Forecasting and Staffing Call Centers with Multiple Interdependent Uncertain Arrival Streams

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1 Forecasting an Staffing Call Centers with Multiple Interepenent Uncertain Arrival Streams Han Ye Department of Statistics an Operations Research, University of North Carolina, Chapel Hill, NC 27599, James Luetke Department of Inustrial an Systems Engineering University of Wisconsin-Maison, Maison, WI 53706, Haipeng Shen Department of Statistics an Operations Research, University of North Carolina, Chapel Hill, NC 27599, We consier forecasting an staffing call centers with multiple interepenent uncertain arrival streams. We first evelop general statistical moels that can simultaneously forecast multiple-stream arrival rates that exhibit inter-stream epenence. The moels take into account several types of inter-stream epenence. With istributional forecasts, we then implement a chance-constraint staffing algorithm to generate staffing vectors an further assess the operational effects of incorporating such inter-stream epenence, consiering several system esigns. Experiments using real call center ata emonstrate practical applicability of our propose approach uner ifferent staffing esigns. An extensive set of simulations is performe to further investigate how the forecasting an operational benefits of the multiple-stream approach vary by the type, irection, an strength of inter-stream epenence, as well as system esign. Managerial insights are iscusse regaring how an when to take avantage of the inter-stream epenence operationally. Key wors : cross training, vector time series, quality of service, chance constraint, stochastic programming History : 1. Introuction Call centers are moern service networks in which agents provie services to customers via telephones. They have become a primary contact channel between service proviers an customers. There are several goo reviews about research relate to call center operations, incluing Gans et al. (2003) an Aksin et al. (2007). Inboun call centers take calls that are initiate by customers, an they use a hierarchical staffing an scheuling system. The process begins with forecasting call arrival rates over a planning horizon, ranging from a ay to several weeks. The current paper focuses on call centers with multiple uncertain call arrival streams, such as from multiple customer classes, that are epenent. For example, the numbers of English-speaking an French-speaking customers that call a major Canaian telecommunication company uring the same time intervals are positively epenent (Ibrahim an L Ecuyer 2013). We aim at answering 1

2 2 the following questions: what statistical moels lea to better forecasts? o the better forecasts translate into improve operational performance? if they o, what kin of systems can benefit the most? To answer the questions, we first evelop forecasting moels that take into account the inter-stream epenence to give more accurate istributional forecasts. In aition, we use a chance-constraint formulation (Gurvich et al. 2010) to investigate the operational benefits of incorporating such epenence on quality of service (QoS) an staffing costs, an offer insights into how the benefits interact with the flexibility of the service system in terms of system esign an cross-training. Traitionally, call centers assume that arrival rate forecasts are accurate an use point forecasts of the arrival rates to rive the staffing an scheuling plans, which very often results in mismatch between the point forecasts an realize arrival volumes an hence improper staffing levels. A lot of progress has been mae recently in the statistics an operations management (OM) literatures to better cope with arrival rate uncertainty. However, almost all the statistical work is about forecasting a single arrival stream, such as Avramiis et al. (2004), Brown et al. (2005), Weinberg et al. (2007), Taylor (2008), Shen an Huang (2008a), Shen an Huang (2008b), Alor-Noiman et al. (2009), Matteson et al. (2011), Taylor (2012), an references therein. The one exception is the recent work of Ibrahim an L Ecuyer (2013) that extens the one-stream moel of Alor-Noiman et al. (2009) to evelop two mixe-effects moels for forecasting bivariate arrival streams. On the other han, OM papers account for uncertainty when making workforce management ecisions. Several papers use stochastic programming (SP) (Birge an Louveaux 1997) to account for arrival rate uncertainty when making staffing an call-routing ecisions, incluing Harrison an Zeevi (2005), Bassamboo et al. (2005, 2006), Bassamboo an Zeevi (2009), Bertsimas an Doan (2010), Gurvich et al. (2010). More recent papers exten the SP formulation to scheuling, such as Robbins an Harrison (2010), Robbins et al. (2010), Liao et al. (2012). To cope with biase initial arrival-rate forecasts, Mehrotra et al. (2010) uses mi-ay recourse actions to ajust pre-scheule staffing levels. Some of the above papers eal with staffing/scheuling when there are multiple arrival streams. For example, Gurvich et al. (2010) proposes a chance-constraint formulation to staff multiple-stream call centers. The above papers have mae important progress aressing the problems cause by arrival-rate uncertainty, although only partially. Statistical forecasting papers have evaluate their methos using traitional forecasting accuracy measures base on realize arrival counts, while ignoring the operational effects the forecasting errors might have on cost an QoS measures. On the other han, OM papers have carefully emonstrate the cost an QoS implications of their proceures,

3 3 assuming that the arrival rate istributions are given, although in practice they have to be estimate from ata. Gans et al. (2012) is the only paper that aims at solving the whole problem, integrating arrival rate forecasting with stochastic programming to illustrate the operational effects of SP with or without recourse using arrival-rate istributions forecaste an upate from real ata. As in almost all the forecasting papers, Gans et al. (2012) only consiers a single arrival stream. Our paper focuses on call centers with multiple uncertain arrival streams, an for the first time, integrates arrival rate forecasting an staffing together for such systems. We exten the univariate forecasting moel of Gans et al. (2012) to the multivariate cases, to simultaneously provie istributional forecasts for multiple arrival streams. Given the istributional forecasts, we then implement the chance-constraint formulation propose by Gurvich et al. (2010) to evaluate the ownstream effects of the improve forecasts on staffing costs an QoS measures. We use a real call center application an extensive simulation stuies to emonstrate the effects of incorporating interstream epenence on both forecasting accuracy an operational efficiency, an how the avantages of our multiple-stream approach vary by the type an strength of inter-stream epenence, as well as staffing esigns. Hence our paper nicely complements the earlier works of Ibrahim an L Ecuyer (2013), Gans et al. (2012), Gurvich et al. (2010). More specifically, we make the following contributions. We introuce our statistical forecasting moels in Section 2. Our general moels take into account three types of inter-stream epenence. The moels impose a multiplicative structure on the arrival rates, an treat them as proucts of aily-total rate an within-ay proportion profile. This formulation reuces the imensionality of the problem. We theoretically an numerically evaluate the forecasting benefits of incorporating inter-stream epenence of ifferent type an strength. Furthermore, to test the operational effects of incorporating inter-stream epenence, we consier the chance-constraint staffing formulation with sample base approximation by Gurvich et al. (2010), which is briefly reviewe in Section 3. We integrate our forecasting metho an the chance-constraint staffing formulation as an entire solution to staff call centers with multiple uncertain inter-epenent arrival streams. We test the forecasting an operational benefits of our forecasting approach using a real call center application in Section 4 an extensive simulation stuies in Section 5. In particular, we consier 225 simulate scenarios of ifferent types an strengths of inter-stream epenence as well as 9 staffing esigns, to unerstan how the forecasting an operational benefits vary by irection an strength of epenence among the streams, as well as the infrastructure of the staffing system. Our results suggest that the multiple-stream approach provies more accurate istributional forecasts; the stronger the epenence on the other streams past information, the better point forecasting improvement the multiple-stream approach has. Operationally, our forecasting approach

