$, we have that$\lambda _{e}^{P}\geq \lambda _{e}^{F}$if$T> \max \{\rho S, S-\sqrt {S}\}$.

Corollary 1 reveals that the priority policy can improve the system throughput compared to the FIFO scheme, especially when the (normalized) travel cost is sufficiently large. Recall from the discussion after Propositions 1–2 that when the normalized travel cost is small, the priority policy gives long-distance customers significant incentives to join the system. However, this is at the cost of losing medium- and short-distance customers (see, e.g., Fig. 6.1a) because these customers would be overtaken by long-distance customers while waiting to obtain service and thereby have reduced incentives to seek service under the priority policy. Overall, the extra throughput brought by long-distance customers under the priority policy cannot compensate for the loss of throughput caused by medium and short-distance customers, making the total throughput less than that of the FIFO benchmark.

In contrast, when the normalized travel cost is sufficiently large, the priority policy (or the FIFO policy) does not offer long-distance customers enough incentives to join the system. However, customers who live at a medium distance from the service location and who would not join the system under the FIFO scheme will now have incentives to join under the priority policy because they are given service priorities (over the short-distance customers) which reduces their expected wait time to obtain service. This may discourage a portion of the short-distance customers who would join the system under the FIFO scheme from seeking service under the priority policy. However, the rest of the short-distance customers, if not all of them, can be retained under the priority policy because their travel costs are low enough to tolerate the increase in the service waiting time. Overall, the extra throughput brought by medium-distance customers under the priority policy exceeds the loss of throughput caused by short-distance customers, making the total throughput exceed that of the FIFO scheme (see, e.g., Fig. 6.1c–d).

As mentioned earlier, increasing system throughput is critical for service providers. As such, we quantify the potential of the priority policy to increase the system throughput in the next result. We define the throughput improvement (of the priority policy over the FIFO policy) as $TI(T, S)=(\lambda _{e}^{P}-\lambda _{e}^{F})/\lambda _{e}^{F}$.

Proposition 4

The service throughput/coverage can be increased by up to 50% by the priority policy.

Proposition 4 suggests that by simply adapting to the priority policy, service providers can increase system throughput by as much as half, which makes the priority policy very enticing. We supplement the result in Proposition 4 using a contour plot (Fig. 6.2) to show how the throughput improvement varies for different normalized travel cost T and service value S (note that $S>1$ due to our model assumption in Sect. 2). In Fig. 6.2, each curve in the contour plot joins points of equal value of the throughput improvement $TI(T,S)$.

Fig. 6.2

Contour plot of throughput improvement, $TI(T, S)$. Note.$\Lambda =\mu =c=B=\bar {x}=1$ while T and S vary ($S>1$ by model assumption in Sect. 2)

We observe from Fig. 6.2 that first, the priority policy increases system throughput compared to the FIFO benchmark if and only if the normalized travel cost T is large (as predicted by Corollary 1). Second, the priority policy is most effective in increasing system throughput when the normalized service value S is small and the normalized travel cost T is about twice as large (e.g., when S is close to 1 and T is close to 2). This is because when the service value is relatively low, the service would not attract any customers who live more than a mere short distance away under the FIFO service discipline, making the priority policy more effective in increasing throughput because of its ability to attract medium- and long-distance customers. Given a value of S, an intermediate T value achieves the largest percentage of throughput improvement under the priority policy over the FIFO policy which corresponds to the right incentive level that attracts the most medium- and short-distance customers to join the system. In contrast, when the value of T is too small, only long-distance customers join the system and throughput cannot be improved by the priority policy compared to the FIFO policy, as explained after Corollary 1. When the value of T is too large, no long-distance and too few medium-distance customers are attracted to the service, again diminishing the effectiveness of the priority policy in terms of improving throughput (also see the graphical illustrations in Fig. 6.1c–d).

The direct consequence of an increase in system throughput when customers’ travel costs are sufficiently large, is higher server utilization and longer average wait time to obtain service. In particular, the wait time for short-distance customers will notably increase because they are de-prioritized for service requests under the (distance-based service) priority policy. To this end, we next compare welfare of the priority policy to that of the FIFO policy when T is sufficiently large.

