Call center staffing: Service-level constraints and index priorities

Transcription

1 Submitted to Operations Research manuscript Call center staffing: Service-level constraints and index priorities Seung Bum Soh College of Business Administration, Sejong University, Itai Gurvich Kellogg School of Management, Northwestern University, May 2, 2016 Call centers attribute different values to different customer segments. These values are reflected in qualityof-service targets. The prevelant Target Service Factor (TSF) formulation requires, for example, that 80% of VIP customers wait less than 20 seconds while setting the target to 30 seconds for non-vip customers. The call center must determine the staffing level together with a prioritization rule that meets these targets at minimal cost. In practice, due to the underlying complexity of these systems, the prioritization rule is often selected in a heuristic manner rather than being systematically optimized. When considering the universe of prioritization policies, index rules provide a customizable and easy to define heuristic and for this reason are implemented in various call-center software packages. We use the TSF formulation as a stepping stone towards a better understanding of index rules. We first construct an asymptotically optimal solution for the TSF problem. The prioritization component of our solution is a tracking policy rather than an index rule. We prove that despite index rules significant flexibility, no instance of these prioritization rules is optimal for the TSF problem. The sub-optimality of index rules follows from an essential characteristic of these: restricting attention to index rules (as is heuristically done in practice) is asymptotically equivalent to requiring that a VIP customer always waits less than a regular (non-vip) customer who arrives at the same time. This, in particular, implies that the use of index rules in practice can be rationalized if (and only if) the manager requires such strong differentiation. 1. Introduction Waiting times are regarded as central to customers service experience and are used as a key performance indicator for call centers. In the spirit of measuring what is managed, call centers use metrics that aggregate waiting-time related information. They are collectively referred to as QoS (Quality-of-Service) metrics. A notable example is the Target-Service-Factor (TSF, the fraction of customers who exceed a target waiting time) ; see Chapter 2 of Koole [2013] for further discussion of call-center performance metrics. TSF is one of the key performance indicators that are traced and targetted in call centers. When the call center serves multiple customer classes, their relative importance is typically reflected in the TSF targets requiring, for example, that 80% of class-1 customers wait less than 20 seconds but 80% of class 2 customers wait less than 30 seconds. Solving the call-center s staffing problem in a multi-class environment requires determining the capacity level and the prioritization rule that, combined, guarantee that the service-level targets are met at a minimal cost. 1

2 2 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities In the two-class model that we consider here, the staffing problem is given by min N (TSF) s.t. s.t. P W N,π 1 > w 1 α1, P W N,π 2 > w 2 α2, N Z +, π Π, where, informally at this stage, W N,π i represents the steady-state waiting time of class i when the number of servers is N and the prioritization policy is π. We seek to find the minimal number of servers N together with a supporting prioritization policy π so that the TSF constraints are met. Here, Π denotes the family of admissible prioritization policies; see Definition 1. To the extent that the TSF constraint reflects relative importance, having α 1 α 2 together with w 1 w 2 corresponds to class 1 being the VIP class. Truly solving this problem to optimality by jointly determining the staffing level and the priority rule is difficult. This paper is the first to offer an asymptotically optimal solution for a relatively simple network. In practice, call centers typically restrict attention to parametric families of prioritization rules and subsequently adjust the parameters and the staffing level to meet the QoS targets. This heuristic approach is understandable given the complexity of the problem. Degrees of freedom in the prioritization policy are built into the software that manages the real time call selection and routing of calls. The software provides sufficiently many adjustable parameters while maintaining simplicity. A family of prioritization heuristics is that of index priorities; see e.g. Chan et al. [2014] or Koole and Pot [2006] where these are referred to as Time Function Scheduling (TFS). This family of policies is also well understood analytically in the context of convex holding cost minimization; see Van Mieghem [1995] and Mandelbaum and Stolyar [2004] A customer is assigned an index upon arrival. As the customer waits, his index increases periodically. The system manager chooses the initial indices for each customer type as well as the update intervals and update-to levels (these are easily defined in a spreadsheet). This leads to an index function as the one in Figure 1 which is used in an Israeli Bank that serves three categories of retail customers. In real time, a server who becomes available serves the customer with the highest index. Somewhat more formally, customer type i is associated with a piecewise-constant index g i ( ) so that a server that becomes available chooses for service the customer at the head of queue i arg max i g i (w i (t)) where w i (t) is the accumulated waiting time of the customers at the head of the class-i queue. This family of policies is intuitive to understand and easy to implement. Moreover, as both the updated periods and levels can be defined by the user it is expected to be rather flexible. Analytically understanding the role of index policies in constrained staffing

3 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities 3 Figure 1 Index functions in practice Figure 2 The non-index structure of optimal prioritization optimization is practically important, due to their use in practice, and intellectually important, due to the attention they have received in the literature on the optimization of queues. We first asymptotically solve the TSF formulation which, despite its prevalence in practice, has not been solved even in the simplest form of multi-class queues. We provide a simple staffing expression based on the M/M/N queue that explicitly captures the parameters of the problem. We prove that this staffing solution, together with a corresponding priority policy, is asymptotically optimal when the volume of arrivals is high. Since we are interested in better understanding the role of index rules, we proceed to show that all asymptotically optimal solutions to the TSF formulation are, in particular, non-index: all optimal prioritization policies share some non-conventional properties: (i) Non-monotonicity of waiting time in congestion: For at least one customer class (be it VIP or Regular), the waiting time has the structure depicted in Figure 2 it is greater when the congestion (specifically the total queue) in the system is moderate relative to when the congestion is high. (ii) Discontinuity of waiting time in congestion: Customers arriving within a short time interval