4 4 that incorporates inter-stream epenence leas to stable QoS performance regarless of staffing esign an type/strength of inter-stream epenence, while the approach that ignores such epenence causes mismatch between eman an staffing level, either unerstaffing or overstaffing, an the severity of the mismatch increases when the staffing system becomes more flexible. This suggests that it is even more crucial for flexible systems to use our multiple-stream forecasting approach, in orer to fully utilize the flexibility of the system. Finally, our stuy suggests that cross-training improves cost-efficiency of the staffing system, an the magnitue of improvement epens on the irection an strength of inter-stream epenence. 2. Forecasting Multiple Arrival Streams We present our general statistical moel for forecasting multiple arrival streams, an iscuss various types of inter-stream epenence that our moel covers. We rigorously illustrate the forecasting benefits of incorporating inter-stream epenence. We then evelop a sequential iterative moel estimation algorithm, an obtain point an istributional forecasts for arrival rates an counts General Statistical Moel Denote the number of customer types (or arrival streams) as I. For the ith arrival stream, i = 1,..., I, we observe the number of call arrivals uring time perio t on ay, an enote it as N (i),t for t = 1,..., T, = 1,..., D. In practice, each time perio can be either quarter hour or half hour. We moel N (i),t as a Poisson ranom variable with a ranom arrival rate λ (i),t that epens on customer type i, time perio t, an ay (most likely through ay of the week, enote as w ) (Brown et al. 2005, Kim an Whitt 2013). Following common practice in call center arrival forecasting (see Ibrahim an L Ecuyer (2013) an references therein), we first apply the following square-root transformation to normalize the arrival counts an stabilize the variance as σ 2 (i) : X (i),t = N (i),t + 1 ( ) 4 N λ (i),t, σ2 (i). We then consier the following statistical moel on the square-root-transforme counts X (i),t : ( ) X (i),t = λ (i),t + ɛ(i),t, i = 1,..., I, ɛ,t = ɛ (1),t, ɛ(2),t,..., ii ɛ(i),t N (0, Σ), θ (i),t λ (i),t = u(i) f T (i) w,t, f (i) w,t 0, f (i) w,t = 1, t=1 u α w = A ( ) ( u 1 α w 1 + z, z = where w is the ay-of-week of ay, u = (u (1), u(2) z (1), z(2),..., z(i) ) ii N (0, Ω), (1),..., u(i) ) is the vector of aily total arrival rates of all customer streams (on the square-root scale), α w = (α (1) w, α (2) w,..., α (I) w ) moels the

5 ay-of-week effect on aily total arrival rate (on the square-root scale), an f (i) w,t is the intraay rate proportion of the tth time interval for customer type i that also epens on ay of week. Moel (1) can be viewe as the multi-arrival-stream extension of the forecasting moel consiere by Gans et al. (2012). It captures inter-ay, intra-ay an inter-stream epenences that have been ientifie in previous call center work an the current stuy (Section 4). We can unerstan the moel in the following way. The square-roote count ata approximately follow a multivariate Gaussian istribution (Brown et al. 2010). On the square-root transforme scale, the arrival rate profile for any customer type i on ay, i.e. θ (i) (θ (i),1, θ(i),2,..., θ(i),t ), is assume to have a multiplicative structure, where it can be moelle as the prouct of the aily total rate u (i) the intraay proportion profile, f (i) w (f (i) w,1, f (i) w,2,..., f (i) w,t ), that epens on the corresponing ay-of-week. Such multiplicative structure has been propose for call centers by as early as Whitt (1999), an shown to work well empirically by Brown et al. (2005), Shen an Huang (2008a,b), Gans et al. (2012). In aition, the vector of aily total rates of all customer types u follows a first-orer vector autoregressive (VAR(1)) time series moel (Reinsel 2003), after being ajuste for the ay-of-week effect α w. Diagnostics suggest that the moel works well for our ata. Furthermore, Moel (1) incorporates three types of inter-stream epenence as we now elaborate: Type (a) epenence: Σ = (Σ sl ) I I 5 an is the covariance matrix that accounts for inter-stream epenence among the (transforme) arrival counts, conitional on knowing the arrival rates; Type (b) epenence: A = (a sl ) I I inter-stream epenence among the aily total arrival rates; Type (c) epenence: Ω = (Ω sl ) I I epenence among the innovations z. is the auto-regressive coefficient matrix that moels the is the covariance matrix that captures the inter-stream An Illustrative Example: Bivariate Arrival Streams To better unerstan the three types of inter-stream epenence, we consier a special case of Moel (1) for bivariate arrival streams, i.e. I = 2. We first illustrate how the inter-stream epenence is moele, an then compare our moel with the two bivariate mixe moels propose by Ibrahim an L Ecuyer (2013). The two-imensional moel is as follows: ( ) (( ) ( )) X (i),t = λ (i),t + ɛ(i),t, i = 1, 2, ɛ (1),t ii 0 Σ 11 Σ 12 ɛ (2) N,,,t 0 Σ 21 Σ 22 θ (i),t λ (i),t = u(i) f T w (i), f w (i) 0, f w (i) = 1, i = 1, 2, ( ) ( ) ( t=1 u (1) α w (1) (1) a 11 a 12 u 1 u (2) = ) ( ) ( ) (( ) ( )) α(1) w 1 z (1) z (1) ii 0 Ω 11 Ω 12 α w (2) a 21 a 22 u (2) 1 + α(2) w 1 z (2), z (2) N,. 0 Ω 21 Ω 22 (2) Accoring to Moel (2), the two arrival streams are epenent in the following three ways:

