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# B.V. Gnedenko anniversary

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### B.V. Gnedenko anniversary

1. 1. Markov modelsfor multi-skill call centers Manfred Schneps-Schneppe Ventspils University College, Latvia Manfreds.sneps@gmail.com Int’l Conf Gnedenko-100 Moscow, June 27, 2012 1
2. 2. Outlines1. Introduction: What is call center2. On Optimality of Gradings3. On Equivalent Random Traffic Method Extension4. On Russian System 112 2
3. 3. 1. Introduction The simplest one-skill call center• When waiting places N=0, then we talk about loss system and probability of blocking (call loss) is determined by Erlang B formula En (A) = (An/n!)/(1 + A + A/2! +…+An/n!)• When N is infinite, then we talk about queuing system and probability of waiting is determined by Erlang C formula An n Pw = n! n − A n −1 Ai A n n ∑ i =0 i! + n! n − A 3
4. 4. Full scale one-skill call centerTeletraffic phenomena:1) waiting calls are impatient and after abandonment could go away (lost calls)or make retrials,2) the same is true in case of “waiting places of ACD busy”,3) the served calls also make retrials (return for additional service), etc. 4
5. 5. Multi-skill call center S1=7 S1,2=6 S2=5 S2,3= S3=5 5 S3,4= 5 S1,…,5 S4=5 =6 S4,5=6 S5=7Therefore, speaking in telephony terms, this scheme with 5 inlets (5 call flows)has 29 individual outlets, 22 pairs and 6 common outlets. 5
6. 6. Numerical example: On optimal gradingTwo limited availability schemes with 4 inlets and 6 outlets. Each inlet can connect to 3 outlets (searching from left to right):b) the grading scheme contains 4 individual and 2 common outlets,c) the scheme with equally distributed outlets: each outlet is available to 2 inlets.According to Wikipedia, gradings are still popular now, and principles of optimal limited availability discussed below are not widely known. 6
7. 7. Traditional gradings are recommended for low load only b) c)Curves cross at the loss probability as low as 0.0025.That is reachable at total load value 0.73,or load per agent equal 0.728/6 = 0.121. 7
8. 8. 2. On Optimality of Gradings (by loss probability expansion in powers of λ at λ → 0 and λ → ∞) Khinchin A.Ya. Works in mathematical queueing theory (Ed.by B.V. Gnedenko), Moscow, 1963, pp 209-220 (in Russian). Beneš V. E. Markov Processes Representing Traffic in Connecting Networks. Bell System Techn. J., 1963, vol 42 Schneps-Schneppe M.A., New principles of limited availability scheme design, Elektrosviaz, Nr 7, 1963 Sedol J., Schneps-Schneppe M. Some qualitative study of limited availability schemes, Problemy peredachi informatsii, 1, Nr 2 (1965)M. Schneps-Schneppe, J. Sedols „Markov models for multi-skill call centers”// International Journal of Networks and Communications (Vol.2, No.4, July2012) . 8
9. 9. We consider rectangular switches:n – inlets (call flows, subscriber groups),d – availability (number of steps),v – outlets (total number of lines).Therefore, n · d contacts (points) divided into v groups (outlets).n Poisson call flows (each of intensity λ), the holding time is exponentially distributed (µ=1).If all d lines available to some call are busy, the call is lost. 9
10. 10. 10
11. 11. Asymptotic expansion of loss probability at λ → 0call loss probability is equal to 11
12. 12. Asymptotic expansion of loss probability at λ → ∞ 12
13. 13. On optimality of gradings at λ → 0At λ → 0 and given switch parameters (n, d, v) the optimal limited availability scheme should follow the principle:• The contact field (n, d, v) divides (as possible) in contact sets with 1 and n contact points, and individuals are available earlier than commons.In case of call center, it means that each agent has 1 or n skills. 13
14. 14. On optimality of equally distributed contact points at λ → ∞In case of call center, it means that each agent has r or r+1 skills. 14
15. 15. 3. On Equivalent Random Traffic Method Extension for multi-skill call centers “Call Admission Control in Cellular Networks” M. Schneps-Schneppe, V.B.Iversen //Chapter in "Mobile Networks", InTech, 2012, ISBN 979-953-307-568-5. 15
16. 16. Kosten’s model 16
17. 17. ERT-method 17
18. 18. On accuracy of the ERT-method 18
19. 19. A new interpretation of Kosten’s results 19
20. 20. Correlation of overflow streams 20
21. 21. Correlated streams: Neal’s formulae 21
22. 22. Extended ERT-method. Numerical example 22
23. 23. 4. On Russian System 112 Call Center ManagementHow to manage the relationship against three customer groups: RG1 : high-value customers, RG2 : marginally profitable customers (with potential), RG3 : unprofitable customerand eight different criteria? 23
24. 24. The paper is financed from ERDFs project SATTEH (No.2010/0189/2DP/2.1.1.2.0/10/APIA/VIAA/019) being implemented inEngineering Research Institute «Ventspils International Radio AstronomyCentre» of Ventspils University College (VIRAC). Thanks for your patience! Manfreds.sneps@gmail.com 24