Staffing and routing in a twotier call centre. Sameer Hasija*, Edieal J. Pinker and Robert A. Shumsky


 Hector Cameron
 1 years ago
 Views:
Transcription
1 8 Int. J. Operational Researc, Vol. 1, Nos. 1/, 005 Staffing and routing in a twotier call centre Sameer Hasija*, Edieal J. Pinker and Robert A. Sumsky Simon Scool, University of Rocester, Rocester 1467, NY, USA Fax: *Corresponding autor Abstract: Tis paper studies service systems wit gatekeepers wo diagnose a customer problem and ten eiter refer te customer to an expert or attempt treatment. We determine te staffing levels and referral rates tat minimise te sum of staffing, customer waiting, and mistreatment costs. We also compare te optimal gatekeeper system (a twotier system) wit a system staffed wit only experts (a directaccess system). Wen waiting costs are ig, a directaccess system is preferred unless te gatekeepers ave a ig skill level. We also sow tat an easily computed referral rate from a deterministic system closely approximates te optimal referral rate. Keywords: staffing; routing in queueing systems; call centres; gatekeeper systems. Reference to tis paper sould be made as follows: Hasija, S., Pinker, E.J. and Sumsky, R.A. (005) Staffing and routing in a twotier call centre, Int. J. Operational Researc, Vol. 1, Nos. 1/, pp.8 9. Biograpical notes: Sameer Hasija is currently a PD candidate in Operations Management at te Simon Business Scool, University of Rocester. He as a BTec (Major: Naval Arcitecture and Ocean Engineering, Minor: Industrial Engineering) from te Indian Institute of Tecnology at Madras and a MS (Management Science Metods) from te University of Rocester. His researc interests include efficiency and flexibility issues in service systems, wit a focus on CallCenter and Healt Care Management. Edieal J. Pinker is an Associate Professor of Computers and Information Systems at te Simon Scool of Business, University of Rocester. He conducts researc on te use of contingent workforces, crosstraining, and experiencebased learning in service sector environments as it applies to work and workflow design. He also studies te use of online auctions in electronic commerce and te issues faced by legacy firms trying to transition into electronic commerce. Pinker as consulted for te United States Postal Service, te financial services industry, and te auto industry. His work as been publised in Management Science, Manufacturing and Service Operations Management, te European Journal of Operational Researc, IIE Transactions and te Communications of te Association of Computing Macinery. He serves on te editorial boards of M&SOM, POMS, Decisions Sciences and IJOR. Professor Pinker earned is MS and PD in Operations Researc from te Massacusetts Institute of Tecnology. Copyrigt 005 Inderscience Enterprises Ltd.
2 Staffing and routing in a twotier call centre 9 Robert A. Sumsky is an Associate Professor of Operations Management at te Simon Scool of Business, University of Rocester. Professor Sumsky as researc and teacing interests in te modelling and control of service systems. Current researc focuses on te dynamic use of flexible capacity, te use of incentives for operational control of service systems, and te application of revenue management under competition. His researc as been publised in Management Science, Operations Researc, Manufacturing & Service Operations Management (M&SOM), and Air Traffic Control Quarterly. He is an Associate Editor for Management Science and Operations Researc, serves on te editorial review boards of M&SOM and te Journal of Revenue and Pricing Management, and is a Senior Editor for Production and Operations Management. He as also conducted researc on te USA air traffic management system and studied transportation operations for te Massacusetts Port Autority and te Federal Aviation Administration. Professor Sumsky earned is MS and PD in Operations Researc from te Massacusetts Institute of Tecnology. 1 Introduction In tis paper we consider te problem of capacity planning and call routing in a twotier service system in wic te first tier consists of gatekeepers wo diagnose te customer s problem, may solve te problem, or may refer te customer to an expert in te second tier. A typical example: call centres for ealtcare services are often staffed by certified nurses wo diagnose te problem and provide advice. If justified by te nature and severity of te call, a nurse may refer a call to a specialist (Bernett 003). Sumsky and Pinker (003) provide additional examples of service systems wit tis twotier arcitecture, but wit te ealtcare motivation in mind, we refer to te resolution of a customer s problem as a successful treatment. Call centres must balance te relatively low cost of lessskilled gatekeepers wit te benefits of te expert s ability to andle difficult calls. A manager of suc a call centre must determine te staffing levels of gatekeepers and experts as well as te optimal referral rate to minimise total costs. Tese costs may include te cost of staffing, te cost of customer waiting time, and mistreatment costs wen a customer must see an expert for successful treatment despite spending time receiving (unsuccessful) treatment from te gatekeeper, as well. Tis paper focusses on a staffing and referral strategy tat is based only on call difficulty and steady state queue lengts and not on realtime queue lengts (te policies are static, not dynamic). We use a squareroot staffing rule to approximate te optimal staffing for bot tiers, given any particular referral rate. Tis rule is asymptotically optimal as system size increases. We ten use te staffing approximation to determine te optimal referral rate and to compare te gatekeeper (or twotier ) system wit a directaccess (or onetier) system in wic customers do not encounter a gatekeeper but instead immediately see an expert. Our main findings are:
3 10 S. Hasija, E.J. Pinker and R.A. Sumsky Te coice of system (gatekeeper or direct access) as a complex relationsip wit te customer s waiting time cost. In particular, as te waiting cost increases, it may or may not be optimal to coose a directaccess system, depending on te oter parameters suc as te gatekeeper s skill level. If it is optimal to coose te directaccess system wen te unit cost of a customer s waiting time is low (or in a deterministic system in wic waiting costs are not assessed), ten it is optimal to coose te directaccess system wen te cost of te customers waiting time is iger. However, tis does not imply tat a ig waiting cost always leads to a onetier system. As in point 1, it is possible to prefer a gatekeeper system, no matter ow ig te waiting cost per unit time. Wen twotier systems are preferred, a simple, deterministic model can be used to coose te referral rate. Te optimal referral rate converges to tis deterministic referral rate as te size of te system grows. If we construct an optimal staffing plan, given te deterministic referral rate, ten we will only see a small increase in cost over te cost of te globally optimal system tat uses te optimal referral rate. In Section, we review te related literature. In Section 3 we introduce our model and in Section 4 we describe an approximation tat generates asymptotically optimal staffing levels and referral rates as te system size increases. In Section 5 we use te approximation to caracterise te beaviour of te referral rate as a variety of parameters cange. In tat section we also use numerical experiments to test te accuracy of te approximation and to confirm te four observations described above. Section 6 contains a discussion of our results and describes additional directions for researc. Literature review Tis paper is related to te literature on capacity planning and routing in queuing networks. Halfin and Witt (1981) establis eavytraffic stocastic limits for multiserver queues in wic te number of servers is allowed to increase along wit te traffic intensity but te steadystate probability tat all servers are busy is eld fixed. Witt (199) sows tat by using te squareroot staffing principle, discussed below in Section 4, one can generate te same limiting regime as in Halfin and Witt (1981). Borst et al. (004) use a similar framework to approximate te waiting time distribution of an M/M/N queue and demonstrate te asymptotic optimality of te square root staffing principle, given a cost function involving bot waiting and staffing costs. We apply teir approximation to a twotier queuing network. We use te squareroot staffing rule to find te number of servers for eac level as a function of te routing strategy. We ten determine te total cost of operating suc a system and minimise tat cost wit te static routing strategy as te decision variable.