Proposition 5

For any given S, there exists$\hat {T}$such that$SW^{P}>SW^{F}$and$CW^{P}<CW^{F}$if$T>\hat {T}$.

Proposition 5 shows that when the normalized travel cost is sufficiently large (i.e., when $T>\hat {T}$), the priority policy not only increases the throughput of the system (Corollary 1) but also achieves higher overall social welfare which is the sum of customer welfare and service revenue. However, the increase in social welfare is driven by the higher service revenue (i.e., $B\lambda _{e}^{P}>B\lambda _{e}^{F}$) because customer welfare is reduced (i.e., $CW^{P}<CW^{F}$). It can be shown that $\hat {T}\rightarrow \infty$ when $B\downarrow 0$. Therefore, when the underlying service provided is free for the customers (i.e., $B=0$), social welfare which is the same as customer welfare will be reduced under the priority policy compared to the FIFO policy (i.e., $SW^{P}=CW^{P}<CW^{F}=SW^{F}$ when $B=0$).

Moreover, Proposition 5, together with Corollary 1, shows that when the normalized travel cost is sufficiently large (i.e., when $T>\hat {T}$), the average customer utility of those who join the system is smaller under the priority policy compared to the FIFO policy (i.e., $CW^{P}/\lambda _{e}^{P}<CW^{F}/\lambda _{e}^{F}$ because $CW^{P}<CW^{F}$ by Proposition 5 and $\lambda _{e}^{P}>\lambda _{e}^{F}$ by Corollary 1). In Fig. 6.3, we illustrate equilibrium customer utility under the FIFO and priority policies when $T>\hat {T}$. It is clear from the figure that the priority policy is making the utility of short-distance customers lower while that of some customers with a longer distance higher. Thus, short-distance customers are asked to subsidize others under the priority policy by giving up their FIFO service rights. Essentially, hassle cost for customers who seek service includes travel cost and service waiting cost, and the priority policy tries to balance it among customers who will join the system—the nearby (resp., distant) customers incur less (resp., more) travel cost and the priority policy assigns them more (resp., less) service wait. As a result, the priority policy makes total utility more evenly distributed among customers, compared to the FIFO policy, generating more beneficiaries and resulting in higher system throughput. However, the per-capita utility among all customers who end up joining the system is reduced.

Fig. 6.3

Illustration of customer utility under the FIFO and priority policies when $T>\hat {T}$

The discussion above demonstrates that while the service provider can improve system throughput and service revenue and the society can increase social welfare by adapting to the priority policy when customers’ travel costs are sufficiently large, customers overall or on average can suffer. In other words, the social welfare is improved at the expense of customer welfare. In what follows, we propose a possible remedy to coordinate service revenue and customer welfare under the priority policy when the normalized travel cost is sufficiently large (i.e., when $T>\hat {T}$). The idea is to have the service provider give back some revenue to the customers under the priority policy so that both the service revenue and customer welfare can be improved compared to the FIFO benchmark. When the service is free of charge (i.e., $B=0$), the service provider does not collect any revenue. Therefore, we focus on the case when $B>0$ and propose a distance-based service rebate $R(x)$ for customers who decide to join the system. That is, a customer with distance x will pay an effective service price of $B-R(x)$ if he or she decides to obtain the service. We denote the corresponding equilibrium system throughput, service revenue, social welfare and customer welfare with the rebate scheme (still under the priority policy) as $\hat {\lambda }_e^{P}$, $\hat {\Pi }^{P}$, $\hat {SW}^{P}$ and $\hat {CW}^{P}$, respectively, where $\hat {SW}^{P}=\hat {\Pi }^{P}+\hat {CW}^{P}$.