6 6 Type (a) epenence: conitional on the arrival rates (θ (1),t, θ(2),t ), the two (square-roote) counts, X (1),t an X(2),t, are epenent as measure by the correlation r Σ 12/ Σ 11 Σ 22 ; Type (b) epenence: there is a bivariate lag-1 epenence among the aily total arrival rates, u (1) an u (2), an the lag-1 cross epenence is measure by a 12 an a 21 ; Type (c) epenence: the innovations in the bivariate lag-1 equations of the aily total arrival rates are epenent, measure by the correlation ρ Ω 12 / Ω 11 Ω 22. Ibrahim an L Ecuyer (2013) propose two aitive mixe moels specifically for bivariate arrival streams. Different from our formulation, they impose an aitive structure on the arrival rates, taking into account ay-of-week an time-of-ay effects. In aition, each of their two moels respectively consiere one type of inter-stream epenence: epenence of the counts between the two streams of the same ay an same time interval, as in our Type (a) epenence, an epenence of the two aily rates of the same ay, similar to our Type (c) epenence. However, they i not consier the Type (b) epenence, which we later argue is crucial in reucing forecasting errors. Hence, our moel is more general an flexible in that it works for any number of arrival streams an allows more general inter-stream epenence. Our numerical stuies (Section 4.4) also suggest that our approach reaches the QoS target better at the ownstream staffing stage, an our estimation algorithm is much faster than the fitting approach use for their mixe moels Forecasting Error We show theoretically that our Moel (1) will not increase the forecasting error over alternative moels that are base on only a subset of the arrival streams. As a result, fitting Moel (1) simultaneously on all arrival streams (in theory) will perform at least as well as the alternative approach that moels each arrival stream separately. Later in Sections 5 we empirically emonstrate such improvement via simulation stuies when the moels have to be estimate from ata. To fix ieas, let y enote the ranom variable that we want to forecast, with mean µ y an variance Γ y, an let ξ n, n = 1, 2,..., enote a series of ranom variables that we use to buil the forecasting moel. Define Γ n Cov(y, ξ n ), Γ (n) = (Γ 1, Γ 2,..., Γ n ), ξ (n) = (ξ 1, ξ 2,..., ξ n ), µ (n) = E(ξ (n) ), m s,l = Cov(ξ s, ξ l ), s, l = 1, 2,..., M (n) = Cov(ξ (n) ) = (m s,l ) n n, m (n+1) = (m 1,n+1, m 2,n+1,..., m n,n+1 ). We assume that M (n) is non-singular, n = 1, 2,.... Consier the forecasting variances of y when using either ξ (n) or ξ (n+1) as the forecaster, enote as Γ n an Γ n+1, respectively. Proposition 1 suggests that using ξ (n+1) never increases the forecasting variance, which makes intuitive sense as there is no less information in ξ (n+1). In the current context, the results suggest that it won t hurt to use all available arrival streams (rather than a

7 subset) to buil the forecasting moel, barring any estimation errors. The proof of the proposition follows naturally from multivariate normal theory, see Section A of the online supplement. Proposition 1. Assuming that ξ n+1 is non-eterministic given ξ (n), then Γ n Γ n+1 = (Γ n+1 Γ T (n)m 1 m (n) (n+1)) 2 m n+1,n+1 m T (n+1) M 1 m 0. (3) (n) (n+1) Consiering Proposition 1 in the context of Moel (1), we can unerstan the forecasting variance reuction obtaine by introucing more arrival streams in our forecasting moel. Suppose we forecast u (1) using u (1) 1, u(2) 1,..., u(i) 1, Proposition 1 suggests that incluing more streams in the forecasting moel will not increase the forecasting variance. The variance reuction, uner the autoregressive structure in Moel (1), is a function of A an Ω, but has a complicate form. However, in the two-imension example, when there is no Type (b) epenence, i.e. a 12 = a 21 = 0, we can show that the variance reuction (3) is zero, suggesting there is no nee to consier a bivariate forecasting approach when the moel oes not incorporate Type (b) epenence. When a 21 0 an a 12 0, the variance reuction (3) is non-zero, an we will unerstan the effects of Type (b) epenence in the simulation stuies of Section 5. The above theoretical stuy suggests that consiering Type (b) epenence is essential for improving point forecast accuracy. This fining provies one plausible explanation for why the two bivariate moels of Ibrahim an L Ecuyer (2013) i not improve point forecast accuracy over the univariate mixe moel, because their bivariate moels in t consier the Type (b) epenence between two arrival streams Moel Estimation Moel (1) is in fact a Hien Markov Moel (HMM) (Cappé et al. 2005), since the autoregressive aily total rates u (i) are unobserve. HMMs can be estimate using maximum likelihoo. However, the very general epenence structures assume in Moel (1) makes irect maximum likelihoo computationally very expensive, hence less practical. Here we propose to use a sequential algorithm to estimate all the parameters involve in Moel (1). We first escribe how to obtain estimates for u (i), f (i) w,t, an Σ for i = 1,..., I, t = 1,..., T, = 1,..., D, an then iscuss estimation of the vector time series parameters α w, A an Ω. For univariate arrival forecasting, Shen an Huang (2008a) have shown that the sequential approach performs similarly to maximum likelihoo. Estimation of u (i), f w (i) an Σ. Consier the following moel equation: X (i),t = u(i) f (i) w,t + ɛ (i),t, i = 1,..., I, ɛ,t = with the constraints f (i) w,t 0 an T t=1 ( ) ɛ (1),t, ɛ(2),t,..., ii ɛ(i),t N (0, Σ), f (i) w,t = 1. This is a multiplicative linear moel that can be estimate using an iterative General Least Squares (GLS) algorithm as we iscuss below. 7

