Supplement to Call Centers with Delay Information: Models and Insights


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1 Supplement to Call Centers with Delay Information: Models and Insights Oualid Jouini 1 Zeynep Akşin 2 Yves Dallery 1 1 Laboratoire Genie Industriel, Ecole Centrale Paris, Grande Voie des Vignes, ChâtenayMalabry, France 2 College of Administrative Sciences and Economics, Koç University, Rumeli Feneri Yolu, SariyerIstanbul, Turkey Manufacturing & Service Operations Management, 13: , 2011 This is a supplement to the main paper Jouini et al. (2011), with the same title. 1. Performance Sensitivity in β This section supports the paragraph Effect of θ on the optimal announcement coverage β in Section 5 of the main paper Jouini et al. (2011). We would like to see how the call center performance measures under the optimal announcement coverage β are different from those under other β values. In other words, does it matter that a manager chooses a β that would be different from the optimal one, and to what extent? We consider Model 2 under the update case (θ = 0) with the same parameters as those in Table 1 of the main paper: µ = 1, γ = 0.5, α 0 = 5%, (s, λ) = (3,3), (5,5), (20,20), and (100,100). We compute the performance measures of Model 2 for different β, β =10%, 20%,..., 90%. Note from Table 1 of the main paper that for all call center sizes, β =90% is approximately optimal. The results are given in Table 1, and in Figures 1(a)1(d). From Table 1, we see that there are considerable changes in the performance measures when we announce delays other than those under the optimal announcement coverage. These changes are particularly apparent when the pooling effect is absent. The pooling effect improves the performance measures thereby reducing the sensitivity to the choice of β. For s = 3, P B is divided by around 10 when moving from β = 10% to β = 90%. However for s = 100, it is only divided by around 2. In summary, for all call center sizes, all performance measures except P S vary considerably in β. When the probability of reneging does matter for a call center manager, choosing the correct β is important, especially 1
2 for small call centers in which customers update their patience to the announcements, as considered here. Table 1 reveals that the probability of service P S does not vary that much in β. As we can see from Figures 1(a)1(d), the speed by which P B is decreasing is close to that with which P R is increasing. Then P S = 1 P B P R would not vary too much. This would make the choice of optimal β less critical when considering a throughput objective. Another observation is that the relative changes of E(X S) and σ(x S) are considerable. This may make a manager (considering a throughput objective) choose a small β that would, on the one hand, lead to a somewhat smaller value of P S, but on the other hand, would considerably improve E(X S) and σ(x S). 0.3 PB PR 0.3 PB PR β β % 10% 20% 30% 40% 50% 60% 70% 80% 90% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% (a) s = λ = 3 (b) s = λ = PB PR PB PR β β % 10% 20% 30% 40% 50% 60% 70% 80% 90% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% (c) s = λ = 20 (d) s = λ = 100 Figure 1: Impact of β on the probabilities of balking and reneging 2. Additional Experiments for a Strict Service Level In Table 2 we give the results for a stricter reneging constraint, δ = 1%. This supplements the paragraph Effect of system size and service level constraint on β in Section 5 of the main paper Jouini et al. (2011). 2
