# 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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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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