8 8 The Iterative GLS Algorithm 1. Initialize: (a) Estimate f (i) w,t with the proportion of the transforme counts of time interval t out of all transforme counts in the same ay-of-week: ˆf (i),ol w,t = { :w =w } X(i),t { :w =w } t X (i),t. (4) (b) Fit Orinary Least Squares (OLS) to the following moel to obtain û (i),ol : X (i) (i),ol,t = ˆf w,t u (i) + ɛ(i),t, ɛ(i) ii,t N (0, σ 2 ), i = 1, 2,..., I, = 1, 2,..., D, t = 1, 2,..., T. (c) Use the empirical covariances among the moel resiuals to get ˆΣ ol. 2. Upate: (a) Fit GLS to the following moel to get the upate estimates ˆf (i),new w,t : X (i),t = û(i),ol f (i) w,t + ɛ (i),t, ɛ,t = (ɛ (1),t, ɛ(2),t,..., ɛ(i),t ) ii N (0, ˆΣ ol ). (5) (b) Obtain the upate estimate ˆΣ new using the moel resiual covariance matrix. (c) Fit GLS to the following moel to get the upate û (i),new : X (i) (i),new,t = ˆf w,t u (i) + ɛ(i),t, ɛ,t = (ɛ (1),t, ɛ(2),t,..., ɛ(i),t ) ii N (0, ˆΣ new ). () Upate ˆΣ new with the moel resiual covariance matrix. 3. Repeat Step 2 setting û (i),ol T ( t=1 = û (i),new an Σ ol = Σ new until convergence such that (i),ol (i),new ˆf w,t ˆf w,t ) 2 /T < 10 8, w, i. At convergence, û (i),new an ˆf (i),new w,t are the final estimates for u (i) an f (i) w,t, respectively. Then we use the resiual covariance matrix to estimate ˆΣ with.f. = DT D w (T 1), where w is the number of working ays in a week. For example, w = 5 for centers that only open on weekays. Note that fitting GLS regression can be time-consuming. Alternatively, we can replace GLS with OLS in the above algorithm, which results in great reuction on computing time. This alternative provies estimates that are quite close to those from the GLS metho. In our call center application (Section 4), the relative istance of {f (i) w,t} t=1,...,t between the two estimates is aroun 0.02%.

9 Estimation of α w, A an Ω. Given the estimates û, ˆf (i) w,t an ˆΣ obtaine from the first stage estimation, we have the following vector time series moel: 9 û α w = A (û 1 α w 1 ) + z, z ii N (0, Ω), = 2,..., D. Uner the Gaussian assumption, the above moel can be fitte using maximum likelihoo to obtain estimates ˆα w,  an ˆΩ Point an Distributional Forecast Given the above parameter estimates, we can obtain the point an istributional forecasts for future arrival rates an counts as follows. For a positive integer h, the h-step-ahea point forecast for the aily total rate on ay D + h, u D+h is given by with the forecasting error as h 1 û D+h = ˆα wd+h + Âh (û D ˆα wd ), (6) h=0 matrix of the forecast error of û D+h is  hz D+ h, where z D+ h ii N (0, ˆΩ). In particular, the covariance h 1 ˆΩ (D+h) =  h ˆΩ( h). (7) h=0 Given the mean in (6) an variance in (7), an uner the Gaussian assumption, the istributional forecast for the aily arrival rate u D+h is u D+h N (û D+h, ˆΩ (D+h) ). Define ˆF (1) (I) D+h,t iag{ ˆf w D+h,t,..., ˆf w D+h,t}. Then the istributional forecast for the arrival rate vector θ D+h,t (θ (1) D+h,t,..., θ(i) D+h,t ) = ˆF D+h,t u D+h is as follows: θ D+h,t N ( ˆFD+h,t û D+h, ˆF D+h,t ˆΩ(D+h) ˆFD+h,t ). In particular, the point forecast for θ (i) (i) D+h,t is ˆf w D+h,t û D+h, i = 1,..., I, an the forecasting covariance between θ (i) D+h,t an ) θ(i (i) (i D+h,t is ˆf w D+h,t ˆf ) w ˆΩ(D+h) D+h,t ii, where ˆΩ (D+h) ii point an istributional forecasts for the arrival rate λ (i) D+h,t naturally follow. is the (i, i )th entry of ˆΩ (D+h).The In terms of the arrival counts, efine X D+h,t (X (1) D+h,t,..., X(I) D+h,t ) an ɛ D+h,t (ɛ (1) D+h,t,..., ɛ(i) D+h,t ) N (0, ˆΣ). Notice that X D+h,t = θ D+h,t + ɛ D+h,t. Then the istributional forecast for the arrival count vector X D+h,t is ˆX D+h,t N ( ˆF D+h,t û D+h, ˆF D+h,t ˆΩ(D+h) ˆFD+h,t + ˆΣ). In particular, the point forecast of X (i) (i) D+h,t is ˆf w D+h,t û D+h, an the forecasting covariance between X (i) D+h,t an ) X(i (i) (i D+h,t is ˆf w D+h,t ˆf ) w ˆΩ(D+h) D+h,t ii + ˆΣ ii.

