Duality in staffing problems: Between holding costs and waiting constraints Seung Bum Soh Northwestern University, [email protected]

Itai Gurvich Northwestern University, [email protected]

There are two alternative ways to capture the tension between capacity expenses and customer-delay costs in staffing problems. The cost paradigm assigns a price tag to customer delay and optimizes the combined costs of staffing and waiting. The constraint paradigm, in contrast, replaces the waiting-time cost with constraints and seeks to minimize staffing costs subject to these constraints. The duality of these two formulations is important for both the implementation of delay costs through constraints (e.g., specifying constraints in a contract) and for the reverse engineering of the dollar value that a provider, solving a given constraint formulation, assigns implicitly to customer delay. In the single-class queue, this duality is a simple matter: the optimal trade-off of capacity and delay can be implemented via a staffing problem with, for example, average waiting constraints. Given the waiting-time constraints that a provider uses, we can figure out the underlying implicit delay costs. In the multiclass case—where one must determine both the optimal staffing and the optimal prioritization—things become more involved. Strictly convex costs can be reliably implemented by any strictly convex constraints. Linear waiting constraints, while common in practice, do not provide a “safe” implementation of any simple cost structure. They can be made safe, however, by augmenting them with a variance constraint. When seeking to reverse engineer constraints to costs, strictly convex constraints are straightforward. Linear constraints are, however, not uniquely reversible, and strictly concave constraints cannot be an implementation of any strictly increasing waiting costs. Finally, since strictly convex costs have multiple implementations through constraints, it is desirable to propose a “best” implementation. We numerically study the robustness of different implementations.

1.

Introduction

Determining capacity (or staffing) is a routine and central task in the day-to-day operations of service systems. Fundamentally, the staffing problem represents a tradeoff between the cost of capacity and the cost (or revenue loss) that is attributed to customer delays. The greater the capacity relative to demand, the less customers have to wait. In modeling the staffing decision as an optimization problem, the capacity cost has an obvious representation: a dollar amount per server-hour is multiplied by the number of servers assigned to that hour. Waiting time, however, can be modeled into the staffing problem through either waiting cost or waiting constraints. 1

2

The traditional cost formulation assigns a dollar value to customers’ waiting time and places this cost in the objective function as a counterbalance to the staffing cost. The problem to be solved is then, Minimize Staffing cost + Waiting cost.

(cost formulation)

This formulation captures explicitly the tradeoff between capacity and waiting costs. As the staffing increases and, consequently, staffing costs increase, waiting costs decrease. It is more common for practitioners, however, to impose a constraint on a waiting-time-related metric, namely, to solve the problem of minimizing capacity subject to a so-called Quality of Service (QoS) constraint. Minimize Staffing cost Subject to waiting-time constraints

(constraint formulation)

There may be multiple reasons for the prevalence of constrained formulations in practice: (i) Customers’ disutilities from waiting are difficult to pin down. It is only recently that empirical papers in operations have estimated such disutilities; see, e.g., Allon et al. (2011) and Aksin et al. (2013); (ii) Industry standards often dictate the “acceptable” waiting time. These standards are expressed as constraints: average speed of answer (ASA) must be less than 1 minute, 80% of customers must wait less than 20 seconds, etc. In some cases, the standards are internally imposed (such as doorto-doctor time limits in emergency rooms) or enforced by government-agency regulations; see, e.g., Allon (2012). In the service-operations literature, the two formulations have been typically studied separately; see §2. Our objective in this paper is to examine the duality of these two in service systems with multiple classes of customers. Conceptually, we ask two questions: (i.) A prescriptive-implementation question (the operations researcher view): Although the provider might have a good idea of the dollar value of customer delay—so that, in principle, it is possible to solve the cost formulation—operations (such as contracting with an outsourcer) might require specifying constraints rather than waiting costs. What form, we ask, should these constraints take to capture the correct tradeoff between staffing and delay costs? And how are these recommendations consistent (or not) with formulations used in practice? In the newsvendor (inventory) setting, for example, the costs of overstock and understock can be used to specify a service level that can, in turn, be contracted. In a similar spirit, we ask which Quality of Service (QoS) constraints should be contracted to reflect the customer-delay costs. (ii.) A reverse-engineering question (the econometrician’s view): What does a choice of constraints—e.g., ASA smaller than 20 seconds—say about the implicit cost structure of a provider, specifically about the dollar value assigned to customer delay relative to the cost of server time?

3 Reverse engineering the M/M/N queue 1

60

0.9

55

cw/cs=5

0.7

Total cost

45

0.6

cw/cs=3 40

0.5

cw/cs=1 

0.4

35

0.3 30

0.13 minutes

Average waiting time (minutes)

0.8

50

0.2

25

0.1

Average wait

0

20 33

34

35

36

37

38

39

40

41

Number of servers 

Figure 1

Reverse engineering for an M/M/N queue

In the newsvendor setting, by observing the service level that the provider chooses, Olivares et al. (2008) impute the implicit overstock and understock costs in a healthcare setting; see also Cohen et al. (2003) for a supply-chain setting. It is useful to turn first to the single class M/M/N queue—a markovian queue with one class of customers served by one group of identical servers. Let W N be the steady-state waiting time when the staffing level is N and consider the two problems minN ≥0 N : s.t. E[f (W N )] ≤ w, ¯

(constraint)

min N + ληE[f (W N )]. (cost) N ≥0

In the first problem, there is a constraint on a metric f of the waiting time, and in the latter, there is a cost—a customer who waits w imposes a cost of ηf (w) and this is multiplied by the number λ of customers who arrive per hour. The implementation and reverse engineering here are as follows: (i) given η, how should we set w ¯ in the constraint formulation to generate the same outcome as the cost formulation? And (ii) given w, ¯ what can we say about η? For illustration, consider the special case f (w) = w. Suppose that the mean service time is six minutes and the arrival rate is λ = 300 customers per hour. Let cw and cs be the cost for one hour of customer wait and for one hour of agent time, respectively. Consider the staffing problem min cs N + cw λE[W ]. N

Figure 1 displays the total cost as a function of the staffing level N for three values of relative cost, η = cw /cs . For cw /cs = 3, the optimal solution is N ∗ = 37, and the average waiting time under this solution is 0.13 minutes (roughly 10 seconds). A service firm that has c2 /cs = 3 and wishes

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to outsource its call-center operations can guarantee that the outside provider’s staffing decision reflects its tradeoff correctly by contracting the Average Speed of Answer (ASA) constraint problem min N s.t. E[W ] ≤ 10 seconds. Suppose, instead, that we do seek to infer the ratio cs /cw , knowing that a service provider uses the ASA formulation with the 10 seconds target. Given the arrival rate λ and the mean service time, we can find this implicit cost ratio by simple trial and error. As the figure shows, for instance, η = 1 is too small (the optimal staffing is 35 and the resulting ASA is greater than 10 seconds) and η = 5 is too large. Due to the integrality of N , there will be a range of values in the neighborhood of η = 3 that do the job. With the exception of this single class queue—in which the duality of cost and constraint is straightforward—the two classes of staffing problems have been typically studied in isolation from each other; see §2. Our paper is fundamentally about this duality—about the prescriptive translation of costs to constraints and about the reverse engineering of constraints to waiting-time costs—going beyond the single class queue. We consider the simplest of such multiclass systems—the so-called V model with a common service rate; see Figure 2. Here, the cost parameters are vectors that determine which class’s waiting time is more costly—class-1 waiting time might cost 1$ per hour while that of class 2 may be 2$ per hour—as are the constraints–requiring, for example, a 20-second average delay for one class of customers, but allowing a 40-second delay for another class.

Figure 2

V-model

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The operational implication of this heterogeneity of costs and constraints is that different cost parameters lead optimally to different prioritization rules: for one cost structure it may be optimal to use a static priority rule, while for another it may be optimal to use a more elaborate dynamic rule. To relate a cost formulation to a constraint formulation, one must compare both components of their solution—the staffing level (the number of servers) and the prioritization rule. Let us revisit the ASA formulation. For a provider with a set I = {1, . . . , I} of customer classes, the immediate generalization is given by the problem on the left of min N s.t.

E[Wi ] ≤ wi , i ∈ I ,

min N + (ASA constraints)

π ∈ Π, N ∈ Z+ ,

P

i ci λi E[Wi ]

π ∈ Π, N ∈ Z+ .

