Cross-Selling in a Call Center with a Heterogeneous Customer Population. Technical Appendix

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1 Cross-Selling in a Call Center with a Heterogeneous Customer opulation Technical Appendix Itay Gurvich Mor Armony Costis Maglaras This technical appendix is dedicated to the completion of the proof of roposition 1, whose sketch was given in A of [1], as well as to the proof of the auxiliary results that were stated in the same section without proof. Completion of roof for roposition 1 Following our proof sketch in A of [1] we show that there exists δ such that for any sequence of initial states ξ n Ξ with : lim n 1 E ξ n[ξξn 1 + δ )] =. A1) Whenever this holds we can always find a positive number K, such that for all ξ > K, 2) holds. To this end let V t) = j 1 v jt) be the amount of residual work in the system. Also, let V s t) and V q t) be, respectively, the residual work for the customers in service at time t, i.e. V s t) = j N v jt), and the residual work for the customers in queue at time t, i.e. V q t) = V t) V s t). ) Set µ = 1 1, + 1 µs µ cs so that 1/ µ is the mean customer handling time in case cross-selling is performed. The following Lemma is a parallel of Lemma 4.3 in Dai [1]. Its proof is hence omitted. Lemma 4 For every sequence of initial conditions with we have that: a) Almost surely uniformly on compact sets lim n V s t) N1 t) +, A2) and lim n V q t) 1 + t, λ µ A3) Columbia Business School, 4I Uris Hall, 322 Broadway, NY, NY 127. ig2126@columbia.edu) Stern School of Business, NYU, 44 West 4th Street, NY, NY 112. marmony@stern.nyu.edu) Columbia Business School, 49 Uris Hall, 322 Broadway, NY, NY 127. c.maglaras@gsb.columbia.edu) 1

2 b) For each t, the sequences Vqt),n 1, are uniformly integrable. Vst),n 1 and Finally, for any fixed t lim n 1 Qt) V q t) 1 µ s ξ n lim n ξ n lim n µ Qt) ξ n, A4) and lim n 1 E ξ n[qt)] E ξ n[v q t)] 1 µ s ξ n lim n ξ n lim n µ E ξ n[qt)] ξ n. A5) For the following, let Dt) be the departure process from the system. Also, let Wt) be the virtual waiting time at time t. Then, we have the following tightness result. Lemma 5 For any sequence of initial conditions ξ n Ξ with as n, and on ) compact subsets of [1, ), the sequence Qt), Dξn t), Vqξn t), Wξn t) is tight and unifomrly integrable. Moreover, any limit point Qt), Dt), V q t), Wt)), satisfies for all t and finally that for all t 1, V q t) Wt) Qt) Nµ s, A6) 1 µ Qt) V q t), A7) [ )] λ + V q 1) + t 1) N. A8) µ s roof: The proof of the tightness of ) Qt), Dξn t) is reminiscent of the proof of Theorem 4.1 in Dai [1] and we omit it. As for the virtual waiting time process Wt), note that it has the representation Wt) = infs : Dt + s) Dt) Qt), A9) 2

3 and in particular, Wt) = inf Considering a convergent subsequence n j of in [25], to conclude that Wξn j t) ξ nj s : Dξn t + s)) Dξ n t)) ξ n Qξn t) ξ n. ) Qt), Dξn t), we can now apply the corollary also converges to a limit Wt) that satisfies Wt) Qt) Nµ s. The inequality A7) follows from Lemma 4. Finally, to establish A8), fix ǫ > and set τ n := inft 1 : V q t) ǫ. Then, for 1 t τ n and by work conservation V q t) + V s t) = V q 1) + V s 1) Nt 1) + At) l=a) s l Wτ l )). A1) Here τ l is the time of the l th arrival. Consider a convergent subsequence Then, uniformly on compact subsets of [1,τ n j ), we claim that Qξ n j t) ξ nj, Dξnj t) ξ nj ), Wξnj t) ξ nj. y n j t) := Aξ n j t) l=aξ nj ) s l Wτ l )) ξ n j λ µ s t 1), in probability, as j. A11) Indeed, for fixed T > 1, consider the set Ω n j := ω Ω : inf 1 t T Wξn j t) 1 2 By inequalities A6) and A7), Ω n j ) as j. Then, µv q ξ n j t). Nµ s 1 t T τ n j yn j t) λ t 1) µ s > ǫ 1 t T τ n j yn j t) λ t 1) µ s > ǫ,ωn j ) c + Ω n j, as j. A12) for any ǫ >, where the convergence 1 t T τ n j yn j t) λ t 1) µ s > ǫ,ωn j ) c, as j, follows from the definition of Ω n j and a careful application of the strong law of large numbers using the fact that on Ω n j ) c, for t τ n j and for any ǫ > there exists j large enough so that for all 3