4 4 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities may experience starkly different waiting times (this corresponds to the discontinuity point in Figure 2). Since, as we also prove, index rules generate waiting time profiles that are continuous and monotone in the total congestion, we conclude that if one heuristically restrict attention to index rules even if one choose the best index functions staffing costs are non-negligibly greater than the minimum necessary. Thus, while the family of index rules is seemingly rich, there is no instance of an index rule that is asymptotically optimal (together with an appropriately chosen staffing level) for the TSF problem. To better understand this gap, it is useful to point to an interesting phenomenon one finds when considering the use of index rules in practice. Figure 3 displays the waiting-time processes 1 on a single day (May 10th, 2007) in a three-class Bank call center that uses the index rule in Figure 1 2. Figure 3 Waiting-time processes in a large Israeli bank A striking fact in this graph is that, sample path by sample path (we see a similar pattern on other days), a perfect real-time ranking between the classes is preserved, that is, Private customers wait less than their Star counterparts, whereas Star customers wait less than their Rainbow counterparts. This reflects a rather strong notion of what it means to be a VIP customer 1 Specifically, for each 2.5 minutes interval the graph depicts the waiting time averaged over customers that arrived over that specific time interval 2 We are thankful to the Technion Service Enterprise Engineering Laboratory (SEE Lab) for the data

5 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities 5 VIP customers wait less at all times (rather than, say, on average over a day or an hour). This requirement can be mathematically formulated as W N,π 1 P w 1 W N,π 2 1. w 2 This is a perfect service-level differentiation. One can view service-level differentiation (SLD) on a continuum parameterized by a constant β [0, 1] and captured by the constraint W N,π 1 P w 1 W N,π 2 β w 2 (SLD(β)) The case of β = 0 corresponds to requiring that differentiation holds only in aggregate by TSF formulation as in: on average at least 80% of VIP customers wait less than 20 seconds and at least 80% of Regular customers wait less than 30 seconds whereas β = 1 reflects a perfect real time differentiation as observed in Figure 3. We label by TSF+SLD(β) a formulation that has both the TSF constraint and the SLD(β) constraints. The observation that perfect SLD is preserved in this data is generalizable. Perfect service-level differentiation is a characterizing property of index policies. That is, requiring perfect service-level differentiation but giving full freedom in the choice of the prioriziation policy is equivalent to restricting attention to index rules in the TSF problem. More formally, the following two formulations are asymptotically equivalent: (TSF + SLD(1)) min N s.t. P W N,π 1 > w 1 α1, P W N,π 2 > w 2 α2, P W N,π 1 w 1 N Z +, π Π. W N,π 2 w 2 = 1, min N s.t. P W N,π 1 > w 1 α1, (TSF + Index) P W N,π 2 > w 2 α2, N Z +, π Π(Index). In words, restricting attention to index rules (as is often heuristically done) is asymptotically equivalent to requiring perfect SLD (i.e, β = 1). Thus, whereas the a priori restriction of some call centers to index rules may be driven by the software packages that they use, their choice is consistent with optimality, provided that perfect SLD is desirable to them. We close this introduction with a few important pointers to index rules in the literature. The use of index-rule based heuristics in the context of staffing subject to QoS targets is not grounded in theoretical foundation. However, it is the opposite case when considering holding cost (or waitingtime cost) minimization for given staffing levels. The best known results in this context pertain to the optimality of the Generalized cµ (Gcµ) rule for the minimization of convex holding costs as pioneered by Van Mieghem [1995]. Waiting-cost minimization is also the subject of subsequent

6 6 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities extensions to Van Mieghem s original work; see e.g. Mandelbaum and Stolyar [2004] and many others that followed. Such rules were also shown to be optimal for some constrained staffing problems. Average Speed of Answer (ASA) constraints give rise (as one possible solution) to a Fixed-Queue- Ratio rule a special case of an index rule; see Gurvich and Whitt [2007]. Thus, in certain contexts, these rules are very well understood. We expand this body of work with some new insights about the role of index rules in constrained staffing problems. Constrained staffing problems in multi-class queues have been also studied and we refer the reader to Aksin et al. [2007] for some references. The typical paper in this literature seeks to identify one optimal (or nearly optimal) solution. To that literature we add our solution to the TSF formulation. However, questions as we raise here, that pertain to structural properties of the family of all asymptotically optimal solutions, do not arise naturally in that line of work. They arise naturally here when we seek to understand the role of index rules in staffing problems. 2. Model and analysis framework We consider the two-class V model. Arrivals of class-i customers follow a Poisson process A i = (A i (t), t 0) with rate i > 0; i = 1, 2. Let = The two Poisson processes are independent. Service times are exponential with rate µ and are independent across customers and independent of the arrival processes. Various metrics of interest will be superscripted by the staffing level N and the prioritization policy π to denote their dependence on these decision variables. The TSF and SLD constraints were defined in the introduction using a virtual waiting time W N,π i. The Poisson Arrivals See Time Averages (PASTA) property guarantees that the fraction of customers who wait more than a target w equals the fraction of time that the virtual waiting time exceeds this value. We substitute the virtual waiting time with a proxy - the local average waiting time, which is based on the actual waiting times. Let w N,π i,k be the waiting time of the k th class-i customer to arrive. The local average waiting time of customers arriving over an interval (t, t + δ] is defined to be, where 0/0 is defined to be 0. W N,π 1 δ,i (t) := A i (t + δ) A i (t) A i (t+δ) k=a i (t)+1 w N,π i,k, (1) Local average is the measure used in Figure 3 where δ is 2.5 minute. If δ is kept sufficiently small, the local average captures the dynamics and variation of waiting time. The use of local average (rather than virtual waiting time) helps us overcome a technical difficulty (see Lemma 5.1). We assume throughout that α 1, α 2 > 0, α 1 α 2 and α 1 + α 2 < 1. (2)