(linear costs)

We prove that the ASA constraint (with w > 0) cannot serve as a perfect implementation of linear or strictly convex costs. A provider with such customer-delay costs that writes a contract with ASA (linear) constraints alone cannot guarantee that the outsourcer decision will represent the desired tradeoff between capacity and delay costs. Thus, the ASA constraint, while common in practice, is not a safe implementation. Suppose, alternatively, that we observe the ASA constraints that a firm uses and seek to identify its (implicit) delay costs. The mathematical intuition that dualization via Lagrange multipliers should allow us to map constraints to costs (reverse engineer) is only partly correct; see Remark 2. It turns out that multiple (very) different cost structures are consistent with the ASA formulation, which means, in particular, that one cannot impute the originating cost structure. The formulation is simultaneously consistent with strictly convex delay costs and some discontinuous and non monotone costs. Let us flesh out our setup. We restrict attention to family of power functions f (x) = xa . The single parameter a captures whether the cost (or constraint) is strictly convex, strictly concave, or linear. Our formalization of the constraint and cost formulations is min N s.t.

E [Wia ]

≤ wi , i ∈ I ,

π ∈ Π, N ∈ Z+ ,

min N + (constraint)

P i∈I

π ∈ Π, N ∈ Z+ .

λi ci E [Wib ]

(cost)

N above is the number of servers and π is the prioritization rule. Note that Wi depends on both of these decisions. With a = 1, we have the ASA constraints. Strictly convex (concave) corresponds to a > 1 (a < 1). Similarly, the exponent b captures the convexity/concavity of the cost. To avoid

6

trivialities, we assume that classes are truly heterogeneous, i.e., wi 6= wj for all i, j, and ci 6= cj for all i 6= j (otherwise the classes can be merged).1 Our main results are as follows: i. Strictly convex constraints are reversible to strictly convex waiting costs, and strictly convex costs are perfectly implementable through convex constraints and have an imperfect implementation as linear constraints. This means, for instance, that contracting convex constraints guarantees that the solution used by the outsourcer will reflect the waiting vs. staffing tradeoff of the provider. Linear (ASA) constraints are not guaranteed to reflect this tradeoff. It turns out they leave too much freedom. ii. A linear (ASA) constraint shares solutions with different (fundamentally different) waitingcost functions—one that is strictly convex and another cost that is not even monotone in the customer waiting time. It is, thus, difficult to pin down the implicit delay costs that support the constraints. In fact, the ASA constraint formulation is a true hybrid. It shares solutions with both strictly convex constraint formulations (a > 1) and with strictly concave formulations (a < 1). Linear waiting costs can be implemented trivially by degenerate version of any of convex, concave, or linear constraints. iii. Concave constraints (a < 1) with w > 0 do not share solutions with either linear, concave, or convex costs and thus cannot be reversed within this family. The econometrician trying to impute the delay costs will have to look elsewhere, possibly to discontinuous and non monotone waiting-cost functions. iv. Choosing the “best” implementation: We prove that a strictly convex cost formulation has multiple implementations through constraints. This multiplicity of implementations begs the question of which constraints to choose. Take, for instance, two implementations of a quadratic cost problem through constraints: one with a = 2 and an appropriately chosen target vector w and the other with a = 3 and a target vector w. ˆ Implementation here means that both constraint problems will have the same solution set as the originating cost problem. Is there a reason to prefer one implementation over the other? To compare these, it makes sense to introduce robustness considerations. We find that the higher the a that is used, the less sensitive is the outcome to mistakes in specifying the target: a 10% error in w ˆ will have a lesser effect on the outcome than a 10% error in w. Similarly, suppose one is comparing two service providers that differ in their actions (staffing and prioritization) and wishes to impute their underlying (implicit) delay costs. If one assumes 1

It is possible also to consider different values of the exponent for different classes, i.e, bi and ai for class i. However, in optimal solutions only the smallest exponent will matter, as classes with higher exponents will be given the static priorities.

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that the delay cost is cubic (b = 3), small differences between the firms’ actions translate into small differences in (reversed) coefficients relative to using a quadratic (b = 2) delay structure: the higher the exponent, the less sensitive are the reversed coefficients.

2.

Literature Review

The staffing of queues with multiple servers is a thoroughly studied topic. The study of staffing and prioritization of multiple customer classes is somewhat more recent. A central challenge in solving these problems is finding the optimal prioritization of customers. Multiple papers study the prioritization of multiple classes, for given staffing levels, with the objective of minimizing various types of delay-related costs. Some examples are Atar et al. (2004), Atar (2005), Tezcan and Dai (2010), Perry and Whitt (2009), Mehrotra et al. (2012), Bassamboo et al. (2006). Some papers in this group study networks with multiple server pools, which requires, in addition to prioritization, specifying the routing to servers. Staffing and prioritization for the constraint formulation is studied, for example, in Wallace and Whitt (2005), Pot et al. (2008), Gurvich et al. (2008), Pang and Perry (2014) and Jouini et al. (2010). The recent paper Chan et al. (2014) lies in the intersection of the cost and constraint formulations. It develops a heuristic for dualizing the constraints to costs and for optimizing the routing given those costs. This first step can subsequently be used for staffing optimization. We use the simplest multiclass queue to study the relationship between costs and constraints. This mapping, as explained in the introduction, is conceptually straightforward for the single-class case. One of the first papers to relate the formulations in this setting (and, indeed, one of the first to study the optimization of the single class queue) is Borst et al. (2004); see Examples 2.1 and 2.2 therein. An earlier exploration of the connection between delay constraints and delay costs appears in Soh and Gurvich (2014). The paper studies the role of the well-known family of Generalized cµ (Gcµ) rules—developed by Van Mieghem (1995) for the minimization of convex holding costs—in solving a typical staffing problem, specifically, the Target Service Factor (TSF) formulation min N s.t. P {Wi > wi } ≤ αi , i ∈ I , π ∈ Π, N ∈ Z+ , where wi , wi are the acceptable waiting time (AWT) targets and α1 , α2 are the Service Levels (SL). The authors prove that no Gcµ rule can be optimal for the constraint problem—only certain non monotone and discontinuous rules work. Thus, the TSF constraint problem, while widely used in practice, does not provide a mechanism to implement monotone increasing costs. The gap, however,

8

can be closed by adding a service-level-differentiation (SLD) constraint: a formulation with both TSF and SLD constraints can serve as an implementation of “reasonable” cost structures. Our result about the augmentation of the ASA constraint formulation with a variance constraint is in a similar spirit; see Remark 1. Finally, there is an important connection between our work and that of Milner and Olsen (2008). Fundamentally, both papers are about how to implement certain “intentions” into quality-of-service constraints. Theirs is a context of outsourcing, where a single customer class is outsourced to an outside provider that caters, in turn, to multiple clients and hence is operating a multiclass queue. They show how certain contracted constraints can lead to undesirable consequences relative, say, to serving the customers in-house. Our setting is different—we compare two multiclass settings (one with delay costs and the other with constraints). In this context of multiclass queues, we seek to formalize what “undesirable” means by explicitly relating delay costs to quality-of-service constraints. Staffing problems or general cost-minimization problems are typically difficult to solve and some of the analytical works cited above resort to asymptotic approximations where one solves the problem in the “limit”; solutions to the limit problem are identified as “asymptotically optimal” if the gap between the real optimal solution and the suggested one becomes negligible as the system size grows. We also follow this method. The remainder of this paper is organized as follows: The model and analysis framework are detailed in §3. The duality results appear in §4. These results build on explicitly solving both the staffing and constraint problems in §5. The proofs of the main duality results appear in §6, and a numerical study of robustness appears in §7.

3.

Model and analysis framework

We consider a set I = {1, . . . , I} of customer classes all requiring processing from a single pool of servers; see Figure 2. Arrivals follow I independent Poisson processes with rate λi for class i and P we let λ = i∈I λi be the aggregate arrival rate. Service time is exponentially distributed with a common mean (normalized for convenience) of 1. The number of servers is denoted by N . The prioritization rule is denoted by π. If steady state exists under the prioritization rule π and the staffing level N , the steady-state class-i waiting time and queue length are denoted as WiN,π and QN,π . We restrict the prioritization rule π to be i admissible in the following sense: Definition 1. (admissible policies) A prioritization rule π is admissible if: 1. There is no blocking: All incoming customers are eventually served. 2. Customers within the same class are served in a First-Come-First-Served (FCFS) manner.