4 k j, E[s l Wτ l ))] 1 µ s + ǫ. Now, Let ỹt) := [ )] λ V q 1) + t 1) N 3 µ 2 ǫ. s Combining A1) and A12) we then have that for any ǫ >, t τ nj T [ Vq ξ n j t) + ξ n ỹt)] > ǫ, as j. j A13) Define now two random times as follows: τ n = inft τ n : V q t) > 2 ǫ and τ n = t τ n : V q t) ǫ. Then, we can extend the arguments we used above to show that for any ǫ >, Since for ǫ < 1 2 ǫ, τ n j T t τ n j T τ n j < T [ Vq ξ n j t) + ξ n ỹt)] > ǫ, as j. j τ n j T t τ nj T we have that τ n j < T, as j. In particular, since [ Vq ξ n j t) + ξ n ỹt)] > ǫ j we can conclude that for arbitrarily small ǫ, t τ nj T [ Vq ξ n j t) + ξ n ỹt)] > ǫ, j [ Vq ξ n j t) + ξ n ỹt)] > ǫ j τ n j < T, [ Vq ξ n j t) + ξ n ỹt)] > ǫ, as j. j The result of the Lemma now follows since ǫ was arbitrary. A14) To complete the proof of roposition 1, note that with δ = 1+λ/ µ N R, and since by Lemma 4, Vq 1) 1 + λ µ, we have that V ) q 1) + δ ) λ N. To conclude the argument, we fix µs an arbitrary sequence of initial conditions ξ n. Consider a convergent subsequence n j j 1, of E[V q1+δ ))]. By Lemma 5 each subsequence n jk n j such that Vqξnj k 1+δ )) ξ n j k converges 4

5 satisfies E[V q ξ n j k 1 + δ ))] lim k ξ n =. j k In particular, lim n E[V q 1 + δ ))] =. Since the sequence ξ n was arbitrary the argument is complete. Before presenting the proofs of some auxiliary results we specify our sample path construction that differs from the one we used for roposition 1 and relies on the oisson nature of most of the processes involved. Assuming throughout that S)-C) is used, we generate the arrivals using a oisson process with rate Λ. Each customer, that arrives to the system will be assigned his arrival time. The customers will be ordered so that the first N Λ where N Λ is the number of agents) customers are those that are in service or cross-selling) and the rest are in queue. The customers in queue will be kept in order of arrival, while the customers in service will be ordered according to the agent that is serving them. Specifically, the customer in place j N Λ is the one being served by agent j. Also, we will keep an array of N Λ variables, n j NΛ j=1, with n j equal to if agent j is idle, 1 if the customer served by agent j is in the service phase and 2 if the agent is performing a cross-selling phase. We then generate phase 1 completions using a non-homogenous oisson process with instantaneous rate µ s Z Λ t) at time t. Given a jump of this process at time t, we will choose the agent that completes service by generating a discrete uniform [1,Z Λ t)] random variable. Let the customer that completes service be customer j. To determine the customer class of the customer that completed service, we will draw a uniform [,1] random variable and set the customer class to i, if i 1 j=1 λ i Λ U i j=1 λ i Λ.1 Assuming the customer if of class i k to determine whether the customer continues into cross-selling we will use an additional uniform [, 1] random variable, Ũ, independent of U, and cross-sell to this customer if Ũ q iw j )1 Qt) ηi. U and Ũ are also independent across customers. If the outcome of this randomization is that the customer continues into cross-selling and that the customer belongs to class i, then Zi,2 Λ will increase by 1. Finally, for i k, we generate completions of cross-selling for class i customers by a non-homogenous oisson process with an instantaneous rate µ cs Zi,2 Λ t) at time t. All of the oisson processes mentioned above can be constructed as time changes of unit rate oisson process. Moreover, using Strong Approximations see [22]), we may write, for example, the 1 Here we used the fact that since all customer types share the same service time distribution and that customers are served FCFS so that it suffices to know the customer s type upon the service completion. 5