4 Staffing and routing in a twotier call centre 11 Tere exists a substantial literature on optimal routing strategies in call centres wit crosstrained servers ( skillbased routing ). For example, Örmeci (004) and Cevalier et al. (004) study loss models wit specialised and fully flexible servers. Wallace and Witt (004) examine systems wit an arbitrary crosstraining pattern (e.g., eac server may be crosstrained in any subset of six skills). Tey use euristics and simulation to find te minimum number of crosstrained servers needed to satisfy performance goals for eac customer type. However, tese models of skillbased routing differentiate calls by type, and not by difficulty level; a server is eiter sufficiently skilled to andle a call or is not. In our model, gatekeepers ave some probability of success wit a particular call. As in our model, de Vèricourt and Zou (004) assume tat eac server as a different call resolution probability (p), and tey also assume tat eac server may ave a different service rate (µ). Tey identify te routing policy of calls to servers (a pµ rule ) tat minimises te total time a call spends in te system, including recalls. Wile tey assume tat te staffing level is given one server of eac type our model considers bot te staffing and routing problem for large systems. Te structure of our service system is also quite different. We assume tat tere are two pools of servers: te expert pool as a resolution probability equal to one and te gatekeeper pool attempts to treat calls or passes tem along te expert pool. Our model is closest to Sumsky and Pinker (003). Tey determine te optimal routing strategy for a deterministic system and ten formulate a principal/agent model to determine te impact of performancebased incentives on te gatekeeper s beaviour. Teir model does not incorporate queuing effects, for tey assume tat te firm maintains a level of staffing sufficient to satisfy exogenous waiting time goals. Here we model queuing effects but we do not consider te incentive problem. We sow ow te costminimising referral rate varies wit canges in parameters related to queuing, i.e., arrival rates and service rates. We also sow tat as te arrival rate increases, te optimal referral rate converges to te optimal referral rate for te deterministic case. 3 Te queueing network model In tis section we describe an open queueing network model of a service centre wit gatekeepers. Te network is essentially two queues in series: n g gatekeepers and n e experts, wit staffing costs c g and c e per unit time, respectively (see Figure 1). Customers (or calls ) arrive to te gatekeepers according to a Poisson process wit rate λ. To te gatekeepers, te calls vary in difficulty and complexity, and we represent te difficulty of eac call wit a random draw from a uniform distribution, U[0, 1]. Tis random variable represents te call s percentile in a ranking of calls by treatment complexity. Given tat a call as complexity x, te probability tat te customer can be treated successfully by te gatekeeper is f(x). As in Sumsky and Pinker (003), we will refer to f(x) as te treatment function. Because complexity increases wit x, we assume tat f (x) 0.
5 1 S. Hasija, E.J. Pinker and R.A. Sumsky Figure 1 Customer flows Wit eac new call, a gatekeeper spends time diagnosing te problem and determining te complexity (te value of x). Te gatekeeper may ten eiter send te call directly to te expert pool or attempt to solve te problem. If te gatekeeper successfully solves, or treats, te problem, te call leaves te system. If te gatekeeper attempts to treat and te treatment fails, we assess a cost m due to te inconvenience to te customer, and te call is sent to te expert pool. Once a call as reaced an expert, it is served and leaves te system. Bot server pools ave unlimited waiting space, and tere is a cost w for eac unit of time spent waiting. Te time required for an expert to treat a call averages 1/µ. Te time for a gatekeeper to diagnose a call averages 1/µ d, wile te average time to diagnose and treat is 1/µ t > 1/µ d. If te gatekeeper follows a static policy and treats a proportion k of calls, ten te gatekeeper s service rate is, 1 k k µ = + µ d µ t 1. We assume tat service times are distributed as independent, exponential random variables, even wen te gatekeeper only diagnoses some calls, and combines diagnosis wit treatment in oter calls. Given tese assumptions, te gatekeeper and expert pools can eac be modelled as M/M/N queueing systems (see Gross and Harris, 1985, Section 4.1), were te arrival rate to te expert pool is te sum of te rate of calls untreated by te gatekeeper and te rate of calls mistreated by te gatekeeper. We will discuss additional implications of te exponential servicetime assumption in Section 5.3. Our objective is to minimise te sum of staffing, waiting, and mistreatment costs. Given te complexity of a call, we must decide weter te gatekeeper sould treat te call or refer it immediately to an expert. Suppose tat te staffing is fixed at (n g, n e ) and te gatekeeper treats all calls in S, were S is a (possibly noncontiguous) subset of [0, 1].
6 Staffing and routing in a twotier call centre 13 Let k be te proportion of te range [0, 1] covered by S. If te gatekeeper replaces te set S wit te set [0, k], we know tat Te gatekeeper s service rate does not cange because te proportion k does not cange. Te rate of untreated calls does not cange. Te rate of mistreated calls and te waiting time stays te same or decrease because f (x) 0. Terefore, given any staffing configuration and treatment set S wit proportion k, te waiting, staffing, and mistreatment costs will not increase if te gatekeeper instead treats calls in [0, k]. Tis argument indicates tat te optimal treatment set S takes te form [0, k], and we will refer to k as te treatment tresold. Given treatment tresold k, te gatekeeper refers a proportion 1 k of calls witout attempting treatment. Te expected k fraction of calls treated successfully by a gatekeeper is Fk ( ) = f( x)dx, te fraction mistreated is k F(k), and te fraction of calls seen by te expert pool is 1 F(k). We now develop te objective function for our problem. Te decision variables are k, te proportion of calls treated by te gatekeeper, and te staffing levels n g and n e. Let q(n, λ, µ) be te expected wait for an M/M/N queueing system wit n servers, arrival rate λ, and service rate µ. Te total cost per unit time is: C (n g, n e, k) = wλ [q(n g, λ, µ ) + (1 F(k))q(n e, (1 F(k))λµ)]+c g n g + c e n e + mλ(k F(k)). () Te subscript indicates tat tis is a cost function for a twotier service system (as opposed to te onetier directaccess system described below). Te first term is te expected cost of waiting in front of te gatekeeper and expert pools, te second and tird terms are te staffing costs, and te last term is te mistreatment cost. Terefore, we consider te following problem: min C ( n, n, k ) (3) ng, ne, k subject to g e n > λ / µ (4) g ne > λ (1 F)/ µ. (5) Te constraints ensure tat te gatekeeper and expert pools are stable. In te following sections we will be comparing tis twotier system wit an allexpert system in wic customers do not see gatekeepers, but instead go directly for treatment in te expert server pool. Tis onetier system is simply an M/M/N system wit n e servers, arrival rate λ and service rate µ. Terefore, te total cost is, C 1 (n e ) = wλq(n e, λ, µ) + c e n e. (6) Wile we can numerically find a routing policy and staffing levels tat minimise C 1 and C, in te next section we will use a squareroot staffing euristic tat will 0