3 Table 1: Model 2 performance measures for different β s = λ = 3 s = λ = 5 β P B P R P S E(X S) σ(x S) P B P R P S E(X S) σ(x S) 10% 2.45% 30.96% 66.59% % 25.83% 72.34% % 3.33% 28.78% 67.88% % 24.41% 73.26% % 4.43% 26.28% 69.29% % 22.74% 74.30% % 5.99% 23.25% 70.76% % 20.66% 75.44% % 8.15% 19.66% 72.20% % 18.10% 76.67% % 10.97% 15.56% 73.47% % 14.96% 77.91% % 14.44% 11.16% 74.40% % 11.29% 79.02% % 18.38% 6.81% 74.81% % 7.28% 79.81% % 22.73% 2.89% 74.38% % 3.30% 79.97% s = λ = 20 s = λ = 100 β P B P R P S E(X S) σ(x S) P B P R P S E(X S) σ(x S) 10% 0.91% 14.72% 84.37% % 7.09% 92.49% % 1.04% 14.28% 84.68% % 6.97% 92.56% % 1.22% 13.73% 85.05% % 6.82% 92.65% % 1.47% 13.04% 85.49% % 6.62% 92.76% % 1.83% 12.13% 86.04% % 6.36% 92.91% % 2.39% 10.90% 86.71% % 5.99% 93.11% % 3.28% 9.19% 87.52% % 5.45% 93.39% % 4.73% 6.82% 88.45% % 4.56% 93.81% % 7.02% 3.65% 89.33% % 2.96% 94.45% Table 2: Results for Model 1, and for Model 2 with δ = 1% Model 2, θ = 0 Model 2, θ = 1/3 s = λ β 96% 97% 97% 98% 98% 98% 90% 89% 84% 75% 50% 20% γ P B 25.86% 20.08% 13.68% 9.74% 5.99% 4.20% 24.08% 18.27% 12.29% 8.32% 5.01% 3.46% P R 0.94% 0.82% 0.98% 0.70% 0.73% 0.72% 0.98% 0.91% 0.94% 0.93% 0.96% 0.98% P S 73.20% 79.09% 85.34% 89.56% 93.28% 95.08% 74.94% 80.82% 86.78% 90.75% 94.03% 95.56% E(X S) σ(x S) Model 2, θ = 2/3 Model 2, θ = 1 s = λ β 85% 81% 72% 55% 16% 3% 81% 75% 62% 41% 9% 1% γ P B 23.13% 17.27% 11.62% 7.81% 4.67% 3.28% 22.55% 16.75% 11.21% 7.55% 4.55% 3.27% P R 0.98% 0.99% 0.97% 0.99% 0.98% 0.97% 0.94% 0.97% 0.97% 0.97% 0.99% 0.78% P S 75.89% 81.73% 87.41% 91.19% 94.35% 95.75% 76.51% 82.29% 87.83% 91.48% 94.46% 95.95% E(X S) σ(x S) Model 2, θ = 5/3 Model 1 s = λ β 72% 63% 45% 24% 2% 1% γ P B 21.37% 15.84% 10.58% 7.22% 4.31% 3.50% 5.93% 5.68% 5.33% 4.94% 4.35% 3.82% P R 0.98% 0.99% 0.98% 0.93% 0.96% 0.48% 14.82% 10.97% 7.07% 4.38% 2.14% 1.15% P S 77.64% 83.17% 88.44% 91.86% 94.73% 96.01% 79.25% 83.36% 87.60% 90.68% 93.51% 95.02% E(X S) σ(x S)
4 3. Additional Experiments for the Announced Delays In this section we also give additional experiments to those of the paragraph Effect of system size and service level constraint on β in Section 5 of the main paper Jouini et al. (2011). In Figure 2, we give the announced delays as a function of the system size, for θ = 5/3. We omit the graphs for θ = 1/3, 2/3 and 1 because they are very similar. The main conclusion here is that the announced delays decrease in θ. The reason is simply that β decreases. 3.0 n=0 n = n=3 n = 3 n=5 n = 5 n=10 n = s = λ Figure 2: Announced delays in Model 2 with θ = 5/3 4. Details for the Analysis of the load Effect on β This section gives the details of Figure 7 of the paragraph Effect of increased system load on β in Section 5 of the main paper Jouini et al. (2011). In Figure 7 of the main paper, we draw the optimal announcement coverage β as a function of the arrival rate λ, while keeping the number of servers s unchanged. This allows us to study the behavior of the system under the efficiency driven regime. The details are given in Table 3. Recall that the parameters are s = 10, µ = 1, α 0 = 5%, γ = 0.5, and δ = 3%. 5. Analysis of Model 2AVG This section is related to the paragraph Comparison of Model 1, Model 2, and Model 2 with mean delay announcement in Section 5 of the main paper Jouini et al. (2011). In what follows, we give the analysis of Model 2AVG. Model 2AVG is a version of Model 2 4
5 Table 3: Details of Figure 7 of the main paper θ = 0 θ = 1/3 λ β 85% 92% 95% 95% 92% 87% 26% 57% 74% 66% 54% 47% γ P B 4.29% 11.60% 32.23% 47.60% 63.77% 77.01% 2.78% 9.69% 31.34% 47.13% 64.41% 77.28% P R 2.93% 2.84% 2.76% 2.54% 2.90% 2.99% 2.96% 2.92% 2.92% 2.90% 2.26% 2.71% P S 92.78% 85.56% 65.02% 49.86% 33.33% 20.00% 94.26% 87.39% 65.74% 49.97% 33.33% 20.00% E(X S) σ(x S) θ = 2/3 θ = 1 λ β 6% 34% 59% 53% 42% 30% 1% 20% 44% 35% 31% 15% γ P B 2.22% 8.86% 31.27% 47.55% 63.90% 77.08% 1.87% 8.29% 30.89% 47.31% 64.21% 77.06% P R 2.92% 2.96% 2.76% 2.47% 2.77% 2.86% 2.83% 2.99% 2.99% 2.70% 2.45% 2.94% P S 94.86% 88.18% 65.96% 49.98% 33.33% 20.00% 95.30% 