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
Transcript
Duality in staffing problems:Between holding costs and waiting constraints
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) Cus-
tomers’ 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 door-
to-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?
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33 34 35 36 37 38 39 40 41
Average waitin
g tim
e (m
inutes)
Total cost
Number of servers
Reverse engineering the M/M/N queue
cw/cs=1
cw/cs=3
cw/cs=5
Average wait
0.13 minutes
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 WN be the steady-state waiting time when
the staffing level is N and consider the two problems
minN≥0 N :
s.t. E[f(WN)]≤ w,(constraint) min
N≥0N +ληE[f(WN)]. (cost)
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
minN
csN + cwλE[W ].
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 trans-
lation 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,
π ∈Π,N ∈Z+,
(ASA constraints)min N +
∑i ciλiE[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 [W ai ]≤wi, i∈ I,
π ∈Π,N ∈Z+,
(constraint)min N +
∑i∈I λiciE [W b
i ]
π ∈Π,N ∈Z+.(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
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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) waiting-
cost 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 ques-
tion 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. 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 consid-
erations. 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 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 staticpriorities.
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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,
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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
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 WN,πi and QN,π
i . We restrict the prioritization rule π to be
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
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
minN,π
N
s.t. E[(WN,πi
)a]≤wi, i∈ I, (1)
π ∈Π, N ∈Z+,
and the cost formulation as
minN,π
N +∑i∈I
λiciE[(WN,πi
)b](2)
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,
QNΣ , 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 >x}≈(
1 +βΦ(β)
φ (β)
)−1
e−βx, (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 QNΣ is invariant to the priority rule, controlling the system requires dis-
tributing this total queue between the I classes in the best way given the problem formulation. In
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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
QNi ≈
1
IQN
Σ .
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) , (4)
where∑
i∈I 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
Qi (t)− pi(QN
Σ (t))>QΣ (t)Qj (t)− pj
(QN
Σ (t))QN
Σ (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] = λiE[Wi]. In many-server analysis, a stronger, sample-path form of Little’s lawQi (t)≈λiWi (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
wi/wj =
(cicj
)− ab−1
; (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
pj =λjc− 1b−1
j∑i∈I λic
− 1b−1
i
(6)
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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0q*
Total queue qΣ
p1(qΣ)
0 Total queue qΣ
p1(qΣ)
Figure 3 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.
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Waitin
g cost
c=c1=c2
Imperfection of an ASA implementation of convex costs
Optimal waiting cost
"Possible'' waiting‐cost outcome
λ1=200, λ2=300
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
1) w(2) = (w(2)i , i∈ I),
minN,π N
s.t. E[(WN,πi
)2]≤w(2)
i , i∈ I,
π ∈Π, N ≥ 0.
(2nd moment)
This problem shares its unique solution with the originating cost problem. Now let w(1)i =E[WN,π
i ]
be the average waiting time under this unique solution, and write the ASA + Variance formulation:
minN,π N
s.t. E[WN,πi
]≤w(1)
i , i∈ I,
V ar(WN,πi )≤ vi :=w
(2)i − (w
(1)i )2, i∈ I,
π ∈Π, N ≥ 0.
(ASA+Var)
Recalling how w(1)i 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 implemen-
tation 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 for-
mulation the remaining solutions in which case the constraint is consistent with two fundamentally
distinct cost structures.
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Theorem 2. (reverse engineering)
1. A strictly convex constraint (a> 1) is reversible to a b-cost formulation for any b > 1 (includ-
ing, 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 a′ (a,a′ > 1) there exists a w′ such that SQ(a,w) = SQ(a′,w′).
Cost: For any (b, c) and b′ (b, b′ > 1), there exists a c′ such that SQ(a,w) = SQ(a′,w′).
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 fixed-
ratio solutions reflects Corollary 3, i.e, that multiple instances of convex problems share identical
solutions.