7 14 S. Hasija, E.J. Pinker and R.A. Sumsky 1 allow us to solve tese problems quickly enable us to caracterise te effects of certain parameters on te optimal solution 3 allow for direct comparison between te onetier and twotier systems. 4 An approximation using a squareroot staffing rule In bot te one and twotier systems, eac server pool is an M/M/N queue wit linear staffing and waitingtime costs. Borst et al. (004) demonstrate tat wen te number of servers is adjusted to minimise total staffing and waiting costs, and wen we allow λ, te ratio of staffing and waiting costs is bounded. Suc a system is described as being in te rationalised regime. Building on te work of Witt, Borst et al. (004) also sow tat for systems in te rationalised regime, a simple squareroot staffing euristic is asymptotically optimal as λ. In tis section we describe te euristic and apply it to our system. In Section 4.1 we sow ow te euristic can be used to generate bot nearoptimal staffing levels and an approximation of te total cost function for a singleserver pool. Section 4.1 is primarily a summary of te work of Borst et al., and tese results can be applied directly to te onetier system. In Section 4. we apply tese staffing results to te twotier system, so tat te routing and staffing problem reduces to a singlevariable optimisation in te treatment tresold, k. 4.1 Approximation for an M/M/N queue Consider an M/M/N queue wit load ρ = λ/µ, staffing cost c per unit time and waiting cost w per unit time. Borst et al. sow tat staffing n servers according to te following squareroot rule is asymptotically optimal (te superscript refers to eiter euristic or Halfin Witt ): n = ρ + y*( c, w) ρ. (7) At least ρ agents are needed to guarantee stability and y*( c, w) ρ is te safety staffing for protection against stocastic variability. Te quantity y*(c, w) can be tougt of as te optimal service level and is found by balancing te staffing and waiting costs. Specifically, y*(c, w) minimises te function were wπ ( y) α ( ycw,, ) = cy+, (8) y yφ( y) π ( y) = 1+ φ( y) 1 and φ(y) and Φ(y) are te unit normal pdf and cdf, respectively. Tat is, y *( cw, ) = arg min α( ycw,, ) (10) y>0 (9)
8 Staffing and routing in a twotier call centre 15 Because te function α(y, c, w) as a finite, unique, and positive minimum value, y*(c, w) can be found quickly by numerical metods. Te function π(y) as an important interpretation tat will be useful for constructing te approximate cost function. It is known as te Halfin Witt delay function, and it is an asymptotically exact approximation of te probability of delay, Pr{wait > 0}, for te M/M/N queue. Let D(ρ, c, w) be an approximation for te total cost per unit time of staffing and waiting under te rationalised regime, given load ρ, unit staffing cost c, and unit waiting cost w. Given tat π(y) is te approximation for te probability of delay under te rationalised regime, wλπ ( y*( c, w)) D( ρ, c, w) = cn + n µ λ (11) = cρ + α( y*( cw, ), cw, ) ρ. Te second expression follows by substitution for n and te definition of α. Because te onetier system is simply an M/M/N queue, our approximation of te optimal total cost for tis directaccess system is C 1 = D( λ / µ, c, w) (1) e were λ is te arrival rate to te system, µ is te service rate of te experts, c e is te cost of experts per unit time, and w is te waiting cost per unit time. In numerical experiments, Borst et al. sow tat tis staffing euristic is remarkably robust, even for offered loads as low as 10. We present similar results in our numerical experiments (See Section 5.3), and we also find tat using te approximate total cost function D(ρ, c, w) allows us to find nearoptimal solutions to te staffing and routing problem in te twotier system wit gatekeepers. 4. Approximation for a twotier system Given te size of te load to eac server pool in te twotier system, we use te squareroot staffing euristic to determine te optimal number of servers for tat pool. In te twotier system, te coice of te treatment tresold k determines te arrival and service rates of te gatekeeper and expert pools, and terefore determines te load for eac pool. Specifically, te load for te gatekeeper pool, ρ g (k) = λ/ µ (k) and te load for te expert pool, ρ e (k) = (1 F(k))λ/µ. Terefore, for a given k, te number of servers in eac pool is, n = ρ + y*( c, w) ρ, i = g, e, (13) i i i i and our approximation for te total cost of te twotier system is C = D( ρ, c, w) + D( ρ, c, w) + mλ( k F). (14) g g e e Because te squareroot staffing rule specifies te number of servers in eac pool, k is te remaining decision variable, and our problem is to find te costminimising value of k 0 k 1 k = arg min C (15)
9 16 S. Hasija, E.J. Pinker and R.A. Sumsky and to compare te optimal twotier cost C ( k ) wit te onetier cost, C 1. Wile k, C ( k ), and C 1 are approximations, in numerical experiments we will see tat tese approximations follow closely te optimal values derived from a more realistic model (Tis alternate model relaxes bot te asymptotic assumptions of te rationalised regime and te markovian assumptions of te original network model presented in Section 3). For an arbitrary treatment function f(k), C ( k ) can take an arbitrary form, e.g., it need not be unimodal. In te next section we assume tat te treatment function is linear. Working wit tese approximations, and wit a linear treatment function, allows us to analytically caracterise te beaviour of te (approximately) optimal treatment tresold and to quickly identify relative advantages of te onetier and twotier systems as te system parameters cange. 5 Analysis and numerical experiments wit a linear treatment function Now assume tat f(k) belongs to a class of linear functions, f(k) = b(1 k), were b [0,1]. Wit tis treatment function, gatekeepers ave a positive cance to successfully treat all calls, altoug te probability approaces 0 for te most difficult calls. Parameter b is a measure of te gatekeeper s skill: as b rises, te gatekeeper as a greater cance to andle all calls. For analytical tractability, we ave cosen a functional form for f so tat te vertical intercept and slope are bot equal to b. A byproduct of tis coice is tat as b increases, te implied variance in call difficulty to te gatekeeper increases as well. For brevity, trougout tis section we use te following notation: ρ = λ / µ ρ = λ / µ t ρ = λ / µ d t d y = y*( c, w) for i = g, e i i α = α( y, c, w) for i = g, e i i i Te following Proposition states tat te total cost functions C ( k ) and C 1 are minimised wit a single, optimal system design and treatment tresold. All proofs are in te appendix. Proposition 1: Wen minimising C ( k ) to find k, and wen comparing C 1 wit C ( k ), tere are two possible outcomes: 1 a twotier system wit a unique treatment tresold k is optimal a onetier system is optimal.
10 Staffing and routing in a twotier call centre 17 A comparison of C ( k ) and C 1 also demonstrates tat a twotier system is favored wen parameters c g, m, and µ are low, and wen c e, µ d and µ t are ig. Before considering ow k canges as te parameters cange, it is convenient to introduce a simple, deterministic model and te deterministic treatment tresold, k d. 5.1 Te deterministic model Consider a deterministic model of te twotier system wit no stocastic variability in te arrival or service rates, so tat te capacity of te gatekeeper and expert pools are set equal to te load. Given te linear treatment function f(k), te total cost of tis system is C k c c m k bk bk d 1 k k λ(1 bk + bk /) ( ) = gλ + + e + λ( + /) µ d µ t µ and te optimal treatment tresold is, (16) k d 1 m+ cg(1/ µ t 1/ µ d) = 1 b m+ ce (1/ µ ) +. (17) Note tat k d is equivalent to te optimal treatment tresold for te model in Sumsky and Pinker (003), wic also focuses on a deterministic gatekeeper system. A onetier deterministic model as total cost, C d 1 c λ e µ =. (18) Te quantity k d will be useful in te following analysis and will also be useful for generating a simple rule of tumb for te system design in te numerical experiments. 5. Analysis of te optimal treatment tresold Here we examine ow te optimal treatment tresold is affected by te system s parameters. In tis section, we limit our attention to cases were bot 0 < k < 1 and 0 < k d < 1. Te proof of Proposition 1 demonstrated tat wen k is an interior solution, C / k > 0. By using tis fact, and applying te implicit function teorem to C ( k ), we find: / c g < 0, / c e > 0, / m < 0, / µ t > 0, / µ d < 0, / µ < 0 and / b > 0. Terefore, for large values of c g, m, µ d,, µ and small values of c e, µ t, b, it is optimal for gatekeepers to treat only te less difficult calls. Te impact of te arrival rate λ and te waiting cost w is more complex. First, we consider λ. We find tat as λ increases, k can eiter fall or rise, and tat it monotonically converges to k d.