6 process of phase 1 completions as µ s ) Z Λ u)du + B µ s Z Λ u)du + OlogmΛt 2)), where B ) is a standard Brownian motion and m ) µ s 1 + z)µ s µ cs ) m is determined so as to ensure that mλ is an upper bound on all the instantaneous transition rates). Since, some system equations that we will use involve more than a single oisson process but always fewer than ) 4), we set from now on m = µ s 1 + z)µ s µ cs ). The multidimensional nature of the system implies a multi-dimensional Brownian motion as a basis for our strong approximation. The strong approximation theory implies that given a unit rate poisson process N ) and an instantaneous rate function λt) λ, for some λ >, we have that N λu)du) λu)du B λu)du) t log2 λt) C, where C is a non-negative random variable with C > γ + βx c 1 e c 2x, for some strictly positive constants γ,β,c 1 and c 2. In all our arguments, rather than dealing directly with the different coordinates of the Brownian motion, it will suffice to use the L 1 norm of BmΛt) given by BmΛt). Furthermore, with respect to the Brownian Motion, we will be using the fact that given values x,t >, BmΛt) x Λ c 3 T x e c 4 x2 T, for some strictly positive constants c 3 and c 4 see for example roblem in [2]). Overall, we may say that for Λ large enough and for all T >, there exist strictly positive constants c 5, c 6, such that for x > and T T, BmΛt) + C logmλt 2) x Λ c 5 1 ) T e c 6 min x2 T, T x. x A15) Denote by Ω Λ,T,x) = ω Ω : BmΛt) + C logmλt 2) x. A16) 6

7 Then, by A15) Ω Λ,T,x ) ) + T λ) 1 c 5 1 e c 6 min x2 T, T x. x We will use N ) to denote a general unit rate oisson process and we will distinguish between the different oisson processes only if two or more appear in the same equation for an example of a construction of system dynamics based on the underlying unit rate oisson processes and Strong Approximation see [22]). The General Approach: All the subsequent proofs share the same basic ideas. Using the strong approximation construction we examine the behavior of the system on a subset of the sample paths such as Ω Λ,T,x)) where the stochastic fluctuations created by the Brownian Motion are bounded. This allows us to examine a deterministic version of the system dynamics. For the deterministic version, and with arguments reminiscent of Lyapunov function tools used for stability proofs, we show that the system is in some sense attracted back into a certain domain. Finally, we remove the conditioning on the stochastic fluctuations and establish Lyapunov Bounds that allow us to apply, almost directly, some results from [14]. We turn now to the actual proofs of the auxiliary results, starting with the following proposition that is not stated in A of [1] but is required in establishing the other results. roposition 6 For each ǫ > there exists Λ ǫ) and strictly positive constants c 7 and c 8 such that for all Λ Λ ǫ) and x >, Z Λ ) R ǫ + x)λ c 7 e c 8x Λ. Moreover, initializing the system with its stationary distribution, we have that for all Λ large enough Z Λ t) R ǫ + x)λ c 9 e c 1x Λ, A17) for some c 9 > and c 1 >. Note that with the exception of the customer cross-selling probability being delay sensitive, above the level of η1 Λ, our system behaves similarly to the system analyzed in [4] when the system there is above the unique threshold. Still, a detailed proof of roposition 6 would be obtained by slightly 7