10 Forecasting Performance Measures We consier two measures to evaluate the accuracy of point forecasts. For any single arrival stream, let λ,t enote the arrival rates in ay an time interval t, an let ˆλ,t enote the corresponing point forecast. Suppose we are intereste in forecasting the arrival rates for one ay. Then the Root Mean Square Error (RMSE) an Mean Relative Error (MRE) for Day are efine as follows: RMSE = 1 T (ˆλ,t λ,t ) T 2, MRE = 100 T ˆλ,t λ,t. T λ,t t=1 Similar measures can be efine for the arrival counts. 3. Operational Performance Recently, operations management papers attempt to account for arrival rate uncertainty while making staffing an scheuling policies. Among those, Gurvich et al. (2010) propose a multiple stream chance-constraine approach proviing staffing levels that meet the uncertain eman at a pre-chosen probability level. In their approach, they assume the existence of a forecasting istribution for the arrival rates before the staffing planning process can be taken. In this paper we consier a simplifie version of their moel to explore the operational effects of simultaneously moeling multiple arrival streams instea of inepenently moeling each of them. We remark, however, that the improve forecasts obtaine from moeling multiple arrival streams simultaneously have potential to improve performance of any staffing proceure that accepts as input a multivariate istribution of arrival rates A Brief Review of the Chance Constraint Formulation Consier a call center with I customer classes an J server pools. Set I = {1,..., I} an J = {1,..., J}. Servers in the same pool have the same skills in terms of the set of customer classes they are capable of serving an the corresponing service rates. Denote J(i) as the set of server pools with skill i, an I(j) as the set of skills that server pool j has. The staffing vector is enote by N = (N 1, N 2,..., N J ) where N j enotes the number of agents from server pool j to have staffe. We consier the call center as a parallel server system, where customers go through a single stage of service before eparting from the system. The staffing process is performe for one time interval an all the iscussion following is focuse on an arbitrary time interval. Gurvich et al. (2010) consiere a ynamic queueing moel that requires etermining staffing levels for the server pools an a control policy for ynamically assigning customers to servers. In the first step of their solution proceure, they approximate the ynamic queueing process with a static moel that only consiers the forecast uncertainty to obtain an t=1

11 initial staffing solution, which they then moify using a simulation-base proceure to meet QoS constraints. To simplify our experiments, we irectly use the static moel as the system we stuy. In this static moel, the number of class-i customers that arrive in a perio is Λ i, where Λ = (Λ 1,..., Λ I ) is a multivariate ranom variable following a known (forecaste) istribution. The staffing vector N must be chosen before knowing the true arrival counts. After N is chosen, the arrival counts are reveale, an then customers are (fractionally) allocate to servers. The service rate of a class-i customer when serve by a pool-j server is µ ij, for j J(i). It is assume that ψ i proportion of class-i customers are allowe to abanon, an hence these customers o not nee to be serve. Thus, in this moel the QoS targets will all be met if the system of inequalities, µ ij v ij Λ i (1 ψ i ), i I, j J(i) i I(j) v R I J + v ij N j, j J, has a feasible solution, where v ij is the number of pool-j servers allocate to serve class-i customers. Let δ > 0 be a given risk-level, representing the proportion of perios in which we are willing to allow our QoS targets to be violate. Then, the Ranom Static-Planning Problem (RSPP) of Gurvich et al. (2010) is as follows: 11 min c N subject to P(Λ B(N)) 1 δ, N R J +, (8) where B(N) = {Λ R I + : v R I J +, with j J(i) µ ij v ij Λ i (1 ψ i ), i I, i I(j) v ij N j, j J }. A vector λ R I + is in B(N) if there exists an allocation of the servers specifie by the staffing solution N such that the QoS target can be met. Therefore, (8) fins the minimum cost staffing solution that meets the quality-of-service target with probability 1 δ. To solve the RSPP, Gurvich et al. (2010) use a iscrete approximation of the ranom arrival rate Λ an formulate the RSPP as a mixe-integer program. They consiere two iscretization methos (fix gri approximation an Monte Carlo sampling), among which we use the Monte Carlo sampling approximation metho. In particular, inepenent samples of size K are generate from the istribution of Λ an each sample point is assigne the same probability 1/K. Denote the k th sample by Λ(k) = (Λ 1 (k),..., Λ I (k)). Hence the sample-base RSPP is given by:

12 12 min subject to c N µ ij v (k) ij y k Λ i (k)(1 ψ i ), i I, k = 1,..., K, j J(i) v (k) ij N j, j J, k = 1,..., K, i I(j) y k K(1 δ), k N R J +, y k {0, 1}, v (k) R I J +, k = 1,..., K. (9) For each sample, a binary variable y k is introuce, where y k = 1 implies the QoS targets are met for that sample, an a set of continuous variables v (k) ij, i I, j J are introuce, representing the allocation of server type i to customer type j in sample k. Then the constraint k y k K(1 δ) ensures that the QoS targets are met in a high proportion of the samples. Luetke an Ahme (2008) show how the sample-base RSPP can be use to construct feasible solutions an statistical optimality bouns for the original, unsample, RSPP. In particular, it is shown that, with high probability, solving a sample-base RSPP with max violation target δ yiels a solution feasible to the RSPP with target δ + γ(k), where γ(k) O( I/K). In aition, errors in the forecast istribution will further lea to a mismatch between the target δ use in the sample-bae RSPP an the actual violation probability of the resulting solution. Thus, while we cannot expect the violation probabilities will exactly match the target, the extent to which a solution obtaine using a forecasting metho matches the target is an important metric for comparing forecasting methos Operational Performance Measures Given a istributional forecast of Λ = (Λ 1,..., Λ I ), a staffing vector N can be obtaine with the above algorithm. Given the realization of Λ, we are able to evaluate the performance of the staffing vector an further the istributional forecast. Given realize counts C an a staffing vector N, we solve the following linear program, which minimizes the amount by which the QoS constraint is violate: min y i i I subject to µ ij v ij + y i C i (1 ψ i ), i I, j J(i) i I(j) v ij N j, j J, v R I J +, y R I +. Let s(c, N) be the optimal value of this linear program, so that s(c, N) measures how much the QoS constraint is violate. Denote v(c, N) as the violation inicator function for the QoS constraint: v(c, N) = 1 inicates that the QoS constraint is violate (s(c, N) > 0), an v(c, N) = 0