Proposition 6

Consider $B>0$ . Let $R(x)=\alpha [V-B-xd-\frac {c}{\mu [1-\rho (\tilde {x}-x)/\bar {x}]^2}]$ for $x\in [0,\tilde {x}]$ where

$$\displaystyle \begin{gathered} \alpha\in \left(\frac{T (T/\rho-S+1) \left(T/\rho+S-\sqrt{(T/\rho-S)^2+4 T/\rho}\right)}{(S-1)(T-(S+1)\rho) \left(\sqrt{(T/\rho-S)^2+4 T/\rho}+T/\rho+S\right)}-1, \right.\\ \left.\frac{\mu B (T/\rho-S+1) \left(\sqrt{(T/\rho-S)^2+4T/\rho}+S-T/\rho-2\right)}{c (S-1)^2 (T/\rho-S-1)}\right). \end{gathered} $$

Then in equilibrium$\hat {\lambda }_e^{P}>{\lambda }_e^{P}$, $\hat {\Pi }^{P}>\Pi ^{F}$, $\hat {SW}^{P}>SW^{F}$and$\hat {CW}^{P}>CW^{F}$for$T>\hat {T}$.

Proposition 6 shows that the proposed rebate scheme in the proposition can enable the priority policy to achieve a win-win-win situation between the service provider, society and customers when when the normalized travel cost is sufficiently large (i.e., when $T>\hat {T}$). We now detail the results by providing more information. First, by performing extensive numerical experiments we can confirm that the rebate $R(x)$ is always smaller than the service fee B, although this remains analytically challenging to prove. Second, for any given $\alpha$, it can be shown that $R(x)$ decreases in x, that is, the rebate is less for customers with longer distance who are already compensated by higher service priority under the priority policy.

Furthermore, the $R(x)$ in Proposition 6 is structured such that it does not change any customers’ decision as to whether to seek service under the priority policy when $T>\hat {T}$. Consequently, the equilibrium with or without the proposed rebate scheme stays the same. Social welfare under the priority policy with or without the proposed rebate scheme also stays the same (i.e., $SW^P=\hat {SW}^P$) because the rebate can be simply regarded as a transfer price between the service provider and customers, which has no impact on the total social welfare. It is worth noting that there can be (possibly many) other distance-based price-adjustment mechanisms that will also enable the priority policy to achieve a win-win-win situation between the service provider, society and customers when $T>\hat {T}$. However, the one outlined in Proposition 6 retains the same equilibrium and social welfare compared to the original no-adjustment case.

Because social welfare (or total customer welfare and service revenue) under the priority policy is higher than that under the FIFO policy when $T>\hat {T}$ (Proposition 5), we can use $R(x)$ to allocate welfare between the service provider and customers (by redistributing the service provider’s revenue to the customers) so that the resultant customer welfare and service revenue under the priority policy are both higher than those under the FIFO policy. This is essentially guaranteed by the specified range of $\alpha$ given in Proposition 6: When $\alpha$ is at the lower limit of the specified range, the service provider will give the smallest rebate to the customers so that the customer welfare under the priority policy (with the rebate) is equivalent to that under the FIFO policy. In contrast, when $\alpha$ is at the higher limit of the specified range, the service provider will give back the largest rebate to the customers so that the service revenue under the priority policy (with the rebate) is equivalent to that under the FIFO policy. Finally, when $\alpha$ falls in the middle of the specified range, both customer welfare and service revenue will end up being strictly higher under the priority policy (with the rebate) than their counterparts under the FIFO policy. Increasing $\alpha$ would redistribute more of the service provider’s payoffs to the customers, leading to lower service revenue but higher customer welfare.

Finally, in Fig. 6.4, we illustrate customer utility before and after the rebate, and compare it to the FIFO benchmark. It is clear that the rebate increases customers’ utility. For the parameters used in the example in Fig. 6.4, it can be computed that the total social welfare under the priority policy and the FIFO policy satisfy $SW^P=\hat {SW}^P=0.5>0.492=SW^F$ (an increase of 1.63%). Moreover, it can be computed that ${\Pi }^P=0.333>\Pi ^{F}=0.271$ but ${CW}^{P}=0.167<CW^{F}=0.221$, so the priority policy without the rebate would increase system throughput (or service revenue) and social welfare at the expense of customer welfare, compared to the FIFO policy. However, ${\hat {\Pi }}^P=0.274>\Pi ^{F}=0.271$ (an increase of 1.11%) and ${\hat {CW}}^{P}=0.226>CW^{F}=0.221$ (an increase of 2.26%) with the rebate, implying that the rebate scheme can properly coordinate service revenue and customer welfare to make the priority policy desirable for all stakeholders including the service provider, customers, and society as a whole.

