Definition 1

Online Advance Admission Scheduling for Services with Customer Preferences

Xinshang Wang, Van-Anh Truong

Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, USA, xw2230@columbia.edu, vatruong@ieor.columbia.edu

David Bank, MD, MBA

Department of Pediatrics, NYPH Morgan Stanley Children’s Hospital, Columbia University Medical Center, New York, NY, USA, deb40@columbia.edu

We study web and mobile applications that are used to schedule advance service, from medical appointments to restaurant reservations. We model them as online weighted bipartite matching problems with non-stationary arrivals. We propose new algorithms with performance guarantees for this class of problems. Specifically, we show that the expected performance of our algorithms is bounded below by times that of an optimal offline algorithm, which knows all future information upfront, where is the minimum capacity of a resource. This is the tightest known lower bound. This performance analysis holds for any Poisson arrival process. Our algorithms can also be applied to a number of related problems, including display ad allocation problems and revenue management problems for opaque products. We test the empirical performance of our algorithms against several well-known heuristics by using appointment scheduling data from a major academic hospital system in New York City. The results show that the algorithms exhibit the best performance among all the tested policies. In particular, our algorithms are more effective than the actual scheduling strategy used in the hospital system according to our performance metric.

 

We study advance admission scheduling decisions in service systems. Advance admission scheduling decisions are those that determine specific times for customers’ arrival to a facility for service. Advance admission scheduling is used in many service industries. Restaurants reserve tables for customers who call in advance. Healthcare facilities reserve appointment slots for patients who request them. Airlines reserve flight seats for those who purchase flight tickets. Advance admission scheduling enables service providers to better match capacity with demand because they control customers’ actual arrivals to service facilities.

We formulate and analyze a model that generally captures such admission scheduling systems. For concreteness, we focus on the example of MyChart, a digital admission scheduling application developed by Epic System. Epic is an electronic medical records company that is managing the records of millions of health care providers and more than half of the patient population in the U.S. (Husain 2014). Epic deploys MyChart to perform online scheduling of appointments through internet portals. The use of applications like MyChart is part of a general trend in healthcare towards providing electronic access to service through web and mobile applications (TechnologyAdvice 2015).

When a patient schedules an appointment over a web portal, MyChart first asks the patient for the type of visit desired, whether it is for a physical exam, a consultation, a flu shot, etc. Next, it asks for the beginning and end of the range of preferred dates. It then shows a menu with a check box for morning and afternoon session for each day in the preferred date range. Patients can select one or more preferred sessions. Finally, MyChart either offers the patient one or more appointments, or states that no appointment can be found. We can conceive of many variations over this basic interface.

Consider the following model of advance admission scheduling that captures MyChart as an example. There are multiple service providers. Each provider offers a number of service sessions over a continuous, finite horizon. We call a session associated with a single provider a resource. Let be the number of resources available over the horizon. Each resource can serve customers. We call the capacity of resource . Each resource must be booked by time or it perishes at time . There are customer types. Patients of type , , arrive according to some known non-homogeneous Poisson process and make reservations through any of the modes made available by the provider, web, phone, or mobile. A patient of type generates a reward of when served with a unit of resource . We assume that the type of customers can be observed at the time that they arrive to make an appointment, through the pattern of preferences that they indicate and any data stored in the system on their profiles. We require that customers arriving at time have weight for all resources that perish at time . The number of customer types can be kept finite by discretizing the horizon but this number can be very large. We will discuss this point shortly. When a customer arrives, a unit of an available resource must be assigned to her, or she must be rejected. Each unit of a resource can be assigned to at most one customer. We allow no-shows and the practice of overbooking to compensate for the effect of no-shows. The objective of the problem is to allocate the resources to the customers to maximize the expected total reward of the allocation.

Our advance reservation model is essentially an online weighted bipartite matching problem. The resources in our model, when partitioned into units, can be seen as nodes on one side of a bipartite graph. All the customers correspond to nodes on the other side that are arriving online. The type of each arriving customer is determined by a time-varying distribution.