88.72% 66.12% 49.99% 33.33% 20.00% E(X S) σ(x S) θ = 5/3 λ β 1% 8% 27% 19% 10% 2% γ P B 2.09% 7.64% 31.01% 47.03% 64.36% 25.47% P R 1.95% 2.92% 2.70% 2.98% 2.31% 2.93% P S 95.95% 89.43% 66.29% 50.00% 33.33% 20.00% E(X S) σ(x S) where instead of announcing the delay ensuring β coverage, the mean delay is announced. As in Model 2, we let customers in Model 2AVG renege even though they have elected to join the queue. This is argued by the fact that we announce the expected delay, so the actual delay may be larger than the announced one. In what follows, we particularly describe the difference in terms of the functioning and the performance evaluation. We use the same notations as in the main paper. Upon arrival, a customer is addressed by one of the available agents, if any. If not, i.e., all agents are busy and n customers are waiting in queue (n 0), then we announce to her the expected value of the random variable D n, denoted by E(D n ). This substitutes the announced delay d n = G 1 n (β) in Model 2. We have n E(D n ) = Also, it is easy to see that the balking probability p B (n) is given by i=0 1 sµ + iγ. (1) p B (n) = P (T < E(D n )) = 1 e γe(d n). (2) Using Equations (1) and (2), it suffices now to substitute d n by E(D n ) in the analysis of Model 2 in Sections 4.2 and 4.3 in the main paper Jouini et al. (2011) in order to get that of Model AVG. For the fixed point method we use to get the new reneging rate γ of Model AVG, we do as follows. We use the same method as the one of Model 2, except that here we don t have a single β, but β AV G,n which depends on the number of customers found in the queue. Note that this is not a parameter that we control. It is given by β AV G,n = P (D n < E(D n )) = G n (E(D n )). (3) 5
6 We know that the cdf G n (.) of D n is ( n n G n (t) = 1 i=0 j=0, j i ) sµ + jγ e (sµ+iγ )t, (4) (j i)γ for t 0, which easily gives β AV G,n. Following similar arguments as those in the main paper Jouini et al. (2011), we obtain r U n = P (E(D n ) < D n T E(D n )) = 1 β AV G,n, (5) for θ = 0 (the update case), and for the general modeling with θ > 0, we have ( n n ) sµ + jγ sµ + iγ r n (θ) = 1 β AV G,n (j i)γ sµ + γ )E(D n ) + e (sµ+iγ. (6) θ iγ Recall that r N n i=0 j=0, j i (the noupdate case) is given using Equation (6) and setting θ = 1. Finally, it remains to apply the fixed point algorithm as given in Section 4.2 in the main paper Jouini et al. (2011). In Table 4, which is related to the paragraph Comparison of Model 1, Model 2, and Model 2 with mean delay announcement in Section 5 of the main paper, Jouini et al. (2011). we give the announced delays in Model 2 and Model 2AVG. Table 4: Announced delays in Model 2 and Model 2AVG θ = 0 θ = 1/3 θ = 5/3 n s Model 2 Model 2AVG Model 2 Model 2AVG Model 2 Model 2AVG Validation of the Exponential Approximation In this section, we investigate the quality of the approximation we consider for new patience times in Model 2. To get a tractable analysis, we assumed in Section 4.2 in the main paper 6
7 Jouini et al. (2011). that new patience times are exponentially distributed with rate γ and we developed a numerical method to compute this parameter. To assess the exponential assumption, we compare in what follows the performance measures of Model 2 derived by the numerical method (exponential approximation) with those given by simulation. We conduct this study for the different cases of customer reactions, i.e., different values of θ. Consider the simulation of Model 2 under a given customer reaction (with parameter θ), which we will call the ideal simulation. A customer who joins the queue will renege as soon as her waiting time in queue reaches θt k + (1 θ)d n, where t k is her initial random threshold of patience and d n is the delay information. We underline a complexity