1. Markov models
for multi-skill call centers
Manfred Schneps-Schneppe
Ventspils University College, Latvia
Manfreds.sneps@gmail.com
Int’l Conf Gnedenko-100
Moscow, June 27, 2012
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2. Outlines
1. Introduction: What is call center
2. On Optimality of Gradings
3. On Equivalent Random Traffic Method
Extension
4. On Russian System 112
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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
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4. Full scale one-skill call center
Teletraffic 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.
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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=7
Therefore, speaking in telephony terms, this scheme with 5 inlets (5 call flows)
has 29 individual outlets, 22 pairs and 6 common outlets.
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6. Numerical example: On optimal grading
Two 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.
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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. 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, July
2012) . 8

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.
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10. 10

11. Asymptotic expansion of loss probability at λ → 0
call loss probability is equal to
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12. Asymptotic expansion of loss probability at λ → ∞
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13. On optimality of gradings at λ → 0
At 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.
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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.
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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.
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16. Kosten’s model
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17. ERT-method
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18. On accuracy of the ERT-method
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19. A new interpretation of Kosten’s results
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20. Correlation of overflow streams
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21. Correlated streams: Neal’s formulae
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22. Extended ERT-method. Numerical example
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23. 4. On Russian System 112
Call Center Management
How to manage the relationship against three customer groups:
RG1 : high-value customers,
RG2 : marginally profitable customers (with potential),
RG3 : unprofitable customer
and eight different criteria?
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24. The paper is financed from ERDF's project SATTEH (No.
2010/0189/2DP/2.1.1.2.0/10/APIA/VIAA/019) being implemented in
Engineering Research Institute «Ventspils International Radio Astronomy
Centre» of Ventspils University College (VIRAC).
Thanks for your patience!
Manfreds.sneps@gmail.com
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