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3. The rule is work conserving. Let Π be the family of admissible policies. Work conservation implies, in particular, that the total number of customers in the system behaves P as the total number in a single-class M/M/N queue with the arrival rate λ = i∈I λi . The constrained formulation is written as min N N,π

s.t. E

h

WiN,π


a i

≤ wi , i ∈ I ,

(1)

π ∈ Π, N ∈ Z+ , and the cost formulation as min N + N,π

X

λi ci E

h

WiN,π


b i

(2)

i∈I

s.t. π ∈ Π, N ∈ Z+ . Since a and b separate between convex, concave, or linear constraints/costs, they will be central to our results. For ease of reference, we use the term “a-constraint problem” to refer to (1) with exponent a and “b-cost problem” when referring to (2) with exponent b. To avoid trivialities, we assume true heterogeneity throughout, i.e, that ci 6= cj and that wi 6= wj for all i 6= j. Many-server analysis: Control and staffing problems for multiclass queues are difficult to solve exactly. Much of the literature resorts to many-server approximations where one optimizes (instead of the original system) an approximate, more tractable setting and justifies this by proving that what is optimal for the approximate system is nearly optimal for the real system. Building on that literature, we conduct our analysis in the approximate system. We use three characteristics of the approximate system: I. Number-in-system distribution: Recall that the steady-state total number of customers, QN Σ,

in our system is identical to that of an M/M/N queue. This distribution is approximated by

a continuous distribution (see Halfin and Whitt (1981)): P



QN Σ

 −1 βΦ (β) >x ≈ 1+ e−βx , φ (β)

(3)

√ where β = (N − λ)/ λ. In particular, we treat capacity as continuous and the requirement N ∈ Z+

is replaced by N ≥ 0. II. Ratio rule: Since QN Σ is invariant to the priority rule, controlling the system requires distributing this total queue between the I classes in the best way given the problem formulation. In

10

large systems, one can, with relative precision, meet targeted ratios. A longest-queue-first rule, for example, guarantees that the total queue QΣ is distributed evenly among the I classes, i.e., that 1 N QN i ≈ QΣ . I Likewise, a simple generalization of the longest-queue-first rule can be used to generate arbitrary fixed ratios pi , i ∈ I , between the individual queues, i.e., (replacing 1/I with pi ) Qi (t) ≈ pi · QΣ (t) , where

P i∈I

(4)

pi = 1. We refer to this as a fixed ratio rule. A further generalization, also used here,

allows for general ratio functions, p(x) = (p1 (x), . . . , pI (x)). Targeted ratios p are implemented in the real system by using a simple tracking rule whereby a server that becomes available at time t chooses queue i to serve where

N N Qi (t) − pi QN Σ (t) > QΣ (t) Qj (t) − pj QΣ (t) QΣ (t) . for all i 6= j. Note that the special case p1 ≡ 1 and pi ≡ 0 for all i 6= 1 results in a static priority rule. The fact that such a simple rule generates (4) is proved for relatively general (including discontinuous) ratios in Soh and Gurvich (2014). Building on this previous literature, we will assume that a tracking rule can be used to achieve targeted ratios. III. Pathwise Little’s law: Little’s law relates the steady-state waiting time to the average queue E[Qi ] = λi E[Wi ]. In many-server analysis, a stronger, sample-path form of Little’s law Qi (t) ≈ λi Wi (t) holds under FCFS-within-class. We will make use of this property as well. Thus, in what follows, we analyze the approximate system, which is one where the characteristics I–III above hold. In particular, a solution to the staffing problems consists of N (and consequently √ β = (N − λ)/ λ) and a ratio function p. Two solutions to a staffing problem are equivalent if their β components are the same and their ratio functions are (almost everywhere) identical. The solution to a problem is then said to be unique if there is a single staffing component β and a single (up to almost everywhere equivalence) ratio function p.

4.

Duality results

Definition 1. (implementation) Given a, w and b, c, we denote by SC (b, c) and SQ (a, w), respectively, the set of optimal solutions for the cost formulation with parameters (b, c) and the constraint formulation with parameters (a, w) (Q stands here for Quality-of-Service constraints). We say that a b-cost problem is perfectly implementable by an a-constraint problem if for each c there exists a w such that SQ (a, w) = SC (b, c). We call it weakly implementable by an a-constraint formulation if for each c there exists a w such that SC (b, c) ⊆ SQ (a, w) but inclusion is strict.

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In other words, perfect implementation guarantees that any optimal solution that the optimizer of the constraint formulation may choose reflects the tradeoffs inherent to the originating cost formulation. Under a weak implementation, this optimizer may choose a solution that is consistent with the originating waiting-capacity tradeoff but also one that is inconsistent with it. Theorem 1. (implementation of costs as constraints) 1. The convex cost formulation (b > 1) has a unique solution (with a fixed ratio rule as its priority component) and is perfectly implementable by a strictly convex constraint (a > 1). It is weakly implementable by linear constraints (a = 1). 2. Optimal solutions to concave or linear cost formulations have a static-priority prioritization component. They are implementable by degenerate (with wi = 0 for all but one class) convex, concave, or linear constraints. In implementing strictly convex (b, c) costs via a constraint formulation with exponent a ≥ 1 (including a = 1), the target w is chosen so that a  − b−1 ci ; wi /wj = cj

(5)

see Propositions 4 and 5 for detailed solutions to the cost and constraint formulations. The unique optimal solution to both the cost formulation and its implementation through constraints has, as its prioritization component, a fixed ratio rule: a class j whose queue exceeds the fraction 1 − b−1

pj = P

λj cj

1 − b−1

(6)

i∈I λi ci

of the total queue has priority over classes whose queues do not; see Proposition 4. In other words the target queue for class j is a linear function of the total queue length, qΣ , as schematically captured on the left-hand side of Figure 3. Static priority (as in item 2 of the theorem) is a special case where the ratio is 0 for the high-priority classes. By Theorem 1, the ASA constraint, while common in practice, is not a safe implementation of a b-cost problem with any b ≥ 1. The ASA formulation has a fixed-ratio solution Gurvich and Whitt (2007, Theorem 5.1) that is shares with the originating strictly convex cost problem. However, it also has additional—and very different–optimal solutions of which the ratio function follows a bang-bang rule, as on the right Figure 3: it assigns the entire queue to one class up to a threshold q ∗ and then switches and assigns all the queue to the other class. Consider, for example, the case of two classes and with quadratic waiting costs (b = 2) and equal coefficients (c1 = c2 = c). For each value of c, we compute the optimal waiting costs W (c) and construct the implementation through ASA constraints by computing the appropriate targets w = (w1 , w2 ). How good is the implementation is captured by whether its solution reflects the

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p1(qΣ)

p1(qΣ)

0

Total queue qΣ

Figure 3

0

Total queue qΣ

q *

Tracking policies: (left) fixed ratio; (right) bang-bang ratio

tradeoff between staffing and delay in the originating cost problem. By the very definition of implementation, the staffing level will be the same as in the originating cost problem. Moreover, if the optimizer of the constraint problem uses the fixed-ratio optimal solution, the resulting waiting cost will be identical to those of the originating cost problem. This is not the case if the optimizer uses the optimal bang-bang solution. In Figure 4, we plot the optimal waiting cost W (c) in the originating cost formulation versus the waiting cost induced by the bang-bang solution to the ASA implementation. Imperfection of an ASA implementation of convex costs 5000

λ1=200, λ2=300 4500

"Possible'' waiting‐cost outcome

4000

Waiting cost

3500 3000 2500 2000

Optimal waiting cost

1500 1000 500 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

c=c1=c2

Figure 4

The possible downside of ASA constraints as a representation for convex waiting costs

In summary, while seemingly reasonable so as to capture the relative importance of customers (via the ratios of wi /wj ), the ASA formulation is heavily “underspecified”. It shares solutions with convex costs and constraints but also has other solutions. These additional solutions (which it also

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shares with the concave constraint problem) do not capture the tradeoffs of any b-cost formulation. If the provider wishes to contract ASA constraints, it must add additional terms to avoid such outcomes. Remark 1. (augmenting ASA with a variance constraint) Item 1 of Theorem 1 shows that quadratic costs can be safely implemented via quadratic constraints. Yet, as mean and standard deviation are more common operational measures (relative to, say, the second moment), it may be more appealing, instead, to use an ASA formulation augmented by a constraint on the waiting-time variance. This, it turns out, is feasible. Fix b = 2 and cost coefficients c = (ci , i ∈ I ). Take the perfect implementation of this cost problem as a constraint problem with a = 2 and the appropriate targets (see the Proof of Theorem (2)

1) w(2) = (wi , i ∈ I ),

minN,π N s.t.

h E

WiN,π


2 i

(2)

≤ wi , i ∈ I ,

(2nd moment)

π ∈ Π, N ≥ 0. (1)

This problem shares its unique solution with the originating cost problem. Now let wi = E[WiN,π ] be the average waiting time under this unique solution, and write the ASA + Variance formulation: minN,π N s.t.