13 inicates that the QoS constraint is satisfie (s(c, N) = 0). Denote c(n) := c N as the staffing cost uner staffing vector N. Suppose we ve teste the performance of the staffing vector for D ays. Let C (,t) an N (,t) enote the observe counts an staffing vector for the t th interval in the th ay, respectively. The violation probability is efine as V.PROB := 1 D T Denote the aily staffing cost on ay as D COST := 4. Real Call Center Application =1 t=1 T v(c (,t), N (,t) ). T c(n (,t) ). t=1 We present a real call center application to emonstrate the practical applicability of our multiplestream forecasting metho, illustrating both the statistical an operational benefits of incorporating inter-stream epenence in forecasting multiple-stream arrivals as well as staffing call centers with multiple customer classes. We first provie the relevant backgroun information for the ata in Section 4.1, an then report the empirical results with some iscussion about managerial insights in Sections 4.2 an 4.3, an lastly compare our methos with the existing ones in Ibrahim an L Ecuyer (2013) in Section Backgroun of the Call Center Data The ata we analyze were collecte at an Israel telecom call center, provie to us as a courtesy from the Technion Service Enterprise Engineering Lab. There are several customer types, with Private an Business being the two major ones, which account for about 30% an 18% of the overall incoming calls respectively. Alor-Noiman et al. (2009) presente an earlier workloa forecasting stuy using a subset of the ata, focusing on the single arrival stream of Private customers. More etails of the ata can be foun in Alor-Noiman et al. (2009). We focus on the two main arrival streams: Private an Business, an emonstrate the effects of incorporating inter-stream epenence on forecasting accuracy an staffing performance measures. Note that our metho in general can be applie to more than two arrival streams. The ata cover 300 ays between 06/19/2004 an 04/14/2005. For both customer types, the call center operates from 7:00 am to minight everyay, while the call volumes are very low on Friays an Saturays compare with the other weekays, so we focus on the weekays from Sunay to Thursay, which result in 215 ays. For each ay we ivie the 17 working hours into 34 half-hour 13

14 14 within-ay arrival volume profiles Private mean arrival volume profiles for each Mean ay-of-the-week Profile Private aily total arrival volumes Private Arrival Volume Sun Mon Tue We Thu Proportion Sun Mon Tue We Thu Daily Total Arrivals Sun Mon Tue We Thu Time of Day Time of Day Day Inex Figure 1 Plots for Private customers on the transforme scale. within-ay arrival volume profiles Business mean arrival volume profiles for eachmean ay-of-the-week Profile Business aily total arrival volumes Business Arrival Volume Sun Mon Tue We Thu Proportion Sun Mon Tue We Thu Daily Total Arrivals Sun Mon Tue We Thu Time of Day Time of Day Day Inex Figure 2 Plot for Business customers on the transforme scale. intervals, an recor the arrival count uring each interval. The square-root transformation is then applie to the arrival counts to stabilize the variance an normalize the ata. Figures 1 an 2 plot the square root transforme arrival counts of Private an Business customers, respectively. Within each figure, the left panel isplays the call volume profiles for each one of the 215 weekays, where ifferent colors are use for ifferent ays of the week; the mile panel shows the average arrival volume profile for each ay-of-the-week, highlighting the ay-of-the-week effect on the within-ay arrival pattern; finally, the right panel epicts the time series plot of the aily-total arrival volumes against the ay inex, with ifferent ays of the week inicate by ifferent symbols an colors. The two arrival streams exhibit similar patterns in both the withinay arrival profiles an the aily total volumes: the within-ay profiles have two peaks aroun 1:00pm an 6:00pm, Sunays have the highest total volume relative to the other weekays, an there are significant ay-of-the-week effects on both the within-ay arrival profile an the aily total volume. Moel (2) is esigne to capture the above empirical features in the ata. Section E

15 Daily Totals Private Business Sun Mon Tue We Thu Interval Resiuals Private Business Sun Mon Tue We Thu Figure 3 Left: scatter plot for aily totals on the transforme scale removing ay-of-week effect. Right: scatter plot of square-roote interval volumes removing ay-of-the-week an interval effects. Type (a): Σ = (Σ sl ) 2 2 Type (b): A = (a sl ) 2 2 Type (c): Ω = (Ω sl ) 2 2 Σ 11 Σ 22 Σ 12 r a 11 a 12 a 21 a 22 Ω 11 Ω 22 Ω 12 ρ Table 1 Moel (2) parameter estimates base on the real ata. of the supplement contains some iagnostic results that suggest Moel (2) fits our ata well. Figure 3 suggests that the two arrival streams are epenent on each other. The left panel shows the scatter plot of the aily total arrival volumes on the transforme scale with the ay-of-the-week effect remove. The correlation between the two time series is aroun 0.72 which is fairly strong. The right panel is the scatter plot for the square-roote interval call volumes between the two customer types after we remove both the ay-of-the-week an the interval effects. Fourteen outliers are exclue from the plot. We observe a moerately strong correlation of We refer to the Private arrivals as Type 1 an the Business arrivals as Type 2. The twoimensional Moel (2) is fitte to the ata. Table 1 reports the corresponing estimates for the inter-stream epenence parameters. We observe that both the aily-total epenence (Type (c)) an the interval epenence (Type (a)) between the two streams are strong, with the correlation being an respectively, while the cross-lag epenence (Type (b)) is relatively weak in this case, being an Forecasting Performance Comparison Competing Methos To examine whether accounting for inter-stream epenence helps improve forecasting performance, we first consier the following two forecasting methos: MU1: fit Moel (2) separately on ata from each arrival stream, which ignores the epenence between the two arrival streams an essentially consiers them as inepenent;