This resource allocation model can be found in many other applications. We summarize three such applications below.

  Ad allocation.

In a typical display ad allocation problem, e-commerce companies aim at tailoring display ads for each type of customers. Each ad, which corresponds to a resource, is often associated with a maximum number of times to be displayed. Knowing the arrival rates of future customers, the task is to make the most effective matching between ads and customers.

  Single-leg revenue management.

A special case of our model is the classic single-leg revenue management problem in which all resources to be allocated are available at the same time. Customers who bring a higher reward correspond to higher-fare classes. The decision is how to admit or reject customers, given the time remaining until the flight and the current inventory of available seats.

  Management of opaque products.

Internet retailers such as Hotwire or Priceline often offer a buyer an under-specified or opaque product, such as a flight ticket, with certain details such as the exact flight timing or the name of the airline withheld until after purchase. We assume that demand for each opaque product is exogenous and independent of the availability of other products. When demand occurs, a decision is made to assign a specific product to that demand unit. Knowing the arrival rates of all demands, we want to maximize the total expected revenue by strategically assigning specific products.

Our contributions in this work are as follows:

  • We provide a general, high-fidelity model of advance admission scheduling that captures customer preferences across different resources. We allow non-stationary arrivals and no-shows. We model the advance admission scheduling problem as an online weighted bipartite matching problem with non-stationary arrivals and propose new algorithms with guarantees on the relative performance.

  • We prove the tightest known performance bound for the online matching problem with non-stationary stochastic arrivals. Specifically, we prove that a primitive algorithm, which we call the Separation Algorithm, has expected performance that is bounded below by times that of an optimal offline algorithm, which knows all future information upfront, where is the minimum capacity of any resource. Our performance bound improves upon the lower bound of Alaei, Hajiaghayi and Liaghat (2012). Moreover, it is close to an upper bound on the performance of the Separation Algorithm that the same authors found.

    We obtain our bound by analyzing a novel bounded Poisson process. This is a Poisson process to which we apply a sequence of reflecting barriers. The process arises in the dual of an optimization problem that characterizes our performance bound. The behavior of this process is very complex, with no known closed-form description. We managed to obtain a closed-form approximate characterization of the process.

  • We improve on the Separation Algorithm by devising a novel bid-price-based algorithm, called the Marginal Allocation Algorithm, that is much more practical. First, the Marginal Allocation Algorithm is non-randomized, therefore more stable. Second, it is fair in the sense that it never rejects a high-priority customer but accepts a low-priority customer, assuming that their arrival times and preferences are the same. We prove that the Marginal Allocation Algorithm has the same theoretical performance guarantee as the Separation Algorithm. In addition, in numerical experiments, we show that it achieves much better practical performance.

  • We test the empirical performance of our algorithm against several well-known heuristics by using appointment scheduling data from a department within a major academic hospital system in New York City. The results show that our scheduling algorithms perform the best among all tested policies. In particular, our algorithm is more effective than the actual scheduling strategy used in the hospital system according to our performance metric.

Our work is related to the literature on appointment scheduling. This area has been studied intensively in recent years (Guerriero and Guido 2011, May et al. 2011, Cardoen et al. 2010, Gupta 2007). A large part of this literature considers intra-day scheduling, in which the number of patients to be treated on each day is given or is exogenous, and the task is to determine an efficient sequence of start times for their appointments. Another part of the literature considers multi-day scheduling, in which patients are dynamically allocated to appointment days. Some works in this literature focus on the number of patients to be served today, with the rest of the patients remaining on a waitlist until the next day. This paradigm is called allocation scheduling. See, for example, Huh et al. (2013), Min and Yih (2010), Ayvaz and Huh (2010), Gerchak et al. (1996). Recently, more works have focused on the problem of directly scheduling patients into future days. This paradigm is called advance scheduling. This paper considers an advance scheduling model with multiple patient classes. In the literature of advance scheduling, Truong (2014) first studies the analytical properties of a two-class advance scheduling model and gives efficient solutions to an optimal scheduling policy. For the multi-class model, no analytical result is known so far. Gocgun and Ghate (2012) and Patrick et al. (2008) propose heuristics based on approximate dynamic programming for these problems, but have not characterized the worst-case performance of these heuristics. We propose online scheduling policies with performance guarantees for a very general multi-class advance scheduling problem.