  (1) E WiN,π ≤ wi , i ∈ I , (2)

(1)

V ar(WiN,π ) ≤ vi := wi − (wi )2 , i ∈ I ,

(ASA+Var)

π ∈ Π, N ≥ 0. Recalling how

(1) wi

was defined, it is evident that this constraint problem has the same optimal

objective-function value as the second-moment constraint problem and, moreover, that they share the (unique) optimal prioritization rule. Thus, the ASA+Var formulation provides a safe implementation for the quadratic cost problem. In fact, since all strictly convex costs are outcome equivalent (see Corollary 3), this formulation provides a safe implementation for any b-cost formulation with b > 1. Definition 2. (reverse engineering) We say that a a-constraint formulation is strongly reversible to a b-cost formulation if for every w there exists a c such that SC (b, c) = SQ (a, w). We say that it is weakly reversible to a b-cost formulation if (i) for every w there exists a c such that SC (b, c) ⊆ SQ (a, w) and the inclusion is strict.

In other words, a constraint formulation is strongly reversible if we can find a cost formulation that leads to exactly the same decisions. It is weakly reversible if there is no cost formulation that shares all of its solutions: there may be another cost formulation that shares with the constraint formulation the remaining solutions in which case the constraint is consistent with two fundamentally distinct cost structures.

14

Theorem 2. (reverse engineering) 1. A strictly convex constraint (a > 1) is reversible to a b-cost formulation for any b > 1 (including, in particular, b = a). 2. The linear constraint (a = 1) is weakly reversible to a b-cost formulation for each b > 1. 3. The concave constraint (a < 1) cannot be reversed (weakly or strongly) to a b-cost formulation (regardless of the value of b). For the econometrician, Theorem 2 says that if the call center uses strictly convex constraints, it will be possible to estimate coefficients for strictly convex costs as there is, for each b, a unique inversion. If, instead, the call center uses linear constraints, imposing a linear or strictly convex cost structure for estimation is not necessarily the correct route—the “implicit” costs might not be convex. Indeed, as we have seen in the discussion following Theorem 1, the linear constraint formulation has some solutions that are consistent with quadratic costs. Its other solutions (which have a bang-bang prioritization component) are consistent with non monotone and discontinuous cost functions. Finally, if the call center uses concave constraints—the econometrician must look beyond simple cost models. The following is a direct corollary of Theorems 1 and 2. Corollary 3 (multiplicity of representations). All a-constraint problems with a > 1 are equivalent to each other as are all b-cost problems with b > 1. Constraint: For any (a, w) and a0 (a, a0 > 1) there exists a w0 such that SQ (a, w) = SQ (a0 , w0 ). Cost: For any (b, c) and b0 (b, b0 > 1), there exists a c0 such that SQ (a, w) = SQ (a0 , w0 ). In all cases, the rule component of the optimal solution is a fixed-ratio function. The multiplicity of implementations through constraint of a single cost problem means that one must make a choice on which formulation to use; we return to this question in §7. Summary: The schematic Figure 5 summarizes our findings thus far. On the left-hand side, we have the “world” of cost formulations—one can think of this as the space of parameters (b, c) in the cost formulation. On the right, we have the world of constraint formulations—the space of parameters (a, w). Each of these can be divided into its convex, linear, and concave subspaces. The diamond shape is the world of outcomes—this is the space of staffing and prioritization rule pairs. A b-cost is implementable by an a-constraint if there is a path between them in the graph, i.e., there is a solution that solves both of them. The fact that there is a path between strictly convex costs and strictly convex constraints (passing through a single outcome) captures graphically item 1 of both Theorems 1 and 2. The double line connecting convex costs and constraints to fixedratio solutions reflects Corollary 3, i.e, that multiple instances of convex problems share identical solutions.

15

Costs

Decisions  (staffing and priorities)

Convex

Convex

Linear

Linear

Concave

Concave

Static priorities

Figure 5

Constraint

Fixed ratio

Bang bang

Summary of implementability and reverse-engineering results

Linear constraints have paths connecting them (through fixed ratio solutions) to convex costs but also to bang-bang solutions (denoted by black circles) that correspond to non monotone and discontinuous costs and, in particular, to none of the b-costs. Concave constraints also have some of these bang-bang solutions. This captures the weak implementability of costs as linear constraints and the weak reversability of the latter. Per item 2 of Theorem 1, linear and concave costs have static priority solutions and are implementable by degenerate versions of convex, linear, or concave constraints (this is captured by the dashed line). Remark 2. (on dualization via Lagrange multipliers) Considering a constraint problem such as min N N,π

s.t. E

h

WiN,π


a i

≤ wi , i ∈ I ,

π ∈ Π, N ≥ 0, a mathematically intuitive way to generate the reverse engineered costs is to dualize the constraints via Lagrange multipliers to obtain a dual problem  X  h
a i g(η) := min N + ηi E WiN,π − wi : π ∈ Π, N ≥ 0, N,π

i∈I

and consider the problem g¯ = maxη≥0 g(η). It can be shown that when a > 1, this dual problem has a unique solution η ∗ and that the consequent waiting-cost problem is h X
a i : π ∈ Π, N ≥ 0, g(η ∗ ) := min N + c∗i λi E WiN,π N,π

i∈I

16

where c∗i = ηi∗ /λi shares the optimal solution (staffing and prioritization) of the original strictly convex constraint problem. Although this is not our method of derivation, our proposed reverse (when one sets b = a) has c∗ as its coefficients. For a ≤ 1, Theorem 2 shows that this straightforward dualization cannot generate the desired results. If a = 1, item 2 of that theorem states that there is no reverse with b = 1 (except for the trivial one with ci ≡ c for all i). If a < 1, even that option is not available. The concave cost formulation shares no solutions with the concave constraint formulation. A central building block in the proof of Theorems 1 and 2 is the full characterization of the optimal solutions to both the cost and constraint formulations. We turn to this task in the next section and complete the proofs of the duality theorems in §6.

5.

Optimal solutions

Proposition 4. Suppose b > 1. The solution for the cost formulation (2) is unique and has the ratio component 1 − b−1

p∗j (qΣ )

=P

λj cj

1 − b−1

· qΣ .

i∈I λi ci

√ The optimal staffing level is N (β ∗ ) = λ + β ∗ λ where β ∗ is the unique solution to √

λ−

I X

1 − b−1

!1−b

λi ci

i=1

o0 √ b n βΦ(β) β/ λ 1 + φ(β) · Γ (b + 1) ·  o2 = 0. √ b n βΦ(β) β/ λ 1 + φ(β) 

Proof: By “pathwise Little’s law”, N+

X

λi ci E

h


b WiN,p

i

i∈I

b # QN,p i =N+ λi ci E λi i∈I h X
i N,p b . =N+ λ1−b c E Q i i i X

"

i∈I

The optimal distribution p(·) of the total queue is derived by solving the following for all values of qΣ : min

X

s.t.

X

λ1−b ci (pi ) i

b

i∈I

pi = qΣ ,

i∈I

pi (qΣ ) ≥ 0, i ∈ I .