Fig. 6.4

The effect of rebate on customer utility when $\alpha =0.35$

5 Two-Dimensional Service Area

In the model above, we assumed that distances between potential customers and the service location were distributed uniformly on the support of $[0,\bar {x}]$, i.e., the probability density function for a potential service request to come from a customer with distance $x\in [0,\bar {x}]$ is $1/\bar {x}$. Implicitly, we were assuming in the main model that customers live uniformly in a linear city with the service provider located at one end of the city.

We now extend the earlier model by considering a two-dimensional circular service area where potential customers are distributed uniformly. We assume the farthest potential customers live at a distance $\bar {x}$ away from the service provider. Because all points in a plane that are equidistant from the service location form a circle, the probability density function for a potential service request to come from a customer with distance $x\in [0,\bar {x}]$ is $2\pi x/\pi \bar {x}^2=2x/\bar {x}^2$ (circumference of the circle with radius x divided by the area of the circle with radius $\bar {x}$). It is clear that the distances between potential customers and the service location are no longer uniformly distributed over $[0,\bar {x}]$. Rather, more customers are located at a larger distance from the service provider than at any smaller distance because there are more customers located on an outer circle than on an inner circle.

We assume all other model assumptions remain the same as before. Then, under the FIFO policy, there continues to exist a threshold distance $x_2^{F}\in [0,\bar {x}]$ such that in equilibrium potential customers will choose to join the system if and only if their distance satisfies $x<x^{F}$. This implies that the utility for a customer with distance x to seek service is

$$\displaystyle \begin{aligned} U_2^{F}(x)\kern-0.5pt=\kern-0.5ptV-B-d\cdot x-c\cdot W_2^F\,\mbox{for}\, x\in[0,\bar{x}]\,\mbox{where}\, W_2^F\kern-0.5pt=\kern-0.5pt[\mu-(x_2^{F})^2\Lambda/\bar{x}^2]^{-1}{} \end{aligned} $$

(6.10)

because the proportion of customers who decide to join is $\pi (x_{2}^{F})^2/[\pi \bar {x}^2]=(x_{2}^{F})^2/\bar {x}^2$, and thus the effective arrival rate is $(x_2^{F})^2\Lambda /\bar {x}^2$. It follows that $x_2^{F}$ can be uniquely determined by

$$\displaystyle \begin{aligned} x_2^{F}=\max\{x\vert U_2^{F}(x)\geq 0, x\in[0,1]\}. \end{aligned} $$

(6.11)

Based on (6.10) and (6.11), we can derive that $x_2^{F}=\min \{\tilde {x}_{2}^{F},\bar {x}\}$, where $\tilde {x}_{2}^{F}$ uniquely solves $U_{2}^{F}(\tilde {x}_{2}^{F})=0$.

For the priority policy, we continue to use $\tilde {x}$ to denote the distance of the farthest potential customer who decides to seek service under an equilibrium strategy. We have $\tilde {x}=\max \{x\vert V-B-d\cdot x-c/\mu \geq 0, x\in [0,\bar {x}]\}$ as in the main model, and it follows that $\tilde {x}=\min \left \{\bar {x}(S-1)/T,\bar {x}\right \}$. When a mixed strategy $q(x)$ is adopted by all customers with distance $x<\tilde {x}$, the proportion of customers who own priority over a tagged customer with distance x is $\pi \left (\int _{x}^{\tilde {x}}q(t)dt^2\right )/[\pi \bar {x}^2]=\left (\int _{x}^{\tilde {x}}q(t)dt^2\right )/\bar {x}^2$. It follows that the expected utility for a customer with a particular distance x ($x\leq \tilde {x}$) to seek service can be derived as

$$\displaystyle \begin{aligned} U_2^{P}(x)&=V-B-d\cdot x-{c}\cdot W_2^P(x) \mbox{ where } W^P(x)\\&={\mu^{-1}\left[1-\rho\left(\int_{x}^{\tilde{x}}q(t)dt^2\right)/\bar{x}^2\right]^{-2}} \end{aligned} $$

(6.12)

based on (6.9). With the utility function shown in (6.12), it is analytically intractable to fully characterize customers’ equilibrium strategy under the priority policy for the entire parameter space. However, we can analytically compare the FIFO and priority policies when the normalized travel cost T is sufficiently large, in the following Proposition 7.

