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Presentation held 3 August 2012 at QTNA 2012 in Kyoto
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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)
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γ enables to study short-term balance
• coefficient of correlation γ ∈ [−1, 1] captures correlation incall 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
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Quantifying the call blending balancein two way communication retrial queues:
analysis of correlation
based on joint work while at Kyoto University
Wouter Rogiestb,∗ & Tuan Phung-Duca,c
aGraduate School of Informatics · Kyoto University · JapanbDept. of Telecomm. & Inf. Processing · Ghent University · Belgium
cDept. of Math. & Comp. Sciences · Tokyo Institute of Technology · Japan
∗presenting
QTNA 2012 · Kyoto · 1–3 August 2012
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Outline
1 Introduction
2 Model & Analysis
3 Numerical examples for constant retrial rate
4 Conclusion
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Outline
1 Introduction
2 Model & Analysis
3 Numerical examples for constant retrial rate
4 Conclusion
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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
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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)
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Outline
1 Introduction
2 Model & Analysis
3 Numerical examples for constant retrial rate
4 Conclusion
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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 orbitδ0,n : Kronecker delta
ν1 service rate incoming call
ν2 service rate outgoing call
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Markov chain
• S(t): server state at time t,
S(t) =
0 if the server is idle,
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
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Correlation coefficient γ
• numbering consecutive events Sk with k
• Sk : incoming (Sk = s1) (↓) or outgoing (Sk = s2) (↑)• assuming steady-state
γm =E[SkSk+m]− (E[Sk ])2
Var[Sk ]; m ∈ Z+
• −1 ≤ γm ≤ 1
• main interest γ1, or γ
• main challenges
1 extracting distrib. (Sk ,Nk) from distrib. (S(t),N(t))2 determining E[SkSk+1]
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From S(t) to Sk : 2 steps
original Markov chain
censor: remove idle periods
discretize: “compensate” for ν1 6= ν2
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In general: from (S(t),N(t)) to (Sk ,Nk)
• original Markov chain: under conditions, unique stochasticequilibrium, with limt→∞ :
πi ,j = Pr[S(t) = i , N(t) = j ], (i , j) ∈ {0, 1, 2} × Z+
• censor, with limt→∞ :
π̃i ,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+
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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 ] ,
T =1
σ1ν1 + σ2ν2,
ηi ,j = πi ,jT
Ti, i ∈ {1, 2} , j ∈ Z+ ,
ηi = σiT
Ti, i ∈ {1, 2} .
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Determining E[SkSk+1] and γ
Choosing{s1, s2} = {1, 0} ,
leads to
E[Sk ] = η1 ,
Var[Sk ] = η1(1− η1) ,
E[SkSk+1] =∞∑j=0
η1,jχj ,
where
A classical: χj different for each j (infinite sum)
B constant: χj = χ1 for j ≥ 1 (finite sum)
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Outline
1 Introduction
2 Model & Analysis
3 Numerical examples for constant retrial rate
4 Conclusion
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correlation positive when outgoing activity limited
γ λ ν
µµ
µµ
µ
µ
µ
outgoing call rate limited (α = 0.1),primary incoming call rate varying (λ ∈ [0, λmax)),call durations matched (ν1 = ν2 = 1)
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correlation positive when time share matched
γ λ ν
µµ
µ
µ
outgoing rate (α) increasing with incoming (λ) such that timeshare incoming/outgoing is matched,call durations matched (ν1 = ν2 = 1)
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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)
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Outline
1 Introduction
2 Model & Analysis
3 Numerical examples for constant retrial rate
4 Conclusion
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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)
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Questions?