On firstcome, firstserved queues with two classes of impatient customers
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Abstract
We study systems with two classes of impatient customers who differ across the classes in their distribution of service times and patience times. The customers are served on a firstcome, firstserved basis (FCFS) regardless of their class. Such systems are common in customer call centers, which often segment their arrivals into classes of callers whose requests differ in their complexity and criticality. We first consider an M/G/1 + M queue and then analyze the M/M/k + M case. Analyzing these systems using a queue length process proves intractable as it would require us to keep track of the class of each customer at each position in the queue. Consequently, we introduce a virtual waiting time process where the service times of customers who will eventually abandon the system are not considered. We analyze this process to obtain performance measures such as the percentage of customers receiving service in each class, the expected waiting times of customers in each class, and the average number of customers waiting in queue. We use our characterization to perform a numerical analysis of the M/M/k + M system and find several managerial implications of administering a FCFS system with multiple classes of impatient customers. Finally, we compare the performance a system based on data from a call center with the steadystate performance measures of a comparable M/M/k + M system. We find that the performance measures of the M/M/k + M system serve as good approximations of the system based on real data.
Keywords
Call centers Impatient customers Virtual queueing time process M/M/k + M queue M/G/1 + M queueMathematics Subject Classification
60K25 68M20 90B221 Introduction
In this paper, we analyze queueing systems with different classes of customers who may abandon (renege from) the system if their waiting time exceeds their patience time, i.e., the maximum amount of time they are willing to wait before abandoning the system. This work is motivated primarily by customer call centers, which often segment their callers into different classes. Since call centers are the prevalent customerfacing service channel of many organizations, they often receive a variety of caller requests which may differ significantly in their service requirements and criticality. For example, banking call centers receive requests as simple as balance inquiries and as complex and critical as dealing with fraudulent activity on a caller’s account. While a service representative can obtain an account balance relatively quickly, handling fraudulent activity takes longer as it involves a higher level of legal expertise and paperwork. Furthermore, because fraudulent activity is usually more critical than obtaining a balance, callers who are calling about fraud may be more patient than callers who are calling to obtain a balance. Because callers’ service requests vary so greatly, call centers often train subsets of their representatives to handle only certain types of service requests. Based on the service the callers request from the phone menu, the automatic call distributor (ACD) segments callers into classes and routes the callers within each class to the appropriately trained subset of representatives. Depending on which types of requests are included in each class, these classes may differ from each other with respect to their distribution of service times and patience times. Consequently, one subset of representatives may serve a queue that receives arrivals from multiple classes that differ from each other in their typical service requirements and their callers’ patience levels. Call centers sometimes give priority to certain classes based on criteria such as the callers’ value to the organization or the criticality of the callers’ service request. However, in an effort to be fair, call centers often serve their callers on a firstcome, firstserved basis (FCFS) regardless of class. Because the FCFS policy is such a common practice, it is important to describe the performance of such systems. In this study, we do this by characterizing the performance of FCFS systems with two customer classes that may differ from each other in both their distribution of service times and their distribution of patience times.
As queue abandonment is a common customer behavior in many service systems, it is not surprising that a large number of studies have been devoted to characterizing queueing systems with impatient customers. Approaches for describing the performance of systems with a single class of impatient customers have included analytical characterizations (Daley [11], Baccelli and Hebuterne [3], Baccelli et al. [2], Stanford [21]) and performance approximations (Garnett et al. [12], Zeltyn and Mandelbaum [25], Iravani and Balcıoğlu [16]). The literature also contains several studies of twoclass systems. In most of these studies, one customer class is prioritized over the other. Choi et al. [10] analyze the underlying Markov process of an M/M/1 queue where one class of customers with constant patience times receives preemptive priority over a second class of customers who have no impatience. The authors obtain the joint distribution of the system size, and the Laplace transform (LT) of the response time of the second class of customers. Brandt and Brandt [6] extend the approach in [10] to include generally distributed patience times for the first class of customers. Iravani and Balcıoğlu [17] use the levelcrossing technique proposed by Brill and Posner [8, 9] to study two M/GI/1 settings. They first consider a preemptiveresume discipline where customers in both classes have exponentially distributed patience times, and then consider a nonpreemptive discipline where customers in the first class have exponentially distributed patience times, but customers in the second class have no impatience. Iravani and Balcıoğlu obtain the waiting time distributions for each class and the probability that customers in each class will abandon.
