In this section, we focus on the email routing problem for the multi-server case. The modeling is the same as described in Section 3, expect that instead of one server, there are s identical, parallel servers. As previously, we consider a call center manager that wants to optimize the email routing as a function of the system parameters (Problem (1)). In other words, we want to either optimize p and q for Model PM, or give the ordering of the extreme cases Models 1 to 4 (easier to use in practice and also efficient).
An exact analysis as that done for the single server case is too complex. We propose an approximation based on the single server results. It consists on replacing the s servers by a single super server. The service rates become sµ0 (for emails), sµ1 (first stage of call service), sµ2(second
Table 3: Validation of the light-traffic approximation λ µ=28%, λ = 0.2, λ µ=20.83%, λ = 0.1, µ1= 1, λ µ=2.00%, λ = 0.01 µ1= µ2 = 5, µ3 = µ0 = 2, µ2 = 2, µ3 = 3, µ0 = 4, µi = 2 for i = 0,· · · , 3 p = 10%, q = 80% p = q = 1 p = q = 50%
Exact Approximation Exact Approximation Exact Approximation P0 65.9406% 67.9873% 0,0000% 0,0000% 49,0025% 49.0093% a0 3.6157% 3.0217% 8,5896% 7,8955% 0,4962% 0.4901% b0 3.4766% 3.0217% 4,0903% 3,9478% 0,4937% 0.4901% b′0 6.3211% 6.0433% 1,9953% 1,9739% 0,2456% 0.2450% c0 7.3267% 7.5541% 2,5745% 2,6318% 0,4900% 0.4901% m0 7.3267% 7.5541% 77,2358% 78,9551% 49,0025% 49.0093% a1 0.3308% 0.1209% 1,2323% 0,7896% 0,0038% 0.0025% b1 0.4518% 0.2417% 0,7816% 0,5922% 0,0062% 0.0049% b′1 1.3961% 1.0878% 0,4299% 0,3454% 0,0043% 0.0037% c1 2.1406% 1.9641% 0,6378% 0,5483% 0.0098% 0.0086% m1 0.6661% 0.7554% 1,8838% 1,9739% 0.2438% 0.2450% a2 0.0451% 0.0048% 0,1559% 0,0790% 0.0000% 0.0000% b2 0.0607% 0.0145% 0,1114% 0,0691% 0,0001% 0.0000% b′2 0.2374% 0.1378% 0,0648% 0,0432% 0.0000% 0.0000% c2 0.4380% 0.3414% 0,1042% 0,0758% 0,0001% 0.0001% m2 0.0606% 0.0755% 0,0459% 0,0493% 0,0012% 0.0012% a3 0.0071% 0.0002% 0,0195% 0,0079% 0.0000% 0.0000% b3 0.0091% 0.0008% 0,0146% 0,0074% 0.0000% 0.0000% b′3 0.0382% 0.0153% 0,0087% 0,0048% 0.0000% 0.0000% c3 0.0787% 0.0499% 0,0146% 0,0089% 0.0000% 0.0000% m3 0.0055% 0.0076% 0,0011% 0,0012% 0.0000% 0.0000%
Table 4: Comparison between approximation and simulation Interval of the call arrival rate λ
s = 10, µ0 = 2, µ1 = µ3 = 1, µ2= 3 s = 10, µ0 = µ1 = µ2= µ3= 2
Constraint on calls Model Simulation Approximation Simulation Approximation
Model 4 0− 3.04 0− 2.96 0− 4.44 0− 4.42 E(W )≤ 1 Model 3 3.04− 3.74 2.96− 2.97 4.44− 4.48 4.42− 4.45 Model 2 3.04− 3.74 2.97− 3.66 4.48− 6.05 4.45− 6.04 Model 1 3.74− 3.85 3.66− 3.69 6.05− 6.07 6.04− 6.06 Model 4 0− 1.2 0− 1.1 0− 1.3 0− 1.2 P (W < 1)≥ 0.8 Model 2 1.2− 1.9 1.1− 1.7 1.3− 2.0 1.2− 1.8 Model 1 1.9− 2.3 1.7− 2.1 2.0− 2.5 1.8− 2.4 study in order to assess the quality of this approximation. Some of the comparison results between approximation and simulation are given in Table 4.
In Table 4 we give as a function of the interval of the call arrival rate value the ordering of Models 1 to 4 with respect to the optimization problem. The intervals are given using the single server approximation and also using a combined simulation and optimization of the multi-server system. The same intervals hold also when considering Model PM. We observe from Table 4 that the approximation provides an appropriate solution for the email routing optimization.
7
Conclusions
We considered a blended call center with calls and emails. The call service is characterized by successive stages where one of them is a break for the agent in the sense that inside the conversation there is no required interaction during a non-negligible time between the two parties. We addressed an important question in the call center practice: how should managers make use of this opportunity in order to better improve performance? We focused on the optimization of the email routing given that calls have a non-preemptive priority over emails. Our objective was to maximize the throughput of emails subject to a constraint on the call waiting time.
We developed a general framework (Model PM) with two probabilistic parameters for the email routing to agents. One parameter controls the routing between calls, and the other does the control inside a call conversation. We have also considered 4 particular cases corresponding to the extreme values of the probabilistic parameters. For these routing models, we have derived various structural results with regard to the optimization problem. We have also numerically illustrated and discussed the theoretical results in order to provide guidelines to call center managers. In
particular, we proved for the optimal routing that all the time at least one of the two email routing parameters has an extreme value.
There are several avenues for future research. It would be interesting to extend the structural results to the multi-server case. It would also be useful but challenging to extend the analysis to cases with an additional channel, in particular the chat which is increasingly used in call centers. Using the chat channel, an agent may handle many customers at the same time, which represent an additional opportunity to efficiently use the agent time.
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