11 18 S. Hasija, E.J. Pinker and R.A. Sumsky Proposition : 1 If k k d ten / λ 0 If k < k d ten / λ > 0 3 k k d as λ. Figures and 3 sow convergence from above and below, respectively. Convergence to k d as an intuitive explanation: for very large λ, waiting costs are relatively small, compared to te sum of staffing and mistreatment costs. Terefore, for very large λ it is optimal to use te treatment tresold from te deterministic model, wic only considers staffing and mistreatment costs. Figure Treatment tresold k vs. λ. Oter parameters are µ t = 0.75, µ d = 5, µ = 1, c g = 1, c e = 4, m = 1,b = 1, w = 5 Figure 3 Treatment tresold k vs. λ. Oter parameters are te same as for Figure except b = 0.8 and w = 0.5
12 Staffing and routing in a twotier call centre 19 To understand te impact of w on te optimal treatment tresold, it is useful to examine te expression for te partial derivative of k wit respect to w: ρb(1 k ) π( y ) ( ) ( ) e ρt ρ π y d g C ( k ) = w 1 bk b( k ) / ye ρ ( ) yg ( k ) d k ρt ρ + + d 1. (19) Given tat k is an interior solution, te denominator is positive. Terefore, te sign of te derivative depends upon te sign of te numerator. If te numerator is multiplied by w, ten te first term is te marginal decline in te cost of waiting at te expert queue as k increases. Te second term is te marginal cost of waiting at te gatekeeper queue as k increases. Terefore, if te marginal cost at te gatekeeper queue is lower, ten k rises wit w, sifting some of te workload to te gatekeepers and reducing te expert queue. Expression 19 allows us to see ow te parameters affect te relationsip between w and k. For example, if b is ig and te gatekeeper is skilled, te first term in te numerator dominates, and k rises as w rises. On te oter and, if ρ t ρ d is large, implying tat treatment by te gatekeepers is slow, ten te second term dominates and k falls as w rises. Figure 4 sows ow k canges wit w for tree different gatekeeper skill levels (oter parameters are te same as for Figure ). Figure 4 Treatment tresold k vs. w for b = 0.7, 0.8, and 1 Intuitively, as w rises, queueing economies of scale become more important. Tese economies of scale imply tat it is more efficient to ave one large and one small server pool, rater tan two pools tat are closer in size. If te parameters give an advantage to te gatekeepers, ten a rising w implies a rise in k, expanding te ranks of te gatekeepers and reducing te expert pool. If te parameters give an advantage to te experts, ten rising w implies tat pooling sould occur on te expert level, dropping k and eventually producing a onetier system. Te following proposition sows tat tis effect is monotonic if k is above k d.
13 0 S. Hasija, E.J. Pinker and R.A. Sumsky Proposition 3: If k > k d, ten / w > 0. In te next section we will see numerical examples of tese effects, and we will compare te one and twotier systems under a variety of system parameters. 5.3 Numerical experiments In tis section we demonstrate te accuracy of te approximation described above, and investigate ow te optimal design of te service centre is influenced by te parameter values. In particular, we see numerically ow te optimal treatment tresold canges, and we compare onetier and twotier systems under a variety of scenarios. We also sow tat te treatment tresold derived from te deterministic model, k d, is an excellent approximation for te optimal treatment tresold in stocastic systems, as long as it is optimal to use a twotier, rater tan a onetier, system. Recall tat in te model introduced in Section 3, we assume tat te gatekeeper s service time is exponentially distributed wit mean µ. However, te gatekeeper s actual service time is a mixture of time spent only diagnosing a customer and time spent bot diagnosing and treating. Because tese two types of services may ave significantly different average times, a more accurate model would use a mixture of two exponential service times: a proportion k wit mean 1/µ t and 1 k wit mean 1/µ d. Given tat te gatekeeper s service times follow suc a distribution, we model te gatekeeper pool as an M/H /N queue and te expert pool as a G/M/N queue. Te arrival process to te expert pool is difficult to caracterise, and we use te approximation suggested by Adan (004). In tis section, we compare our euristic solution, using te squareroot staffing rule, wit te optimal solution determined by numerically solving a model based on te more general queueing systems described above. We use a software package tat implements te G/G/N approximations by Witt (1993) to find te optimal combination of k*, n g *, n e * tat minimises te total staffing, waiting, and mistreatment cost. Hencefort we will call tis solution te optimal staffing and routing strategy and we will call te values k, n, n, determined by equations (7), (10), and (15) te euristic staffing and routing g e strategy. We first verify te accuracy of te squareroot staffing rule in te twotier setting. Te accuracy of tis approximation will be driven by te sensitivity of te results to te assumption tat te gatekeeper s service times are exponential and tat arrivals to te experts are Poisson. Terefore, te larger te difference between µ d and µ t, te worse te performance of te euristic. However, we find tat even wit extremely large differences (µ d /µ t as large as 100), te cost of a system operated according to te euristic is witin 1% of te optimal cost, as long as λ > 0. We also observe tat te referral rates and staffing levels generated from te euristic converge quickly to te optimal levels as λ increases. In tis section, we will focus on more reasonable examples tan µ d /µ t = 100; we set µ t = 0.75 and µ d = 5, wile varying oter parameters, suc as te skill level b and te waiting cost w. For example, Figure sows ow k*converges to k as te arrival rate increases, and ow k converges to k d from above, as implied by Proposition. Most of te variation of k*around k is due to te integrality of n g * and n e *. Figure 3 sows a similar pattern, altoug wit a lower value of b and w, and ere k converges to k d from below. In Figure 5, we see tat using te euristic does not significantly increase system
14 Staffing and routing in a twotier call centre 1 costs, given large λ, and in Figure 6 we see tat te staffing levels n g *, n e *and n g, n e are nearly identical. (Figures 5 and 6 use te set of parameters tat led to Figure.) Figure 5 Percentage cost penalty for using euristic solution rater tan te optimal solution Figure 6 Staffing levels vs. λ
15 S. Hasija, E.J. Pinker and R.A. Sumsky In all remaining figures, we do not sow te optimal solution, but in every case te euristic and optimal solutions are nearly identical, and te difference in total cost wen using eac is negligible. We will also be using as a baseline te parameter values for te system sown in Figures, 5 and 6. Wile we will only be presenting a subset of our experiments, we observed tat te euristic solution was nearly optimal over a wide range of parameter values for labor and waiting costs, service times, and gatekeeper skills. Figure 7 plots te staffing levels of eac server pool as a function of b, a measure of te gatekeepers skills. Below a certain skill level a directaccess system is optimal and above tat skill level a twotier system is optimal. As te skill level continues to increase, te gatekeeper pool grows and te expert pool srinks. Figure 7 Staffing levels vs. b Figure 8 sows k and k d as a function of te skill level. Te steep, initial increase in eac tresold represents te transition from one to twotier systems; note tat k d rises at a lower value of b tan k. We consistently observed tis penomenon in all numerical experiments we conducted. To understand wy k d sould rise before k, it is useful to interpret k d as te optimal treatment tresold wen w is extremely low (a deterministic system essentially ignores waiting costs). To justify aving gatekeepers in systems wit ig waiting costs, tey must ave iger skills tan needed to justify gatekeepers in systems wit lower waiting costs. In oter words, if it is optimal to coose te directaccess system wen w is very low (as in te deterministic system), ten it is optimal to coose te directaccess system wen te customer s cost of waiting is iger. Tis effect can be explained by te fact tat a onetier system offers benefits from pooling and tat tese benefits are more powerful wen waiting costs are ig. However, tis does not imply tat a ig waiting cost always leads to a onetier system. As in Proposition 3, it is possible to prefer a gatekeeper system, no matter ow ig te value of w (we saw an example of tat in Figure 4).