8 expanding the proofs of Lemma B.1 and roposition B.1 in [4] with the unique threshold there replaced by η1 Λ. The proof is hence omitted. roposition 7 lim Λ E[Q Λ ) η Λ k ) + ] Λ =, and lim ΛE[W Λ )] <. Λ A18) Moreover, initializing the system with its stationary distribution, we have that there exist c 11,c 12, M 1,M 2 > such that for all Λ large enough and for all x >, Q Λ t) η Λ k ) + M 1 + x c 11 e c 12x, and W Λ t) M 2 + x c 11 e c12x. Λ A19) Note that Lemma 1 is covered as a special case of roposition 7. roof: The proof contains two main steps. In the first step we use the system dynamics to establish a Lyapunov type of attraction of the queue length to η k. In the second step we use this attraction to establish exponential bounds on the steady state excess queue above η Λ k. Assume, then, that Q Λ ) > η Λ k + M, for some constant M >. Fix < η M/2 and let τ Λ = inf t : Q Λ t) Q Λ ) η. Note that on [,τ Λ T], the queue length process Q Λ t) satisfies Q Λ t) > η Λ k and Q Λ t) = Q Λ ) + A Λ t) D Λ 2 t) DΛ 1, t), where AΛ t) is the oisson arrival process, D2 Λt) is the process of cross-selling completions and DΛ 1, t) is the process of service completions not followed by a cross-selling phase. In particular, D2 Λ t) is a non-homogenous poisson process with instantaneous rate µ cs Z2 Λt) at time t, and the instantaneous rate of DΛ 1, t) at time t can be bounded on [,τ Λ T]) from below by µ s 1 q))z Λ t), where q) = k 1 i=1 λ i Λ q i). Now, fix ǫ > and using A16) define the set ˆΩΛ) = Ω ω Ω : Z Λ t) R ǫλ Λ,T/Λ,δ), for some constant δ >. Then, on ˆΩΛ) and for t τ Λ T Λ, Q Λ t) Q Λ ) + Λt µ cs Z2 Λ u)du µ s 1 q)) 8 Z Λ u)du + δ.

9 In particular, since as long as the queue is positive Z2 Λ = NΛ Z Λ, recalling that N Λ R = k λ i q i i=1 µ cs, we have that on ˆΩΛ) after straightforward algebra), Q Λ t τ Λ ) Q Λ ) + µ s 1 q)) + µ cs )Λǫt τ Λ ) λ kq k) t τ Λ + δ, for t T Λ. We can now choose ǫ so that Q Λ t τ Λ ) Q Λ ) + δ 1 2 Λ λ kq k) t τ Λ, A2) where for all i K, λ i := λ i Λ. In particular, on ˆΩΛ) we have that Λτ Λ t := 1 2 η+δ λ kq k). Choosing δ η/2, the same argument shows that on ˆΩΛ) and for all τ Λ t t /Λ, Q Λ t) Q Λ ) η/2. Indeed, along the arguments leading to A2) one can establish that on ˆΩΛ), the queue length is a linearly decreasing process as long as Q Λ t) > η Λ k. Overall, if Q Λ ) η Λ k + M, then on ˆΩΛ) e QΛ t /Λ) η Λ k ) + e η/2, e QΛ ) η Λ k ) + and one can always choose M and, in turn, η large enough so that e η/2 < 1. To complete the argument, since Q λ t) Q λ ) + A λ t) we have that if Q Λ ) η Λ k + M, E [e QΛ t /Λ) η Λ k ) +] e QΛ ) η Λ k ) + e η/2 + E [ ] e AΛ t /Λ) 1 ˆΩΛ)) c. Using the Cauchy-Schwartz inequality, noting that for all Λ, E[e 2AΛ t /Λ) ] c for some constant c, and that ˆΩΛ) 1 as Λ, we can choose Λ large enough so that if Q λ ) > η Λ k + M: E [e QΛ t /Λ) η Λ k ) +] e η/4. A21) e QΛ ) η Λ k ) + Moreover, using again the fact that Q Λ t) Q Λ ) + A Λ t), we have that c can be re-defined so that regardless of the initial value Q Λ ) E[e QΛ t /Λ) η Λ k ) + ] e QΛ ) η Λ k ) + c. A22) So far we have established a Lyapunov type attraction of the queue length to the domain [,η Λ k ]. We now turn to the second step of the proof where we follow the arguments from the proof 9