15
Concave Concave
Linear
Convex
Costs ConstraintDecisions (staffing and priorities)
Convex
Linear
Static priorities Fixed ratio Bang bang
Figure 5 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
minN,π
N
s.t. E[(WN,πi
)a]≤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
g(η) := minN,π
N +∑i∈I
ηi
(E[(WN,πi
)a]−wi) : π ∈Π, N ≥ 0,
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
g(η∗) := minN,π
N +∑i∈I
c∗iλiE[(WN,πi
)a]: π ∈Π, N ≥ 0,
16
where c∗i = η∗i /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
p∗j (qΣ) =λjc− 1b−1
j∑i∈I λic
− 1b−1
i
· qΣ.
The optimal staffing level is N(β∗) = λ+β∗√λ where β∗ is the unique solution to
√λ−
(I∑i=1
λic− 1b−1
i
)1−b
·Γ(b+ 1) ·
((β/√λ)b{
1 + βΦ(β)
φ(β)
})′((
β/√λ)b{
1 + βΦ(β)
φ(β)
})2 = 0.
Proof: By “pathwise Little’s law”,
N +∑i∈I
λiciE[(WN,pi
)b]= N +
∑i∈I
λiciE
[(QN,pi
λi
)b]= N +
∑i∈I
λ1−bi ciE
[(QN,pi
)b].
The optimal distribution p(·) of the total queue is derived by solving the following for all values
of qΣ:
min∑i∈I
λ1−bi ci (pi)
b
s.t.∑i∈I
pi = qΣ, (7)
pi (qΣ)≥ 0, i∈ I.
17
Denote the solution (as a function of qΣ) as p∗(qΣ). Using the KKT condition, it is easily verified
that the optimal solution (which is unique due to convexity) must satisfy that(λ1−bi ci (p
∗i (qΣ))
b)′
=
bλ1−bi ci (p
∗i (qΣ))
b−1are identical for all i ∈ I. The following function p∗ uniquely satisfies this
requirement (within the family of positive functions):
p∗j (qΣ) =
(λ1−bj cj
)− 1b−1∑
i∈I
(λ1−bi ci
)− 1b−1
· qΣ =λjc− 1b−1
j∑i∈I λic
− 1b−1
i
· qΣ.
Then
bλ1−bj cj
λjc− 1b−1
j∑i∈I λic
− 1b−1
i
· qΣ
b−1
= b
1∑i∈I λic
− 1a−1
i
b−1
(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=
{qΣ ≥ 0 : p∗ (qΣ) 6= p (qΣ)} is nonzero. For qΣ ∈A,∑
i∈I λ1−bi ci (p
∗i (qΣ))
b −∑
i∈I λ1−bi ci (pi (qΣ))
b> 0
and vice versa, because p∗ (qΣ) is the unique solution for problem (7).
We will show that there exists ε > 0 such that m (Bε)> 0 for
Bε =
{qΣ :
∑i∈I
λ1−bi ci (p
∗i (qΣ))
b−∑i∈I
λ1−bi ci (pi (qΣ))
b> ε
}
Define Bm (m= 1,2, ...) as follows:
Bm =
{qΣ :
1
2m+1<∑i∈I
λ1−bi ci (p
∗i (qΣ))
b−∑i∈I
λ1−bi ci (pi (qΣ))
b ≤ 1
2m.
}.
Also define B0 as
B0 =
{QΣ :
1
2<∑i∈I
λ1−bi ci (p
∗i (qΣ))
b−∑i∈I
λ1−bi ci (pi (qΣ))
b
}.
Evidently,⋃∞m=0Bm =A and the sets Bm’s are disjoint.
By countable additivity of the Lebesgue measure,
∞∑m=0
m (Bm) =m (A)> 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∗+1makes ε > 0 such that m (Bε)> 0 holds.
18
Now let δ :=m (Bε). Let q be a positive number that satisfies
m [0, q]>δ
2.