16 Staffing and routing in a twotier call centre 3 Figure 8 Treatment tresold k vs. b Figure 9 sows te contribution of mistreatment, waiting, and staffing costs to te total cost as a function of gatekeeper skill. Tis plot sows te actual costs, calculated from te G/G/N models, given te euristic solution. Te staffing cost decreases as b increases because we substitute gatekeepers for experts, as seen in Figure 7. An increase in b coupled wit an increase in k also implies tat a iger fraction of calls leave te system after successful treatment by te gatekeeper, tus reducing te queues and te waiting time in te system. It is somewat counterintuitive tat total mistreatment cost increases wit b. On te one and, increasing b reduces te probability tat gatekeepers mistreat eac call tat tey address. On te oter and, increasing k increases te number of calls treated by te gatekeeper, and terefore increases te mistreatment rate. In all of our experiments, we observed tat te second effect dominates te first, so tat rising b always increases total mistreatment costs. Figure 9 Cost of staffing, waiting and mistreatment for te euristic solution vs. b
17 4 S. Hasija, E.J. Pinker and R.A. Sumsky In Section 5, we saw tat te response of te optimal treatment tresold to canges in w is complex. In Figure 4 we plot k and k d for different values of b as a function of waiting cost. We see tat for ig values of b, k increases wit waiting cost wile for low values of b te optimal treatment tresold decreases until a directaccess system is preferable. As we discussed in Section 5, te optimal location to pool resources as w increases depends upon te skill level of te gatekeepers and te cost parameters. In all of our experiments, we noticed tat wen a twotier system is preferred to a directaccess system, k and k d are remarkably close (see, for example, Figures 4, 8). Terefore using k d as an estimate of te optimal treatment tresold does not increase total costs significantly (see Figure 10). However, te coice of a one or twotier system sould be based on a cost comparison tat takes optimal staffing and waiting costs into account, as was seen in Figures 4 and 8. Terefore, we propose te following rule of tumb for coosing te optimal system: Calculate k d using equation (17). Using k d as te treatment tresold, use te squareroot staffing rule to determine te number of gatekeepers and experts in a twotier system. Given tese staffing levels, calculate te total cost d C ( k ). Also using te squareroot staffing rule, determine te number of experts in te directaccess system and calculate te cost C 1. If d C ( k ) < C 1, coose a twotier system using k d as te treatment tresold. Oterwise, coose a directaccess system. Tis rule of tumb does not require managers to find k* or k, bot of wic require significant computational effort compared to finding k d. Figure 10 Percentage cost penalty for using k d and te squareroot staffing rule rater tan te optimal solution
18 Staffing and routing in a twotier call centre 5 6 Conclusions In practice most call centres ave multiple tiers, were te tiers are distinguised by teir abilities to serve te customers. Differing abilities typically imply differing compensation rates and service rates as well. Managers must determine staffing at eac tier in conjunction wit routing rules to balance customer queueing delay costs, mistreatment costs and staffing costs. In tis paper we ave developed an approac tat greatly simplifies tis complex managerial problem. By drawing upon recent results sowing te asymptotic optimality of squareroot staffing rules for standalone queues, we ave sown tat te optimal design of a twotier system can be reduced to determining an optimal routing rule. Furter, we ave sown tat te easily computed routing rule from a deterministic system can be used wenever a twotier system is preferred to a onetier system. It is well known in te queueing literature tat pooling resources can create economic benefits by reducing variability. In a twotier system in wic te second tier is staffed wit iger skilled and more expensive servers, it is not clear ow to take advantage of pooling. We find tat wen waiting costs are iger, gatekeepers need a iger skill level to be wortwile. Tat is, pooling economies are acieved using te experts only. However, we also see tat if te gatekeepers skills are ig enoug, it is optimal to acieve pooling economies at te firsttier for even very ig values of te waiting costs, w. So we see tat depending on te combination of (b, w) we may seek pooling economies at different locations in te system. Muc of our analysis was restricted to te case of a linear treatment function. Furter researc is necessary to test te validity of our results for more general treatment functions. Oter possible areas for future researc include extending te model and analysis to tree or more tiers of servers, considering dynamic routing policies, and incorporating incentive systems for controlling gatekeeper referral beaviour into te model, as is done in Sumsky and Pinker (003). Acknowledgement We would like to tank Harry Groenevelt for supplying us wit is software package, QMacros, and for patiently answering our questions about te software. QMacros includes an implementation of te G/G/N approximation proposed by Witt (1993), and we used te software for te numerical experiments in Section 5. References Adan, I. (004) Teacing Note on MultiMacine Systems, available at ttp://www.win.tue.nl/~iadan/sdp/11.pdf. Bernett, H.(003) Healtcare call centers: a tecnology migration, orizons, Perspectives in Healtcare Management and Information Tecnology, September, pp Borst, S., Mandelbaum, A. and Reiman, M.I. (004) Dimensioning large call centers, Operations Researc, Vol. 5, No. 1, pp Cevalier, P., Sumsky, R.A. and Tabordon, N. (004) Routing and Staffing in Large Call Centers wit Specialized and Fully Flexible Servers, working paper, Simon Scool, University of Rocester, Rocester, NY.
19 6 S. Hasija, E.J. Pinker and R.A. Sumsky de Véricourt, F. and Zou, Y.P. (004) A Routing Problem for Call Centers wit Customer Callbacks after Service Failure, Working Paper, Fuqua Scool of Business, Duke University, Duram, Nort Carolina. Gross, D. and C.M. Harris (1985) Fundamentals of Queueing Teory, Second Edition, Wiley, New York. Halfin, S. and Witt, W. (1981) Heavytraffic limits for queues wit many exponential servers, Operations Researc, Vol. 9, No. 3, pp Örmeci, E.L. (004) Dynamic admission control in a call center wit one sared and two dedicated service facilities, IEEE Transactions on Automatic Control, Vol. 49, No. 7, pp Sumsky, R.A., Pinker, E.J. (003) Gatekeepers and referrals in service, Management Science, Vol. 49, No. 7, pp Witt, W. (199) Understanding te efficiency of multiserver service systems, Management Science, Vol. 38, No. 5, pp Witt, W. (1993) Approximations for te GI/G/m queue, Production and Operations Management, Vol., No., pp Wallace, R.B. and Witt, W. (004) Resource Pooling and Staffing in Call Centers wit SkillBased Routing, Working Paper, Columbia University, ttp://www.columbia.edu/ ~ww040/pooling.pdf. Appendix: Proofs Proof of Proposition 1: For te given treatment function, ( ) ρt ρd αg = cg( ρt ρd) + ρceb(1 k) k ρ + k( ρ ρ ) d t d ρb(1 k) αe + mλ(1 b+ bk). 1 bk + bk / and, C ( ρt ρd) αg ρb( b) αe ρcb 3/ e 3/ k 4[ ρd + k( ρt ρd)] 4[1 bk+ bk / ] = mλb. (0) (1) Te total cost functions ave te following properties: P1: k = 1 >0, C P: is an increasing function in k, P3: Te cost of staffing no gatekeepers is less tan te cost of staffing gatekeepers wo only do a diagnosis of te incoming calls. Te cost of aving no gatekeepers is given by, C 1 = ρc e + ρα e. ()
20 Staffing and routing in a twotier call centre 7 Explore all possible cases. For eac case, we see tat eiter te directaccess system is optimal, or te twotier system as a unique costminimising solution, k. I II C From P k = 0 >0. C >0 for all values of k, C is convex in te domain k [0,1]. From P1 observe tat two subcases are possible. k = 0 >0. Here, ( k ) = 0 as no root on te interval [0, 1]. In tis case it is optimal for te centre to staff only experts. k = 0 <0. Here, ( k ) = 0 as one root in te interval [0, 1]. k is te unique point wic minimises C ( k ). If C ( k )< C 1, ten staff generalists wo treat incoming calls of difficulty level <k, else only staff specialists. C k = 0 <0 and C k = 1 >0. P k (0,1) suc tat C ( k ) is concave for k < k and is convex for k > k. Tere will be four subcases ere: No root for ( k ) = 0 in te interval [0, 1]. It is optimal to staff only experts in tis case. One root for ( k ) = 0 in te interval [0, 1]. It is optimal to staff only experts in tis case. k = 0 <0. Tis case will also ave one root in [0, 1] for ( k ) = 0. Compare te total cost at tat root (k ) wit C 1 to determine wic system is optimal. ( k ) = 0 as two roots in [0,1]. Tis case is sown in Figures 4 and 9. Compare te total cost at te larger root (k ) wit C 1.