A handful of papers have dealt with priority queues with two classes of impatient customers in a multiserver setting. Motivated by call centers where callers may leave a voicemail, Brandt and Brandt [5] consider a multiserver system where callers from the first class are impatient and receive priority over callers from the second class, who have no impatience. Callers from the first class who renege may join the second class by leaving a voicemail and are only contacted when the number of idle servers in the system exceeds some threshold. The authors obtain the exact distribution of the number of callers in service from the first class, and approximations of the moments of the number of callers in service from the second class. Jouini and Roubos [18] consider an M/M/s + M queueing system where all customers have the same mean service times and mean patience times, but callers in one of the classes receive nonpreemptive priority. Within each class, customers may be served according to a FCFS or lastcome, firstserved (LCFS) discipline. Jouini and Roubos obtain the mean unconditional waiting times, the mean waiting times conditional on receiving service, and the mean waiting times conditional on abandoning the system for both classes under several policy permutations.
The studies that are most pertinent to our work are those in which classes of impatient customers are served on a FCFS basis, regardless of their class. Gurvich and Whitt [13] approximate the performance of a multiclass call center with impatient callers under the qualityandefficiencydriven (QED) regime introduced by Halfin and Whitt [14]. They analyze how the call center performs under a class of asymptotically optimal routing policies, of which the FCFS policy is a special case. Talreja and Whitt [23] rely on a deterministic fluid model to approximate the performance of a multiserver, multiclass FCFS system with impatient customers in an overloaded, efficiencydriven (ED) regime. Adan et al. [1] design heuristics to determine the staffing levels required to meet target service levels in an overloaded FCFS multiclass system with impatient customers. Van Houdt [24] considers an MAP/PH/1 multiclass queue where customers in each of the classes have a general distribution of patience times. Van Houdt develops a numerical procedure for analyzing the performance characteristics of the system by reducing the joint workload and arrival processes to a fluid queue, and expresses the steadystate measures using matrix analytical methods. His method produces an exact characterization of the waiting time distribution and abandonment probability under a discrete distribution of patience times, and approximations of the same performance measures under a continuous distribution of patience times. Sakuma and Takine [19] study the M/PH/1 system and assume that customers within each class have the same deterministic patience time. In addition to obtaining the waiting time distribution and abandonment probabilities, they obtain the joint queue length distribution. Finally, Sarhangian and Balcıoğlu [20] study two multiclass FCFS systems. The first system is an M/G/1+M queue where customers across classes may differ in their service time distribution, but all customers share common exponentially distributed patience times. The second system is an M/M/c+M queue where customers across classes may differ in their exponentially distributed patience times, but all customers share the same exponentially distributed service times. For both systems, the authors obtain the LT of the virtual waiting time for each of the k classes by exploiting the levelcrossing technique in [8] and [9]. They then relate the virtual waiting time to the actual waiting time to compute steadystate performance measures such as the mean waiting times and the percentage of customers who renege from each class.
In this study, we analyze two systems in which two classes of impatient customers are served according to an FCFS policy. The distinction between our setting and the settings in previous studies is that in our setting customers across classes may differ in their distributions of their service times and patience times, while in previous settings customers across classes may only differ in one of the two distributions. This distinction is crucial in characterizing the performance of multiclass systems with customers whose service times and patience may vary based on the complexity and criticality of their requests. We first consider an M/G/1+M queue and then analyze an M/M/k+M queue. To characterize the performance of these systems, we introduce a virtual waiting time process as described in Benes [4] and Takács et al. [22] (see Heyman and Sobel [15], pp. 383–390 for details). In a virtual waiting time process, the service times of customers who will eventually abandon the system are not considered. By analyzing this process, we obtain performance characteristics such as the percentage of customers who receive service in each class, the expected waiting times of customers in each class, and the average number of customers waiting in queue from each class. Note that although a related formula for the virtual waiting time in a single class M/G/1+PH queue is reported in [7], it is not suitable for direct computation as it consists of an exponentially growing number of terms. We next perform a numerical analysis of the M/M/k+M system under various arrival rates, mean service times, and mean patience times. Our analysis demonstrates that accounting for differences across classes in the distribution of customers’ service times and patience times is critical, as the performance of our system differs considerably from a system where only the service time distribution varies across classes. The results of our numerical analysis have broad managerial implications including service level forecasting, revenue management, and the evaluation of server productivity. As a final exercise, we compare the simulated performance of a system based on data from a multiclass call center with the performance measures of a comparable M/M/k+M system. To construct our simulated system, we select two classes from the data that differ in their distribution of service times and caller patience times. We find that the performance measures from the M/M/k+M system serve as good approximations of the performance of the simulated system based on the call center data.