(7)

17

Denote the solution (as a function of qΣ ) as p∗ (qΣ ). Using the KKT condition, it is easily verified  0 b ∗ that the optimal solution (which is unique due to convexity) must satisfy that λ1−b c = (p (q )) i Σ i i b−1

bλ1−b ci (p∗i (qΣ )) i

are identical for all i ∈ I . The following function p∗ uniquely satisfies this

requirement (within the family of positive functions): p∗j (qΣ )

= P

λ1−b cj j

i∈I

λ1−b ci i

Then  bλ1−b cj  P j

1 − b−1

1
− b−1

· qΣ = P 1
− b−1

1 − b−1 i∈I λi ci

· qΣ 

1 − b−1

· qΣ .

i∈I λi ci

b−1

− 1 λj cj b−1

λj cj

b−1

 = bP

1

1 − a−1 i∈I λi ci



(qΣ )

b−1

,

is all the same for j ∈ I . Hence, p∗ is optimal, but our derivation of this function does not mean that it is the unique optimal ratio function (there might be a function that is, for specific values of qΣ , not optimal for (7)). Next, we establish that this can happen only over a set of values of qΣ with measure 0. Let p˜ be another ratio function that is not equivalent to p∗ . Then the measure of the set A = P P b b ci (˜ pi (qΣ )) > 0 ci (p∗i (qΣ )) − i∈I λ1−b {qΣ ≥ 0 : p∗ (qΣ ) 6= p˜ (qΣ )} is nonzero. For qΣ ∈ A, i∈I λ1−b i i and vice versa, because p∗ (qΣ ) is the unique solution for problem (7). We will show that there exists  > 0 such that m (B ε ) > 0 for ( ) X X b b  1−b ∗ 1−b B = qΣ : λi ci (pi (qΣ )) − λi ci (˜ pi (qΣ )) > ε i∈I

i∈I

Define Bm (m = 1, 2, ...) as follows: ( ) X X 1 1 b b 1−b ∗ 1−b Bm = qΣ : m+1 < λi ci (pi (qΣ )) − λi ci (˜ pi (qΣ )) ≤ m . . 2 2 i∈I i∈I Also define B0 as (

) X 1 X 1−b b b ∗ 1−b B0 = QΣ : < λ ci (pi (qΣ )) − λi ci (˜ pi (qΣ )) . 2 i∈I i i∈I Evidently,

S∞

m=0 Bm

= A and the sets Bm ’s are disjoint.

By countable additivity of the Lebesgue measure, ∞ X

m (Bm ) = m (A) > 0,

m=0

and it cannot be the case that m (Bm ) = 0 for all m. Let m∗ be the minimum m with m (Bm ) > 0. ∗

Then 1/2m

+1

makes  > 0 such that m (B  ) > 0 holds.

18

Now let δ := m (B  ). Let q be a positive number that satisfies δ m [0, q] > . 2 The cost difference between p∗ and p˜ satisfies the following (where fN is pdf of QN Σ ):

(cost difference) =

X

λ1−b ci E i



∗ QN,p i

b 

−

X

i∈I

=


i p˜ b QN, i

λ1−b ci p∗i i



b
N X QN fN QN Σ Σ dQΣ −

λ1−b ci p∗i i



b
N X QN fN QN Σ Σ dQΣ −

0

XZ B

i∈I

≥

h

i∈I ∞

XZ i∈I

≥

λ1−b ci E i

B  ∩[0,q]

i∈I

Z

≥  · fN (q) ·

p∗i

QN Σ



b

fN

QN Σ




dQN Σ

−

λ1−b ci p˜i QN i Σ

B

i∈I

λ1−b ci i

λ1−b ci p˜i QN i Σ



b


N fN QN Σ dQΣ

0

i∈I

XZ

∞

Z

XZ i∈I

B  ∩[0,q]



b


N fN QN Σ dQΣ

λ1−b ci p˜i QN i Σ



b


N fN QN Σ dQΣ

δ > 0, 2

where, for the last inequality, we used the fact that fN (q) > 0 for all q ≥ 0; recall (3). We conclude that p˜ cannot be optimal and, in particular, that p∗ is unique. For the optimal staffing level, notice that, for a given N , with the optimal function p∗ , the objective function value is given by N+

λ1−b ci E i

h

QN,π (∞) i


b i

=N+

X j∈I

i∈I

=N+

λ1−b cj  P j

X

1 − b−1

λi ci

b

1 − b−1

 X

λj cj

h

1 − b−1

 E

i∈I λi ci !1−b

h

E

QN Σ (∞)

QN Σ (∞)


b i


b i

i∈I

√

= λ+β λ+

X

− 1 λi ci b−1

!1−b

i∈I



Γ (b + 1) o. √ b n βΦ(β) β/ λ 1 + φ(β)

The first-order condition is then given by √

λ−

X i∈I

1 − b−1

λi ci

!1−b

o0 √ b n βΦ(β) β/ λ 1 + φ(β) · Γ (b + 1) ·  o2 = 0. √ b n βΦ(β) β/ λ 1 + φ(β) 

√ The optimal β (and the optimal staffing level N = λ + β λ) is characterized by the unique

solution to this equation supposing that the function is convex in β. To see this define the function g (β) as

o0 √ b n βΦ(β) β/ λ 1 + φ(β) g (β) :=  o2 . √ b n βΦ(β) β/ λ 1 + φ(β) 

19

Then g 0 (β) = g1 (β) /g2 (β), where

g1 (β) := −4 · β −2−b (2 b + b2 + 2 (b − 1) β 2 + β 4 + β 2b2 + 4b + (4b − 1) β 2 + 2β 4 Z √
∞
2 2 2exp β /2 exp −x2 dx √ −β/ 2

+β

2

2

2

4

2 + b + β + β + b 3 + 2β

2





4exp β

2




·

2

∞

Z

2

√ −β/ 2

exp −x




dx

),

and Z √
2 g2 (β) := 2 + 2 2βexp β /2 

∞ 2

√ −β/ 2

exp −x




 dx 3 .

Clearly, g1 (β) < 0 and g2 (β) > 0 for b > 1, so that g 0 (β) < 0. We conclude that the second-order condition for the objective function value is satisfied, as required, since X

−

− 1 λi ci b−1

!1−b · Γ (b + 1) · g 0 (β) > 0,

i∈I

which concludes the proof.



Proposition 5. Suppose a > 1. The solution for (1) is unique and has the ratio component 1/a

λj wj

· qΣ , 1/a λi wi √ and the optimal staffing is given by N (β ∗ ) = λ + β ∗ λ, where β ∗ is the unique solution to

pj (qΣ ) = P

(8)

i∈I

n

X Γ (a + 1) 1/a  √ a n o= λi wi . βΦ(β) β/ λ 1 + φ(β) i Proof: By the pathwise Little law (see §3), (1) is equivalent to min N N,p

s.t. E

h


a QN,p i

i

≤ λai wi , i ∈ I ,

(9)

π ∈ Π, N ≥ 0. N (β ∗ ) and p∗ satisfy h E


a QN,p j

i

!a

1/a

λj wj

=

∞

Z

1

=


a 

 √  √  a · β/ λ exp −xβ/ λ x dx !a

1/a

λj wj P

QN Σ





1 + βΦ(β) φ(β)

0

=

E

1/a i∈I λi wi

P

n X

1/a

i∈I λi wi

i

!a 1/a

λi wi

= λaj wj

20

To prove that (N (β ∗ ), p∗ ) is optimal and unique, suppose, towards contradiction, that there exist ˜ ≤ N (β ∗ ) that satisfy the constraints in (9). p˜ and N We are first going to show that (N (β ∗ ), p∗ ) are optimal for the cost formulation min N +

X

λ1−a ci E [Qai ] ,

(10)

i∈I

with the specific coefficients − a−1 a

cj = K · wj

,

(11)

where K is K =−

 o2  !a−1  √ a n βΦ(β) λ 1 + β/ 1 φ(β)   λi wia   √ a n o0  j∈I β/ λ 1 + βΦ(β) φ(β)

1 · Γ (a + 1)

X

.