Verifying Numerical Convergence Rates
1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and
More informationCan a LumpSum Transfer Make Everyone Enjoy the Gains. from Free Trade?
Can a LumpSum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lumpsum transfer rules to redistribute te
More informationComparison between two approaches to overload control in a Real Server: local or hybrid solutions?
Comparison between two approaces to overload control in a Real Server: local or ybrid solutions? S. Montagna and M. Pignolo Researc and Development Italtel S.p.A. Settimo Milanese, ITALY Abstract Tis wor
More informationStrategic trading in a dynamic noisy market. Dimitri Vayanos
LSE Researc Online Article (refereed) Strategic trading in a dynamic noisy market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt and Moral
More informationGeometric Stratification of Accounting Data
Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual
More informationThe EOQ Inventory Formula
Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of
More informationAn inquiry into the multiplier process in ISLM model
An inquiry into te multiplier process in ISLM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) 0062763074 Internet Address: jefferson@water.pu.edu.cn
More information2.23 Gambling Rehabilitation Services. Introduction
2.23 Gambling Reabilitation Services Introduction Figure 1 Since 1995 provincial revenues from gambling activities ave increased over 56% from $69.2 million in 1995 to $108 million in 2004. Te majority
More informationUnemployment insurance/severance payments and informality in developing countries
Unemployment insurance/severance payments and informality in developing countries David Bardey y and Fernando Jaramillo z First version: September 2011. Tis version: November 2011. Abstract We analyze
More information2 Limits and Derivatives
2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line
More information2.28 EDGE Program. Introduction
Introduction Te Economic Diversification and Growt Enterprises Act became effective on 1 January 1995. Te creation of tis Act was to encourage new businesses to start or expand in Newfoundland and Labrador.
More information7.6 Complex Fractions
Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are
More informationStrategic trading and welfare in a dynamic market. Dimitri Vayanos
LSE Researc Online Article (refereed) Strategic trading and welfare in a dynamic market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt
More informationSchedulability Analysis under Graph Routing in WirelessHART Networks
Scedulability Analysis under Grap Routing in WirelessHART Networks Abusayeed Saifulla, Dolvara Gunatilaka, Paras Tiwari, Mo Sa, Cenyang Lu, Bo Li Cengjie Wu, and Yixin Cen Department of Computer Science,
More informationSAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY
ASA Section on Survey Researc Metods SAMPLE DESIG FOR TE TERRORISM RISK ISURACE PROGRAM SURVEY G. ussain Coudry, Westat; Mats yfjäll, Statisticon; and Marianne Winglee, Westat G. ussain Coudry, Westat,
More informationOptimized Data Indexing Algorithms for OLAP Systems
Database Systems Journal vol. I, no. 2/200 7 Optimized Data Indexing Algoritms for OLAP Systems Lucian BORNAZ Faculty of Cybernetics, Statistics and Economic Informatics Academy of Economic Studies, Bucarest
More informationCHAPTER 7. Di erentiation
CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.
More informationInstantaneous Rate of Change:
Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over
More informationTangent Lines and Rates of Change
Tangent Lines and Rates of Cange 922005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims
More informationDistances in random graphs with infinite mean degrees
Distances in random graps wit infinite mean degrees Henri van den Esker, Remco van der Hofstad, Gerard Hoogiemstra and Dmitri Znamenski April 26, 2005 Abstract We study random graps wit an i.i.d. degree
More informationAn Interest Rate Model
An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal
More informationTRADING AWAY WIDE BRANDS FOR CHEAP BRANDS. Swati Dhingra London School of Economics and CEP. Online Appendix
TRADING AWAY WIDE BRANDS FOR CHEAP BRANDS Swati Dingra London Scool of Economics and CEP Online Appendix APPENDIX A. THEORETICAL & EMPIRICAL RESULTS A.1. CES and Logit Preferences: Invariance of Innovation
More informationDerivatives Math 120 Calculus I D Joyce, Fall 2013
Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te
More informationON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE
ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE Byeong U. Park 1 and Young Kyung Lee 2 Department of Statistics, Seoul National University, Seoul, Korea Tae Yoon Kim 3 and Ceolyong Park
More informationWe consider the problem of determining (for a short lifecycle) retail product initial and
Optimizing Inventory Replenisment of Retail Fasion Products Marsall Fiser Kumar Rajaram Anant Raman Te Warton Scool, University of Pennsylvania, 3620 Locust Walk, 3207 SHDH, Piladelpia, Pennsylvania 191046366
More informationCollege Planning Using Cash Value Life Insurance
College Planning Using Cas Value Life Insurance CAUTION: Te advisor is urged to be extremely cautious of anoter college funding veicle wic provides a guaranteed return of premium immediately if funded
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationComputer Science and Engineering, UCSD October 7, 1999 GoldreicLevin Teorem Autor: Bellare Te GoldreicLevin Teorem 1 Te problem We æx a an integer n for te lengt of te strings involved. If a is an nbit
More informationReferendumled Immigration Policy in the Welfare State
Referendumled Immigration Policy in te Welfare State YUJI TAMURA Department of Economics, University of Warwick, UK First version: 12 December 2003 Updated: 16 Marc 2004 Abstract Preferences of eterogeneous
More informationMath 113 HW #5 Solutions
Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten
More informationA system to monitor the quality of automated coding of textual answers to open questions
Researc in Official Statistics Number 2/2001 A system to monitor te quality of automated coding of textual answers to open questions Stefania Maccia * and Marcello D Orazio ** Italian National Statistical
More informationThe modelling of business rules for dashboard reporting using mutual information
8 t World IMACS / MODSIM Congress, Cairns, Australia 37 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,
More informationWhat is Advanced Corporate Finance? What is finance? What is Corporate Finance? Deciding how to optimally manage a firm s assets and liabilities.