The remainder of the paper is organized as follows: In Sect. 2, we analyze the M/G/1+M queue. In Sect. 3, we analyze the M/M/k+M queue, including a special case where the two classes share a common mean service time. In Sect. 4, we derive steadystate performance measures. In Sect. 5, we present our numerical analysis, and in Sect. 6 we compare the performance of the simulated system based on real data and our analytical characterization of the M/M/k+M system.
2 M/G/1+M system
It seems natural to study this model through its queue length process. However, keeping track of the number of customers from each class who are in the queue is not sufficient. We would also need to keep track of the class of each customer at each position in queue, since patience times depend on customer class. This renders the Markov process intractable. Thus, we introduce the virtual queueing time process below. This process appears to be tractable.
Sarhangian and Balcıoğlu [20] have derived this equation by a different method; however, they could solve it only in the special case \(\theta _1 = \theta _2\). Here, we develop an efficient method to solve Eq. (2) even when \(\theta _1 \ne \theta _2\). We have included the detailed derivation of this equation, since the same steps can be used to derive the equations in the multiserver setup studied in the next section.
Remark 1
One can also show that our results reduce to known results (see Daley [11] for example) when specialized to a system with a single class of customers.
3 M/M/k+M system
3.1 Model
3.2 Virtual queueing time process
As in the singleserver case, we study the virtual queueing time process, augmented by a supplementary variable to construct a Markov process. Let W(t) be the virtual queueing time at time t in this system. This is the queueing time that would be experienced by a virtual customer arriving at time t. Let \(N_i(t)\) be the number of servers serving a class i customer just after time \(t+W(t)\) but before the next customer (if there is one) entering service at time \(t+W(t)\). This means that \(N_i(t)\) is the number of servers busy with a class i customer just before a customer arriving at time t enters service. So \(N_1 (t) + N_2 (t)\) is always at most \(k1\).
This unusual definition enables us to determine the size of the upward jump of the virtual queueing time if an arriving customer at time t decides to join the queue (since his patience exceeds W(t)). The jump is the minimum of the service time of the arriving customer and the residual service times of the customers in service at the moment he enters service at time \(t+W(t)\). As we will explain below, \(\{(W(t),N_1(t),N_2(t)), t \ge 0\}\) is a Markov process with upward jumps, the size of which depend on W(t), and a continuous downward deterministic drift of rate 1 between jumps.
A sample path of the Markov process \(\{(W(t),N_1(t),N_2(t)), t \ge 0\}\) is shown in Fig. 3. It describes the following events: At time \(t_0 = 0\), the system is in state (0, 1, 1), which means that the virtual queueing time is 0, two servers are busy, one with a class 1 customer and the other with a class 2 customer. Then, a class 2 customer arrives at time \(t_1\), so the system states change to state (0, 1, 2). At time \(t_2\), a class 1 customer arrives and W(t) jumps an Exp(\(2\mu _1+2\mu _2\)) amount. At time \(t_2+W(t_2)\), a class 2 customer will complete service (from the two class 1 and two class 2 customers in service), so \((N_1(t),N_2(t))\) jumps to (2, 1) at time \(t_2\). The next customer arrives at time \(t_3\). He is from class 1, and his patience exceeds \(W(t_3)\). So this customer joins the queue and W(t) jumps an Exp(\(3\mu _1+\mu _2\)) amount. At time \(t_3+W(t_3)\), the class 2 customer will complete service before one of the three class 1 customers in service, so \((N_1(t),N_2(t))\) jumps to (3, 0) at time \(t_3\). At time \(t_4\), a class 2 customer arrives, but his patience is less than \(W(t_4)\), and thus he leaves the system without receiving service. At time \(t_5 = t_3+W(t_3)\), the class 2 customer completes service and the virtual queueing time reaches 0. The last event in Fig. 3 occurs at time \(t_6\). A class 1 customer completes service, and the system state jumps to (0, 2, 0).
We next determine the Laplace–Stieltjes transform (LST) of the virtual queueing time.