(12)

β=β ∗

Then N+

X

λ1−a cj E i

h

QN,p j


a i

=N+

− a−1 a

X

λ1−a · K · wj

h E

QN,p j


a i

.

j∈I

j∈I

By Proposition 4, this problem has a unique ratio solution, given exactly by p∗ in (8), 1 − a−1

pj (qΣ ) = P

λj cj

1 − a−1

i∈I λi ci

1 −a

1

· qΣ = P

1/a

K − a−1 λj wj

1 −a

1

− a−1 λi wi i∈I K

· qΣ = P

λj wj

1/a

i∈I

λi wi

1/a

a · E

· qΣ ,

and the objective function value under this optimal decision is N+

X

λ1−a cj E j

h

QN,p j


a i

=N+

X

P

j∈I

j∈I

=N+

n X j=1

λaj wj

− a−1 a

λ1−a Kwj j

i∈I

QN Σ

λj wj

1/a i∈I λi wi

X

1/a λi wi

i∈I

a · E [QaΣ ]

·

Γ (a + 1) o. √ a n β/ λ 1 + βΦ(β) φ(β)

√ The solution N = λ + β λ for the problem √

λ+β λ+K

!1−a X i∈I

1/a

λi wi


a 

1/a

K P

!1−a = N +K

λi wi



·

Γ (a + 1) o √ a n β/ λ 1 + βΦ(β) φ(β)

must satisfy the first-order condition if the second-order condition is met:  0 !1−a X √ 1 1/a o  = 0. λ+K λi wi · Γ (a + 1) ·   √ a n βΦ(β) β/ λ 1 + φ(β) i∈I

21

√ Plugging in K from (12), we see that β ∗ (from N ∗ = λ + β ∗ λ) satisfies this first-order condition.

The second-order condition is easy to see as in Proposition 4. Also define g (β) as in Proposition 4 except for the change of the exponent b to a. Since g 0 (β) < 0, the second-order condition is met. ˜ , p˜) is feasible for Now let us return to the constraint formulation. By our assumption, that (N ˜ ≤ N ∗ . Then (10) is now (9) with N h X X
i − a−1 − a−1 p˜ a ˜+ N λ1−a Kwj a E QN, ≤ N∗ + λ1−a Kwj a λaj wj j j j j∈I

j∈I ∗

= N +K

X

1/a

λi wi

i∈I

= N∗ + K 

Γ (a + 1) o √ a n β/ λ 1 + βΦ(β) φ(β)

But the right-hand side is the optimal cost with the staffing level N ∗ and the ratio function p∗ , which means that there are two different solutions to the cost-minimization problem (10), contradicting Proposition (4).



Proposition 6. Suppose a ≤ 1. The optimal staffing for (1) is the same as that for the singleclass problem: min N N

s.t. E



QN Σ


a 

≤

X

λai wi .

(13)

i∈I

N ≥ 0.

Proof: By the pathwise Little’s law (1) is equivalent to min N N,p

s.t. E

h

QN,p i


a i

≤ λai wi , i ∈ I .

(14)

π ∈ Π, N ≥ 0. Next, by the triangle inequality for p-norm

P i∈I

xpi


(1/p)

≤

P i∈I

xi , so that for all q ≥ 0,

!a X i∈I

qi

≤

X

qia .

i∈I

Therefore, if N is feasible for (14), it must be the case that " !a # X N,p  N
a  E QΣ =E Qi " i∈I # X N,p
a X h N,p
a i X ≤E Qi = E Qi ≤ λai wi . i∈I

i∈I

i∈I

22

It is then evident that the staffing solution for (14) is greater than or equal to the one for √ (13). Let us denote it by N (β ∗ ) = λ + λβ ∗ . Thus, if we can find a ratio function that, with the staffing solution N (β ∗ ), satisfies the constraints in (14), it must be that N (β ∗ ) is optimal for this constrained problem. We claim that the following function p∗ does the job:   0 ≤ qΣ < xi−1 , 0, ∗ pi (qΣ ) = qΣ , xi−1 ≤ qΣ ≤ xi ,  0, xi < qΣ ,

(15)

where x0 = 0 and xi ’s are defined recursively by Z xi   √  √  a 1 ∗ · β / λ exp − xβ/ λ x dx = λai wi . ∗ ∗ β Φ(β ) xi−1 1 + φ(β ∗ ) The existence of xi ’s follows from the fact that β ∗ , by its definition as the optimal solution to (13), satisfies Z 0

∞

1 1+

β ∗ Φ(β ∗ ) φ(β ∗ )

  h √  √ 
a i X a · β ∗ / λ exp −xβ ∗ / λ xa dx = E QN,p = λi wi . Σ i∈I

Applying the definition of p∗ in (15), we have that   h ∗ ∗ a i Z ∞ √  √  1 a ,p ∗ · β / λ exp − xβ/ λ (pi (x)) dx E QN = i β ∗ Φ(β ∗ ) 1 + 0 φ(β ∗ ) Z xi   √  √  a 1 ∗ = · β / λ exp − xβ/ λ x dx ∗ ∗ β Φ(β ) xi−1 1 + φ(β ∗ ) = λai wi , √ and we conclude that, with a ≤ 1, N ∗ = λ + β ∗ λ and p∗ are feasible for (14) and, in particular,

optimal. Since (14) is equivalent to (1), (N (β ∗ ), p∗ ) is optimal for the latter with a ≤ 1.



Proposition 7. Suppose b ≤ 1. An optimal ratio rule for h X
b i min N + λi ci E WiN,p N,p

i∈I

s.t. π ∈ Π, is given by the following ratio functions: pi∗ (qΣ ) = qΣ , and pi (qΣ ) = 0, i 6= i∗ , 1−b where i∗ is a class that satisfies λ1−b cj . The optimal staffing is given by N (β ∗ ) = i∗ ci∗ = minj∈I λj √ λ + β ∗ λ, where β ∗ is the optimal solution to  b  Γ (b + 1) N (β) 1−b min N (β) + λi∗ ci∗ E QΣ = N (β) + λ1−b o. i∗ ci∗  √ b n β≥0 β/ λ 1 + βΦ(β) φ(β)

23

Proof: By the pathwise Little’s law, N+

X

λi ci E

h

b # QN,p i =N+ λi ci E λ i i∈I h X
i N,p b =N+ λ1−b c E Q . i i i


i N,p b

"

X

Wi

i∈I

i∈I

For fixed N , the optimal allocation of a total queue length qΣ is derived by solving for each qΣ : min

X

s.t.

X

λ1−b ci pbi i

i∈I

pi = q Σ ,

i∈I

pi ≥ 0, i ∈ I . It is easy to solve the problem using KKT conditions from the Lagrangian: ! X X X 1−b b ti pi . pi − L= λi ci pi + s qΣ − i∈I

i∈I

i∈I

The use of the KKT method is justified by the regularity of linear constraints. s and ti ’s denote the Lagrangian multipliers (these may depend on the parameter qΣ ). The conditions are 1. bλ1−b ci pb−1 − s − ti = 0, i ∈ I , i i P 2. qΣ − i∈I pi = 0, 3. ti pi = 0, 4. ti ≥ 0. = s. By condition 2, ci pb−1 Let I ∗ be the set of i’s with pi > 0. Then for i ∈ I ∗ , ti = 0 and bλ1−b i i X

pi =

X

pi =

i∈I ∗

i∈I

X i∈I ∗

1  b−1

s

and s is calculated to be  s= so that  pi =

bλ1−b ci i s

1  1−b

1

bλ1−b ci 1−b i


P

i∈I ∗

= qΣ ,

bλ1−b ci i 1−b 

qΣ

= bλ1−b ci i

,

qΣ

1
1−b

P i∈I

bλ1−b ci i

1
1−b

for all i ∈ I ∗ . Thus, the objective function is then  X i∈I ∗

λ1−b ci pbi = i

X i∈I ∗




λ1−b ci  P i

1

bλi1−b ci 1−b

j∈I ∗

bλ1−b cj j

1
1−b

b  qΣb =

X i∈I ∗

λ1−b ci  i

bλ1−b ci i P

j∈I ∗

b
1−b

bλ1−b cj j

1 b
1−b

qΣb

24

P =  P

i∈I ∗

j∈I ∗

X

=


1
1 P 1−b ci 1−b bb λ1−b ci 1−b b i i∈I ∗ λi q = qb 1 b Σ 1 b Σ

P 1−b bλ1−b cj 1−b cj 1−b j j∈I ∗ λj !1−b 1

qΣb .

λi ci1−b

i∈I ∗

 Notice that

P

1 1−b

i∈I ∗

1−b

λi ci

only increases if the number of elements in I ∗ increases. Hence,

the cost is minimized when only one queue is positive (at least one must be positive when qΣ > 0 P 1−b ci for all i ∈ I , to meet the condition that i pi = qΣ ). By choosing i∗ that satisfies λ1−b i∗ ci∗ ≤ λi the cost is minimized, which proves that the specified ratio function is optimal.

