TRAFFIC ENGINEERING OF DISTRIBUTED CALL CENTERS: NOT AS STRAIGHT FORWARD AS IT MAY SEEM. M. J. Fischer D. A. Garbin A. Gharakhanian D. M.

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1 TRAFFIC ENGINEERING OF DISTRIBUTED CALL CENTERS: NOT AS STRAIGHT FORWARD AS IT MAY SEEM M. J. Fischer D. A. Garbin A. Gharakhanian D. M. Masi January 1999 Mitretek Systems 7525 Colshire Drive McLean, VA reference to the Erlang-B and C models as well as other queueing models see Gross and Harris (1998). Abstract As of 1994, AT&T estimated that 350,000 businesses employed 6.5 million people in call centers (Brigandi et al., 1994). In 1997, call center revenue was estimated at $900M, with annual spending on call centers growing at 12 percent each year (Kay, 1998). Accurate performance analyses are essential in determining staffing levels and trunk requirements in call centers, because poor performance means lost business opportunities. The purpose of this paper is to show that as the complexity of these systems increases, traditional methods, like Erlang B and C table lookup, can result in poor evaluation of the call center performance. We start by examining the simplest of all call centers and show traditional methods can result in poor estimates of system performance and then present a more accurate model for this call center configuration. As the complexity of call centers increases more advance methods are required. This is demonstrated by considering two more complex systems: distributed systems of multiple interdependent call centers and a virtual call center configuration. We also discuss methods to analytically solve each of these systems. Introduction The increased focus on customer service has made call centers a critical component of today s business environment. The industry shows all of the signs of continued growth in the future. While call centers have proliferated in number, they have also evolved from a local to a distributed system. Call center managers must determine the optimal mix of system components in order to minimize cost while maintaining an acceptable level of customer service. This requires a mix of disciplines that are not typically found in organizations (voice, data communications, information systems and personnel). Appropriate sizing and traffic engineering of a call center is not trivial, and increases in complexity as the system transitions to a distributed environment. Sizing a call center requires knowledge on how various call center components interact with one another along with expertise in traffic engineering to address these interactions quantitatively. Businesses usually rely on their switch vendor, outside consultants or in-house expertise, including the call center manager, to perform this task. Traditionally, this has involved the use of simplified look-up tables based on the Erlang-B and Erlang-C traffic engineering models to calculate the required number of resources to meet a desired level of customer service, see Sharma (1986), Cravis (1990) and Hills (1994). For a This paper looks at three possible configurations of call centers and discusses methods to evaluate their performance. As the configurations become more complex so do the analysis methods; our discussion not only presents methods for the evaluation of the system performance but also discusses how traditional table lookup methods fall short. A Simple Call Center Configuration. Figure 1 shows a configuration of a simple call center. The center is composed of an Automatic Call Distributor () which takes calls from 50 incoming trunks and distributes them among 35 agents. If all agents are busy, calls are queued for the next available agent. Incoming calls that do not find a free trunk are lost. Otherwise, a call is accepted and occupies the line until it completes service by one of the agents and then is dropped. We compare the results of using a combination of M/M/c/c (Erlang-B model) and M/M/c (Erlang-C model), against the M/M/c/K model in characterizing the performance of the call center. This comparison presents the results of the above queueing models against that of a computer generated simulation. Table 1 presents the performance characteristics as represented by the blocking probability and customer waiting times derived from the M/M/c/K model, the simulation, and the traditional Erlang-B and Erlang-C methods. The Erlang-B blocking probabilities are calculated by assuming that the customer waiting time is known and equal to the simulation results. The Erlang-C waiting times were calculated by using the blocking probability derived through the Erlang-B model to determine the effective calling rate (calls per minute that are not blocked). The expected service time with an agent is 8 minutes. The M/M/c/K model is an effective predictor of the two key performance measures: the probability of blocking and expected customer waiting time. The Erlang- B calculations are close to the simulation s results, but yield lower blocking probabilities. The underestimated Erlang-B blocking probability is due to the fact that Erlang-B assumes that the call is being served for the entire duration that it is in the system; it does not distinguish between the two states of a call waiting for service and being served by an agent. At any given instant there is a probability that a call will complete its service and depart. In the simulation, calls that are waiting for an agent and are not yet receiving service cannot depart the call center (the departure probability is zero).