7 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities 7 If α 1 +α 2 1, the staffing problem trivializes and, even for small values of w 1, w 2 > 0, any staffing level N > /µ is feasible; see the discussion on page page 285 of Gurvich et al. [2008], We allow the controller to use information about the length of the queues, the elapsed service times of customers that are in service and the accumulated waiting times of the all customers in the queues and we let X(t) be the state descriptor that captures all this information; see the appendix for the formal definition. Definition 1 (admissible policies): A prioritization policy π is admissible if: 1. It does not use admission control all customers are served. 2. It serves customers in a first-come first-served (FCFS) fashion within each class. 3. It is work conserving and stationary with respect to X. Let Π be the family of admissible policies. Non-preemption and FCFS within each class are both natural assumptions for our application. The literature does have cases in which non-work-conserving policies are shown to be optimal or asymptotically optimal (see Gurvich et al. [2008]). The results in Gurvich et al. [2008] are driven, however, by order-of-magnitude differences between the probability targets of the various classes (say α 1 = 0.01 but α 2 = 0.2) that lead inherently to solutions that induce perfect SLD rendering our main research questions irrelevant. Work conservation has the following immediate consequence which we state formally and repeatedly refer to. Lemma 2.1 Fix the number of servers N, the service rate µ and the arrival rates 1, 2. Then, under any admissible policy π, the total number of customers in the system X Σ (t) (respectively the total queue Q Σ (t)) follows the distribution of the total number of customers (respectively the queue length) in an M/M/N queue with arrival rate = 1 + 2, service rate µ and N servers. Many-server analysis: The problem TSF+SLD(β) is difficult to solve exactly. Instead, we resort to identifying solutions that are asymptotically optimal as the system size grows. A series of solutions is asymptotically optimal if the induced optimality gap is negligible in a central-limit-theorem (CLT) scaling. This is consistent with many studies in the context of call-center optimization; see e.g. Armony [2005], Armony and Mandelbaum [2011], Borst et al. [2004]. We consider a sequence of V models with the total arrival rate = increasing along the sequence but keeping the service rate µ fixed. We assume that there exists a i > 0, i = 1, 2 so that a 1 + a 2 = 1 and i = a i for i = 1, 2 and all. All relevant quantities are superscripted by to make the dependence on explicit. As is standard in the literature, the targets w 1 and w 2 scale with : wi = w i /, i = 1, 2 where w i s are fixed strictly positive constants. This guarantees that the system is in the Halfin-Whitt regime (see

8 8 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities Lemma 5.5 below). The Halfin-Whitt regime facilitates a CLT type of analysis. In our numerical experiments we illustrate how our proposed solutions perform for given parameters (rather than asymptotically); see 4. To simplify the notation, we denote a staffing-policy pair (N, π) by ξ and let N(ξ) and π(ξ) be, respectively, the staffing and prioritization components of ξ. The problem TSF+SLD(β) is now re-written as: min N(ξ ) ξ s.t. P W ξ, > w 1 α 1, P W ξ, > w 2 α 2, (3) W ξ, P W ξ, β, w1 w2 ξ Z + Π. W ξ, δ,i should be read as the the local average (as in (1)) when the system is initialized (at t = 0) with its steady-state distribution, the customer incoming rate is and the staffing-policy pair ξ is used. Definition 2 (asymptotic feasibility): A sequence of staffing-policy pairs ξ is asymptotically feasible, if ξ Z + Π for all and for each ɛ > 0, lim sup δ 0 lim sup P W ξ, δ,i > w i (1 ɛ) α i (1 + ɛ), i = 1, 2, and lim sup δ 0 lim sup P W ξ, w 1 W ξ, + ɛ β(1 ɛ). w2 We say that a sequence N is an asymptotically feasible sequence of staffing levels for (3) if N = N (ξ ) for a sequence ξ of asymptotically feasible staffing-policy pairs. Definition 3 (asymptotic optimality): A sequence of staffing-policy pairs ξ is asymptotically optimal if it is asymptotically feasible and [ ] + ( ) N(ξ ) N( ξ ) = o as, for any other sequence ξ of asymptotically feasible staffing-policy pairs. We say that a sequence N is an asymptotically optimal sequence of staffing levels for (3) if N = N (ξ ) for an asymptotically optimal sequence ξ of staffing policy pairs.

9 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities 9 3. The proposed solution to TSF+SLD and its implications This section contains our main results. We identify an asymptotically optimal staffing and prioritization solution. The two components of our solution are inter-dependent. The proposed staffing level is feasible if one uses it together with our proposed prioritization policy. For simplicity of exposition, we discuss them separately (each in a dedicated subsection), starting with the staffing component. Theorems stated in this section are proved in 5 and in the technical supplement Soh and Gurvich [2015] Optimal staffing and the cost of SLD To specify our staffing recommendation, let Q N, be a random variable whose distribution is that of the steady-state queue length in an M/M/N queue with arrival rate, service rate µ and N servers (as µ is fixed throughout, we omit it from the superscript). For each, let N be the solution of the following M/M/N staffing problem N = min N Z + : P Q N, 1 w w 2 min α1, 1 β + α 2. (4) Notice that, by (2), min α 1, 1 β + α 2 < 1 holds Theorem 1 (asymptotically optimal staffing) The sequence N is an asymptotically optimal sequence of staffing levels for (3). The prioritization component π (ξ ) of ξ, constructed in the next subsection, guarantees that all the inequalities W ξ, w 1, W ξ, w 2 and W ξ, /w 1 W ξ, /w 2 are asymptotically satisfied when the total queue length is smaller than 1 w w 2. Thus, the choice of the staffing level (4) guarantees that the total violation is bounded by minα 1, 1 β + α 2. The challenge in designing the priorities is to make sure that this violation budget is distributed correctly between the three constraints; for example, the TSF constraint for class 2 absorbs at most α 2 of the total violation budget. Remark 1 (aggregate-based staffing) Theorem 1 maps TSF+SLD(β) into a staffing problem for a single class queue. The fact that this solution works for the multiclass queue will be supported by our careful choice of the prioritization rule. Such a reduction to a one dimensional model is a recurring theme in recent literature on staffing; see e.g. Armony and Mandelbaum [2011], Gurvich et al. [2008], Gurvich and Whitt [2007]. In the latter two papers, a global Average Speed of Answer (ASA) constraint allows for a relatively simple mapping of the parameters into the single-class staffing problem. This mapping is more subtle in our setting as is reflected in the non-trivial way in which the constants α 1, α 2 and β appear on the right-hand side of (4).