Proposition 7

Assuming customers are distributed uniformly over a two-dimensional circular service area, there exists$\bar {T}$such that$\lambda _{e}^{P}>\lambda _{e}^{F}$, ${\Pi }^{P}>\Pi ^{F}$, $CW^{P}>CW^{F}$and$SW^{P}>SW^{F}$if$T>\bar {T}$.

Proposition 7 shows that when the service area is two-dimensional rather than one-dimensional which was assumed in the main model, the priority policy continues to improve system throughput, service revenue, and social welfare compared to the FIFO policy when the normalized travel cost is sufficiently large. However, unlike the main model, where customer welfare was lower under the priority policy than under the FIFO policy, even customer welfare is improved under the priority policy. This is because the priority policy reduces the utility of customers with shorter distances and improves that of customers with longer distances. With the two-dimensional service area under consideration, there are now significantly more customers who are farther away from the service provider, and the priority policy can improve the utility of these customers at the cost of reducing the utility of a smaller number of customers, thereby increasing overall customer welfare. This result further demonstrates the potential of the priority policy, which can lead to a win-win-win solution on its own between the service provider, customers, and society as a whole, especially when customers’ travel costs are significant.

6 Optimal Service Fee B

Thus far, when we compare the FIFO and priority policies, we assume that the service fee B is exogenously given. However, service providers can charge different service fees under the FIFO and priority policies. This may be especially true for for-profit service providers whose goal is to maximize revenue. In this section, we consider the situation by assuming that the service provider will charge optimal service fees, denoted by $B^{F}$ and $B^{P}$, under the FIFO and priority policies, respectively, such that $B^{F}=\mathop{\text{argmax}} \limits _{B\geq 0}B\lambda _{e}^{F}(B)$ and $B^{P}=\mathop{\text{argmax}} \limits _{B\geq 0}B\lambda _{e}^{P}(B).$ We assume $\rho \geq 1$ to focus on the nontrivial case where not all potential customers can be served, namely, an overloaded system. All other model assumptions remain the same as the main model in Sects. 2–4. The following two lemmas characterize $B^{F}$ and $B^{P}$, and show that they are unique for any given system parameters ($\Lambda , \mu , V, c, d, \bar {x}$).

Lemma 1

The optimal fee of the service provider under the FIFO policy, $B^{F}$ , is the unique solution of B that solves

$$\displaystyle \begin{aligned} T/\rho+S-B \left(1+\frac{T/\rho-S}{\sqrt{4 T/\rho+(T/\rho-S)^2}}\right)\mu/c-\sqrt{4 T/\rho+(T/\rho-S)^2}=0. \end{aligned}$$

where $S=\mu (V-B)/c$ as before.

Lemma 2

The optimal fee of the service provider under the priority policy, $B^{P}$ , is given by

$$\displaystyle \begin{aligned} {} B^{P}=\left\{ \begin{array}{ll} \hat{B}, & \mathit{\mbox{if}}\ T<{T}_{1}; \\ \frac{c\sqrt{1+4 T/\rho}+2 V\mu-c-2 cT/\rho}{2\mu}, & \mathit{\mbox{if}}\ {T}_{1}\leq T<{T}_{2}, \\ \frac{V \mu-c}{2 \mu }, & \mathit{\mbox{if}}\ T\geq {T}_{2}, \end{array} \right. \end{aligned}$$

where$\hat {B}\in (0,V)$is the unique solution of B that solves$2S\left (\sqrt {S}-1\right )-B\mu /c=0$, ${T}_{1}$is the unique solution of T which solves$\frac {c\sqrt {1+4 T/\rho }+2 V\mu -c-2 cT/\rho }{2\mu }=\hat {B}$, and${T}_{2}=\frac {\rho [1+V \mu /c-\sqrt {2(1+V \mu /c)}]}{2}$.

What is clear from the above results is that while $B^F$ is continuous in the normalized travel cost T, $B^P$ is a piecewise function of it. This is because the system throughput is continuous in T under the FIFO policy but piecewise in T under the priority policy. Based on these results, we can derive the following comparison results on system throughput, service revenue, and social welfare in equilibrium between the FIFO and priority policies.