2 11/1/02 2 Unlike M/M/c/K, Erlang-C radically overestimates the delay experienced by the simulation. In Table 1, for the case of 4.7 calls per minute, the number of agents required by the Erlang-C model to provide a minutes of customer waiting time (calculated by the simulation) is 37; for the case of 6 calls per minute the required number of agents increases to 50. This characteristic is mainly driven by the fact that Erlang-C assumes an infinite customer waiting room, where in reality the maximum number of customers that wait for service at any point in time is 15 (50 incoming lines minus 35 agents). This comparison indicates that the M/M/c/K model is a good subsystem building block for a distributed call center model. The Erlang B model for determining the blocking on the incoming is not too bad but one has to know the overall system call waiting time before it can be used. Obviously, if the number of incoming trunks is very large then the Erlang C model can be used because the blocking probably would be zero. A Distributed Call Center Configuration. Call centers are configured depending on the business application. Traditionally, call centers were physically centralized and the would route and distribute incoming calls among a pool of customer service agents. Today, call centers are becoming more distributed. They are networked to one another to transfer customer calls from local to remote centers. At each call center, the may be servicing calls to local groups of agents, an Interactive Voice Response (IVR) unit, or another system at a remote call center. Optimizing a call center involves balancing cost with customer service level. This requires determining the appropriate number of telecommunications components and personnel while insuring that the overall system provides the desired levels of customer rejection rate (blocking probability), customer service time (time in the system) and customer waiting times (time spent in the queue). Unacceptable customer service levels lead to loss of business. Figure 2 shows a generic configuration for a distributed call center network. In this configuration incoming calls are answered by the, which first distributes them to a pool of local customer service agents, and then to a remote call center for service by a set of remote agents. In this example, once a call is accepted by the, agents at the local call center collect basic data from the caller and then may transfer the call to a remote call center for service by agents with specific expertise. A customer could wait for service at the local for the next available local agent and at the remote s for the next available remote agent. In traffic engineering terms, this network is a combination of dependent subsystems. Some subsystems behave as loss systems (if an arriving call does not find a free trunk it is dropped). Others behave as delay systems with a finite waiting room, which allows waiting for the next available agent. A distributed call center is composed of a combination of loss and delay subsystems that depend on each other for the duration of each call. As the call center manager expands the system to meet business needs its complexity grows. A distributed call center network has a significant amount of dependency, which complicates the task of determining the congestion within the network. Bottlenecks further down the system (e.g., remote sites) impact customer service levels back at the local site. The complicated structure and subsystem dependencies of such networks do not allow a particular subsystem to be traffic engineered in isolation. The entire network must be considered as a single integrated system, which makes the use of simple look-up tables inappropriate. Accurate modeling of a networked call center, like the one shown in Figure 2 must take into account the concepts of load reduction and balancing subsystem dependencies. Load reduction is the iterative process of reducing the traffic load that is expected to be present on a given trunk group to account for any additional blocking that may be experienced downstream. Some distributed call centers may have tandem loss subsystems. For these systems, the calls that may be lost due to downstream blocking (second loss system) must be eliminated from the initial traffic load (incoming to the first loss system). For an in-depth discussion of way to model the systems shown in Figure 2, see Fischer et al. (1998). In a distributed call center network, the subsystems are not independent but function in a highly dependent fashion. The total number of calls in each subsystem must equal the number of calls on the incoming lines. For example, in the system shown in Figure 2, the number of calls on the 800 and local lines must equal the number of calls at all the other centers in the system added together and so all the subsystems are statistically dependent. An integrated network model must address these concepts to accurately predict network performance. A comparison between the traditional traffic engineering methods that utilize Erlang-B and Erlang-C, and a modified method that incorporates both Erlang-B and M/M/c/K to address the performance of distributed call center is presented in Table 2. Figure 2 presents the specific topology of the distributed system; it consists of 50 incoming 800 lines, 45 incoming local lines, 15 agents at the main call center (CC1), 50 tie lines connecting the main call center to one remote call center (RC1) with 45 agents, and 30 tie lines connecting the main call center to another remote call center (RC2) with 27 agents. An incoming 800 or local call is first transferred to the local agents for service, and once the service with the local agent is completed, 65% of the calls are transferred to the first remote center (RC1) and the remaining 35% are transferred to the second remote center (RC2). Each caller spends 4 minutes with the agents at the local center and 22 minutes with the agents at either remote center. Customer waiting is performed at the local call center s for the local agents and at each remote call center s s for the remote call center agents. The incoming call arrival rate is 1.75 calls per minute on the 800 lines and 1.5 calls per minute on the local lines. From Table 2 the results of the modified method using Erlang-B and M/M/c/K are very close to that of the simulation. The traditional method radically overestimates customer-waiting times (three times as high) which then overestimates the number of required agents. The number of agents at Remote Center 2 (RC2) required to provide a customer waiting time equal to that of the simulation (0.28 min.) is 30. This is an 11% increase in the number of personnel. If the agent pool consists of personnel in a highly skilled area, such as nurses in a health care organization, this 11% can easily represent hundreds of thousands of dollars in annual recurring cost. Virtual Call Center Configuration Figure 3 shows the configuration of a virtual call center composed of two isolated centers. Incoming calls for