3.3 Steadystate analysis
Lemma 1
For each \(\delta >0\), the series \(\sum _{i=0}^\infty \sum _{j=0}^\infty C_{i,j}(s)\) is absolutely and uniformly convergent for all \(s>\delta \).
Proof
Since \(D_{ij} (s)\) are uniformly bounded for all \(s>\delta >0\) and \(i+j \ge 0\), we immediately get the following.
Corollary 1
The series \(\sum _{i=0}^\infty \sum _{j=0}^\infty D_{i,j} (s) C_{i,j}(s)\) is absolutely and uniformly convergent for all \(s>\delta >0\).
Lemma 2
For each \(\delta >0\), the series \(\sum _{i=0}^\infty \sum _{j=0}^\infty C'_{i,j}(s)\) is absolutely and uniformly convergent for all \(s>\delta \).
Proof
Lemma 3
Proof
Theorem 1
3.4 Special case \(\mu _1 = \mu _2 = \mu \)
4 Performance measures
5 Numerical analysis
Recall that the previous analytical models of multiclass FCFS systems have allowed for customers across classes to differ in either their service time distributions or their patience time distributions, but not in both distributions [19, 20, 24]. A contribution of our work is that we allow customers across classes to differ in both distributions. This enables us to model service systems such as call centers that segment their arrivals into classes of callers whose requests may differ greatly in their complexity and criticality. We are therefore interested in using our characterization to compare the performance of a system where customers across classes differ in only one of the distributions with the performance of a system where the customers across classes differ in both distributions. To do this, we conduct a numerical analysis by using the performance measures that we derived for the M/M/k+M system in Sect. 4.

% All customers receiving service (Fig. 5a) The percentage of all customers who receive service is lowest in the positive system and highest in the negative system over all arrival rates. Recall that class 1 customers are more patient than class 2 customers in the positive system and vice versa in the negative system. Consequently, a greater (smaller) proportion of the customers who receive service are from class 1 in the positive (negative) system. Also recall that class 1 customers have longer average service times than class 2 customers. Since servers in the positive (negative) system spend a greater (smaller) percentage of their time serving the customers with longer service times, their aggregate service rate decreases (increases), which decreases (increases) the percentage of all customers who receive service. This result has ramifications for service level forecasting as one of the common measures of service level is the percentage of customers who receive service. This demonstrates that managers who do not account for differences in the distribution of customers’ service times and patience times across classes may produce inaccurate service level forecasts.

Mean waiting time of all customers (Fig. 5b) Irrespective of the arrival rate, customers have the highest mean waiting times in the positive system. However, the mean waiting times of customers in the base system and the negative system are nearly the same. This result was surprising as we expected the average waiting times of the customers in the negative system to be lowest since the average waiting times of customers are highest in the positive system. Because this trend was puzzling to us, we calculate the mean waiting times conditional on receiving service and on reneging for both systems. We find two trends. First, in both of the systems the average waiting times of customers who receive service are higher than the average waiting times of customers who renege. Second, both of these waiting time measures are lower in the negative system than in the base system. So, if both of these measures are lower in the negative system, why are the average waiting times of all customers nearly equal between the two systems? The answer lies in the fact that a higher percentage of customers receive service in the negative system. One can think of the average waiting time of all customers as a weighted average of the waiting times of the customers who receive service and of the customers who renege, where the respective weights are the percentage of customers who receive service and the percentage of customers who renege. Recall from Fig. 5a that the percentage of customers who receive service is highest in the negative system. This means that in the negative system there is a greater weight from a subset of customers who tend to wait longer, since the average waiting time for customers who receive service is greater than the average waiting time of customers who renege. The result is that the average waiting times of all customers in the base system and the negative system are nearly equal even though the average waiting times of customers who receive service and the average waiting times of customers who renege are both lower in the negative system.^{1} This result again has ramifications for service level forecasting as the average waiting time of all customers is another common measure of service level.