associated with the computation of d n in simulation. It is very hard to derive in simulation the exact distribution of D n, based on which we determine the value d n with coverage probability β. This requires indeed the computation of the convolution of the distributions of the remaining patience times of the customers waiting ahead in queue. To simplify the analysis, we simulate an intermediate Model 2 in which we assume that D n is exponentially distributed with rate γ. In order to determine the value of γ, one would think to apply a fixed point algorithm using simulation at each step. This is again very heavy to do. Instead we use the value of γ derived from the numerical computation using the fixedpoint algorithm as shown in Section 4.2 in the main paper Jouini et al. (2011). We label the latter as the approximate simulation model. In summary, we will simulate Model 2 such that we announce the same delay d n as in the exponential approximation model. Customers who join the queue will then renege exactly as soon as their waiting time in queue reaches θt k + (1 θ)d n (if they do not start service before that epoch). The resulting Model 2 that we simulate is intermediate between the ideal simulation and the exponential approximate model. We believe that the comparison between the approximate simulation model and the approximate exponential one (developed in Section 4 of the main paper) will give good indications about the quality of the exponential approximation made in the latter. We run simulations for several sets of the system parameters, with different system sizes, different coverage probabilities, and different types of customer reaction to delay announcement. For all cases, we choose µ = 1, γ = 0.5, and α 0 = 5%. The results are given in Table 5. For each set of parameters, we compare between the performance measures derived by simulation and those derived using the numerical approximation method. In what follows, we give some information about the accuracy of the simulation. For a given set of parameters, we consider a single replication that we run for a sufficiently long time. The lengths of the confidence intervals for the different performance measures derived by simulation are in the order of
8 Distribution of times before reneging in the simulation models: For each one of the various sets of parameters considered here, we have recorded the actual times before reneging. We then investigated whether their observed distribution fits with an exponential distribution. This is an approximate analysis, as reneging times will not be i.i.d. If the system is busy with many waiting customers, the generated times before reneging would tend to be large. And if the system has a few waiting customers in queue, the generated times before reneging would tend to be small. In our models with delay announcement as well, we make such an approximation that times before reneging are i.i.d., even though in reality they are not. We do not report the details of the results here, however we summarize them below. The best fitting function was typically a Pearson6 distribution followed sometimes by Gamma, Lognormal, or Inverse Gaussian, suggesting that there is no clear alternative to the exponential distribution we have assumed. We see that an exponential distribution fits best when we have θ values 1, followed by 5/3, especially for the larger examples (50 servers). Even though the fit deteriorates slightly as we decrease θ, even a very small θ value (1/3) suggests a relatively good fit by an exponential. The QQ plots are mostly linear, which lends support to the idea that for most quantiles the exponential distribution provides a good approximation. As θ decreases, the QQ plots become slightly steeper than x = y, implying that the exponential is