Our advance scheduling model captures the preferences of patients in a general way. Patient preferences are an important consideration in most out-patient scheduling systems. In the literature considering patient preferences, Gupta and Wang (2008) consider a single-day scheduling model where each arriving patient picks a single slot with a particular physician, and the clinic accepts or rejects the request. Our model can be seen as a multi-period generalization of their work. We also characterize the theoretical performance in an online setting, whereas they use stochastic dynamic programming as the modeling framework and develop heuristics. Feldman et al. (2014) study how to offer sets of open appointment slots to a stream of arriving patients over a finite horizon of multiple days, given that patients have preferences for slots that can be captured by the multinomial logit model. Their work is strongly influenced by assortment planning problems. An important observation, which was first made by Gupta and Wang (2008), is that there is a fundamental difference between many advance admission scheduling problems and assortment planning problems. In admission scheduling, we can often work with revealed preferences, whereas in assortment planning problems, decisions are made with knowledge only of a distribution of customer preferences. Working with revealed preferences allows for a more efficient allocation of service compared to working with opaque preferences. It also leads to more analytically tractable models.

Our work is closely related to works on online matching problems. Traditionally, the online bipartite matching problem studied by Karp et al. (1990) is known to have a best competitive ratio of for deterministic algorithms and for randomized algorithms. For the online weighted bipartite matching problem that we consider, the worst-case competitive ratio cannot be bounded below by any constant. Many subsequent works have tried to improve performance ratios under relaxed definitions of competitiveness.

Specifically, three types of assumptions are commonly used. The first type of assumption is that each demand node is independently and identically (i.i.d.) picked from a known set of nodes. Under this assumption, Jaillet and Lu (2014), Manshadi et al. (2012), Bahmani and Kapralov (2010), Feldman et al. (2009) propose online algorithms with competitive ratios higher than for the cardinality matching problem, in which the goal is to maximize the total number of matched pairs. Haeupler et al. (2011) study online algorithms with competitive ratios higher than for the weighted bipartite matching problem. Our definition of competitive ratio is the same as theirs. Our model is also similar, but we allow a more general arrival process of demand nodes in which the distribution of nodes can change over time. Previous analyses depend crucially on the fact that demand nodes are i.i.d. in order to simplify the expression for the probability that any demand node is matched to any resource node. The expression becomes much more complex, and the arguments break down in the case that demand arrivals are no longer i.i.d.

The second type of assumption is that the sequence of demand nodes is a random permutation of an unknown set of nodes. This random permutation assumption has been used in the secretary problem (Kleinberg 2005, Babaioff et al. 2008), adword problem (Goel and Mehta 2008) and the bipartite matching problem (Mahdian and Yan 2011, Karande et al. 2011). Kesselheim et al. (2013) study the weighted bipartite matching problem with extension to combinatorial auctions. Our work is different from all of these in that the non-stationarity of arrivals in our model cannot be captured by the random permutation assumption.

The third type of assumption made is that each demand node requests a very small amount of resource. The combination of this assumption and the random-permutation assumption often leads to polynomial-time approximation schemes (PTAS) for problems such as adword (Devanur 2009), stochastic packing (Feldman et al. 2010), online linear programming (Agrawal et al. 2014), and packing problems (Molinaro and Ravi 2014). Typically, the PTAS proposed in these works use dual prices to make allocation decisions. Under this third assumption, Devanur et al. (2011) study a resource allocation problem in which the distribution of nodes is allowed to change over time, but still needs to follow a requirement that the distribution at any moment induce a small enough offline objective value. They then study the asymptotic performance of their algorithm. In our model, the amount capacity requested by each customer is not necessarily small relative to the total amount of capacity available. Therefore, the analysis in these previous works does not apply to our problem.