System throughput (Fig. 5c) We observed in Fig. 5a that the percentage of customers receiving service is lowest (highest) under the positive (negative) system. It is therefore not surprising that system throughput is lowest (highest) under the positive (negative) system. However, what is surprising is that in the positive system, throughput is first increasing in the arrival rate, but is then decreasing after some threshold arrival rate. Initially, increasing the arrival rate increases server utilization and hence system throughput. However, the gains in throughput due to the increase in server utilization diminish and are eventually offset by a reduction in the effective service rate of the system. The reason that the service rate is decreasing in system load is that the percentage of time the servers spend with customers from class 1 (who take longer to serve) is increasing in load. This is because a higher proportion of class 2 customers renege as load increases since they are less patient than class 1 customers. The interesting takeaway is that in a system where customers’ service times and patience times are positively correlated, increasing traffic can actually decrease throughput. This result has potential implications for systems with limited service capacity that generate revenue based on system throughput, for example, restaurants. Mangers of such systems may attempt to increase traffic through marketing efforts in order to generate additional revenue but may instead reduce their revenue if the customers in their system who take longer to serve are also more patient.

Average service time Fig. 5d In the base system, the average service time of customers who receive service remains the same regardless of the arrival rate. This is because customers in each class are equally patient, which makes the proportion of customers receiving service who are from each class invariant to the system load. However, in the positive (negative) system, the average service time increases (decreases) as the arrival rate increases. In the positive system, the class 1 customers are more patient. Thus, as the load on the system increases, the proportion of customers receiving service who are from class 1 increases. Because class 1 customers take longer to serve, the average service time increases as the arrival rate increases. The reverse is true in the negative system, where class 2 customers are more patient but take less time to serve. This relationship between system load and average service time has managerial implications. Servers in customerfacing systems are often evaluated and incentivized based on their average service times. For example, a common practice in call centers is to provide financial rewards for agents to keep their average service time under some threshold. Our analysis shows that managers may reach false conclusions regarding their servers’ productivity if they evaluate their servers based solely on their average service times. In the negative system, managers may receive the impression that servers are speeding up when the arrival rate is higher. Consequently, managers may wrongly reward their servers for their supposed efforts to work faster under heavy loads. Conversely, in the positive system, managers may falsely reprimand their servers for allegedly slowing down as arrival rates increase. Our model demonstrates than an observed correlation between average service times and system load, such as we see in Fig. 5d, may have no correlation with the servers’ efforts. Rather, the observed correlation may be entirely due to a correlation across classes between customers’ service times and patience times.
Overall, our numerical analysis demonstrates the importance of accounting for differences across classes in the distribution of customers’ service times and patience times. We see that differences in these distributions may substantially affect key performance measures, which have a wide range of managerial implications, including service level forecasting, revenue management, and the evaluation of server performance. Consequently, a contribution of our work is that our analytical characterizations may be used to demonstrate to managers some of the implications of administering a multiclass FCFS queue.
6 Model as performance approximation
Performance comparison of simulated system with real data and M/M/k+M (analytical) system
System  \(\lambda \) (h)  AWT (s)  %RS  AQ  

1  2  1  2  1  2  Util %  AST (all)  
Simulation  36  26.98  31.43  93.23%  96.74%  0.13  0.16  64.44%  339.32 
\(\mu _1\ne \mu _2\)  36  27.92  32.56  92.92%  96.56%  0.14  0.16  64.15%  338.56 
Error  3.48%  3.59%  0.34%  0.19%  3.66%  3.76%  0.45%  0.23%  
\(\mu _1=\mu _2\)  36  26.24  30.26  93.34%  96.80%  0.13  0.15  63.96%  336.40 
Error  2.75%  3.73%  0.12%  0.06%  2.58%  3.57%  0.73%  0.86%  
Simulation  45  53.63  63.47  86.58%  93.39%  0.34  0.40  76.70%  341.16 
\(\mu _1\ne \mu _2\)  45  54.84  65.37  86.08%  93.09%  0.34  0.41  76.33%  340.79 
Error  2.26%  3.00%  0.57%  0.32%  2.25%  2.99%  0.49%  0.11%  
\(\mu _1=\mu _2\)  45  50.99  59.92  87.06%  93.67%  0.32  0.37  76.00%  336.40 