10 of Theorem 5.1 in Gamarnik and Zeevi [14], to establish exponential bounds for Q Λ ) η Λ k ) +. Specifically, let ν Λ ) be the process stationary distribution function. Then, initializing the Λ th system with its stationary distribution, we have that E[e QΛ ) η Λ k ) + ] = E[e QΛ t /Λ) η Λ k ) + ]. A23) and in particular, letting ςξ) be the queue length coordinate of the state ξ, = ξ Ξ Λ ) e ςξ) ηλ k ) + E[e QΛ t /Λ) η Λ k ) + ] ν λ dξ). A24) Combining equations A21) and A22) we have that for any initial condition E[e QΛ t /Λ) η Λ k ) + ] max e η/4 e QΛ ) η Λ k ) +, ce M 1 Q Λ ) η Λ k +M, and in particular that e QΛ ) η Λ k ) + E[e QΛ t /Λ) η Λ k ) + ] 1 e η/4 )e QΛ ) η Λ k ) + ce M. lugging back into A24), we then have that ξ Ξ Λ 1 e η/4 )e ςξ) ηλ k ) + ce M) ν λ dξ). Overall, we have established that for all Λ large enough E[e QΛ ) η Λ k ) + ] cem, A25) 1 e η/4 Through Jensen s inequality, A25) implies that there exists a constant ĉ such that E[Q Λ ) η Λ k ) + E[Q ] ĉ, for all Λ large enough and in particular that lim Λ ) η Λ k ) + ] λ Λ =, so that the first part of A18) is established. Note that we have actually established that E[Q Λ ) η Λ k ) + ] = O1), which is stronger than the statement of the roposition which requires only that E[Q Λ ) η Λ k ) + ] = o Λ). The statement about the steady state waiting time follows by a straightforward application of Little s Law. Before proceeding note that, by Markov s inequality, A25) implies the existence of constants c 1, c 2 and M so that for all Λ large enough and for any x > Q Λ ) η Λ k + M + x c 1 e c 2x. 1

11 To analyze the behavior of the queue length process over the interval [,T], re-define the set ˆΩΛ) = ω Ω : Q Λ ) η Λ k + M + x 2 Ω ω Ω : Z Λ t) R ǫλ Λ,T,δΛ), τ Λ = t T : Q Λ t) η Λ k + M + x 2,for x >, and τ Λ = inf t τ Λ : Q Λ t) η Λ k + M + x. Now, by the same arguments as before, and choosing ǫ and δ appropriately, we have that on ˆΩΛ), Q Λ ) is strictly decreasing as long as it is above η Λ k + M. In particular, on ˆΩΛ) we must have that τ Λ > T and in particular that Q Λ t) η Λ k + M + x. Hence, and the result follows. Q Λ t) η Λ k + M + 2x ˆΩΛ)) c, The proof of the second part of A19), uses the following representation for the virtual waiting time: W Λ t) = infs : D Λ t + s) D Λ t) Q Λ t), where D Λ t) is the number of departures from the system up to time t. Let τ Λ = infs : Q Λ t + s) =. Then, on [t,τ Λ T] and on ˆΩΛ), we clearly have that D Λ t + s) D Λ t) µ cs Z2 Λ u)du δλ µ cs zr ǫλ)t δλ, s and in particular, W Λ t) Q Λ t) µ cs zr ǫλ) δλ, and the result follows. In order to analyze the number of cross-sold customers from each class a finer analysis is required. In particular, we need a handle of the waiting time of the customers that are present in the system at time which is assumed to be distributed according to the stationary distribution). This handle is provided in the following Lemma, whose proof is omitted due to its simplicity and great similarity to our previous arguments. For the following results, we let Z Λ t) be the number of customers in the first phase of service or in queue that found upon arrival a queue that is longer than 2η Λ k. We then have the following Lemma: 11