The cost difference between p∗ and p satisfies the following (where fN is pdf of QNΣ ):
(costdifference) =∑i∈I
λ1−bi ciE
[(QN,p∗
i
)b]−∑i∈I
λ1−bi ciE
[(QN,pi
)b]=∑i∈I
∫ ∞0
λ1−bi ci
(p∗i(QN
Σ
))bfN(QN
Σ
)dQN
Σ −∑i∈I
∫ ∞0
λ1−bi ci
(pi(QN
Σ
))bfN(QN
Σ
)dQN
Σ
≥∑i∈I
∫Bελ1−bi ci
(p∗i(QN
Σ
))bfN(QN
Σ
)dQN
Σ −∑i∈I
∫Bελ1−bi ci
(pi(QN
Σ
))bfN(QN
Σ
)dQN
Σ
≥∑i∈I
∫Bε∩[0,q]
λ1−bi ci
(p∗i(QN
Σ
))bfN(QN
Σ
)dQN
Σ −∑i∈I
∫Bε∩[0,q]
λ1−bi ci
(pi(QN
Σ
))bfN(QN
Σ
)dQN
Σ
≥ ε · fN (q) · δ2> 0,
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 +∑i∈I
λ1−bi ciE
[(QN,πi (∞)
)b]= N +
∑j∈I
λ1−bj cj
λjc− 1b−1
j∑i∈I λic
− 1b−1
i
b
E[(QN
Σ (∞))b]
= N +
(∑i∈I
λic− 1b−1
i
)1−b
E[(QN
Σ (∞))b]
= λ+β√λ+
(∑i∈I
λic− 1b−1
i
)1−bΓ(b+ 1)(
β/√λ)b{
1 + βΦ(β)
φ(β)
} .The first-order condition is then given by
√λ−
(∑i∈I
λic− 1b−1
i
)1−b
·Γ(b+ 1) ·
((β/√λ)b{
1 + βΦ(β)
φ(β)
})′((
β/√λ)b{
1 + βΦ(β)
φ(β)
})2 = 0.
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
g (β) :=
((β/√λ)b{
1 + βΦ(β)
φ(β)
})′((
β/√λ)b{
1 + βΦ(β)
φ(β)
})2 .
19
Then g′ (β) = g1 (β)/g2 (β), where
g1 (β) := −4 ·β−2−b(2(b+ b2 + 2(b− 1)β2 +β4
)+β
(2b2 + 4b+ (4b− 1)β2 + 2β4
)2√
2exp(β2/2
)∫ ∞−β/√
2
exp(−x2
)dx
+β2(2 + b2 +β2 +β4 + b
(3 + 2β2
))4exp
(β2)·(∫ ∞−β/√
2
exp(−x2
)dx
)2
),
and
g2 (β) :=
(2 + 2
√2βexp
(β2/2
)∫ ∞−β/√
2
exp(−x2
)dx
)3.
Clearly, g1 (β)< 0 and g2 (β)> 0 for b > 1, so that g′ (β)< 0. We conclude that the second-order
condition for the objective function value is satisfied, as required, since
−
(∑i∈I
λic− 1b−1
i
)1−b
·Γ(b+ 1) · g′ (β)> 0,
which concludes the proof. �
Proposition 5. Suppose a> 1. The solution for (1) is unique and has the ratio component
pj (qΣ) =λjw
1/aj∑
i∈I λiw1/ai
· qΣ, (8)
and the optimal staffing is given by N(β∗) = λ+β∗√λ, where β∗ is the unique solution to
Γ(a+ 1)(β/√λ)a{
1 + βΦ(β)
φ(β)
} =n∑i
λiw1/ai .
Proof: By the pathwise Little law (see §3), (1) is equivalent to
minN,p
N
s.t. E[(QN,pi
)a]≤ λaiwi, i∈ I, (9)
π ∈Π,N ≥ 0.
N(β∗) and p∗ satisfy
E[(QN,pj
)a]=
(λjw
1/aj∑
i∈I λiw1/ai
)aE[(QN
Σ
)a]=
∫ ∞0
1
1 + βΦ(β)
φ(β)
·(β/√λ)
exp(−xβ/
√λ)xadx
=
(λjw
1/aj∑
i∈I λiw1/ai
)a( n∑i
λiw1/ai
)a= λajwj
20
To prove that (N(β∗), p∗) is optimal and unique, suppose, towards contradiction, that there exist
p and N ≤N(β∗) that satisfy the constraints in (9).