a bit more dispersed than the data. Indeed, as θ decreases, customers are updating more, implying more clustering around the announced delays. For the particular case with updatetype behavior (θ= 0), the reneging times will take the values of the announcements we are making as a function of the queue state. Thus we will observe clusters of reneging times around these delay announcements for n=1, 2, etc. As a result, the reneging time distribution we obtain will be discrete. Our model will approximate this discrete distribution by a continuous distribution. As such, the fit of the exponential distribution is clearly worse in this case compared to the above analyzed cases with θ > 0. Conclusions: In summary, the exponential approximation for the distribution fit of times before reneging seems quite good for some settings, and a rough approximation for others. When comparing the performance measures given by simulation and those given by the exponential approximation (see Table 5 below), it would seem that for the rougher cases, the distortion that is introduced is quite small. One can then conclude that the performance measures are not sensitive to the reneging time distribution. This is coherent with work by Avi Mandelbaum showing that the Erlang A is a robust model even though reneging 8
9 times are not necessarily exponential in practice. This can be explained using the results derived in Zeltyn and Mandelbaum (2005). Under the QualityEfficiencyDriven regime where delays are short, it is pointed out that the patience distribution near the origin determines the behavior of the system. Similar results are found for patience distributions with a positive density at the origin (Theorem 4.1 in Zeltyn and Mandelbaum (2005)), as well as for patience distributions with an atom at the origin (Theorems 4.2 and 4.3 in Zeltyn and Mandelbaum (2005)). In Theorem 4.3 of Zeltyn and Mandelbaum (2005), the authors consider a scaled balking increasing in the queue state, which is somewhat similar to what we consider in this paper, p B (n). They conclude that approximate performance measures depend only on the patience distribution near the origin. For small systems as those in our experiments, waiting times are also relatively short thus giving the main importance to the behavior of the patience distribution at the origin rather than in the tail. Since in addition to that the balking behavior is the same for the simulated and exponential approximated models, the exponential approximation has a good quality. Recall that for both the noupdate and the update cases the simulated model takes the random variable D n to be the same as that in the approximation (based on the same value of γ ). So, p B (n) is the same in the simulated and approximated models. Then, the patience distributions in both models have the same atom at the origin. 7. The case of Heterogeneous Customer reactions In the main paper Jouini et al. (2011), customers are assumed to react in the same way to delay information. We capture this by a single value for θ. In this section, we extend the analysis to the case of heterogeneous customer reactions. We capture this by substituting the real parameter θ by a random variable, denoted by Θ. A new arrival would react based on an outcome of Θ, denoted by θ. The objective of this section is to study the impact of having different customer reactions. More specifically, we want to answer the question: Is there any difference between having deterministic and random Θ? For simplicity and ease of interpretation, we consider examples of Θ with discrete distributions. We assume that Θ has a finite number k (k 0) of outcomes denoted by θ 1, k..., θ k (belonging to R + ) which occur with probabilities q 1,..., q k (with q i = 1), respectively. The analysis of the heterogenous case is similar to the homogeneous one developed in the main paper Jouini et al. (2011). The only difference is in the way of computing the i=1 9