In our model, the arrival rates, or the distribution of demand nodes, are allowed to change over time. This non-stationarity poses new challenges, because it cannot be analyzed with existing methods. At the same time, it is an essential feature in our model because it allows us to capture the perishability of service capacity in the applications that we consider. When a resource perishes within the horizon, the demand for that resource drops to . Such a demand process must be time-varying. This important feature has received only limited attention so far. Ciocan and Farias (2012) consider an allocation model with a very general arrival process, but their allocation policy has performance guarantee only when the arrival rates are uniform. In this paper, we allow arrival processes to be non-homogeneous Poisson processes with arbitrary rates.

Our algorithms solves a linear program and uses its optimal solution to make matching decisions. The idea of using optimal solutions to a linear program is natural and has been used by several previous works mentioned above. For example, Feldman et al. (2009), Manshadi et al. (2012), Haeupler et al. (2011), and Kesselheim et al. (2013) have used similar algorithms to obtain constant competitive ratios, albeit for different demand models.

The paper of Alaei et al. (2012) solves an online matching problem with non-stationary arrivals in a discrete-time setting. They propose an algorithm similar to our Separation Algorithm, which is a primitive algorithm that we analyze initially and later improve upon. They prove that this algorithm achieves a competitive ratio of at least and at most approximately , where is the minimum capacity of a resource. Compared to Alaei et al. (2012), we prove a stronger lower bound of on the competitive ratio for our Separation Algorithm, using a few of the same ideas but largely different techniques, as we will elaborate on in Section id1. Thus, our lower bound is more similar to their upper bound. We also point out that the Separation Algorithm is not practical because it might route customers to resources that are already exhausted, while there are still other available resources. More importantly, because of randomization, it might reject a high-priority customer, but accept a low-priority customer at nearly the same time. For this reason, we propose a new “bid-pricing” algorithm, based on the Separation Algorithm, that avoids all of the above problems. We prove that the improved algorithm has the same theoretical performance guarantee, and has much better computational performance as tested on real data.

Our work is also related to the revenue management literature. We refer to Talluri and van Ryzin (2004) for a comprehensive review of this literature. Traditional works in this area assume that demands for products are exogenous and independent of the availability of other products (Lautenbacher and Stidham 1999, Lee and Hersh 1993, Littlewood 1972). The decision is whether to admit or reject a customer upon her arrival. Our model reduces to this admission control problem in the special case that the resources are identical and are available at the same time.

When customers are open to purchase one among a set of different resources, our model controls which resource to assign to each customer. Thus, our model captures the problem of managing opaque products. Sellers of an opaque product conceal part of the products’ information from customers. Sellers have the ability to select which specific product to offer after the purchase of opaque product. This enables the seller to more flexibly manage their inventory. Opaque products are often sold at a discount compared to specific products, making them attractive to wider segments of the market. These products are common in internet advertising, tour operations, property management (Gallego et al. 2004) and e-retailing. Customers purchase an opaque product if the declared characteristics fit their preferences. The buyer agrees to accept any specific product that meets the opaque description. In our model, a specific product corresponds to a node on the right side of a bipartite graph. A unit of demand for an opaque product corresponds to a node on the left that connects to all of the specific products contained in the opaque product. The weight of an edge corresponds to the revenue earned by selling the opaque product.

Previous works related to opaque products include Gallego and Phillips (2004), Fay and Xie (2008), Petrick et al. (2010), Chen et al. (2010), Lee et al. (2012), Gönsch et al. (2014) and Fay and Xie (2015). Due to the problem of large state space, most analyses focus on models with very few product types. For systems with many product types, some pricing and allocation heuristics are known. There is numerical evidence that much of the benefit of opaque products can be obtained by having two or three alternatives (Elmachtoub and Wei 2013). However, when a retailer has a large number of alternative products, it is unclear how to design such an opaque product. Our work focuses on online allocation policies with constant performance guarantees for the management of an opaque product with an arbitrary number of alternatives.