We are first going to show that (N(β∗), p∗) are optimal for the cost formulation
min N +∑i∈I
λ1−aciE [Qai ] , (10)
with the specific coefficients
cj =K ·w−a−1a
j , (11)
where K is
K =− 1
Γ(a+ 1)·
(∑j∈I
λiw1ai
)a−1((β/√λ)a{
1 + βΦ(β)
φ(β)
})2
((β/√λ)a{
1 + βΦ(β)
φ(β)
})′β=β∗
. (12)
Then
N +∑j∈I
λ1−ai cjE
[(QN,pj
)a]= N +
∑j∈I
λ1−a ·K ·w−a−1a
j E[(QN,pj
)a].
By Proposition 4, this problem has a unique ratio solution, given exactly by p∗ in (8),
pj (qΣ) =λjc− 1a−1
j∑i∈I λic
− 1a−1
i
· qΣ =K−
1a−1λjw
− 1a
j∑i∈IK
− 1a−1λiw
− 1a
i
· qΣ =λjw
1/aj∑
i∈I λiw1/ai
· qΣ,
and the objective function value under this optimal decision is
N +∑j∈I
λ1−aj cjE
[(QN,pj
)a]= N +
∑j∈I
λ1−aj Kw
−a−1a
j
λajwj(∑i∈I λiw
1/ai
)a ·E [(QNΣ
)a]= N +
n∑j=1
Kλjw
1/aj(∑
i∈I λiw1/ai
)a ·E [QaΣ]
= N +K
(∑i∈I
λiw1/ai
)1−a
· Γ(a+ 1)(β/√λ)a{
1 + βΦ(β)
φ(β)
} .The solution N = λ+β
√λ for the problem
λ+β√λ+K
(∑i∈I
λiw1/ai
)1−a
· Γ(a+ 1)(β/√λ)a{
1 + βΦ(β)
φ(β)
}must satisfy the first-order condition if the second-order condition is met:
√λ+K
(∑i∈I
λiw1/ai
)1−a
·Γ(a+ 1) ·
1(β/√λ)a{
1 + βΦ(β)
φ(β)
}′ = 0.
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′ (β)< 0, the second-order condition is met.
Now let us return to the constraint formulation. By our assumption, that (N , p) is feasible for
(9) with N ≤N∗. Then (10) is now
N +∑j∈I
λ1−aj Kw
−a−1a
j E[(QN,pj
)a] ≤ N∗+∑j∈I
λ1−aj Kw
−a−1a
j λajwj
= N∗+K∑i∈I
λiw1/ai
= N∗+KΓ(a+ 1)(
β/√λ)a{
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 single-
class problem:
minN
N
s.t. E[(QN
Σ
)a]≤∑i∈I
λaiwi. (13)
N ≥ 0.
Proof: By the pathwise Little’s law (1) is equivalent to
minN,p
N
s.t. E[(QN,pi
)a]≤ λaiwi, i∈ I. (14)
π ∈Π, N ≥ 0.
Next, by the triangle inequality for p-norm(∑
i∈I xpi
)(1/p) ≤∑
i∈I xi, so that for all q≥ 0,(∑i∈I
qi
)a≤∑i∈I
qai .
Therefore, if N is feasible for (14), it must be the case that
E[(QN
Σ
)a]=E
[(∑i∈I
QN,pi
)a]
≤E
[∑i∈I
(QN,pi
)a]=∑i∈I
E[(QN,pi
)a]≤∑i∈I
λaiwi.
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:
p∗i (qΣ) =
0, 0≤ qΣ <xi−1,
qΣ, xi−1 ≤ qΣ ≤ xi,0, xi < qΣ,
(15)
where x0 = 0 and xi’s are defined recursively by∫ xi
xi−1
1
1 + β∗Φ(β∗)φ(β∗)
·(β∗/√λ)
exp(−xβ/
√λ)xadx= λaiwi.
The existence of xi’s follows from the fact that β∗, by its definition as the optimal solution to (13),
satisfies ∫ ∞0
1
1 + β∗Φ(β∗)φ(β∗)
·(β∗/√λ)
exp(−xβ∗/
√λ)xadx = E
[(QN,p
Σ
)a]=∑i∈I
λaiwi.