Proposition 8

When service fees are set optimally by the service provider, there exists${\bar {\bar {T}}}$such that$\lambda _{e}^P>\lambda _{e}^{F}$, $B^{P}\lambda _{e}^{P}(B^{P})>B^{F}\lambda _{e}^{F}(B^{F})$and$SW^{P}>SW^{F}$if$T>\bar {\bar {T}}$.

Proposition 8 indicates that when the normalized travel cost is sufficiently large, the priority policy improves system throughput, service revenue and social welfare compared to the FIFO policy. In addition, we have verified through extensive numerical experiments that when T is sufficiently large, customer welfare is lower under the priority policy than under the FIFO policy. In other words, the results shown in Propositions 1 and 5 from the main model are robust even when the service provider sets optimal service fees instead of having fixed service fees.

7 Comparison to Price Discrimination Strategy

The priority policy we propose here awards residents who live farther away from the service provider with a higher service priority, but all customers are charged the same service price. In this section, we compare the priority policy to the traditional price discrimination strategy (PDS) from the revenue management literature where for our context customers traveling different distances are charged different service prices but no one receives service priority over others.

We assume the service price decreases with the distance that customers must travel, in order to provide customers who live farther away with more incentive to seek service. In particular, we assume the service price $P(x)\geq 0$ is given by $\bar {P}-\delta \cdot x$ where $\bar {P}$ is the intercept and $\delta$ is the slope. In the rest of this section, we use the superscript D to denote the price discrimination strategy. The expected utility of a customer with distance x is therefore given by

$$\displaystyle \begin{aligned} U^{D}(x)=V-P(x)-d\cdot x-c\cdot W^{F} \mbox{ for } x\in[0,\bar{x}] \mbox{ with } W^F=(\mu-x^{F}\Lambda/\bar{x})^{-1}, \end{aligned}$$

where the waiting time expression is standard for a FIFO $M/M/1$ system. We focus on the case $\delta \leq d$ to ensure that $U^{D}(x)$ decreases in x (otherwise the customers will join if and only if their distance is sufficiently large). It follows that there exists a threshold distance $x^{D}\in [0,\bar {x}]$ such that in equilibrium potential customers will choose to join the system if and only if their distance satisfies $x<x^{D}$ where $x^{D}$ can be uniquely determined by

$$\displaystyle \begin{aligned} x^{D}=\max\{x\vert U^{D}(x)\geq 0, x\in[0,\bar{x}]\}. \end{aligned}$$

It can be derived that $x^{D}={\left [T_{D}/\rho + S_{D}-\sqrt {4T_{D}/\rho +(T_{D}/\rho - S_{D})^2}\right ]\bar {x}}/{(2T_{D})}$, where $S_{D}=\mu (V-\bar {P})/c$ and $T_{D}=\mu (d-\delta )\bar {x}/c$. Furthermore, we derive the system throughput, service revenue, customer welfare and social welfare under the price discrimination strategy. In what follows, we first compare the price discrimination policy to the (non-price-discrimination) FIFO benchmark, and then to our proposed priority policy in this chapter.

7.1 Comparing PDS to the FIFO Benchmark

Consider the (non-price-discrimination) FIFO policy with service price B as a benchmark (see Sect. 2.1). It is clear that the price discrimination policy has the potential to achieve higher system throughput, or higher service revenue, or higher social welfare, or higher customer welfare one at a time with the right choices of $\bar {P}$ and $\delta$ because the benchmark is simply a special case with $\bar {P}=B$ and $\delta =0$. However, our next result demonstrates that with carefully selected parameters, the price discrimination policy can actually achieve higher system throughput, higher service revenue, higher social welfare and higher customer welfare all at the same time, compared with the FIFO policy.