6. 6.1.



Proofs of dualization theorems Proof of Theorem 1

Proof of 1. Consider the problems (18) and (17). We will show that given a, b > 1 and c, we can find w such that SQ (a, w) = SC (b, c). Let the optimal ratio functions pcost and pconst be again as in Propositions (4) and (5); see (19). Suppose a, b, and ci ’s are given. Define wi ’s to be a − b−1

wi = K const · ci

.

(16)

Then pconst (qΣ ) = P j

1/a λj wj 1/a

i∈I

λi wi

 1/a − a λj K const · cj b−1 · QΣ = 1/a · QΣ  P − a const · c b−1 λ K i i∈I i

− 1

λj · cj b−1 · QΣ = pcost = P (QΣ ) . 1 j − b−1 λ c i i i∈I It remains to show that with this choice of w, the staffing levels are the same. The optimal staffing for (17) is given by N (β cost ), where β cost is the unique solution to  0 !1−b X − 1 1   1+ λi ci b−1 · Γ (b + 1) ·   √ b n o  = 0. i∈I β/ λ 1 + βΦ(β) φ(β) See Proposition 4. Define K const = 

1 Pn i

1 − b−1

λi · ci

a · 

Γ (a + 1)  . √ a β cost Φ(β cost ) β cost / λ 1 + φ β cost ( )

By Proposition (5), the optimal staffing for (18) is the unique solution β to !a n X Γ (a + 1) 1/a  √ a n o= λi wi . β/ λ 1 + βΦ(β) i φ(β)

25

We claim that β cost is this solution when wi is set as in (16). Indeed, !a !a n n X X 1 1 − b−1 1/a const a λi wi = λi (K ) · ci i

i

=K

const

n X

·

− 1 λi · ci b−1

!a =

i

Γ (a + 1)  . √ a β cost Φ(β cost ) cost 1 + φ β cost β / λ ( )

Thus, we find that with this choice of wi , both the ratio and staffing components are identical as required. Proof of 2. Any a-constraint formulation trivially lets static priority solutions by letting wi = 0 for all i except for one class.  6.2.

Proof of Theorem 2

Proof of 1. Using the pathwise Little’s law, Wi is replaced by Qi /λi , so that the cost formulation (2) is equivalent to min N +

X

λ1−b ci E i

h


b QN,p i

i

i∈I

s.t. p ∈ P , N ≥ 0.

(17)

Similarly, (1) is equivalent to min N N,p

s.t. E

h

QN,p i


a i

≤ λai wi , i ∈ I ,

(18)

p ∈ P , N ≥ 0, and we focus on these two problems. Consider the case a, b > 1. Let the optimal ratio functions be pcost and pconst for (17) and (18), respectively. Then, by Propositions 4 and 5, we have the optimal ratio functions 1 − b−1

pcost (QΣ ) j

=P

λj cj

1 − b−1

1/a

· QΣ and

pconst (QΣ ) j

i∈I λi ci

=P

λj wj

i∈I

1/a

λi wi

· QΣ .

(19)

We will first show that for an arbitrary choice of a > 1, b > 1, and wi ’s (wi > 0), there exist ci ’s such that SC (a, w) = SQ (b, c). Define ci ’s as ci = K

cost

 ·

1 wi

 b−1 a ,

(20)

26

where K cost will be defined in (21) below. Then

pcost (QΣ ) = P j

K cost ·

λj

1 − b−1

λj cj

P

λi K cost ·

i∈I



= P

1 wi

λi ·

i∈I

−1/a



1 wi

1 wi

 b−1 a

1 !− b−1

· QΣ =

1 − b−1

i∈I λi ci

λj ·





1 wi

 b−1 a

1 !− b−1

1/a

−1/a = P

λj · wj

1/a i∈I λi · wi

= pconst (QΣ ) . j

Both formulations share the same ratio function and we have found the desired c (recall that this ratio function is unique by Propositions 4 and 5). It remains to show that the staffing levels are the same. By Proposition 5, the staffing for (1) (and, in particular, for the equivalent (18)) is given by √ N const (β const ) = λ + β const λ, where β const is the unique solution to !a n X Γ (a + 1) 1/a  √ a n o= λi wi . β/ λ 1 + βΦ(β) i φ(β) Define o2  √ b n βΦ(β) 1 + φ(β) β/ λ     ·    0  n o b √   β/ λ 1 + βΦ(β) φ(β)  

√

K

cost

= Γ (b + 1) ·

λ P n

1/a

j=1 λj wj

1−b

.

(21)

β=β const

The objective function value of (17) with ci as in (20) and with the optimal ratio function pcost = pconst is given for a given staffing N by N+

n X

λ1−b K cost j

 ·

j=1

=N+

n X



λ1−b K cost · j

j=1

=N+

n X



λ1−b K cost · j

j=1

=N+

n X

1 wj 1 wj 1 wj

 b−1 a ·E

h

QN,p j

 b−1 a ·

 b−1 a

K

j=1

= N + K cost ·

·

1/a

P i∈I

n X j=1

λi wi !1−b

1/a

λj wj

λi wi 1/a

·

λj wj

1/a i∈I λi wi

P

b · 

· 

·E

1/a

P i∈I

λj wj

!b

1/a

λj wj

1/a

cost


b i

h

!b · 

QN Σ


b i

Γ (b + 1) o √ b n 1 + βΦ(β) β/ λ φ(β)

Γ (b + 1) o √ b n β/ λ 1 + βΦ(β) φ(β)

Γ (b + 1) o. √ b n βΦ(β) β/ λ 1 + φ(β)

27

The second-order condition is verified as in the proof of Proposition 4:  o0 √ b n βΦ(β) ! 1−b 1 + φ(β) β/ λ n X √ 1/a λ − K cost · λj wj · Γ (b + 1) ·  o2 = 0. √ b n βΦ(β) j=1 1 + φ(β) β/ λ It can be now verified that β const is a solution (and hence the unique solution) to this first-order condition and thus also optimal for the cost formulation as required. Proof of 2. Suppose that more than one of the wi ’s are strictly positive. From Propositions 6 and 7, only static priority policies are optimal and hence only one customer class should have a positive queue in b-cost formulations with b ≤ 1. Then no linear constraint formulation can be an implementation of b-cost formulation with b ≤ 1 as the optimal solutions for the linear constraint formulation have the expected queue of each class positive. By Proposition 6, the optimal staffing for a-constraint formulation with a ≤ 1 is given by the solution for (13). For a = 1, " !a # X N,π X h N,π
a i X  N
a  = Qi = E Qi λai wi . =E E QΣ i∈I

i∈I

i∈I

By the constraints, h

X

E

QN,π j


a i

X

≤

λai wi

j∈I,j6=i

j∈I,j6=i

and h E

QN,π i


a i

≥ λai wi .

This applies to all i ∈ I , and hence no customer class queue can have a zero expected queue; thus, the linear constraint formulation is not an implementation of b-cost formulation with b ≤ 1. In b-cost formulation with b > 1, the unique solution is FQR as is shown in Proposition 4. The linear constraint formulation also has FQR as a solution (see Gurvich et al. (2008)) and hence is an implementation of b-cost formulation with b > 1. Proof of 3. For a < 1 we show that all the ratio policies are bang-bang rules: at each value of qΣ (except maybe for a set of measure zero), the queue of one class is positive while others are 0. Suppose this is not the case, i.e., one the optimal prioritization policy for min N N,π

s.t. E

h

WiN,π


a i

≤ wi , i ∈ I ,

(22)

π ∈ Π, N ∈ Z+ , has a tracking function pi such that pi (qΣ ) > 0 and pj (qΣ ) > 0 for some qΣ with positive measure. Define A, M1 and M2 as A := {qΣ : pi (qΣ ) > 0, pj (qΣ ) > 0} ,

28

Z

1

Mi := qΣ ∈A

Z

1 + βΦ(β) φ(β) 1

Mj := qΣ ∈A

1 + βΦ(β) φ(β)

  β β a · √ · exp − √ qΣ (pi (qΣ )) dqΣ , λ λ   β β a · √ · exp − √ qΣ (pj (qΣ )) dqΣ . λ λ

Then define B ⊂ A to satisfy   Z 1 β β a Mi := · √ · exp − √ qΣ (pi (qΣ ) + pj (qΣ )) dqΣ . βΦ(β) λ λ qΣ ∈B 1 + φ(β) Now define a new ratio function p0 (qΣ ), which is a modification of p (qΣ ) as follows: p0i (qΣ ) = pi (qΣ ) + pj (qΣ ) for qΣ ∈ B, p0j (qΣ ) = 0 for qΣ ∈ B, p0i (qΣ ) = 0 for qΣ ∈ A ∩ B c , p0j (qΣ ) = pi (qΣ ) + pj (qΣ ) for qΣ ∈ A ∩ B c . For the original ratio function p (qΣ ),   Z 1 β β a a · √ · exp − √ qΣ ((pi (qΣ )) + (pi (qΣ )) ) dqΣ = M1 + M2 . βΦ(β) λ λ qΣ ∈A 1 + φ(β) But for p0 (qΣ ), Z

=

=

=
(x1 + x2 ) where a < 1 and x1 , x2 > 0.