Applying the definition of p∗ in (15), we have that
E[(QN∗,p∗
i
)a]=
∫ ∞0
1
1 + β∗Φ(β∗)φ(β∗)
·(β∗/√λ)
exp(−xβ/
√λ)
(pi (x))adx
=
∫ xi
xi−1
1
1 + β∗Φ(β∗)φ(β∗)
·(β∗/√λ)
exp(−xβ/
√λ)xadx
= λaiwi,
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
minN,p
N +∑i∈I
λiciE[(WN,pi
)b]s.t. π ∈Π,
is given by the following ratio functions:
pi∗ (qΣ) = qΣ, and pi (qΣ) = 0, i 6= i∗,
where i∗ is a class that satisfies λ1−bi∗ ci∗ = minj∈I λ
1−bj cj. The optimal staffing is given by N(β∗) =
λ+β∗√λ, where β∗ is the optimal solution to
minβ≥0
N(β) +λ1−bi∗ ci∗E
[(QN(β)Σ
)b]= N(β) +λ1−b
i∗ ci∗Γ(b+ 1)(
β/√λ)b{
1 + βΦ(β)
φ(β)
} .
23
Proof: By the pathwise Little’s law,
N +∑i∈I
λiciE[(WN,pi
)b]= N +
∑i∈I
λiciE
[(QN,pi
λi
)b]= N +
∑i∈I
λ1−bi ciE
[(QN,pi
)b].
For fixed N , the optimal allocation of a total queue length qΣ is derived by solving for each qΣ:
min∑i∈I
λ1−bi cip
bi
s.t.∑i∈I
pi = qΣ,
pi ≥ 0, i∈ I.
It is easy to solve the problem using KKT conditions from the Lagrangian:
L=∑i∈I
λ1−bi cip
bi + s
(qΣ−
∑i∈I
pi
)−∑i∈I
tipi.
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−bi cip
b−1i − s− ti = 0, i∈ I,
2. qΣ−∑
i∈I pi = 0,
3. tipi = 0,
4. ti ≥ 0.
Let I∗ be the set of i’s with pi > 0. Then for i∈ I∗, ti = 0 and bλ1−bi cip
b−1i = s. By condition 2,
∑i∈I
pi =∑i∈I∗
pi =∑i∈I∗
(s
bλ1−bi ci
) 1b−1
= qΣ,
and s is calculated to be
s=
∑i∈I∗(bλ1−b
i ci) 1
1−b
qΣ
1−b
,
so that
pi =
(bλ1−b
i cis
) 11−b
=(bλ1−b
i ci) 1
1−b qΣ∑i∈I
(bλ1−b
i ci) 1
1−b
for all i∈ I∗. Thus, the objective function is then
∑i∈I∗
λ1−bi cip
bi =
∑i∈I∗
λ1−bi ci
(bλ1−b
i ci) 1
1−b∑j∈I∗
(bλ1−b
j cj) 1
1−b
b
qbΣ =∑i∈I∗
λ1−bi ci
(bλ1−b
i ci) b
1−b(∑j∈I∗
(bλ1−b
j cj) 1
1−b)b qbΣ
24
=
∑i∈I∗
(bbλ1−b
i ci) 1
1−b(∑j∈I∗
(bλ1−b
j cj) 1
1−b)b qbΣ =
∑i∈I∗
(λ1−bi ci
) 11−b(∑
j∈I∗(λ1−bj cj
) 11−b)b qbΣ
=
(∑i∈I∗
λic1
1−bi
)1−b
qbΣ.
Notice that
(∑i∈I∗ λic
11−bi
)1−b
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
to meet the condition that∑
i pi = qΣ). By choosing i∗ that satisfies λ1−bi∗ ci∗ ≤ λ1−b
i ci for all i ∈ I,
the cost is minimized, which proves that the specified ratio function is optimal. �
6. Proofs of dualization theorems6.1. 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
wi =Kconst · c− ab−1
i . (16)
Then
pconstj (qΣ) =λjw
1/aj∑
i∈I λiw1/ai
·QΣ =λj
(Kconst · c
− ab−1
j
)1/a
∑i∈I λi
(Kconst · c
− ab−1
i
)1/a·QΣ
=λj · c
− 1b−1
j∑i∈I λic
− 1b−1
i
·QΣ = pcostj (QΣ) .