Error  4.93%  5.59%  0.56%  0.30%  4.94%  5.60%  0.92%  1.40%  
Simulation  60  112.64  138.75  71.55%  85.43%  0.94  1.15  90.58%  346.57 
\(\mu _1\ne \mu _2\)  60  114.06  141.66  71.06%  85.03%  0.95  1.18  90.13%  346.46 
Error  1.26%  2.10%  0.68%  0.47%  1.47%  2.31%  0.50%  0.03%  
\(\mu _1=\mu _2\)  60  104.76  127.56  73.42%  86.52%  0.87  1.06  89.67%  336.40 
Error  6.99%  8.06%  2.61%  1.28%  6.81%  7.88%  1.01%  2.94%  
Simulation  120  294.91  432.12  25.22%  54.23%  4.92  7.21  99.98%  377.06 
\(\mu _1\ne \mu _2\)  120  293.92  434.13  25.42%  54.13%  4.90  7.24  99.96%  376.98 
Error  0.33%  0.46%  0.77%  0.18%  0.44%  0.36%  0.02%  0.02%  
\(\mu _1=\mu _2\)  120  274.74  389.50  30.28%  58.85%  4.58  6.49  99.95%  336.40 
Error  6.84%  9.86%  20.06%  8.52%  6.94%  9.96%  0.03%  10.78% 
In all of the systems, we assume that calls from each class arrive according to independent Poisson processes with equal arrival rates. To compare the performance of the systems across different loads, we hold the number of servers in the system at 5 and vary the total arrival rate to be 36, 45, 60, and 120 calls per hour. In Table 1, we compare the performance of the three systems, where “Simulation” corresponds to the simulated system, “\(\mu _1\ne \mu _2\)” corresponds to the analytical characterization of the system where service rates are allowed to differ across classes and “\(\mu _1=\mu _2\)” corresponds to the analytical characterization where we restrict the service rates to be the same across classes. For each class, we present the average waiting time in seconds (AWT), the percentage of callers who receive service (%RS), and the average number of callers waiting in queue (AQ). We also present the server utilization (Util %), and the average service time of callers who receive service (AST(All)). We measure how close our analytical characterizations come to the simulated system using relative error, which is given by \(\text {simulated}  \text {analytical}/\text {simulated}.\) Overall, we find that the performance measures of the \(\mu _1\ne \mu _2\) system serve as good approximations of the simulated system, with relative errors of no greater than 3.76%. Of particular note is the high accuracy in predicting the percentage of callers who receive service (%RS) and the average service time of callers who receive service (AST(All)), with relative errors typically less than 1%. Under the lowest arrival rate (\(\lambda =36\)), the \(\mu _1=\mu _2\) characterization provides nearly the same approximation accuracy as the \(\mu _1\ne \mu _2\) characterization. However, as the arrival rate increases, the accuracy of the \(\mu _1 = \mu _2\) characterization decreases relative to the \(\mu _1\ne \mu _2\) characterization. In particular, the \(\mu _1 = \mu _2\) characterization underforecasts AWT, AQ, Util% and AST(All) while overforecasting %RS. This is due to the fact that, in this system, callers who tend to be more patient also tend to have longer service times. While the \(\mu _1\ne \mu _2\) characterization accounts for this correlation, the \(\mu _1 = \mu _2\) characterization does not. Overall, these results demonstrate that managers of twoclass FCFS service systems may use our analytical characterization of the \(\mu _1\ne \mu _2\) system to produce good approximations of the performance of their systems by collecting the mean service time and the mean patience time of customers in each class.
Footnotes
 1.
For example, in the base system with \(\lambda =20\), the average waiting time of customers who receive service is 0.654 and the average waiting time of customers who renege is 0.337. In the negative system, the respective averages are 0.641 and 0.324. However, in the base system the percentage of customers receiving service is 33.4%, while it is 37.2% in the negative system. Hence, the total average waiting time in the base system is \(0.334\times 0.654 + (10.334)\times 0.337 = 0.443\), and the total average waiting time in the negative system is \(0.372\times 0.641 + (10.372)\times 0.324 = 0.442\), which are nearly equal.
 2.
Because we do not know the patience times of the callers in our data who received service, the patience time data are rightcensored. Thus, we estimate the patience time distribution using the Kaplan–Meier estimator, which accounts for this form of censoring. However, since waiting times in this call center were not long enough to reveal the entire distribution of patience times, the Kaplan–Meier estimator only estimates a portion of the distribution. To fill in the remainder of the distribution, we assume that callers’ patience times are exponentially distributed with rate parameters equal to the callers’ reneging rate over the estimated portion of the distribution.
 3.
The mean service times of callers from class 1 and class 2 are 223.97 and 448.82 s, respectively. The mean patience times are 394.08 and 946.53 s, respectively.
Notes
Acknowledgements
The authors would like to thank Marko Boon for carefully checking all calculations with Mathematica.
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