Our model assumes independent demands, i.e., the demand for each product is exogenous and independent of the availability of other products. Many recent works in revenue management consider endogenous demands, which means that customers who find their most preferred product unavailable might turn to other products. Examples of works on dependent demands include Gallego et al. (2004), Zhang and Cooper (2005), Liu and van Ryzin (2008) and Gallego et al. (2015). One of the main characteristics of these models is that customer preferences cannot be observed until purchase decisions are made. In such situation, sellers only have a distributional information of customer preferences. This phenomenon does not apply to admission scheduling systems. In these systems, customer preference can be revealed before a unit of a resource is assigned. In MyChart, for example, the system is able to customize the appointment to offer to each patient after knowing the patient’s profile and availability. We assume that each customer’s preference is observed before a resource is assigned. Knowledge of preferences gives service providers the ability to improve the efficiency of the resource allocation process by tailoring the service offered to each customer.

Our work is related to the still limited literature on designing policies for revenue management that are robust to the distribution of arrivals. Ball and Queyranne (2009) analyze online algorithms for the single-leg revenue management problem. Their performance metric is the traditional competitive ratio that compares online algorithms with an optimal offline algorithm under the worst-case instance of demand arrivals. They prove that the competitive ratio cannot be bounded below by any constant when there are arbitrarily many customer types. In our work, we relax the definition of competitive ratio, and show that our algorithms achieve a constant competitive ratio (under our definition) for any number of customer types and for a more general multi-resource model. Qin et al. (2015) study approximation algorithms for an admission control problem for a single resource when customer arrival processes can be correlated over time. They use as the performance metric the ratio between the expected cost of their algorithm and that of an optimal online algorithm. Our performance metric is stronger than theirs as we compare our algorithms against an optimal offline algorithm, instead of the optimal online policy. Qin et al. (2015) prove a constant approximation ratio for the case of two customer types, and also for the case of multiple customer types with specific restrictions. They allow only one type of resource to be allocated. In our model, we assume arrivals are independent over time, but we allow for multiple customer types and multiple resources without additional assumptions.

Throughout this paper, we let denote the set of positive integers. For any , let denote the set .

There are resources and customer types. Customers of each type randomly arrive over a continuous horizon according to a known non-homogeneous Poisson process with rate , for . Let be the expected total number of arrivals of type- customers. Each resource has a capacity of units.

When a customer arrives, one unit of capacity of an available resource must be assigned to the customer, or the customer must be rejected. A customer of type earns a reward if assigned to resource . The objective is to allocate the resources to the customers to maximize the expected total reward from all of the allocated resources.

This model captures the expiration of resources in the following sense. Suppose we assign an expiration time to each resource . Then, for any customer type such that for some , we require . In this way, the reward from assigning resource to any customer who arrives after the expiration time is .

Let be the actual total number of arrivals of type customers. We must have , for all . An offline algorithm knows at the beginning of the horizon. Let OPT be the optimal offline reward given the number of arrivals . Note that an optimal offline algorithm does not need to know the time of each arrival, as the algorithm essentially solves a maximum weighted matching problem, between the customers and resources. An online algorithm, however, does not know the entire sample path of future arrivals, but only knows the arrival rates , . In this paper, we define the competitive ratio as the ratio between the expected reward of an online algorithm and the expected reward of an optimal offline algorithm.

Definition 1

An online algorithm is -competitive if its total reward ALG satisfies

where the expectation is taken over the sample path of customer arrivals (the random vector is determined by the sample path of arrivals).

Before introducing our online algorithms, we first characterize an optimal offline algorithm and an upper bound on the optimal offline reward.

In the offline case, the total number of arrivals of each customer type is known, and the exact arrival time is irrelevant. Given the ’s, the maximum offline reward OPT can be computed by solving a maximum weighted matching problem, which can be formulated as the following LP:

(1)

where the decision is the number of type- customers who are assigned to resource . Let be an optimal solution to this LP. Then OPT.