Wat is? Spring 2008 Note: Slides are on te web Wat is finance? Deciding ow to optimally manage a firm s assets and liabilities. Managing te costs and benefits associated wit te timing of cas in and outflows
More informationWorking Capital 2013 UK plc s unproductive 69 billion
2013 Executive summary 2. Te level of excess working capital increased 3. UK sectors acieve a mixed performance 4. Size matters in te supply cain 6. Not all companies are overflowing wit cas 8. Excess
More informationFinite Volume Discretization of the Heat Equation
Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te onedimensional variable coefficient eat equation, wit Neumann boundary conditions u t x
More informationResearch on the Antiperspective Correction Algorithm of QR Barcode
Researc on te Antiperspective Correction Algoritm of QR Barcode Jianua Li, YiWen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic
More informationImproved dynamic programs for some batcing problems involving te maximum lateness criterion A P M Wagelmans Econometric Institute Erasmus University Rotterdam PO Box 1738, 3000 DR Rotterdam Te Neterlands
More information1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution
1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis
More informationWelfare, financial innovation and self insurance in dynamic incomplete markets models
Welfare, financial innovation and self insurance in dynamic incomplete markets models Paul Willen Department of Economics Princeton University First version: April 998 Tis version: July 999 Abstract We
More informationOPTIMAL FLEET SELECTION FOR EARTHMOVING OPERATIONS
New Developments in Structural Engineering and Construction Yazdani, S. and Sing, A. (eds.) ISEC7, Honolulu, June 1823, 2013 OPTIMAL FLEET SELECTION FOR EARTHMOVING OPERATIONS JIALI FU 1, ERIK JENELIUS
More informationPretrial Settlement with Imperfect Private Monitoring
Pretrial Settlement wit Imperfect Private Monitoring Mostafa Beskar University of New Hampsire JeeHyeong Park y Seoul National University July 2011 Incomplete, Do Not Circulate Abstract We model pretrial
More informationCatalogue no. 12001XIE. Survey Methodology. December 2004
Catalogue no. 1001XIE Survey Metodology December 004 How to obtain more information Specific inquiries about tis product and related statistics or services sould be directed to: Business Survey Metods
More informationLecture 10. Limits (cont d) Onesided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)
Lecture 10 Limits (cont d) Onesided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define onesided its, were
More informationFree Shipping and Repeat Buying on the Internet: Theory and Evidence
Free Sipping and Repeat Buying on te Internet: eory and Evidence Yingui Yang, Skander Essegaier and David R. Bell 1 June 13, 2005 1 Graduate Scool of Management, University of California at Davis (yiyang@ucdavis.edu)
More informationCyber Epidemic Models with Dependences
Cyber Epidemic Models wit Dependences Maocao Xu 1, Gaofeng Da 2 and Souuai Xu 3 1 Department of Matematics, Illinois State University mxu2@ilstu.edu 2 Institute for Cyber Security, University of Texas
More informationMath Test Sections. The College Board: Expanding College Opportunity
Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt
More informationA strong credit score can help you score a lower rate on a mortgage
NET GAIN Scoring points for your financial future AS SEEN IN USA TODAY S MONEY SECTION, JULY 3, 2007 A strong credit score can elp you score a lower rate on a mortgage By Sandra Block Sales of existing
More informationA.4 RATIONAL EXPRESSIONS
Appendi A.4 Rational Epressions A9 A.4 RATIONAL EXPRESSIONS Wat you sould learn Find domains of algebraic epressions. Simplify rational epressions. Add, subtract, multiply, and divide rational epressions.
More informationSimultaneous Location of Trauma Centers and Helicopters for Emergency Medical Service Planning
Simultaneous Location of Trauma Centers and Helicopters for Emergency Medical Service Planning SooHaeng Co Hoon Jang Taesik Lee Jon Turner Tepper Scool of Business, Carnegie Mellon University, Pittsburg,
More informationOPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS
OPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS ERIC T. CHUNG AND BJÖRN ENGQUIST Abstract. In tis paper, we developed and analyzed a new class of discontinuous
More informationDetermine the perimeter of a triangle using algebra Find the area of a triangle using the formula
Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using
More informationf(a + h) f(a) f (a) = lim
Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )
More informationAreaSpecific Recreation Use Estimation Using the National Visitor Use Monitoring Program Data
United States Department of Agriculture Forest Service Pacific Nortwest Researc Station Researc Note PNWRN557 July 2007 AreaSpecific Recreation Use Estimation Using te National Visitor Use Monitoring
More informationBonferroniBased SizeCorrection for Nonstandard Testing Problems
BonferroniBased SizeCorrection for Nonstandard Testing Problems Adam McCloskey Brown University October 2011; Tis Version: October 2012 Abstract We develop powerful new sizecorrection procedures for
More informationGovernment Debt and Optimal Monetary and Fiscal Policy
Government Debt and Optimal Monetary and Fiscal Policy Klaus Adam Manneim University and CEPR  preliminary version  June 7, 21 Abstract How do di erent levels of government debt a ect te optimal conduct
More informationNote nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1
Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te
More informationAnalyzing the Effects of Insuring Health Risks:
Analyzing te Effects of Insuring Healt Risks: On te Tradeoff between Sort Run Insurance Benefits vs. Long Run Incentive Costs Harold L. Cole University of Pennsylvania and NBER Soojin Kim University of
More informationThe Dynamics of Movie Purchase and Rental Decisions: Customer Relationship Implications to Movie Studios
Te Dynamics of Movie Purcase and Rental Decisions: Customer Relationsip Implications to Movie Studios Eddie Ree Associate Professor Business Administration Stoneill College 320 Wasington St Easton, MA
More informationTorchmark Corporation 2001 Third Avenue South Birmingham, Alabama 35233 Contact: Joyce Lane 9725693627 NYSE Symbol: TMK
News Release Torcmark Corporation 2001 Tird Avenue Sout Birmingam, Alabama 35233 Contact: Joyce Lane 9725693627 NYSE Symbol: TMK TORCHMARK CORPORATION REPORTS FOURTH QUARTER AND YEAREND 2004 RESULTS
More informationM(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)
Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut
More informationFINANCIAL SECTOR INEFFICIENCIES AND THE DEBT LAFFER CURVE
INTERNATIONAL JOURNAL OF FINANCE AND ECONOMICS Int. J. Fin. Econ. 10: 1 13 (2005) Publised online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/ijfe.251 FINANCIAL SECTOR INEFFICIENCIES
More informationDigital evolution Where next for the consumer facing business?
Were next for te consumer facing business? Cover 2 Digital tecnologies are powerful enablers and lie beind a combination of disruptive forces. Teir rapid continuous development demands a response from
More informationDynamic Competitive Insurance
Dynamic Competitive Insurance Vitor Farina Luz June 26, 205 Abstract I analyze longterm contracting in insurance markets wit asymmetric information and a finite or infinite orizon. Risk neutral firms
More informationAn Orientation to the Public Health System for Participants and Spectators
An Orientation to te Public Healt System for Participants and Spectators Presented by TEAM ORANGE CRUSH Pallisa Curtis, Illinois Department of Public Healt Lynn Galloway, Vermillion County Healt Department
More informationResearch on Risk Assessment of PFI Projects Based on Gridfuzzy Borda Number
Researc on Risk Assessent of PFI Projects Based on Gridfuzzy Borda Nuber LI Hailing 1, SHI Bensan 2 1. Scool of Arcitecture and Civil Engineering, Xiua University, Cina, 610039 2. Scool of Econoics and
More informationOutsourcing a TwoLevel Service Process
Outsourcing a TwoLevel Service Process HsiaoHui Lee Edieal Pinker Robert A. Shumsky School of Business, University of Hong Kong, Hong Kong The Simon School of Business, University of Rochester, Rochester,
More informationGlobal Sourcing of Complex Production Processes
Global Sourcing of Complex Production Processes December 2013 Cristian Scwarz Jens Suedekum Abstract We develop a teory of a firm in an incomplete contracts environment wic decides on te complexity, te
More informationACT Math Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as
More informationA FLOW NETWORK ANALYSIS OF A LIQUID COOLING SYSTEM THAT INCORPORATES MICROCHANNEL HEAT SINKS
A FLOW NETWORK ANALYSIS OF A LIQUID COOLING SYSTEM THAT INCORPORATES MICROCHANNEL HEAT SINKS Amir Radmer and Suas V. Patankar Innovative Researc, Inc. 3025 Harbor Lane Nort, Suite 300 Plymout, MN 55447
More informationTHE IMPACT OF INTERLINKED INDEX INSURANCE AND CREDIT CONTRACTS ON FINANCIAL MARKET DEEPENING AND SMALL FARM PRODUCTIVITY
THE IMPACT OF INTERLINKED INDEX INSURANCE AND CREDIT CONTRACTS ON FINANCIAL MARKET DEEPENING AND SMALL FARM PRODUCTIVITY Micael R. Carter Lan Ceng Alexander Sarris University of California, Davis University
More informationTraining Robust Support Vector Regression via D. C. Program
Journal of Information & Computational Science 7: 12 (2010) 2385 2394 Available at ttp://www.joics.com Training Robust Support Vector Regression via D. C. Program Kuaini Wang, Ping Zong, Yaoong Zao College
More informationDEPARTMENT OF ECONOMICS HOUSEHOLD DEBT AND FINANCIAL ASSETS: EVIDENCE FROM GREAT BRITAIN, GERMANY AND THE UNITED STATES
DEPARTMENT OF ECONOMICS HOUSEHOLD DEBT AND FINANCIAL ASSETS: EVIDENCE FROM GREAT BRITAIN, GERMANY AND THE UNITED STATES Sara Brown, University of Leicester, UK Karl Taylor, University of Leicester, UK
More informationLecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function
Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between
More informationSimilar interpretations can be made for total revenue and total profit functions.