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(βcost), where βcost is the unique solution to
1 +
(∑i∈I
λic− 1b−1
i
)1−b
·Γ(b+ 1) ·
1((β/√λ))b{
1 + βΦ(β)
φ(β)
}′
= 0.
See Proposition 4. Define
Kconst =1(∑n
i λi · c− 1b−1
i
)a · Γ(a+ 1)(βcost/
√λ)a{
1 +βcostΦ(βcost)φ(βcost)
} .By Proposition (5), the optimal staffing for (18) is the unique solution β to
Γ(a+ 1)((β/√λ))a{
1 + βΦ(β)
φ(β)
} =
(n∑i
λiw1/ai
)a.
25
We claim that βcost is this solution when wi is set as in (16). Indeed,(n∑i
λiw1/ai
)a=
(n∑i
λi (Kconst)
1a · c
− 1b−1
i
)a
= Kconst ·
(n∑i
λi · c− 1b−1
i
)a=
Γ(a+ 1)(βcost/
√λ)a{
1 +βcostΦ(βcost)φ(βcost)
} .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 +∑i∈I
λ1−bi ciE
[(QN,pi
)b]s.t. p∈P, N ≥ 0. (17)
Similarly, (1) is equivalent to
minN,p
N
s.t. E[(QN,pi
)a]≤ λaiwi, 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
pcostj (QΣ) =λjc− 1b−1
j∑i∈I λic
− 1b−1
i
·QΣ and pconstj (QΣ) =λjw
1/aj∑
i∈I λiw1/ai
·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 =Kcost ·(
1
wi
) b−1a
, (20)
26
where Kcost will be defined in (21) below. Then
pcostj (QΣ) =λjc− 1b−1
j∑i∈I λic
− 1b−1
i
·QΣ =
λj
(Kcost ·
(1wi
) b−1a
)− 1b−1
∑i∈I λi
(Kcost ·
(1wi
) b−1a
)− 1b−1
=λj ·(
1wi
)−1/a
∑i∈I λi ·
(1wi
)−1/a=
λj ·w1/aj∑
i∈I λi ·w1/ai
= pconstj (QΣ) .
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(βconst) = λ+βconst√λ, where βconst is the unique solution to
Γ(a+ 1)((β/√λ))a{
1 + βΦ(β)
φ(β)
} =
(n∑i
λiw1/ai
)a.
Define
Kcost =
√λ
Γ(b+ 1) ·(∑n
j=1 λjw1/aj
)1−b ·
(((
β/√λ))b{
1 + βΦ(β)
φ(β)
})2
(((β/√λ))b{
1 + βΦ(β)
φ(β)
})′β=βconst
. (21)
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∑j=1
λ1−bj Kcost ·
(1
wj
) b−1a
·E[(QN,pj
)b]
= N +n∑j=1
λ1−bj Kcost ·
(1
wj
) b−1a
·
(λjw
1/aj∑
i∈I λiw1/ai
)b·E[(QN
Σ
)b]
= N +n∑j=1
λ1−bj Kcost ·
(1
wj
) b−1a
·
(λjw
1/aj∑
i∈I λiw1/ai
)b· Γ(b+ 1)((
β/√λ))b{
1 + βΦ(β)
φ(β)
}= N +
n∑j=1
Kcost ·λjw
1/aj(∑
i∈I λiw1/ai
)b · Γ(b+ 1)((β/√λ))b{
1 + βΦ(β)
φ(β)
}= N +Kcost ·
(n∑j=1
λjw1/aj
)1−b
· Γ(b+ 1)((β/√λ))b{
1 + βΦ(β)
φ(β)
} .
27
The second-order condition is verified as in the proof of Proposition 4:
√λ−Kcost ·
(n∑j=1
λjw1/aj
)1−b
·Γ(b+ 1) ·
(((β/√λ))b{
1 + βΦ(β)
φ(β)
})′(((
β/√λ))b{
1 + βΦ(β)
φ(β)
})2 = 0.
