Approximating the Performance of Call Centers with Queues using Loss Models

1. Approximating the Performance of Call Centers with Queues using Loss Models Ph. Chevalier, J-Chr. Van den Schrieck Université catholique de Louvain

2. Observation • High correlation between performance of configurations in loss system and in systems with queues Ph. Chevalier, J-C Van den Schrieck, UCL

3. Loss models are easier than queueing models • Smaller state space. • Easier approximation methods for loss systems than for queueing systems. (e.g. Hayward, Equivalent Random Method) Ph. Chevalier, J-C Van den Schrieck, UCL

4. Main assumptions • Multi skill service centers (multiple independant demands) • Poisson arrivals • Exponential service times • One infinite queue / type of demand • Processing times identical for all type Ph. Chevalier, J-C Van den Schrieck, UCL

5. Building a loss approximation • Queue with infinite length • Incoming inputs with infinite patience • No queues • Rejected if nothing available Rejected inputs Ph. Chevalier, J-C Van den Schrieck, UCL

6. Building a loss approximation • Server configuration • Use identical configuration in loss system • Routing of arriving calls • Can be applied to loss systems • Scheduling of waiting calls • No equivalence in loss systems • Difficult to approximate systems with other rules than FCFS Ph. Chevalier, J-C Van den Schrieck, UCL

7. Type X-Calls Type Y-Calls X Y X-Y Building a loss approximation • multiple skill example Type Z-Calls Z X-Y-Z Lost calls Ph. Chevalier, J-C Van den Schrieck, UCL

8. Building a loss approximation • performance measures of Queueing Systems: • Probability of Waiting: Erlang C formula (M/M/s system): With • « a » = λ / μ, the incoming load (in Erlangs). • « s » the number of servers. Ph. Chevalier, J-C Van den Schrieck, UCL

9. Building a loss approximation • performance measures of Queueing Systems: • Average Waiting Time (Wq) : Finding C(s,a) is the key element Ph. Chevalier, J-C Van den Schrieck, UCL

10. Erlang formulas • Link between Erlang B and Erlang C: Where B(s,a) is the Erlang B formula with parameters « s » and « a » : Ph. Chevalier, J-C Van den Schrieck, UCL

11. Approximations • We try to extend the Erlang formulas to multi-skill settings • Incoming load « a »: easily determined • B(s,a) : Hayward approximation • Number of operators « s » : allocation based on loss system Ph. Chevalier, J-C Van den Schrieck, UCL

12. Approximations • Hayward Loss: Where: • ν is the overflow rate • z is the peakedness of the incoming flow, Ph. Chevalier, J-C Van den Schrieck, UCL

13. Approximations • Idea: virtually allocate operators to the different flows i.o. to make separated systems. Sx Sy Sx Sy + + Sxy’ Sxy Sxy’’ Sxy Operators: allocated according to their utilization by the different flows. Sx Sy Ph. Chevalier, J-C Van den Schrieck, UCL

14. Simulation experiments • Description • Comparison of systems with loss and of systems with queues. Both types receive identical incoming data. • Comparison with analytically obtained information. • analysis of results Ph. Chevalier, J-C Van den Schrieck, UCL

15. 5 Erlangs 5 Erlangs X = 3 Y = n X-Y = 7 Simulation experiments Experiments with 2 types of demands n from 1 to 10 Ph. Chevalier, J-C Van den Schrieck, UCL

16. Simulation experiments Ph. Chevalier, J-C Van den Schrieck, UCL

17. Simulation experiments Ph. Chevalier, J-C Van den Schrieck, UCL

18. Simulation experiments Ph. Chevalier, J-C Van den Schrieck, UCL

19. Simulation experiments Ph. Chevalier, J-C Van den Schrieck, UCL

20. Average Waiting Time • The interaction between the different types of demand is a little harder to analyze for the average waiting time. • Once in queue the FCFS rule will tend to equalize waiting times • Each type can have very different capacity dedicated => One virtual queue, identical waiting times for all types => Independent queues for each type, different waiting times Ph. Chevalier, J-C Van den Schrieck, UCL

21. Average Waiting Time • We derivate two bounds on the waiting time: • A lower bound: consider one queue ; all operators are available for all calls from queue. • An upper bound: consider two queues ; operators answer only one type of call from queue. Ph. Chevalier, J-C Van den Schrieck, UCL

22. Simulation experiments Ph. Chevalier, J-C Van den Schrieck, UCL

23. Simulation experiments Ph. Chevalier, J-C Van den Schrieck, UCL

24. Limits and further research • Service time distribution : extend simulations to systems with service time distributions different from exponential • Approximate other performance measures • Extention to systems with impatient customers / limited size queue Ph. Chevalier, J-C Van den Schrieck, UCL