29

Since   β β a · √ · exp − √ qΣ (p0i (qΣ )) dqΣ βΦ(β) λ λ 1 + qΣ ∈A φ(β)   Z β β 1 a √ √ · · exp − qΣ (p0i (qΣ )) dqΣ = βΦ(β) λ λ qΣ ∈B 1 + φ(β)   Z 1 β β a · √ · exp − √ qΣ (p0i (qΣ )) dqΣ + βΦ(β) λ λ qΣ ∈A∩B c 1 + φ(β)   Z 1 β β a 0 √ √ = · · exp − q Σ (pi (qΣ )) dqΣ + 0 βΦ(β) λ λ qΣ ∈B 1 + φ(β)   Z β β 1 a · √ · exp − √ qΣ (pi (qΣ ) + pj (qΣ )) dqΣ = βΦ(β) λ λ qΣ ∈B 1 + φ(β) Z

1

= Mi , the following holds by (23): Z qΣ ∈A

1

1 + βΦ(β) φ(β)

 
a β β · √ · exp − √ qΣ p0j (qΣ ) dqΣ < Mj λ λ

Recalling the definition of Mj and that p (qΣ ) = p0 (qΣ ) for qΣ ∈ / A, we have   Z ∞
a β 1 β 0 √ √ · exp − q p dqΣ · (q ) Σ Σ j λ λ 1 + βΦ(β) 0 φ(β)   Z ∞ 1 β β a < · √ · exp − √ qΣ (pi (qΣ )) dqΣ , βΦ(β) λ λ 1 + 0 φ(β) and, therefore, E

h

{j }c . Let wj0 := E

h

0

a i

0

a i

QN,p j

QN,p j

1. Its optimal solution is

31 Sensitivity of staffing (β) to perturbations of the target w 1.0015

a=2 1.0005

a=3

0.9995

0.9985

0.9975 0.993

0.995

Figure 6

Sensitivity of staffing (β) to perturbations of the target w

0.997

0.999

1.001

1.003

1.005

1.007

given by a staffing level and a fixed ratio rule (see item (i) of Theorem 2). The “econometrician” observes these actions—the staffing level and the prioritization rule—and chooses a cost structure, specifically, a b value. He then seeks to identify the coefficients ci , i ∈ I such that the b-cost problem has the provider’s action as its solution—this is the action of reverse engineering the costs. Since there are two customer classes, the service provider’s decision is fully specified by the staffing level N and the ratio of class 1, p1 (the ratio of class 2 is then p2 = 1 − p1 ). Given the pair (N, p1 ) and fixing b, we can find cost coefficients c1 and c2 such that (N, p1 ) is the optimal solution to the cost-minimization problem: min N + N,π

X

λi c∗i E

h

WiN,π


b i

: π ∈ Π, N ≥ 0.

i∈I

˜ , p˜), The coefficients ci , i = 1, 2 depend, of course, on b: given two different actions, (N, p1 ) and (N and fixing b, we can compute the differences in the cost coefficients. By subsequently varying b, we can see how changes in providers’ actions map into changes in the “dualized” cost coefficients c1 and c2 . For this experiment, we take the initial firm’s action to be the ratio p¯1 = 0.4 (and p¯2 = 1 − 0.4 = √ ¯ = 500 + 0.4 · 500 ≈ 508). The dualized 0.6), and the staffing level β = 0.4 (with λ = 500, we have N coefficients are given by c˜ = (1.024, 0.682) for b = 2 and c¯ = (4.291, 1.907) for b = 3. That is, the cost formulations min N +

P i∈I

π ∈ Π, N ∈ Z+

λi c˜i E [Wi2 ]

min N + and

P i∈I

λi c¯i E [Wi3 ]

π ∈ Π, N ∈ Z+

have the same solution given by the fixed ratios p¯1 , 1 − p¯1 and by N = 508.

32

We now (multiplicatively) perturb the actions of the provider—specifically, the values of β and p1 —to see how this affects the reversed coefficients. We perturb β by values in the range [0.98, 1.02] so that βα = αβ (again, β¯ = β1 ) and we do the same for p1 . We re-compute the corresponding parameters for each combination of β and p1 . That is, for each combination (β, p1 ), we compute the reverse coefficients (c1 , c2 ) for b = 2 and b = 3. We then capture the change relative to the base by the metric max{c1 /¯ c1 , c2 /¯ c2 , c¯1 /c1 , c¯2 /c2 }. The results, for the cross-sections p1 ∈ {0.98p¯1 , p¯1 , 1.02¯ p1 }, are displayed in Figure 7. When there ¯ the value of the metric is 1. is no perturbation (i.e., p1 = p¯1 and β = β)

b=2

b=3

1.16

1.16

1.14

1.14

1.12

1.12

Ratio (p1) perturbation

Ratio (p1) perturbation

1.1

0.98 1.02

1.08

1

1.06

1.02

1.1 1 1.08 1.06

1.04

1.04

1.02

1.02

1

1

0.98

0.98

0.98 0.98

0.99

1

1.01

1.02

Staffing (β) perturbation

Figure 7

0.98

0.99

1

1.01

1.02

Staffing (β) perturbation

Sensitivity of reversed coefficients to actions with varying β

One can observe that the sensitivity is consistently greater for b = 3. That is, changes in the provider’s actions (staffing and prioritization solutions) lead to greater changes in the dualized cost coefficients when b is larger. We repeated these experiments for various base values β¯ and p¯1 and observed the same pattern. When building a structural model and seeking to impute the waiting-cost coefficients, the reversed coefficients are more sensitive to provider actions when the assumed structure has a higher value for the cost exponent b. Put differently, if one compares two (otherwise identical) providers whose actions N, p differ only slightly, differences in costs will be difficult to identify with small values of b.

8.

Concluding remarks

The literature on staffing and prioritization of service systems has traditionally been focused on solving given problems. One specifies costs or constraints and seeks to find optimal capacity and priority-and-routing prescriptions.

33

Arguably, before asking how we might solve a given problem, we should be asking which problem we should be solving in the first place. In this paper, we try to contribute to the study of this largely open question. Practitioners typically solve constraint formulations, and these, it seems, should be grounded in some beliefs about the cost of delaying customers. In this paper we take a family of formulations— both cost and constraint—and try to understand (a) how one should implement given costs as constraints and (b) if there is a unique way to reverse engineer: given constraints that a provider is using, can we figure out the implicit costs that the provider assigns to customer delay? We find that, in the presence of multiple customer classes, the answers to both of these questions are subtle. Fundamentally, the challenge lies in the complex structure of optimal prioritization solutions to the staffing problem. The questions of duality require going beyond looking at the surface of the formulation and examining these solutions in detail. Only when one fully maps the spectrum of optimal solutions for the different problems, can one understand why the reasonable ASA formulation (as well as strictly concave constraints) can generate solutions that cannot be associated with (and hence cannot be the implementation of) any reasonable costs—convex or concave. While we believe that the essence of formulation duality is captured by the simple power functions we studied here, one may wish to consider more general families. In fact, expanding the scope of cost functions that are well understood is not a purely mathematical pursuit. To give a fuller view of the relationship between costs and constraints one must first understand what the reasonable cost structures are for customer delay—rooted in the psychology of wait and in empirical evidence. With these cost functions in hand, one can then ask (i) how does one implement these through constraints and (ii) how do constraints that are typical in practice reflect such delay costs?

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