It can be now verified that βconst 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,
E[(QN
Σ
)a]=E
[(∑i∈I
QN,πi
)a]=∑i∈I
E[(QN,πi
)a]=∑i∈I
λaiwi.
By the constraints, ∑j∈I,j 6=i
E[(QN,πj
)a]≤ ∑j∈I,j 6=i
λaiwi
and
E[(QN,πi
)a]≥ λaiwi.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
minN,π
N
s.t. E[(WN,πi
)a]≤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
Mi :=
∫qΣ∈A
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(pi (qΣ))
adqΣ,
Mj :=
∫qΣ∈A
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(pj (qΣ))
adqΣ.
Then define B ⊂A to satisfy
Mi :=
∫qΣ∈B
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(pi (qΣ) + pj (qΣ))
adqΣ.
Now define a new ratio function p′ (qΣ), which is a modification of p (qΣ) as follows:
p′i (qΣ) = pi (qΣ) + pj (qΣ) for qΣ ∈B,
p′j (qΣ) = 0 for qΣ ∈B,
p′i (qΣ) = 0 for qΣ ∈A∩Bc,
p′j (qΣ) = pi (qΣ) + pj (qΣ) for qΣ ∈A∩Bc.
For the original ratio function p (qΣ),∫qΣ∈A
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)((pi (qΣ))
a+ (pi (qΣ))
a)dqΣ =M1 +M2.
But for p′ (qΣ),∫qΣ∈A
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)((p′i (qΣ))
a+ (p′i (qΣ))
a)dqΣ
=
∫qΣ∈B
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)((p′i (qΣ))
a+ (p′i (qΣ))
a)dqΣ
+
∫qΣ∈A∩Bc
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)((p′i (qΣ))
a+ (p′i (qΣ))
a)dqΣ
=
∫qΣ∈B
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(pi (qΣ) + pj (qΣ))
adqΣ
+
∫qΣ∈A∩Bc
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(pi (qΣ) + pj (qΣ))
adqΣ
=
∫qΣ∈A
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(pi (qΣ) + pj (qΣ))
adqΣ
<
∫qΣ∈A
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)((pi (qΣ))
a+ (pi (qΣ))
a)dqΣ
= Mi +Mj. (23)
The inequality above follows from the simple inequality xa1 + xa2 > (x1 +x2)a
where a < 1 and
x1, x2 > 0.
29
Since ∫qΣ∈A
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(p′i (qΣ))
adqΣ
=
∫qΣ∈B
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(p′i (qΣ))
adqΣ
+
∫qΣ∈A∩Bc
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(p′i (qΣ))
adqΣ
=
∫qΣ∈B
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(p′i (qΣ))
adqΣ + 0
=
∫qΣ∈B
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(pi (qΣ) + pj (qΣ))
adqΣ
= Mi,
the following holds by (23):∫qΣ∈A
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(p′j (qΣ)
)adqΣ <Mj
Recalling the definition of Mj and that p (qΣ) = p′ (qΣ) for qΣ /∈A, we have∫ ∞0
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(p′j (qΣ)
)adqΣ
<
∫ ∞0
1
1 + βΦ(β)
φ(β)
· β√λ· exp
(− β√
λqΣ
)(pi (qΣ))
adqΣ,
and, therefore, E[(QN,p′
j
)a]< E
[(QN,pj
)a]. Note that E
[(QN,p′
k
)a]= E
[(QN,pk
)a]for all k ∈ I ∩
{j}c. Let w′j :=E[(QN,p′
j
)a]. Then p′ satisfies the constraints of the problem:
minN,π
N
s.t. E[(WN,πi
)a]≤wi, i∈ I ∩{j}c,E[(WN,πj
)a]≤w′j,π ∈Π, N ∈Z+.
By Proposition 6, the optimal staffing for the above problem is lower than the one for (22). This
contradicts to the assumption that p (qΣ) is an optimal solution for (22) since p′ (qΣ) can satisfy
the constraints at a lower staffing costs. We conclude that any optimal policy should be bang-bang
rules.