16 16 Table 2 RMSE MRE Private Business Private Business Metho Meian Mean Meian Mean Meian Mean Meian Mean HA MU MU1-AR MU MU2G Rolling point forecast comparison base on RMSE an MRE. MU2: fit Moel (2) simultaneously on the two arrival streams. Since the only ifference between the two methos is whether the inter-stream epenence is taken into account, any observe ifference between the resulting forecasting istributions can then be attribute to the inter-stream epenence. In aition, we also consier the following three forecasting methos: HA: use the historical average of the same ay-of-week as the forecast. This metho has been repeately use as the benchmark in, for example, Weinberg et al. (2007), Shen an Huang (2008a), Alor-Noiman et al. (2009), Ibrahim an L Ecuyer (2013). MU1-AR: fit Moel (2) separately on the two streams, an at the same time impose a withinay AR(1) correlation structure to {ɛ,t } t=1,2,...,t, similar to Alor-Noiman et al. (2009) an Ibrahim an L Ecuyer (2013). MU2G: fit Moel (2) simultaneously on the two streams, an use the GLS estimation Point Forecast Comparison The following rolling forecasting exercise is performe: for each ay between Day 101 an Day 215, we use its previous 100 consecutive ays to generate the istributional forecast of the arrival profiles for that ay using the five methos, an calculate the RMSE an MRE as efine in Section 2.5. As a result, for each forecasting metho, we have 115 values of RMSE an MRE, the meian an mean of which are summarize in Table 2. In Table 2, the best performer uner each measure is highlighte in bolface. The comparison suggests that, in terms of point forecasting accuracy, the historical average metho performs the worst, an there is no strong evience that a single forecasting metho gives the best point forecast since the forecasting accuracy epens on the performance measure an customer type. Paire t-tests base on RMSE an MRE further confirm that HA performs significantly worse than any one of the other four methos, which again supports the incorporation of epenence; MU1 an MU1-AR perform similarly, which offers no evience that imposing the within-ay AR(1) structure helps improve the point forecasts;

17 MU1, r=0 x MU2, r=0.67 x Y Y X X 0 Figure 4 Exemplary forecasting istributions obtaine by MU1 an MU2. There is no soli evience to conclue that MU2 gives more accurate point forecast than MU1, which is consistent with the theoretical iscussion in Section 2.2 since the particular real ata possesses very weak Type (b) epenence. MU2G slightly improves over MU2, even though MU2G takes much longer to compute Distributional Forecast Comparison Although MU1 an MU2 perform comparably in terms of point forecast accuracy, the istributional forecasts between MU1 an MU2 are significantly ifferent. In the 115 rolling forecasting experiments, MU2 estimates the Type (a) correlation parameter r to be between 0.6 an 0.7, an MU2G gives similar correlation estimates. As an illustration, Figure 4 shows the mesh plots of the forecasting istributions offere by MU1 an MU2 for one ranomly-chosen time interval in the forecasting exercise, with the parameter estimates appropriately roune. Note that the shape of the contours is affecte by the ifferent correlation estimates. Such ifference between the forecasting istributions helps to explain the operational benefits of MU2, reporte below in Section Operational Performance Comparison We now compare the operational effects of using the forecasts generate by MU1 an MU2, via extensive staffing experiments base on the 115-ay rolling forecasting exercise in Section 4.2. For illustration purposes, we implement the chance-constraint staffing formulation of Gurvich et al. (2010) as reviewe in Section 3, an fee the forecasting istributions into it to obtain the staffing costs an the associate quality of service (QoS) measures. More specifically, we consier three system structures with a range of system parameters, an obtain managerial insights about how system esign an cross-training affect staffing cost an QoS performance. The comparisons clearly emonstrate the operational benefits of accounting for inter-stream epenence in the forecasting stage, an reveal an important message for the managers - the more flexible a staffing esign is, the more crucial it is to appropriately incorporate inter-stream epenence.

18 18 II-esign M-esign X-esign Λ 1 Λ 2 Λ 1 Λ 2 Λ 1 Λ 2 ψ 1 ψ 2 ψ 1 ψ 2 ψ 1 ψ 2 μ 11 μ 22 μ 11 μ 12 μ 22 μ 23 μ 11 μ 12 μ 21 μ 22 $c 1 $c 2 $c 1 $c 2 $c 3 $c 1 $c 2 Figure 5 Staffing esigns Effects of System Designs Note that the staffing ecision given by the chanceconstraint program epens on the specific structure of the staffing system an the corresponing parameters. With two arrival streams, we consier three interesting system esigns as shown in Figure 5, which are referre as the I-esign, (or the II-esign in our case), the M-esign, an the X-esign (Gans et al. 2003). These staffing esigns cover a wie range of system complexity an flexibility, as we now iscuss: the II-esign: there are two eicate server pools, each one serving one customer class. the M-esign: there are two eicate server pools, one for each customer class. There is also a flexible server pool, that serves both arrival streams. In comparison, the M-esign as a flexible server pool to the II-esign. the X-esign: there are two separate pools of cross-traine servers: the servers in each pool primarily serve one particular customer class, although they can serve the other customer class if neee. Compare with the II-esign, the X-esign allows resource sharing between the two classes, for example, when there are overloas in one or both classes (Perry an Whitt 2009). We expect that the flexibility level of a staffing esign interacts with the comparison between MU1 an MU2. A more flexible system shall be more capable of taking avantage of the benefits from incorporating inter-stream epenence; hence MU2 shall lea to more operational benefits in a more flexible system. In aition to the various esign structures in Figure 5, we consier two more factors that affect system flexibility level: salary of the servers who serve more than one customer classes, i.e. c 2 of the M-esign in Figure 5; service rate of the cross-traine servers when they hanle their non-primary customer class, i.e. µ 12 an µ 21 of the X-esign in Figure 5.