3 11/1/02 3 either center that find an unoccupied incoming trunk are first offered to their parent center. If an agent is free that agent services the call. If an agent is not free, the call is placed in a queue at the parent center. The s at each center monitor their agents: when an agent becomes free the checks to see if there is a call waiting in the parent queue. If there is, the one waiting the longest is serviced by the agent. If there are no calls waiting in the parent queue the checks to see if there is a free interconnecting line to the other system and a call waiting at the other system. If those conditions are met, the free agent services the call from the other system. This call occupies a trunk at its parent system and an interconnecting line between the systems. A direct application of Erlang B or C tables cannot be used to solve this configuration. More sophisticated methods are required. We used matrix-geometric techniques (Neuts, 1981) to solve for the desired system performance measures. In this approach, a continuous time Markov chain whose transition probability matrix has a specific structure is developed. The stationary probability distribution of the number of calls at each center is represented by the vector π = (π 0, π 1, π 2, ) having a solution π k = π 0 R k (k 1), where the matrix R is a rate matrix found as the solution to a nonlinear matrix equation. The steady-state distribution is then used to compute expected measures of system performance. For details on application of this technique to virtual call center configurations, see Masi, Numerical investigations and system comparisons were conducted in Masi (1998) for single-agent call centers with one interconnecting line, using the matrix-geometric technique. Another rule called the pre-delivery rule was also considered. In that rule, the decision as to the system to join is made immediately and no jockeying is allowed. Upon entrance into the parent system the call is first offered to its agents if they are all busy; then the agents at the other centers are checked to see if one is free. If one is free, then that agent serves the call. If one is not free, then the call waits in his parent queue until one of his agents becomes free. In Masi et al. (1998), the rule as shown by Figure 4 is called the post-delivery rule. We denote by Q i, i = 1,2 as the number of calls who entered the system for center i, and ρ i, i=1,2, as the associated load. The load is the call arrival rate times the expected length of time the call spends with an agent. In Figure 4, we plot the expected total number of calls in the network (E[Q 1 +Q 2 ]) versus ρ 1 for various ρ 2 for both the pre-delivery and post-delivery systems. The matrixgeometric method was used to generate these curves for the case of a single agent at each center and one interconnecting line. Also, we assumed that there was not any blocking on the incoming trunks. We note here that the post-delivery system yields the best performance as indicated by E[Q 1 +Q 2 ]. We also observe that the networked systems demonstrate even larger percent reductions in expected total number in the system over an isolated system when the offered loads to the two centers are unbalanced (e.g., for ρ 2 =.9, ρ 1 =.1, the pre- and post-delivery systems both give a 98% reduction in E[Q 1 +Q 2 ] over the isolated system). The point ρ 1 = 1, ρ 2 =.9 is not indicated on Figure 5 for the two isolated M/M/1 systems because it violates the existence condition for M/M/1 queues. One variation of the post-delivery system can be analyzed with Erlang-C tables. A post-delivery system with load ρ i = 0 to one of the centers (say, i = 2) becomes an M/M/2 queue. The agent at center 2 will always service a type 1 call if one is waiting because the interconnecting line is always available simultaneously with the center 2 agent, and switching to center 2 can occur at any time. The predelivery system with load ρ i = 0 to one of the centers (say, i = 2) does not become an M/M/2 queue, because calls can be routed to center 2 at arrival times only. The number of type 1 calls in the post-delivery system with ρ 2 = 0 is a lower bound for the number of type 1 calls when ρ 2 > 0. For systems with very small ρ 2, then, the Erlang-C tables with c = 2 can be used to give approximate results. Figure 5 shows the expected number of type 1 calls in the system versus ρ 1 and ρ 2. For ρ 1 =.95 and ρ 2 =.1, using the lower bound of ρ 2 = 0 to estimate the expected number of type 1 calls and type 1 waiting times gives only an 8% error; while for ρ 1 =.95 and ρ 2 =.5, using the lower bound of ρ 2 = 0 yields estimates that are 79% lower than the true values. Thus, Erlang-C tables can be used for this post-delivery system with ρ i = 0 (i = 1 or 2), and perhaps to approximate post-delivery systems with ρ i very small for one of the centers i. Summary Traffic engineering a call center is not a trivial task, and requires more than simple look-up tables to accurately predict performance. In this paper we have demonstrated this statement by looking at three potential call center configurations. We have presented methods that can be used to model these complexities, but each new configuration requires a new modeling exercise. Another very strong method that can be used is simulation methods (Law and Kelton, 1991). References Brigandi, A.J., D.R. Dargon, M.J. Sheehan, and T. Spencer (1994), AT&T s Call Processing Simulator (CAPS) Operational Design for Inbound Call Centers, Interfaces 24(1), Cravis, H. (1990), Traffic Engineering with an, Telephone Engineering and Management, pp 56-59, July. Fischer, M.J., D.A. Garbin, and A. Gharakhanian (1998), Performance Modeling of Distributed Automated Call Distribution Systems, Telecommunications Systems, Vol. 9, No. 2,, pp Gross, D. and C. M. Harris (1998), Fundamentals of Queueing Theory, 3 rd Ed., New York: Wiley. Hills, M.T. (1994), Management Reports: Why are they of no use with my traffic tables, Service Level Newsletter, July. Kay, A.S. (1998), Local and Long-Distance Telcos See Opportunity to Sell Their Expertise Carriers Take Aim at Call Centers, InternetWeek, 2 March, pp Law, A.M. and W.D. Kelton (1991), Simulation Modeling and Analysis, 2 nd Ed., New York: McGraw-Hill. Masi, D.M.B. (1998), Resource-Sharing Call-Center Queueing Systems, Ph.D. Dissertation, George Mason University School of Information Technology and Engineering.