As no solution for the b-constraint formulation has this extreme bang-bang rule, the a-constraint
formulation with a< 1 cannot be an implementation of any b-cost formulation. �
30
7. A numerical study of robustness
Recall (Corollary 3) that, with strictly convex constraints or strictly convex waiting costs, multiple
formulations are equivalent in terms of their solutions. If one wants to implement given costs
through constraints, there is a plethora of options to choose from. Similarly, a given constraint can
be an implementation of multiple “originating” costs, in which case results of reverse engineering
may differ depending on the model the econometrician chooses to estimate. We next study the
sensitivity of formulations to the cost and target parameters c and w as a way to guide to the choice
of constraint (specifically, that of a) for implementation and that of cost structure (specifically, the
value of b) for reverse engineering. In our examples, we use a system with two customer classes,
i.e., with I = {1,2}.
Implementation through constraints
By Corollary 3, we can choose w and w so that the following two problems (with a= 2 and a= 3),
min N
s.t. E[W 2i ]≤ wi, i∈ I,
π ∈Π,N ∈Z+,
min N
s.t. E[W 3i ]≤ wi, i∈ I,
π ∈Π,N ≥ 0,
have the same unique solution. To obtain a meaningful comparison, we turn to a question of
robustness. We ask which formulation is more sensitive to misspecification of the targets w and w.
Our base case has w= (0.01,0.02) and w= (0.027,0.008)
The staffing for these two equivalent formulations is given by β = 0.4, so that, with λ = 500,
N = λ+ 0.4√λ = 508. For each of the values of δ ∈ {2,3}, we perturb the targets w1 and w1 by
multiplying both by values δ ∈ [0.93,1.07]. We recompute the optimal value of β corresponding
to the perturbed parameters—that is, we solve the staffing problem for the δ-perturbed values to
arrive at the staffing constant βδ (β = β1).
Figure 7 displays βδ against δ for each of the cases a = 2 and a = 3. Evidently, the cubic for-
mulation is less sensitive to the specification of the target, and the formulation with a= 2 is more
sensitive to “mistakes” in specifying the targets.
We have repeated this experiment for various initial values of w (and hence β), obtaining the
same qualitative insight. When choosing between two equivalent constraint formulations, picking
a higher (rather than smaller) power is “safer” in terms of robustness to misspecification of the
constraints.
Reverse engineering
Next, we consider the effect of the exponent b in the cost formulation. Our departure point is
a constraint formulation that a service provider is using and has a > 1. Its optimal solution is
31
0.9975
0.9985
0.9995
1.0005
1.0015
0.993 0.995 0.997 0.999 1.001 1.003 1.005 1.007
Sensitivity of staffing (β) to perturbations of the target w
a=2
a=3
Figure 6 Sensitivity of staffing (β) to perturbations of the target w
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:
minN,π
N +∑i∈I
λic∗iE[(WN,πi
)b]: π ∈Π, N ≥ 0.
The coefficients ci, i= 1,2 depend, of course, on b: given two different actions, (N,p1) and (N , p),
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 p1 = 0.4 (and p2 = 1− 0.4 =
0.6), and the staffing level β = 0.4 (with λ= 500, we have N = 500+0.4 ·√
500≈ 508). The dualized
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 +∑
i∈I λiciE [W 2i ]
π ∈Π,N ∈Z+
andmin N +
∑i∈I λiciE [W 3
i ]
π ∈Π,N ∈Z+
have the same solution given by the fixed ratios p1,1− p1 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, c1/c1, c2/c2}.
The results, for the cross-sections p1 ∈ {0.98p1, p1,1.02p1}, are displayed in Figure 7. When there
is no perturbation (i.e., p1 = p1 and β = β) the value of the metric is 1.
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
0.98 0.99 1 1.01 1.02Staffing (β) perturbation
b=2
1
0.981.02
Ratio (p1) perturbation
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
0.98 0.99 1 1.01 1.02Staffing (β) perturbation
b=3
0.98
1.02
1
Ratio (p1) perturbation
Figure 7 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
p1 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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