19 19 Design δ ψ i µ ij c j Setting Inex c 1 = c 3 = 1, c 2 = 1.1 M1.1 c 1 = c 3 = 1, c 2 = 1.3 M1.3 M 0.05 ψ 1 = ψ 2 = 0.04 µ 11 = µ 12 = µ 22 = µ 23 = 1 c 1 = c 3 = 1, c 2 = 1.5 M1.5 c 1 = c 3 = 1, c 2 = 1.7 M1.7 c 1 = c 3 = 1, c 2 = 2.0 M2.0 µ 11 = µ 22 = 1, µ 12 = µ 21 = 0.8 X0.8 X 0.05 ψ 1 = ψ 2 = 0.04 µ 11 = µ 22 = 1, µ 12 = µ 21 = 0.6 c 1 = c 2 = 1 X0.6 µ 11 = µ 22 = 1, µ 12 = µ 21 = 0.4 X0.4 II 0.05 ψ 1 = ψ 2 = 0.04 µ 11 = µ 22 = 1 c 1 = c 2 = 1 II Table 3 System parameter settings for the II-esign, M-esign an X-esign. It is natural to consier these three factors. First, managers in large-scale call centers often cross train servers to increase system flexibility, which results in a more complex staffing structure. Secon, cross-traine servers with more skills usually get higher pay, which on the other han makes the system less cost-efficient. Thir, cross-traine servers may be slower in serving their non-primary customers compare with the eicate servers, which to some egree reuces the system operational efficiency. In summary, the system becomes more flexible, when one increases the number of server pools, or ecreases the cost for cross-traine servers, or increases the nonprimary service rate of cross-traine servers. Below we perform two numerical comparisons to present the effects of these three factors on forecasting an staffing performance. We then generalize some managerial insights from the two comparisons. Comparison 1: II vs. M. We consier five M-esigns, with ifferent flexible server costs. The parameter settings are presente in Table 3, where the M-esigns are arrange in the orer of ecreasing system flexibility. The violation probability target δ is 0.05, the allowe abanonment proportion ψ i is 0.04 for both customer classes, an the service rate µ ij is 1 across all server pools an customer classes. Each eicate server costs 1, while the flexible server cost c 2 ranges between 1.1 an 2. The various settings are chosen in a way so that the II-esign can be viewe as the limit of the various M-esigns. More specifically, when a single flexible server costs as much as two eicate servers, the M2.0-esign is basically the II-esign. Figure 6 isplays the results of the staffing experiment uner the various settings. The left panel compares the aily mean of the realize violation probability, obtaine from averaging over the 115 out-of-sample forecasting ays, while the right panel compares the corresponing aily mean of the staffing cost between MU1 an MU2. The following observations can be mae: MU2 is more stable than MU1 in violation probability across all the settings; MU1 s violation probability increases (more severe unerstaffing) when the system becomes more flexible (that is,

20 20 when the flexible server cost ecreases). An intuitive explanation is that MU1 pays more penalty for ignoring inter-stream epenence in a more flexible system which is better at exploiting the benefits of inter-stream epenence. More etaile comparison between MU2 s an MU1 s violation probabilities is given in the bullet point below. Uner the most flexible settings - M1.1 an M1.3, the violation probabilities of MU2 are smaller an closer to the target value 0.05 than MU1 s. Uner the inflexible II-esign, MU1 has a smaller violation probability than MU2, an the reason is as follows. When the esign is inflexible with two separate queues, the staffing ecision is riven only by the marginal forecasting istributions instea of the joint istribution of the two arrival streams, an this fact will later be emonstrate in the simulation stuies of Section 5. MU1 an MU2 perform similarly regaring point forecast accuracy as shown in Table 2. However, we observe that MU1 prouces wier marginal confience intervals than MU2 an hence makes the staffing program account for a larger region of arrival rates. Therefore, MU1 has a smaller violation probability uner the II-esign since it generates similar point forecasts but wier marginal confience intervals, compare to MU2. For each forecasting metho, the M2.0-esign an the II-esign have very similar staffing performance, because these two esigns are basically the same. When the system becomes more flexible (i.e. from II to M1.7 to M1.5, etc.), the staffing cost of MU2 ecreases with the violation probability staying stable, inicating improve cost-efficiency of the service system after cross-training with stable QoS performance. Comparison 2: II vs. X. We consier three X-esigns where the rate at which an agent serves a non-primary customer class varies, an stuy how the varying cross-service rate affects the operational benefits of incorporating inter-stream epenence using MU2. We use the II-esign as the benchmark, because when the cross-service rates are 0, the X-esign reuces to the II-esign. We choose to compare the X-esign with the II-esign in this staffing experiment, since Perry an Whitt (2009) have carefully stuie the X-esign as a potential remey to unexpecte overloa uner the II-esign. Furthermore, note that Perry an Whitt (2009) consier two inepenent arrival streams, while we are intereste in the effects of inter-stream epenence. The parameter settings for the various X-esigns are liste in Table 3, in the orer of ecreasing system flexibility. The values of δ, ψ i, µ ii, c j are fixe across all the X-esigns, while only the crossservice rates µ 12 = µ 21 change from 0.8 to 0.6 to 0.4. It makes sense that the cross-service rates are less than the eicate service rates, which satisfy the strong inefficient-sharing conition (Perry an Whitt 2009).

21 21 Violation Probability Cost Violation Probability MU1 MU2 Cost MU1 MU2 M 1.1 M 1.3 M 1.5 M 1.7 M 2.0 II M 1.1 M 1.3 M 1.5 M 1.7 M 2.0 II Design & Flexible Server Cost Design & Flexible Server Cost Figure 6 Real ata: aily staffing comparison among the II-esign an various M-esigns with ifferent flexible server costs. Violation Probability Cost Violation Probability MU1 MU2 Cost MU1 MU2 X 0.8 X 0.6 X 0.4 II Design & Cross Service Rate X 0.8 X 0.6 X 0.4 II Design & Cross Service Rate Figure 7 Real ata: aily staffing comparison among the II-esign an various X-esigns with ifferent cross service rates. Figure 7 presents the aily staffing comparison between MU2 an MU1 uner the X-esign with varying cross-service rates. We can unerstan Figure 7 similarly as Figure 6: they present very analogous messages. Therefore, etaile explanations are omitte an instea we state the following two major messages: MU2 s violation probabilities are closer to the target δ = 0.05 across all the X-esigns. MU1 unerstaffs with violation probabilities greater than 0.05 in all the X-esigns. When the system becomes more flexible (i.e. from II-esign to X0.4 to X0.6, etc.), the staffing cost ecreases an the violation probability also ecreases for MU2. In light of the two comparisons reporte above, the following managerial insights are observe base on the particular call center ata: When the service system is flexible: incorporating inter-stream epenence ensures stable performance in QoS; ignoring the epenence results in unerstaffing ue to the positive epenence between the two classes, an the severity increases when the system becomes more flexible. When