We are interested in finding an upper bound on the expected optimal offline reward . We next show that LP (2), which uses instead of as the total demand, gives such an upper bound:

(2)
Theorem 1

The optimal objective value of (2) is an upper bound on .

\@trivlist

Since and , we must have and . Thus, is a feasible solution to the LP (2). It follows that the optimal objective value of (2) is an upper bound on

Similar techniques have been used in revenue management to prove similar results (Gallego and van Ryzin 1997).  \@endparenv

In this section, we propose the Separation Algorithm. The algorithm works by solving the LP (2) once, routing the customers to the resources according to an optimal solution to the LP (2). Then, for each resource separately, the algorithm optimally controls the admission of customers who have been routed to that resource. Using the LP information with respect to the expected number of arrivals (or sometimes, an estimate of the expected number of arrivals) is natural and has been used in several previous results (for example, Feldman et al. (2009), Manshadi et al. (2012), Haeupler et al. (2011), and Kesselheim et al. (2013)).

Let be an optimal solution to the linear program (2). Whenever a customer of type arrives, the Separation Algorithm randomly and independently picks a candidate resource with probability , regardless of the availability of resources. We say that this customer is routed to resource . According to the Poisson thinning property, the arrival process of type- customers who will be routed to resource is a non-homogeneous Poisson process with rate

(3)

Viewing the random routing process as exogenous, each resource receives an independent arrival process with split rate from each customer type . Then for each resource , the Separation Algorithm optimally controls the admission of customers who will be routed to resource . That is, when a type- customer is routed to resource at time , the algorithm compares with the marginal cost of taking one unit away from resource , where the marginal cost is computed based on the future customers who will be routed to resource . The customer is accepted and offered resource if is higher than or equal to the marginal cost. The customer is rejected if is smaller than the marginal cost or if resource has no remaining capacity.

For each resource , let denote the amount of resource that remains at time . Given the exogenous random routing process, we define as the optimal future reward of the admission control problem for resource . is governed by the Hamilton-Jacobi-Bellman equation

(4)

The boundary conditions are for all , and for all . According to properties of the HJB equation, is decreasing in , which captures the fact that resources are expiring over time. We call the reward function for resource . In practice, the continuous-time dynamic programming (4) can often be solved by discretizing the horizon (for example, see Arslan et al. (2015)).

Below are the detailed steps of the Separation Algorithm:

  1. Solve LP (2). Let be any optimal solution.

  2. For each resource , compute the reward function according to (4).

  3. Upon an arrival of a type- customer at time , randomly pick a number with probability . Assign resource to the customer if resource has positive remaining capacity and .

The following proposition states that the total expected reward of the Separation Algorithm is given by the reward functions. We omit the proof as this result directly follows from properties of the HJB equation.

Proposition 1

Conditioned on the state , the Separation Algorithm earns reward from resource in time in expectation. In particular, the expected total reward of the Separation Algorithm is .

In this section, we show that if is the minimum capacity of any resource, then the competitive ratio of the Separation Algorithm is . This result is stated in Theorem 4.

To prove the competitive ratio, we fix a resource and focus on the ratio

(5)

where is the expected reward that the Separation Algorithm earns from resource , and is an upper bound on the optimal expected offline reward from resource (see LP (1)).

We want to lower-bound (5) by examining all possible inputs and . As we will prove a lower bound that increases in the capacity value, the worst case for our analysis is , where is the minimum capacity of any resource. Thus, for the rest of the section, we suppose and analyze the ratio .

We apply Jensen’s inequality to the HJB equation (4) to obtain

Thus, the performance of the Separation Algorithm can be lowered by replacing the problem instance with one in which there is only one type of customer arrival with arrival rate and reward rate , so that the worst-case instance has one customer type, and time-dependent reward value . This observation has also been made by Alaei et al. (2012).