10 10 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities Figure 4 η (β) as a function of β for α 1 = 0.3, α 2 = 0.2, w 1 = 2 and w 2 = 4 Theorem 2 The sequence N satisfies N = µ + η µ + o( ), where η is convex increasing in the SLD degree β. It is strictly increasing for β 1 α 1. Increasing the degree of SLD beyond a certain level (in particular, setting β = 1) is thus costly: the optimal staffing level for the TSF formulation is too small for TSF+SLD(1). Conversely, relaxing the SLD requirement reduces the staffing costs Prioritization and the optimality of index policies The asymptotically optimal staffing N is coupled with a carefully chosen sequence of policies π such that the sequence of staffing-policy pairs ξ = (N, π ) is asymptotically optimal in the sense of Definition 3. The policy that we construct in this section is an instance of so-called tracking policies. Tracking policies defined below are also admissible policies (customers are served in a FCFS manner within the same class). The queue length of class i at time t for i = 1, 2 is denoted by Q i (t) and Q Σ (t) is the total queue length, i.e., Q Σ (t) = Q 1 (t) + Q 2 (t). From these, normalized queue lengths are defined as follows: Q i (t) := Q i(t), QΣ (t) := Q 1 (t) + Q 2 (t). Definition 4 (queue tracking policies) Let f = (f 1, f 2 ) : R + R 2 + be a non-negative function such that f 1 (x) + f 2 (x) = x for all x 0. A server that becomes available at time t chooses the queue to serve according to the following criterion:

11 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities 11 Figure 5 Tracking: (Left) General description (Right) Proposed tracking function under sub-perfect SLD (β < 1) 1. If one queue is empty but the other is not, admit to service the customer at the head of the non-empty queue. 2. If both queues are non-empty and Q 1 (t) f 1 ( Q Σ (t)) Q 2 (t) f 2 ( Q Σ (t)), choose the class i with the positive value of Q i (t) f i ( Q Σ (t)). 3. If both queues are non-empty and Q 1 (t) f 1 ( Q Σ (t)) = Q 2 (t) f 2 ( Q Σ (t)), choose class 1. An arriving customer is immediately assigned to an available server if there are such servers. If all servers are busy upon this customer s arrival, the customer is placed in the queue. Figure 5(Left) illustrates the tracking function mechanism. It plots the targeted value for Q 1 (f 1 ( Q Σ )) vs. the value of Q Σ. If at time t, Q 1 (t) > f 1 ( Q Σ (t)) (as in point A in the figure) the rule prioritizes class 1 so as to pull it back toward f 1 ( Q Σ (t)). If Q1 (t) < f 1 ( Q Σ (t)) as in point B in the graph, the rule prioritizes class 2 over class 1 so as to push queue 1 towards f 1 ( Q Σ (t)). The tracking policy targets Q i (t) f i ( Q Σ (t)). The family of tracking functions is immense. The challenge is to carefully choose the tracking function f so that together with our proposed staffing the solution is asymptotically optimal (in particular, asymptotically feasible). Optimal tracking functions: We use the tracking function f 1 Figure 5(Right): defined below, as plotted also in 0, 0 x < a 1 w 1 +a 2 w 2 a 1 w 1 κ, ( ) a 1 w 1 a a 1 w 1 +a 2 w 2 x κ, 1 w 1 +a 2 w 2 a 1 w 1 κ x < a 1 w 1 + a 2 w a 1 w 1 a 2 w 2 κ, ( ) f 1 (x) = 2a 1 w 1 +a 2 w 2 x a 2 w 2 2(a 1 w 1 +a 2 w 2, a ) 2 1 w 1 + a 2 w a 1 w 1 a 2 w 2 κ x < a 1 w 1 + a 2 w 2, x a 2 w 2 + κ, a 1 w 1 + a 2 w 2 x < q, a 1 w 1 κ, q x. and f 2 (x) = x f 1 (x). (5)

12 12 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities The threshold q is given by q = a 1 w 1 + a 2 w 2 + α log 2 +minα 1,1 β α 2 η (β) (6) where η (β) is the unique solution (keeping other parameters constant) to ( 1 + ηφ(η) ) 1 exp( η(a 1 w 1 + a 2 w 2 )) = minα 1, 1 β + α 2, (7) φ(η) Here, φ( ), Φ( ) are, respectively, the standard normal density and distribution functions. The constant κ can be any 0 < κ a 1 w 1 a 2 w 2 2a 1 w 1 + a 2 w 2. Figure 5 (Right) considers the case of sub-perfect SLD (β < 1). The dashed line is the function f 1 (x Σ ) which is the target value for x 1. The thick solid line is where x 1 = a 1 w 1 x Σ /(a 1 w 1 + a 2 w 2 ). When ( Q ξ, 1 (t), Q ξ, 2 (t)) is on this line the waiting times satisfy W ξ, (t)/w 1 W ξ, (t)/w 2 for small δ because W ξ, (t) w 1, Qξ 1 (t) 1 w1, = Qξ 1 (t),, Σ, = Q ξ 1 (t) = Q ξ (t) = Q ξ 2 (t) a1 w 1 a 1 w 1 a 1 w 1 + a 2 w 2 a 2 w 2 W ξ, (t) w2 where W ξ, δ,i (t) (i = 1, 2) are local averages of waiting times when the staffing in (4) and the policy defined above are used and Q ξ, i (t) (i = 1, 2) are the queue lengths when the same staffing-policy pair is used. The first and the last signs follow from the sample path version of Little s law: Q ξ, i (t) i W ξ, δ,i (t), (8) that we will rigorously establish in Lemmas 5.1 and 5.3. As long as queues are below this line we expect to have W ξ, (t)/w 1 < W ξ, (t)/w 2. It makes sense, then, to refer to this line as the SLD line. The key in constructing the function f is figuring out how to divide the real line [0, ) to segments that differ with respect to their position relative to SLD line (below or above) and relative to the values w1 and w2. Suppose Q ( ) ξ, i (t) fi, Qξ Σ (t) (i = 1, 2), i.e., that the is con- ) tracking works (for rigorous proof on asymptotics see Proposition 10). The function f1 structed so that if Q ( ) ξ, Σ (t) a 1 w 1 + a 2 w 2 holds, f1, Qξ Σ (t) Q ξ, 1 (t) a 1 w 1, f2, (, Qξ Σ (t) Q ξ, 2 (t) a 2 w 2 and Q ξ, 1 (t) /a 1 w 1 Q ξ, 2 (t) /a 2 w 2 hold. By (8), W ξ, (t) w 1, W ξ, (t) w 2 and W ξ, (t) /w1 W ξ, (t) /w2 also hold. But if Q ξ, Σ (t) > a 1 w 1 +a 2 w 2, it is impossible to satisfy all the inequalities and one has to choose which constraints to sacrifice while others are met. Our choice of the function is such that, when Q ξ, Σ (t) [a 1 w 1 + a 2 w 2, q), the inequalities W ξ, (t) w 1 and W ξ, (t) /w1 W ξ, (t) /w2 are violated. If t is such that Q ξ, Σ W ξ, (t) w 2 is violated. (t) [ q, ), then the inequality

