Presentation held 3 August 2012 at QTNA 2012 in Kyoto

1. 1
How to distinguish between A and B?
call center · single server · examples of call sequences (A & B)
• two call types (call blending): incoming (↓) or outgoing (↑)
• sequences generated by different Markov chains (A vs. B)
• according to some blending balance: 2 time scales
1 long-term: overall frequency (↓ vs. ↑)
2 short-term: call type correlation (γ)

2. 2
γ enables to study short-term balance
• coefficient of correlation γ ∈ [−1, 1] captures correlation in
call sequence, from one call to the next (definition see further)
• γ in [0, 1]: call type likely repeated (↓↓ & ↑↑ prevail)
• γ in [−1, 0]: call type likely swapped (↓↑ & ↑↓ prevail)
• the larger |γ|, the stronger the correlation

3. 3
Quantifying the call blending balance
in two way communication retrial queues:
analysis of correlation
based on joint work while at Kyoto University
Wouter Rogiestb,∗ & Tuan Phung-Duca,c
a Graduate School of Informatics · Kyoto University · Japan
b Dept. of Telecomm. & Inf. Processing · Ghent University · Belgium
c Dept. of Math. & Comp. Sciences · Tokyo Institute of Technology · Japan
∗ presenting
QTNA 2012 · Kyoto · 1–3 August 2012

4. 4
Outline
1 Introduction
2 Model & Analysis
3 Numerical examples for constant retrial rate
4 Conclusion

5. 5
Outline
1 Introduction
2 Model & Analysis
3 Numerical examples for constant retrial rate
4 Conclusion

6. 6
Context: retrial queue with call blending
• retrial queue, well-known model
• customers not served upon arrival enter orbit and request for
retrial after some random time
• applied to call center with single server
• retrial queue for incoming calls (↓)
• typically assigned by the Automatic Call Distributor (ACD)
• no queue for outgoing calls (↑)
• initiated after some idle time by the ACD, or by operator

7. 7
Call blending: A vs. B
earlier/ongoing work
[A] for classical retrial rate
→ J. R. Artalejo & T. Phung-Duc, QTNA 2011.
[B] for constant retrial rate
→ T. Phung-Duc & W. Rogiest, ASMTA 2012.
findings on blending balance
• long-term: identical for A and B
• short-term: (to be studied!) (no answer from steady-state
expressions alone) (intuitive: should be quite different)

8. 8
Outline
1 Introduction
2 Model & Analysis
3 Numerical examples for constant retrial rate
4 Conclusion

9. 9
Assumptions: {α, λ, µ, ν1 , ν2 }
all: rates of exponential distributions
α outgoing call rate
• when server turns idle, outgoing call after exp. distr. time
λ primary incoming call rate (Poisson arrivals)
• finding idle server: receive service immediately
• finding busy server: enter orbit
µ retrial rate (within orbit)
A classical: nµ,
B constant: µ(1 − δ0,n ), with
n : number of customers in orbit
δ0,n : Kronecker delta
ν1 service rate incoming call
ν2 service rate outgoing call

10. 10
Markov chain
• S(t): server state at time t,

0 if the server is idle,

S(t) = 1 if the server is providing an incoming service,

2 if the server is providing an outgoing service,

• N(t): number of calls in orbit at time t
• {(S(t), N(t)); t ≥ 0} forms a Markov chain
• state space {0, 1, 2} × Z+
• steady-state distribution obtained ([A] & [B])
• input for calculation of γ

11. 11
Correlation coefficient γ
• numbering consecutive events Sk with k
• Sk : incoming (Sk = s1 ) (↓) or outgoing (Sk = s2 ) (↑)
• assuming steady-state
E[Sk Sk+m ] − (E[Sk ])2
γm = ; m ∈ Z+
Var[Sk ]
• −1 ≤ γm ≤ 1
• main interest γ1 , or γ
• main challenges
1 extracting distrib. (Sk , Nk ) from distrib. (S(t), N(t))
2 determining E[Sk Sk+1 ]

12. 12
From S(t) to Sk : 2 steps
original Markov chain
censor: remove idle periods
discretize: “compensate” for ν1 = ν2

13. 13
In general: from (S(t), N(t)) to (Sk , Nk )
• original Markov chain: under conditions, unique stochastic
equilibrium, with limt→∞ :
πi,j = Pr[S(t) = i , N(t) = j], (i, j) ∈ {0, 1, 2} × Z+
• censor, with limt→∞ :
πi,j = Pr[S(t) = i, N(t) = j|S(t) ∈ {1, 2}], (i, j) ∈ {1, 2}×Z+
˜
• discretize, with limk→∞ :
ηi = Pr[Sk = i] , ηi,j = Pr[Sk = i, Nk = j] ,
with
(i, j) ∈ {1, 2} × Z+

14. 14
In general: from (S(t), N(t)) to (Sk , Nk )
• censor and discretize: expressions
T1 = 1/ν1 , T2 = 1/ν2 ,
σi = Pr[S(t) = si ] ,
1
T = ,
σ1 ν1 + σ2 ν2
T
ηi,j = πi,j , i ∈ {1, 2} , j ∈ Z+ ,
Ti
T
ηi = σi , i ∈ {1, 2} .
Ti

15. 15
Determining E[Sk Sk+1 ] and γ
Choosing
{s1 , s2 } = {1, 0} ,
leads to
E[Sk ] = η1 ,
Var[Sk ] = η1 (1 − η1 ) ,
∞
E[Sk Sk+1 ] = η1,j χj ,
j=0
where
A classical: χj different for each j (infinite sum)
B constant: χj = χ1 for j ≥ 1 (finite sum)

16. 16
Outline
1 Introduction
2 Model & Analysis
3 Numerical examples for constant retrial rate
4 Conclusion

17. 17
correlation positive when outgoing activity limited
γ λν
µ
µ µ
µ µ
µ
µ
outgoing call rate limited (α = 0.1),
primary incoming call rate varying (λ ∈ [0, λmax )),
call durations matched (ν1 = ν2 = 1)

18. 18
correlation positive when time share matched
γ λν
µ
µ
µ
µ
outgoing rate (α) increasing with incoming (λ) such that time
share incoming/outgoing is matched,
call durations matched (ν1 = ν2 = 1)

19. 19
correlation strictly negative in some cases
γ λ
µ
µ
µ
outgoing call rate fixed (α = 1),
primary incoming call rate varying (λ ∈ [0, λmax )),
call durations strongly differing (ν1 = 100, ν2 = 1) (and thus,
ρ = λ/ν1 always < 0.01 in the figure)

20. 20
Outline
1 Introduction
2 Model & Analysis
3 Numerical examples for constant retrial rate
4 Conclusion

21. 21
Conclusion
• distinguishing A from B with correlation coefficient γ
• focus: retrial queue model for call center with call blending
• from continuous-time result to discrete sequence:
censor and discretize
• numerical results constant retrial rate (B)
illustrate variability of γ
• currently working on comparison with classical retrial rate (A)

22. 22
Questions?