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?