12 Lemma 6 There exists K 1 >, such that for all x >, ZΛ ) > K 1 + x) Λ c 13 e c14x. A26) and moreover, initializing the system with its stationary distribution, there exists K 2, so that Z Λ t) > K 1 + K 2 T + x) Λ c 13 e c 14x. A27) The proof of Lemma 6 follows the same essential steps used in the roof of roposition 7. While the dynamics of the queue length process are not trivial to analyze, the process Z Λ t) is much simpler. Specifically, when limiting the attention to the subset of the sample space where the queue is bounded on [,T] by 3 2 η k, the dynamics of Z Λ t) is very simple it is an exponentially decreasing process). The proof can now be completed along the same lines as the proof of roposition 7. The details are omitted. Note that the constants K 2, c 13 and c 14 in Lemma 6 are independent of T. roposition 8 There exists K > such that for all x > Z Λ ) Λ K + x) Λ c 11 e c12x, µ s A28) and for all i k 1, Zi,2 Λ ) λ iq i µ cs K + x) Λ c 11 e c12x. A29) Moreover, initializing all processes with their stationary distributions, there exist c 13 >, c 14 > and K > such that for all Λ large enough and for all x > : ZΛ t) Λ µ K + x) Λ c 13 e c14x, s A3) and for all i k 1, ZΛ i,2 t) λ iq i K + x) Λ c 13 e c14x. µ cs A31) 12

13 Note that roposition 8 implies that for i k 1 E[Zi,2 Λ )] = λ iq i µ cs + O Λ). In particular, since E[Z Λ )] = R + O Λ) and N = R + k λ i q i i=1 µ cs, it suffices to show that E[I λ )] = O Λ), where I λ t) is the number of idle agents at time t, to conclude that also for k: E[Z Λ k,2 )] = λ k q k µ cs + O Λ). The required result for I Λ ) is established in roposition 9, which together with roposition 7 establish 21) and complete the proof of Theorem 1. roposition 9 As Λ grows large E[I Λ )] = O Λ), and E[Z Λ i,2 )] = λ iq i µ cs + O Λ), i k. A32) roof of roposition 8: The arguments are very similar to the arguments in the proof of roposition 7. Rather that giving the full argument we specify only the steps that are essentially different. First note that, focusing on the customers in the first phase of service or waiting for it in queue, the following must hold: Q Λ t) + Z Λ t) = Q Λ ) + Z Λ ) + A Λ t) N µ s Using Strong Approximation we may write Q Λ t) + Z Λ t) = Q Λ ) + Z Λ ) + Λt µ s ) Z Λ u)du. ) Z Λ u)du + Λt B + µ s Z Λ u)du + OlogmΛt)), for some standard one dimensional) Brownian motion B ). Consider, on the other hand, the differential equation initialized at Z Λ )) Z Λ t) = Z Λ ) + Λt µ s Z Λ u)du. Then, using Gronwall s inequality, we have that e Z Λ t) Z Λ t) Λ e c QΛ )+ Q Λ t)+ BmΛt)) Λ, for some strictly positive constant c. Fixing T >, and using roposition 7 and basic calculations 13

14 for Brownian motion, it is easy to show that initializing the system with its stationary distribution, for Λ large enough and for all t T, [ E e Z Λ t) ZΛ t) ] Λ e M 1 + M 2 T, A33) for some constant M 1, M 2 > independent of T. We choose now η = 2 M 1 + M 2 T). Whenever Z Λ ) R 2η Λ, it is straightforward that there exists t, for which Z Λ t ) R = Z Λ ) R η Λ. In particular, we have by A33) that for all Λ large enough E [ e Z Λ t ) R ] Λ e ZΛ ) R Λ e η 2. A34) From here the argument towards establishing A28) is very similar to the argument at the end of the proof of roposition 7 and is omitted. Having the exponential bounds on the steady state distribution, the bounds on the transient behavior are obtained by modifying the argument in roposition 7 as well as our argument above to establish A3). In passing, note that using the independence of M1, M 2 from T one could strengthen the result of roposition 8 to the following: there exist K 1, K 2, c 11 and c 12, independent of T, so that Z Λ ) Λ K 1 + K 2 T + x) Λ c 11 e c12x. µ s We are now ready to prove the steady state and transient bound for Zi,2 Λ ). Here Lemma 6 will play a key role. To this end, re-define the set ˆΩΛ) = Ω Λ,T,x Λ) ω Ω : W Λ t) M 2 + x w Ω : Z Λ t) K + x) Λ Λ w Ω : Z Λ t) R K + x) Λ, A35) where, based on our previous results, we will just choose K and M 2 large enough so that for any Λ large enough we have two constants c 1 and c 2 with ˆΩΛ)) c c 1 e c 2x. Now, note that Zi,2 Λ t) = ZΛ i,2 ) DΛ i,2 t) + DΛ 1,i t), where DΛ i,2 t) is the process of cross-selling completions with class i customers, which is, in turn, a oisson process with instantaneous rate µ cs Zi,2 Λ t) at time t. D1,i Λ t) is the process that counts the number of class i customers that completed service and 14