13 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities 13 The choice of the parameter q, combined with the optimal staffing in Theorem 1, guarantees that P, Qξ Σ (t) [a 1 w 1 + a 2 w 2, q) min α 1, 1 β and P, Qξ Σ (t) [ q, ) α 2. Then, P W ξ, (t) > w 1 P W ξ, (t) /w1 > W ξ, (t) /w2 min α 1, 1 β and P W ξ, (t) > w 2 α 2 hold and the solution is feasible. In the following theorem, πf denotes the tracking policy which applies the tracking function f. Theorem 3 (asymptotic optimality) Let N pairs ξ = (N, πf ) is asymptotically optimal. be as in (4). Then, the sequence of staffing-policy Remark 2 (on the interplay of cost reduction and prioritization) There are degrees of freedom in choosing the tracking function there are various choices that generate the same asymptotic optimality result. We chose f so that SLD is violated only when the total queue is small (specifically, less than q). With this choice, f has the appealing property that class-1 (the VIP) customers get superior service when the total congestion in the system is large. Thus, reducing β (from β < 1) allows for a cost reduction in terms of staffing (recall Theorem 2) while making sure that the VIP customers get superior service when it really matters. One thing that may strike the reader as non-standard in Figure 5(Right) is the discontinuity and non-monotonicity of the tracking function f. This is not a consequence of the specific way in which we choose the functions as the next theorem shows it cannot be avoided without compromise to optimality. For the formal statement, we say that a tracking policy is non-monotone if at least one of the functions f 1 ( ) and f 2 ( ) is non-monotone. A function f is defined to be piecewise Lipschitz if it has a finite number of discontinuity points x 1,..., x m, and is Lipschitz continuous on each interval (including [0, x 1 ) and [x m, )). Theorem 4 (non-monotonicity and discontinuity) Assume β < 1 (sub-perfect SLD) and let f be a piecewise Lipschitz tracking function such that the sequence (N, πf ) is an asymptotically optimal sequence of staffing-policy pairs. Then, f is discontinuous and non-monotone. The optimal staffing allows for a limited budget that must be carefully allocated to the three different constraints (the two TSF constraints and the SLD constraints). For different values of Q Σ one must choose which constraints are sacrificed in favor of meeting the others. What we prove is that the only way to do that while minimizing the total budget is to alternate the priorities and this, in particular, introduces the non-monotonicity and the discontinuity that are stated in the theorem.

14 14 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities On the equivalence of index policies and perfect SLD: Theorem 4 proves that, asymptotically, optimal tracking functions must be discontinuous and non-monotone for β < 1. That is, imposing monotonicity (which is a property of index policies; see below) is related to requiring perfect SLD (β = 1). We next show that requiring perfect SLD and restricting the tracking functions to be monotone is indeed equivalent in the asymptotic sense. More importantly, proving that monotone tracking rules are equivalent to index policies, allows us to conclude the equivalence between imposing a perfect SLD constraint and restricting the optimization to use only index policies. To formalize this result, recall that an index policy is one that, upon service completion at time t, admits to a customer from class i = arg max g i (Q i (t)), i where g 1 and g 2 are non-negative increasing functions. We denote by Π (index) Π the family of index policies. Theorem 5 (equivalence of index policies and SLD(1) ) There exists a sequence of staffingpolicy pairs ξ = (N, π ) that is asymptotically optimal for both formulations: min N(ξ ) ξ s.t. P W ξ, > w1 α 1, (TSF + SLD(1)) P W ξ, > w2 α 2, P W ξ, w 1 ξ Z + Π. W ξ, w 2 = 1, (TSF + Index) minn(ξ ) ξ s.t. P W ξ, > w1 α 1, P W ξ, > w2 α 2, ξ Z + Π(index). In particular, letting N, 1 be the objective function value of TSF+SLD(1) and N, 2 be that of TSF+Index, we have that N, 1 N, 2 = o( ). Theorem 5 is argued in three steps stated in Propositions 6-8. The first, Proposition 6, shows that index policies and monotone tracking policies are separate mathematical representations of the same policy and the proof is in Proposition 6 (equivalence of index policies and monotone tracking policies ) Given a monotone tracking function f, there exists an index function g such that the monotone tracking policy with f is equivalent to the index policy with index g. Similarly, given an index function g, there exists a monotone tracking function f such that the index policy with g is equivalent to the tracking policy with f. The equivalence is in the sense that at any given state of (Q 1 (t), Q 2 (t)), both policies take the same action.