10 Table 5: Simulation vs exponential approximation θ = 0, s = λ = 5 θ = 0, s = λ = 50 β 50% 80% 50% 80% γ App Sim App Sim App Sim App Sim P S P R P B E(X S) σ(x S) θ = 1/3, s = λ = 5 θ = 1/3, s = λ = 50 β 50% 80% 50% 80% γ App Sim App Sim App Sim App Sim P S P R P B E(X S) σ(x S) θ = 2/3, s = λ = 5 θ = 2/3, s = λ = 50 β 50% 80% 50% 80% γ App Sim App Sim App Sim App Sim P S P R P B E(X S) σ(x S) θ = 1, s = λ = 5 θ = 1, s = λ = 50 β 50% 80% 50% 80% γ App Sim App Sim App Sim App Sim P S P R P B E(X S) σ(x S) θ = 5/3, s = λ = 5 θ = 5/3, s = λ = 50 β 50% 80% 50% 80% γ App Sim App Sim App Sim App Sim P S P R P B E(X S) σ(x S)
11 conditional probability for a new arrival to renege, given the system state she finds upon k arrival. It is given by r n (Θ) = q i r n (θ i ), where r n (θ i ) is 1 β for θ i = 0, and is given i=1 by Equation (26) in the main paper Jouini et al. (2011) for θ i > 0. In what follows, we study the impact of having different customer reactions by considering various scenarios for the random variable Θ. All scenarios have the same expected value of Θ. We choose E[Θ]=2/3. The four scenarios are described in Table 6. Note that Scenario 1 is a particular case of a deterministic Θ. It corresponds to the case θ = 2/3, given in Table 1 in the main paper Jouini et al. (2011). Table 6: Description of the scenarios of Θ, E[Θ]=2/3 Scenarios Θ: (θ i, q i ) Coefficient of variation 1 (2/3, 1) 0 2 (1/2, 1/2); (5/6, 1/2) (1/3, 1/3); (2/3, 1/3); (1, 1/3) (0, 1/4); (1/3, 1/4); (1, 1/4); (4/3, 1/4) We consider now the same optimization problem as formulated in (29) of the main paper. We choose the same values of the other parameters as those in Table 1 in the main paper Jouini et al. (2011). The results are given in Table 7. Table 7: Results for Model 2 with heterogeneous customer reactions Scenario 1 Scenario 2 λ β 64% 54% 34% 13% 2% 1% 65% 56% 36% 14% 3% 1% γ P B 19.13% 13.89% 8.86% 5.62% 3.09% 2.86% 19.23% 14.04% 8.93% 5.63% 3.31% 2.25% P R 2.98% 2.97% 2.96% 2.93% 2.90% 1.40% 2.99% 2.93% 2.96% 2.96% 2.63% 2.57% P S 77.89% 83.14% 88.18% 91.45% 94.01% 95.74% 77.79% 83.03% 88.11% 91.41% 94.06% 95.18% E(X S) σ(x S) Scenario 3 Scenario 4 λ β 67% 59% 40% 18% 4% 1% 78% 77% 75% 73% 69% 63% γ P B 19.44% 14.24% 9.07% 5.77% 3.34% 3.03% 20.73% 15.53% 10.38% 6.91% 3.90% 2.44% P R 2.97% 2.91% 2.96% 2.93% 2.69% 1.13% 2.95% 2.95% 2.98% 2.95% 2.98% 2.96% P S 77.59% 82.85% 87.97% 91.30% 93.98% 95.84% 76.32% 81.52% 86.64% 90.14% 93.12% 94.60% E(X S) σ(x S) Consider the results in Table 7 here, and in Table 1 in the main paper Jouini et al. (2011). As one would expect, we observe that β for a random customer reaction (Θ) has a value in between the β s of two systems with deterministic customer reactions. The two systems correspond to the extreme cases with the lowest and highest outcomes of Θ. Also, the conclusions drawn in the main paper are still valid here. For example, it is important to increase the reliability of announcements for cases where some customers are likely to strongly update their patience. Also, pooling improves performance and as 11