Proposition 9

For fixed$\delta$, the price discrimination policy achieves higher system throughput, higher service revenue, higher social welfare and higher customer welfare at the same time if and only if$P_{1}(\delta )<\bar {P}<\min \{P_{2}(\delta ), P_{3}(\delta )\}$, where${P}_{1}(\delta )$and$P_{2}(\delta )$are the smaller and larger roots of the equation$x^{D}[\bar {P}-\delta x^{D}/2]-Bx^{F}=0$with respect to$\bar {P}$, and$P_{3}(\delta )=V-{c \left (T_{D} \sqrt {d/(d-\delta )}x^{F}-\frac {\bar {x}^2}{\rho \sqrt {d/(d-\delta )} x^{F}-\bar {x}}\right )}/{\mu \bar {x}}$.

7.2 Comparing PDS to the Priority Policy

According to Propositions 6 and 9, the distance-based service priority policy and the price discrimination strategy can both be superior in all aspects compared to the FIFO policy. Nevertheless, how do they compare to each other? The next result reveals that they each have their own advantages. Denote by $\bar {\lambda }_{e}^{X}$, $\overline {\Pi }^{X}$ the maximal system throughput and service revenue that can be achieved, for $X=P, D$.

Proposition 10

$\overline {\Pi }^{D}>\overline {\Pi }^{P}$and$\bar {\lambda }_{e}^{D}<\bar {\lambda }_{e}^{P}$.

Proposition 10 shows that while the price discrimination strategy has the potential to achieve higher service revenue, the distance-based service priority policy has the potential to achieve higher system throughput. It is well known that price discrimination allows a producer to extract most, if not all, of customer surplus, therefore it is not surprising that the price discrimination strategy works especially well to capture revenue. In contrast, the distance-based service priority policy is more efficient in improving the system throughput compared to the price discrimination policy.

8 Concluding Remarks

In this chapter, we introduced an innovative distance-based service priority policy (shortened as “priority policy” below). The idea is to assign higher service priority to customers who must travel farther for a physical service that requires waiting. As a result, customers who are located far away from the service location can save waiting time to obtain service, which provides them with a new incentive to consider seeking service—these customers would not have been interested in seeking the service under the first-come-first-served policy due to their high travel costs and the high waiting costs to obtain service.

We demonstrated that the priority policy can significantly increase system throughput by attracting more customers to seek service, and the increase can be up to 50% compared to the ordinary first-come-first-served service discipline. We then showed that the priority policy can increase social welfare while benefiting the service provider. This, however, may come at the cost of customer welfare. We proposed a possible remedy to coordinate service revenue and customer welfare when the situation happens, making the priority policy beneficial to all stakeholders.

We now discuss some practical and fairness issues related to implementing the priority policy to conclude this chapter. For the policy to work, it is important that the service provider be transparent about the policy and disclose its procedure to all potential customers so that they can make informed decisions. The service provider also needs to be able to verify the residence information of any customers who choose to seek service and maintain a proper service sequence based on their distance-based service priorities. Verification of the residence information can be done at the facility check-in or by installing self-service check-in kiosks with ID and address authentication capability. Such technology is widely available at airports, casinos and government agencies. Customers can then sit in an open waiting area and they will be called by name (or by their ticket number if tickets are issued from the check-in kiosks) when it is their turn for service. The service provider can then ask customers to go up for service in the order of their service priority, effectively achieving a continuum of priority levels. In practice, it may also be more feasible to implement a few discrete distance-based service priority classes. This would be similar to airlines’ boarding lines for numbered groups (e.g., Groups 1–5). Group 1 can include customers who live farthest from the service location, and they will be given the highest service priority. Group 2 is the second farthest with the next highest service priority, followed by Group 3, and so on. There should be clear signs to guide arriving customers to their designated line.

Finally, we argue that the associated fairness concerns for the distance-based service priority policy may be less protruding compared to the traditional price discrimination strategy. This is because many people consider price discrimination unfair, and price discrimination can even be unlawful under certain circumstances (e.g., gender-based insurance premium prices). The priority policy, however, is more about helping disadvantaged people—that is, providing kindness and convenience to those who live in remote areas and have to travel long distances to reach the service provider. In general, people experience happiness and a sense of fulfillment when helping those in need, and recent research in service operations further suggests that customers are willing to sacrifice their own utility to help others waiting in line (Ülkü et al. 2023). This means that the priority policy is unlikely to receive push-backs among de-prioritized customers who live near the service provider.