15 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities 15 Given this equivalence, to prove Theorem 5, it remains to shows that TSF+SLD(1) is equivalent to the following: min ξ N ( ξ ) (TSF + MT) s.t. P W ξ, > w 1 α 1, P W ξ, > w 2 α 2, ξ Z + Π (MT), where Π (MT) Π is the family of monotone tracking policies. Note that a monotone tracking function must also be continuous by x = f 1 (x) + f 2 (x). It is easy to prove. For an arbitrary point x 1 > 0 and ɛ > 0, f 1 (x 1 ) f 1 (x 1 + ɛ) f 1 (x 1 ) + ɛ. The last inequality is from x 1 + ɛ f 1 (x 1 + ɛ) = f 2 (x 1 + ɛ) f 2 (x 1 ) = x 1 f 1 (x 1 ). We see as ɛ 0, f 1 (x 1 + ɛ) f 1 (x 1 ). The cases for ɛ < 0 and for f 2 are verified similarly. The equivalence is proved in the next two propositions. The first, Proposition 7, proves that there exists a monotone tracking policy which, with the staffing level from Theorem 1, satisfies the constraints in TSF+SLD(1). Proposition 7 (nearly optimal monotone solutions for TSF+SLD(1) ) For any ϑ > 0, there exists a monotone tracking function f ϑ and a sequence N ϑ such that (N ϑ, π f ϑ ) is asymptotically feasible for TSF+SLD(1) and N ϑ N ϑ. The nearly optimal tracking function, whose existence is established in Proposition 7, is depicted in Figure 6 and explicitly specified as follows: 0, 0 x < a 1 w 1 +a 2 w 2 a 1 w 1 κ ϑ, f ϑ a 1 (x) = 1 w 1 a 1 w 1 +a 2 w 2 x κ ϑ a, 1 w 1 +a 2 w 2 a 1 w 1 κ ϑ x < a 1 w 1 + a 2 w 2, a 1 w 1 κ ϑ, a 1 w 1 + a 2 w 2 x. In the figure the constant q ϑ b is the point of intersection between the tracking function f ϑ 1 (x) and the line x a 2 w 2. By (8), W ξ, (t) w 1, W ξ, (t) w 2 and W ξ, (t) /w 1 W ξ, (t) /w 2 hold at times in which Q ξ, Σ (t) qϑ b and only the constraint W ξ, (t) w 2 is violated when Q ξ, Σ (t) > qϑ b. Notice in Figure 6 that f ϑ 2 (x) = x f ϑ 1 (x) > a 2 w 2 when x > q ϑ b. Proposition 8 closes the equivalence argument toward Theorem 5 in showing that, in terms of staffing, requiring monotonicity is as demanding as requiring SLD(1). Since we identified in Proposition 7 that a monotone tracking solution is asymptotically optimal for TSF+SLD(1), we conclude that TSF+MT and TSF+SLD(1) are asymptotically equivalent and, by Proposition 6, so are TSF+index and TSF+SLD(1).

16 16 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities Figure 6 Proposed tracking function under perfect SLD (β = 1) Proposition 8 (equivalence of monotone tracking and SLD(1) under TSF constraints) Let ξ 1 be a series of asymptotic solutions for TSF+SLD(1) and ξ 2 be one for TSF+MT. Then, lim inf N (ξ 2 ) N (ξ 1 ) 0. Staffing TSF+Index TSF+Gcµ TSF+SLD(β) TSF+SLD(β) β Figure 7 Optimal staffing levels

17 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities 17 Remark 3 (the cost of index policies) Figure 7 displays the minimal staffing requirement to satisfy the TSF constraints P W 1 10 sec. 0.2 and P W 2 10 sec. 0.2, when the arrival rate of each class is 50 customers per minute and the average service time is 5 minutes. The asymptotic optimal staffing is derived using Theorem 1. When adding the SLD constraint, the cost changes with the value of β in the SLD constraint as β varies between 0.65 and 1. This is is captured by the solid line. When, instead of having the SLD constraint, one restricts attention to index policies the staffing level is always 517 (captured by the dashed line). The gap between the two lines is the cost of the restriction to index policies. For β = 1, consistent with our equivalence result, there is no gap. For small value of β the gap is 7 servers, corresponding to a non-negligible gap relative to the square root of the system size. 4. Numerical experiments The purpose of this section is to illustrate the performance of the solutions we derived via the asymptotic analysis. We use this opportunity to underscore some important aspects of the proposed (nearly optimal) staffing rule. Altogether, we consider 40 parameter combinations (4 sets, each consisting of 10 cases). Across cases in each set, we vary the arrival rates 1, 2 and the target parameters w 1, w 2 as in Table 1. Note that these parameter combinations are the same across sets, i.e., the parameter combinations for Scenario k of Set i is the same with that of Scenario k of Set j. We also vary α 1, α 2 for the TSF constraints and the SLD target β for the SLD constraint as in Table 2. Set 1 and Set 2 differ in that Set 1 has no SLD constraint (or β = 0). Set 1 and Set 3 differ in that Set 3 has smaller α 1 s (tighter TSF for class 1). Lastly, Set 3 and Set 4 differ in that Set 4 requires perfect SLD). Scenario w w Table 1 Common parameters for the whole sets The mean service time is set to 1 time unit throughout (all parameters are specified in the same time units, so that = 125 for example corresponds to a rate of 125 per time unit, be it an hour or a minute). The number of servers (staffing) is determined via our formula (4) N ( 1, 2, w 1, w 2, α 1, α 2, β) = min N Z + : P Q N, 1 w w 2 minα1, 1 β + α 2.

18 18 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities Scenario α Set 1 α β 0 (No SLD constraint) α Set 2 α β α Set 3 α β 0 (No SLD constraint) α Set 4 α β 1 (Perfect SLD constraint) Table 2 Probability targets for each set Given N (different for each parameter combination), we compute η = (N ( )/µ)/ and use it to design the propopsed tracking functions as in (5). This gives us the tracking policy which, as before, we denote by π. We built a simulation model in ARENA. For each parameter combination, we simulate a sample path of the two-class queue with the proposed solution (N, π)( 1, 2, w 1, w 2, α 1, α 2, β) for T = time units corresponding to 8-40 million arrivals depending on the parameter combination. 1 Ai (T ) At the end of each simulation run, we record A i (T ) k=1 1w i,k > w i (1 ɛ) with A i (T ) being the number of class i customers that arrived over the simulation horizon and w i,k is the actual waiting time of the k th class i customer processed. Since A i (T ) 1 1w i,k > w i (1 ɛ) PW i > w i (1 ɛ) as T, A i (T ) k=1 this statistic provides a proxy for the performance of the policy with respect to the TSF constraints. Our asymptotic theory says that under the proposed solution it holds that lim sup PWi > wi 1 Ai (T ) (1 ɛ) α i (1 + ɛ)(i = 1, 2) and hence we check whether A i (T ) k=1 1w i,k > w i (1 ɛ) α i (1 + ɛ) holds or not for each i. For the SLD constraint, we fix δ = 0.05 and compute at intervals of size δ, W δ,i (t) as in equation (1). Specifically, for each m = 1,..., M = 40000/0.05 = , we compute W δ,i (mδ) := With the fixed δ, the long run average 1 M M m=1 1 A i (mδ + δ) A i (mδ) A i (mδ+δ) k=a i (mδ)+1 w i,k. W (mδ) 1 W (mδ) W (1 + ɛ) P W (1 + ɛ) w 1 w 2 w 1 w 2 as M.