15 began cross-selling by time t. It is clear that we can then write, Z Λ i,2 t) ZΛ i,2 ) µ cs and Z Λ i,2 u)du + µ sq W Λ t) Zi,2t) Λ Zi,2) Λ µ cs Zi,2u)du Λ + µ s q) ) Z Λ u) Z Λ u)du + inf B λt)), Z Λ u) Z Λ u)du + B λt)), where λt) is the corresponding time argument that we do not need to know explicitly but is guaranteed to be smaller than mλt for all t, Z Λ ) is defined as in Lemma 6. Consider now the differential equation initialized at Z Λ i,2 )): Z i,2t) Λ = Zi,2) Λ λ t i + µ s Λ q i)r µ cs Z Λ i,2u)du. A36) Then, for any given T, we can use our previous results [ to establish the existence of a constant M 3 >, so that for Λ large enough and for all t T E e ] Z i,2 Λ t) Z i,2 Λ t) Λ e M 3 T. Note that this last result does depend heavily on the delay sensitivity function. In particular, we used here the fact that under our delay sensitivity model: q i ) q i W Λ t) ) a i W Λ t). We can now complete the proof using an analogous argument to the one used for Z Λ ). roof of roposition 9: Fix x >, T > and re-define the set ˆΩΛ,x) = Ω Λ,T,x Λ) ω Ω : ZΛ t) Λ x Λ; µ s ZΛ i,2 t) λ iq i µ cs x Λ, i k 1; Z Λ t) x Λ; W Λ t) x/ Λ A37) By our previous results, we can choose x large enough so that for Λ large enough and for all y > x, ˆΩΛ,y)) c c 15 e c 16y, for some strictly positive constants c 15 and c 16. Assume that I Λ ) K 2 Λ for some K2 and set τ Λ = inft : I Λ t) I Λ ) η Λ, for 15

16 η K 2 /2. Then, on ˆΩΛ,x), k 1 τ Λ I Λ t τ Λ ) I Λ ) Λt τ Λ ) + µ cs Zi,2u)du Λ i=1 + µ s 1 q x/ )) τ Λ Λ Z Λ u) Z Λ u)du τ Λ k 1 + µ cs N I Λ u) Z Λ u) Zi,2u) Λ du + x Λ. i=1 A38) A simplification of the above equation using the definition of τ Λ and ˆΩΛ,x) shows that for any Λ large enough I Λ t τ Λ ) I Λ ) µ cs K 2 η) Λt τ Λ ) + K 6 x Λt τ Λ ) + x Λ, where K 6 is a real number that might be negative). In particular, we may choose K 2 large enough so that on ˆΩΛ,x), τ Λ t for some finite t and for all τ Λ t t, I Λ t) I λ ) η 2 Λ. To bound the behavior on the complement of ˆΩΛ,x), note that [ E e I Λ t ) Λ 1ˆΩΛ,x)) c ] c 15 x e I Λ ) +y+k 6 t Λ e c16y dy, for some positive constant c 17. So that given ǫ > we can increase x and K 6 if necessary, to [ ] guarantee that e IΛ t ) 1 ˆΩΛ,x)) c ǫ. Overall then E[e IΛ t ) Λ ] 1 c15 e c16x )e IΛ ) η/2 Λ + ǫ. The argument is now straightforward to complete along the lines of the roposition 7. References [1] I. Gurvich, M. Armony, and C. Maglaras, Cross-Selling in a Call Center with a Heterogeneous Customer opulation. reprint