12 a consequence makes the reliability of announcements less important. A new interesting observation from Table 7 is that although β varies as a function of random customer reaction, the performance of the optimized systems are quite insensitive to the variability in customer reactions. Thus, only E(Θ) seems to matter. 8. Numerical Computation In this section we describe the numerical methods used for the computation of the quantities involved in Section 5 of the main paper Jouini et al. (2011). State space truncation: For both Models 1 and 2, the expressions of the stationary probabilities, as well as those of the various performances measures, include infinite summations. These infinite summations are due to the infinite state spaces of our models. In order to compute an infinite summation, we truncate the state space, and transform the original model with infinite queue into one with a finite queue. We choose the truncation point by trial and error. This consists of increasing the truncation point up to the one for which the performance measures do no vary any more, with a sufficiently high precision (six digits beyond the decimal point). Computation of d n : We have d n = G 1 n (β). (7) Since G n (t) is strictly increasing in t, we use a bisection algorithm. We choose the following relative condition for the convergence of this algorithm: d n (i) d n (i 1) d n (i 1) < 10 6, where d n (i) is the value found at iteration i of the bisection algorithm. In each iteration, we need to compute G n (t) for a given t > 0. To do so we use the uniformization method. Let us denote by {Y (t)} the pure death process shown in Figure 4 of the main paper Jouini et al. (2011). Computing G n (t) is equivalent to computing the transient probability, denoted by π s (t), that {Y (t)} is in state s at time t given that it is in state s + n + 1 at time 0. In order to compute π s (t), we use the uniformization method. We define ν as the maximum of the rates of {Y (t)}, i.e., ν = sµ + nγ. We thereafter transform the process {Y (t)} into a probabilistically equivalent one, using the Poisson process with parameter ν. We refer the reader to Tijms (2003), for example, for the details of this standard method. Computation of r n (θ): For the computation of r n (θ) for θ > 0 (given in Equation (26) of the main paper Jouini et al. (2011)), we use some basic transformations in order to avoid 12
13 numerical instabilities. One major issue comes from the product in Equation (26) of the main paper. To fix this issue, we compute for an iteration i, for i 1, the logarithm of the absolute value of the product by relating it to that of iteration i 1. We have ( ) n sµ + jγ n sµ + jγ sµ + (i 1)γ = n i + 1. (8) (j i)γ (j i)γ sµ + iγ i Thus ln n j=0, j i j=0, j i sµ + jγ (j i)γ = ln n j=0, j i 1 j=0, j i 1 sµ + jγ ( ) sµ + (i 1)γ (j i)γ + ln + ln sµ + iγ ( n i + 1 i ), (9) where x denotes the absolute value of x, for x R. We then come back to the original product terms by applying the exponential. Another useful computational idea, suggested by an anonymous referee, is as follows. The numerator of r n (θ) as given in Equation (22) of the main paper Jouini et al. (2011) can be written as P (d n T < D n (1 θ)d n ) = θ = = d n d n d n P (t < D n (1 θ)d n )γe γt dt (10) θ P (D n > θt + (1 θ)d n )γe γt dt (1 π s (θt + (1 θ)d n ))γe γt dt, where π s (t) is the same transition probability as defined in the previous paragraph, and can again be computed using uniformization. To compute the integral, one can use numerical integration. We observed from the numerical experiments that we conducted, that our implementation is suitable for small and medium call centers. However, for large call centers with more than 200 agents for example, numerical problems of convergence of our algorithms may occur. In those cases, the truncation point needs to be very high which leads to death rates that are too large. References Jouini, O., Akşin, Z., and Dallery, Y Call Centers with Delay Information: Models and Insights. Manufacturing & Service Operations Management, 13: , Tijms, H.C A First Course in Stochastic Models. John Wiley and Sons. New Jersey, U.S.A. Zeltyn, S. and Mandelbaum, A Call Centers with Impatient Customers: Many Servers Asymptotics of the M/M/n+G Queue. Queueing Systems, 51:
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