19 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities 19 Figure 8 Set 1(TSF Formulation (β = 0)): (Left) Performance with recommend staffing; (Right) With one server added for each violating scenario serves as a proxy for the SLD and we check, in the simulation, whether it exceeds β(1 ɛ) or not. We set ɛ = 0.1 which is a reasonably ambitious criterion. The math says that as grows large with appropriate scaling we could take ɛ to be increasingly smaller. Figure 8 depicts the results for Set 1. We plot two bar series corresponding to Ai (T ) w 1 1 the realized TSFs ( A i (T ) k=1 1w i,k > w i (1 ɛ)(i = 1, 2)) and the realized SLD ( 1 M 1 W (mδ) M m=1 W (mδ) w 2 (1 + ɛ) ) is captured by the black-circle series for each of the scenarios. Notice that there is no target on the SLD but, still, we can compute the outcome. The labels report the values (the TSF series correspond to the left y-axis and the SLD to the right y-axis). Colored in white are the metrics that do not meet the feasibility criteria α i (1 + ɛ) of the TSF constraints. The performance is rather impressive. There are 5 violations (out of 20 constraints) and with the exception of the violations of class 1 in scenario 6 and of class 2 in scenario 7 (0.345 instead of 0.33 and instead of 0.22), the others are truly small (as in instead of the targetted 0.33 in scenario 5 or instead of 0.22 in scenario 10). Scenarios 5-8 have the smallest w i targets (e.g. w 1 = and w 2 = 0.03) and hence the most sensitive. The violations are resolved with the addition of a single server as plotted in the graph on the right of the figure. Notably, the SLD is very high (indeed, around 0.7 or higher) even though we imposed no SLD constraint. This is an opportunity to recall Figure 7 where the required number of servers is invariant to β as long as 1 β α 1. With a target of α 1 = 0.3 this means one gets for free an SLD of up to β = 0.7 (when α 1 = 0.3) and up to β = 0.8 (when α 1 = 0.2 as in Scenario 8 and 10 of Table 2). Our policy uses this allowance. We run 10 simulations with newly computed staffing levels and the policies for Set 2 to consider

20 20 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities b0 Set 1 ( ) bg0 Set 2 ( ) Normalized safety capacity = Figure 9 (Left) Performance with proposed staffing for Set 2 (TSF+SLD(β)); (Right) The cost of SLD TSF+SLD(β) with positive β s. The results are reported in Figure 9. In this case, the TSF criteria are met in all 10 scenarios with the exception of minor violations in the first two scenarios (that disappear with the addition of one server). The SLD is mostly around 0.9 which is close to the target and always above the criterion of β(1 ɛ). We notice again that α 1 matters only through minα 1, 1 β: For a given β target the solution can afford to have α 1 as small as 1 β and the policy uses some of this allowance. Thus, even though α 1 > α 2, the performance obtained is such that PW 1 > w 1 < PW 2 > w 2 in all scenarios. Finally, on the right of Figure 9 we visualize the cost difference of TSF vs. TSF+SLD(β) by comparing the staffing level for TSF that underlies the graph on the right of Figure 8 and the staffing level of TSF+SLD(β) that underlies the left graph of Figure 9. We plot the normalized safety capacity (i.e., the servers required above the offered load /µ.) These numbers are proxies for the square-root coefficient η in Theorem 2 and capture the cost of SLD. The parameter combinations in Set 3 are obtained from Set 1 by introducing more demanding values of α 1 (e.g instead of 0.3). It underscores further the interaction between α 1 and β. The results are reported in Figure 10. There are negligible violations (0.166 instead of 0.165) for class 1 in scenarios 5 and 9. The more substantial violations are in scenarios 3, 7. The violations are concentrated in scenarios with the smallest w i (w 1 = and w 2 = 0.03). An extra staffing corrects for some of this but leaves slight violations for class 1. This should not be surprising: with very small w and α values the system is pushed away from heavy-traffic and hence the approximations are somewhat less precise. Finally, even though β = 0, our policy again extracts as much SLD as it can get for free which is 1 α 1 (0.85 or 0.9 depending on the scenario). In Set 4, we consider the case of perfect SLD (β = 1): we use the same data from Set 3 but put β = 1 as a requirement in all scenarios. The policy is a monotone policy as in Figure 6. The

21 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities 21 Figure 10 Set 3 (TSF (β = 0), tight α): (Left) Performance with recommended staffing; (Right) With one server added for each violating scenario performance is reported in Figure 11. The TSF targets are met with excess under the recommended staffing and the SLD is very close to 1 for all values. The graph on the right is a visualization of SLD and the workings of the asymptotic arguments. One informally expects (and our proofs are built on formally establishing that) W δ,i (t)/w i W i (t)/w i Q i (t)/ i w i because of the sample path Little s law and since δ is small. With SLD close to 1 we expect Q 1 (t) Q 2(t), t 0. 1 w 1 1 w 2 The graph on the right nicely visualizes this asymptotic argument. We simulate Scenario 1 ( 1 = 2 = 125) of Set 4 and zoom-in on the time interval [460, 480] to obtain a clear view Figure Simulation time Set 4 (TSF + Perfect SLD (β = 1)): (L) Performance with recommended staffing (R) samplepath SLD