4 11/1/02 4 Masi, D.M.B., M.J. Fischer, and C.M. Harris (1998), Numerical Analysis of Routing Rules for Call Centers. The Telecommunication Review. Mitretek Systems, Vol.9. Neuts, M.F. (1981), Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Baltimore: Johns Hopkins University Press. Sharma, R.L. (1986), A Good System Model or Problem Solving Method Takes the Confusion out of Traffic Engineering, Communication Week, pp 24-27, July.

5 11/1/02 5 Incoming Lines (50) 35 Figure 1: A Simple Call Center Configuration Blocking Probabilities Customer Waiting Times Calls per (minutes) Minute M/M/c/K Erlang-B M/M/c/K Erlang-C Simulation Simulation Table 1. Comparison of Queuing Models in a Simple Call Center 1.75 Calls/min 800 Lines Calls/min Local Lines 45 (CC1) Tie Lines 50 (RC1) Talk Time = 4 min 45 Tie Lines 30 (RC2) 27 Talk Time = 22 min Figure 2. Three-Center Distributed Call Center

6 11/1/02 6 Blocking Probabilities Average Customer Waiting Times (min) Calculation Method 800 Traditional : Erlang-B & Erlang-C lines local Lines Tie lines to RC1 Tie lines to RC2 At Call Cntr 1 At Remote Cntr 1 At Remote Cntr Simulation Modified: Erlang-B & M/M/c/K Table 2. Comparison of Queuing Models in a Distributed Call Center Incoming Calls to sys 1 Incoming Calls to sys Interconnecting Lines 2.. Figure 3. A Virtual Call Center Expected Total Number in System 30 Expected Total Number in System, E[Q1+Q2] post:rho2=.1 post:rho2=.5 post:rho2=.9 pre:rho2=.1 pre:rho2=.5 pre:rho2=.9 No Switching: rho2= rho1 Figure 4. Expected Total Number in System for Single-Agent Centers

7 11/1/ Expected Number of Type 1 Customers, E[Q1] Lower Bnd rho2=.1 rho2=.5 rho2=.9 Upper Bnd rho1 Figure 5. Expected Number of Type 1 Customers

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