Furthermore, by (3) and definition of , we can obtain

Thus, to characterize the worst-case performance ratio for the fixed resource , we only need to lower-bound

(6)

where is the new reward function defined based on and :

(7)

with boundary conditions for all and for all .

Note that the HJB equation (7) is different from (4), as (7) is defined by a different arrival rate and reward rate. Moreover, we use to denote the consumed inventory in (7), whereas in (4), we used to denote the remaining inventory. This change is convenient for our analysis.

In order to lower-bound (6), we need to examine all possible reward rates and arrival rates such that the constraints of (2) are satisfied. The first constraint of (2) is satisfied by definition of and . The second constraint of (2) requires

(8)

Without loss of generality, we can change the horizon length, the arrival rate and the reward rate as follows, while keeping the ratio (6) unchanged:

  1. If the inequality (8) is not tight, i.e., , we can extend the horizon to length by adding more arrivals with reward . Thus, we can equivalently assume .

  2. Let be the length of the (possibly extended) horizon such that . Define a virtual time

    We must have for all . Using this new time variable , we can define new reward functions as

    where we interpret as the first time that satisfies . Similarly, we can define . Then we can equivalently transform the HJB equation for as follows

    This equation can be viewed as another HJB equation with arrival rate and reward rate , with boundary conditions for and for . Furthermore, the upper bound on the expected offline reward can be transformed as

    In summary, we can equivalently transform the problem into one whose arrival rate is uniformly and whose time horizon is .

After applying the above transformations, we can write an optimization problem that reveals the competitive ratio as follows

(9)
s.t.

Here the second constraint normalizes the upper bound on the expected offline reward. By using and replacing with linear constraints, we can write the above problem equivalently as (note that )

s.t.

Let be a dual variable for the first constraint, for all and . Let be a dual variable for the second constraint. The dual problem is

(10)

This dual problem tries to maximize the minimum value of with respect to . The optimal is a lower bound on the competitive ratio that we seek to characterize.

We first show that a feasible solution to the dual problem (10) can be constructed based on a modification of a Poisson process. As we shall explain shortly, this is a Poisson process to which we apply a control, using a sequence of bounding barriers. We will use the solution obtained via this derived process to obtain a lower bound on the optimal value of the bound-revealing optimization problem (9). We will refer to the process as a bounded Poisson process. Alaei et al. (2012) also prove their bound by working with a dual-feasible solution. However, we construct our dual-feasible solution differently using a novel method. Because our bound has to be tighter, our analysis of this solution is also much more involved.

Let be a sequence of time points such that .

Figure 1: Illustration of the bounded Poisson process. Dashed red line is the barrier. Solid blue line is the process. The barrier is active when the two lines overlap.

Let be a (counting) Poisson process with rate . We apply an upper barrier to to obtain a new bounded process . Figure 1 illustrates this process. Starting with an initial value at time , the barrier increases by at times . At these time points, the new bounded process has values

with . And for , we have

Eventually,

Theorem 2

There exists a feasible dual solution , for , , such that

(11)

for the bounded Poisson process as constructed above.

Given that and are dual-feasible, we will next attempt to lower-bound objective by analyzing the process .

First we show that the times at which the barriers are applied are bounded above by .

Theorem 3

The time points constructed in the proof of Theorem 2 satisfy , for .

Before proving Theorem 3, we first prove Lemmas 1 to 5, which further characterize the behavior of the process . These lemmas collectively show that when the barriers are applied at regular points starting at some time of the horizon, i.e., for all for some integer , the time spent at the barriers must be monotone decreasing in the index for all .

For ease of notation, let

be the total time that the bounded process stays at the barrier during the interval , for . Note that . Let be the probability that a Poisson random variable with mean is equal to . Let and denote and , respectively.

First, assuming that the barriers are applied at regular points , we can quantify the difference in the expected time spent at each barrier, given different starting points for the process .

Lemma 1

Given any , if and , we must have

for all .

Next, assuming that the barriers are applied at regular points , we can characterize the differences in the expected time spent at each barrier for successive pairs of starting points.