22 22 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities 5. Proofs This section is dedicated to the proofs of our main results. 5.1 proves Theorem 3 which is on the asymptotic optimality of our proposed staffing and policy. 5.2 contains proofs of the theorems that deal with the equivalence of index policies and perfect SLD. The proofs of other theorems and auxiliary lemmas are in Soh and Gurvich [2015] Proof of Theorem 3: We divide the lengthy proof into sub-modules as follows: we first prove an equivalence between the waiting time formulation (3) and a queue-based formulation. This may be somewhat expected given (8). The challenge, however, is to establish such a Little s law result at the formulation level (rather than for a given policy as is typically the case). It is in this step where considering local averages is instrumental. Building on this asymptotic equivalence, a lower bound for the local-average-queue-based formulation provides an asymptotic lower bound for the waiting-time formulation. We identify the staffing recommendation N in (4) as such a bound; see In we proceed to show that the tracking policy with our proposed function f indeed has the desired asymptotic properties in that, when used, it results in Q ξ, i (t) fi ( Q ξ, (t)). Whereas Σ this type of so-called state-space collapse is, by now, a standard result, a challenge arises here from the possible discontinuity of the tracking function f. We combine the pieces in to have the proof of Theorem 3. We prove there that our proposed solution is asymptotically feasible. Since its staffing component (and, in particular, its cost) is a lower bound on any such solutions, we conclude that our solution is asymptotically optimal From waiting times to queues Before directly showing the relationship of (8), we introduce local average queue lengths which connect local average waiting times and queue lengths: Q ξ, δ,i (t) = 1 δ t+δ t Q ξ, i (s) ds. (9) The following lemma shows that if a sequence N is asymptotically feasible for (3) it must (asymptotically) satisfy the appropriate constraints in terms of queues and vice versa. We let Q ξ, δ,i be the δ average when the queue length is initialized (at time t = 0) with its stedy-state distribution. Lemma 5.1 Let ξ be a sequence of admissible staffing-policy pairs. Then the followings are equivalent. 1. For each ɛ 1 > 0, the following holds. lim sup δ 0 lim sup P W ξ, > w 1 (1 ɛ 1 ) α 1 + ɛ 1,

23 Soh and Gurvich: Call center staffing: Service-level constraints and index priorities lim sup δ 0 lim sup δ 0 lim sup P lim sup P W ξ, > w 2 (1 ɛ 1 ) α 2 + ɛ 1, W ξ, w 1 W ξ, ɛ w2 1 β ɛ For each ɛ 2 > 0, the following holds. lim sup δ 0 lim sup δ 0 lim sup δ 0 lim sup P Q ξ, > 1 w 1 (1 ɛ 2 ) α 1 + ɛ 2, lim sup P Q ξ, > 2 w 2 (1 ɛ 2 ) α 2 + ɛ 2, Q ξ,, Qξ ɛ 1 w1 2 w2 2 β ɛ 2. lim sup P Lemma 5.1 proves that replacing the waiting-time constraints with queue-based constraints should yield (asymptotically) identical results in terms of optimality. Thus, we turn to study a queue-based formulation starting with identifying a lower bound. To that end, given ɛ, consider the problem min N ( ξ ) ξ s.t. P Q ξ, > 1 w 1 (1 ɛ) α 1 + ɛ, P Q ξ, > 2 w 2 (1 ɛ) α 2 + ɛ, (10) Q ξ,, P Qξ ɛ β ɛ. 1 w1 2 w2 We next state our lower bound result. Recall that Q N, is a random variable with the stationary distribution of the queue length in an M/M/N queue with arrival rate, service rate µ and N(> /µ) servers. As before, the variable Q N, δ is the local average on [0, δ] when the M/M/N queue is initialized with its steady-state distribution. Proposition 9 Fix, δ, ɛ > 0 and let ξ δ,ɛ be a feasible solution for (10). Define N (δ, ɛ) = min N Z + : P Q N, δ 1 w w 2 min α1, 1 β + α 2 + 2ɛ. Then, N (δ, ɛ) N(ξ δ,ɛ), where N ( ξ δ,ɛ) is the staffing component of ξ δ,ɛ. Proof: Define the following events on the underlying probability space: A := B :=, Q ξ, + Q ξ, 1 w w 2 Q ξ, + Q ξ, > 1 w w 2, Q ξ, > 2 w 2,

24 24 C := Soh and Gurvich: Call center staffing: Service-level constraints and index priorities Q ξ, + Q ξ, > 1 w w 2, Q ξ, 2 w 2. These events form partitions of the space. Further, on C, Q ξ, > 1 w w 2 Q ξ, ( ) 1 w1 + 2 w2 = ( 2 w2 2 w 2 Q ξ, ) + 1w1 Q ξ, 2 w2 1w1 Q ξ, 2 w2. Also it is evident that Q ξ, > 1 w2 on C. In turn, Q ξ,, C > Qξ Q ξ, 1 w1 2 w2 > 1 w 2. (11) Using these definitions and the assumed feasibility of the staffing-policy pair, P B = P Q ξ, + Q ξ, > 1 w 1 + w 2, Q ξ, > 2 w 2 P Q ξ, > 2 w 2 P Q ξ, > 2w 2 (1 ɛ) α 2 + ɛ,. and P C min min P P Q ξ, 1 w1 Q ξ, 1 w 1 minα 1 + ɛ, 1 β + ɛ.., > Qξ 2 w 2, > Qξ ɛ 2 w2, P Q ξ, > 1 w 2, P Q ξ, > 1 w 2 ɛ. where the first inequality follows from (11). Hence P Q ξ, + Q ξ, > 1 w w 2 = P B C = P B + P C minα 1, 1 β + α 2 + 2ɛ. Since the policy is work conserving, Q ξ, +, Qξ has the law of QN, δ with N = N(ξ ). N (δ, ɛ) is the minimum staffing among the ones that satisfy above and hence N (δ, ɛ) N(ξ δ,ɛ). Note that N (δ, ɛ) is closely related to our proposed staffing component N in (4). The following lemma guarantees that the staffing level N in (4), combined with a feasible policy, is asymptotically optimal. Lemma 5.2 lim sup ɛ 0 N (δ, ɛ) N lim sup lim sup 0. δ 0