EXERCISE 37 Tings to remember: 1. MARGINAL COST, REVENUE, AND PROFIT If is te number of units of a product produced in some time interval, ten: Total Cost C() Marginal Cost C'() Total Revenue R() Marginal
More informationPredicting the behavior of interacting humans by fusing data from multiple sources
Predicting te beavior of interacting umans by fusing data from multiple sources Erik J. Sclict 1, Ritcie Lee 2, David H. Wolpert 3,4, Mykel J. Kocenderfer 1, and Brendan Tracey 5 1 Lincoln Laboratory,
More informationRISK ASSESSMENT MATRIX
U.S.C.G. AUXILIARY STANDARD AV044 Draft Standard Doc. AV 044 18 August 2004 RISK ASSESSMENT MATRIX STANDARD FOR AUXILIARY AVIATION UNITED STATES COAST GUARD AUXILIARY NATIONAL OPERATIONS DEPARTMENT
More informationOperation golive! Mastering the people side of operational readiness
! I 2 London 2012 te ultimate Up to 30% of te value of a capital programme can be destroyed due to operational readiness failures. 1 In te complex interplay between tecnology, infrastructure and process,
More information2.12 Student Transportation. Introduction
Introduction Figure 1 At 31 Marc 2003, tere were approximately 84,000 students enrolled in scools in te Province of Newfoundland and Labrador, of wic an estimated 57,000 were transported by scool buses.
More informationEquilibria in sequential bargaining games as solutions to systems of equations
Economics Letters 84 (2004) 407 411 www.elsevier.com/locate/econbase Equilibria in sequential bargaining games as solutions to systems of equations Tasos Kalandrakis* Department of Political Science, Yale
More information2.13 Solid Waste Management. Introduction. Scope and Objectives. Conclusions
Introduction Te planning and delivery of waste management in Newfoundland and Labrador is te direct responsibility of municipalities and communities. Te Province olds overall responsibility for te development
More informationOn a Satellite Coverage
I. INTRODUCTION On a Satellite Coverage Problem DANNY T. CHI Kodak Berkeley Researc Yu T. su National Ciao Tbng University Te eart coverage area for a satellite in an Eart syncronous orbit wit a nonzero
More informationTo motivate the notion of a variogram for a covariance stationary process, { Ys ( ): s R}
4. Variograms Te covariogram and its normalized form, te correlogram, are by far te most intuitive metods for summarizing te structure of spatial dependencies in a covariance stationary process. However,
More informationPioneer Fund Story. Searching for Value Today and Tomorrow. Pioneer Funds Equities
Pioneer Fund Story Searcing for Value Today and Tomorrow Pioneer Funds Equities Pioneer Fund A Cornerstone of Financial Foundations Since 1928 Te fund s relatively cautious stance as kept it competitive
More information1 Derivatives of Piecewise Defined Functions
MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.
More informationDesign and Analysis of a FaultTolerant Mechanism for a ServerLess VideoOnDemand System
Design and Analysis of a Faultolerant Mecanism for a ServerLess VideoOnDemand System Jack Y. B. Lee Department of Information Engineering e Cinese University of Hong Kong Satin, N.., Hong Kong Email:
More informationSurface Areas of Prisms and Cylinders
12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of
More informationHeterogeneous firms and trade costs: a reading of French access to European agrofood
Heterogeneous firms and trade costs: a reading of Frenc access to European agrofood markets CevassusLozza E., Latouce K. INRA, UR 34, F44000 Nantes, France Abstract Tis article offers a new reading of
More informationHow doctors can close the gap
RCP policy statement 2010 How doctors can close te gap Tackling te social determinants of ealt troug culture cange, advocacy and education Acknowledgements We would like to tank te Department of Healt
More informationPretrial Settlement with Imperfect Private Monitoring
Pretrial Settlement wit Imperfect Private Monitoring Mostafa Beskar Indiana University JeeHyeong Park y Seoul National University April, 2016 Extremely Preliminary; Please Do Not Circulate. Abstract We
More informationKeskustelualoitteita #65 Joensuun yliopisto, Taloustieteet. Market effiency in Finnish harness horse racing. Niko Suhonen
Keskustelualoitteita #65 Joensuun yliopisto, Taloustieteet Market effiency in Finnis arness orse racing Niko Suonen ISBN 9789522192837 ISSN 17957885 no 65 Market Efficiency in Finnis Harness Horse
More informationSHAPE: A NEW BUSINESS ANALYTICS WEB PLATFORM FOR GETTING INSIGHTS ON ELECTRICAL LOAD PATTERNS
CIRED Worksop  Rome, 1112 June 2014 SAPE: A NEW BUSINESS ANALYTICS WEB PLATFORM FOR GETTING INSIGTS ON ELECTRICAL LOAD PATTERNS Diego Labate Paolo Giubbini Gianfranco Cicco Mario Ettorre Enel DistribuzioneItaly
More information1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion
Density functions, cummulative density functions, measures of central tendency, and measures of dispersion densityfunctionsintro.tex October, 9 Note tat tis section of notes is limitied to te consideration
More informationSAT Subject Math Level 1 Facts & Formulas
Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses
More informationTHE ROLE OF LABOUR DEMAND ELASTICITIES IN TAX INCIDENCE ANALYSIS WITH HETEROGENEOUS LABOUR
THE ROLE OF LABOUR DEMAND ELASTICITIES IN TAX INCIDENCE ANALYSIS WITH HETEROGENEOUS LABOUR Kesab Battarai 1,a and 1, a,b,c Jon Walley a Department of Economics, University of Warwick, Coventry, CV4 7AL,
More information1. Case description. Best practice description
1. Case description Best practice description Tis case sows ow a large multinational went troug a bottom up organisational cange to become a knowledgebased company. A small community on knowledge Management
More informationSAT Math MustKnow Facts & Formulas
SAT Mat MustKnow Facts & Formuas Numbers, Sequences, Factors Integers:..., 3, 2, 1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas
More informationOn Distributed Key Distribution Centers and Unconditionally Secure Proactive Verifiable Secret Sharing Schemes Based on General Access Structure
On Distributed Key Distribution Centers and Unconditionally Secure Proactive Verifiable Secret Saring Scemes Based on General Access Structure (Corrected Version) Ventzislav Nikov 1, Svetla Nikova 2, Bart
More informationRewardsSupply Aggregate Planning in the Management of Loyalty Reward Programs  A Stochastic Linear Programming Approach
RewardsSupply Aggregate Planning in te Management of Loyalty Reward Programs  A Stocastic Linear Programming Approac YUHENG CAO, B.I.B., M.Sc. A tesis submitted to te Faculty of Graduate and Postdoctoral
More informationNew Vocabulary volume
. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding
More informationThe differential amplifier
DiffAmp.doc 1 Te differential amplifier Te emitter coupled differential amplifier output is V o = A d V d + A c V C Were V d = V 1 V 2 and V C = (V 1 + V 2 ) / 2 In te ideal differential amplifier A c
More information