Lemma 2

Given any , if for all , we must have

for all .

Using the previous result, we relax the assumption that all the barriers are applied at regular points . We assume now that the barriers are applied at regular times beyond a point. Under this condition, we show that the differences in the expected time spent at successive barriers are increasing with the starting point of the process.

Lemma 3

Given any , if for , and for , we must have

for all .

Next, assuming that the barriers are applied at regular points , we show that the expected time spent by the process at each barrier is decreasing with the index of the barrier.

Lemma 4

If for all , we must have for all .

\@trivlist

It is obvious that for any ,

because when the starting position becomes lower, it is harder for the random process to reach the barrier at any later time. Since , and by symmetry, , we have for all .  \@endparenv

Finally, we relax the requirement of Lemma 4. We require only that the barriers be applied at regular points only after some time. We show that the expected time spent at the barriers are still decreasing.

Lemma 5

Given any , if for , and for , we must have

for all .

The idea of the proof of Theorem 3 is as follows. We will start by setting the barriers at times . We then successively reduce the values , , until the expected time spent at each barrier is no more than . By the monotonicity shown in Lemma 5, this procedure must stop with the expected time spent at each barrier bounded above by .

If we change the value of , the time points that result from the above procedure must change continuously in . We simply choose such that, when the procedure ends, the expected time spent at the last barrier is , which implies that the expected time spent at all barriers is exactly .

First, we prove an inequality, which will be useful in computing our bound.

Lemma 6

For any and such that , we must have for any ,

Finally, we derive our lower bound on . The bound is simply a reduction of the equation

which follows from Theorem 2. is strictly greater than for . For example, when , satisfies

from which we can obtain .

Proposition 2
Theorem 4

If is the minimum capacity of any resource, the competitive ratio for the Separation Algorithm is at least

In this section, we propose the Marginal Allocation Algorithm, which improves on the Separation Algorithm by converting it to a bid-price algorithm.

The Separation Algorithm, when carried out in practice, has several problems. First, it might route customers to unavailable resources when they can be better matched to other resources. Second, because of the random routing, it might unfairly accept a lower-priority customer after rejecting a higher-priority customer. In this section, we present the Marginal Allocation Algorithm which resolves these issues by converting the Separation Algorithm into a bid-price algorithm. We will prove that the Marginal Allocation Algorithm has theoretical performance no worse than that of the Separation Algorithm.

The Marginal Allocation Algorithm uses the marginal reward as a bid price for resource . When a customer of type arrives, the Marginal Allocation Algorithm rejects the customer if for all available resources ; otherwise, it assigns this customer to resource

To carry out this algorithm, we only need to compute the reward functions at the beginning of the horizon. Thus the space requirement is polynomial in and . At any time , we only need to know the reward functions , for , so as to make a decision.

The following theorem states that the Marginal Allocation Algorithm performs at least as well as the Separation Algorithm:

Theorem 5

The expected total reward of the Marginal Allocation Algorithm is no less than that of the Separation Algorithm.

As a result, when is the minimum capacity, the competitive ratio of the Marginal Allocation Algorithm is . When tends to infinity, the competitive ratios tends to , so the Marginal Allocation Algorithm is asymptotically optimal.

In settings in which customers have similar preferences for all the resources, the Marginal Allocation Algorithm utilizes resources more effectively than the Separation Algorithm, because the latter restricts each customer to only one resource but the former can allocate any available resource. We focus on such settings in this section, and lower-bound the expected reward that the Marginal Allocation can earn more than the Separation Algorithm.

Proposition 3

The expected reward earned by the Marginal Allocation Algorithm can be as much as times that earned by the Separation Algorithm.

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Suppose for all and . Suppose for all . Suppose . In this way, the total expected number of arrivals is equal to the total capacity.

The optimal LP solution must satisfy .

Since for any resource , the reward values are the same for all customer types , the expected future reward earned from future customers must be no more than the reward values, i.e., for all .