Convexity properties of queueing systems with applications to call centers

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Convexity properties of queueing systems with
applications to call centers

• Ger Koole
• Vrije Universiteit Amsterdam
• Ecole Centrale de Paris
• 10 February 2005

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Planning and call centers

• Large socio-technical systems with sometimes
  100s of agents
• Personnel costs ?70 of total costs
• Efficient workforce planning is crucial
• Many different planning tools
• But Tools do not go far beyond Erlang C!
• Opportunities for real OR

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Big WFM issues

• Unpredicted fluctuations in parameters
• Multiple skills
• Multiple communication channels
• Science is rapidly advancing on these issues ?
• Soon implementation in WFM tools

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But

• how about current practice?
• Are the standard methods that good?
• Do we really understand the baseline cc?
• Most use Erlang C, the law of diminishing
  returns, etc.
• But what is a good call center model?
• And given the model what are the right
  optimization methods?

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Call center models

• Baseline performance model
• "stationary independent period by period" (SIPP)
• Erlang C ? MMs queueing model
• Works well in well-dimensioned cc
• Gives nonsense in overload (0 SL, ? queue)

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Better model

• Include abandonments (Erlang A) or even redials
• Consider a transient model

SL in Erlang A for exponential patience with
average ?, 5 and 1
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How and what to optimize?

• Trade-off between costs and SL (definition!)
• WFM consists of multiple steps
• Forecasting
• Staffing
• Scheduling
• Focus on staffing How many agents should be
  scheduled at any time?

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Multi-period model

• Arrival rate lk in interval k
• Erlang C
• How many agents to schedule in interval k?
• Standard approach for every interval determine
  minimal s such that P(W(s)lt t)gtd
• Algorithm increase s until P(W(s)lt t)
• Uses increasingness of P(W(s)lt t)

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Alternative approach

• Alternative minimize åksk such that
• åklkP(Wk(sk)ltt) /ålllgtd
• Leads to lower costs
• SL varies over time
• Standard or alternative is choice
• Algorithm greedy method, start with minimal sk
  such that skmgtl and add agent that adds most to
  objective
• Optimal if SL concave in s ? diminishing returns

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Recapitulation

• Standard (simplest) call center model
• Optimization over multiple intervals
• Simple greedy optimization procedure is optimal
  if SL is concave
• Questions
• Is SL concave in s for Erlang C?
• And how about Erlang A??

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Concavity of SL for Erlang C

• Jagers Van Doorn 1991 P(W(s) t) concave for
  smgtl
• Greedy algorithm (implementation by Arnoud
  Wattel, www.math.vu.nl/awattel)

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Greedy algorithm, Erlang C
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Erlang A

• Not concave!!!

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Greedy algorithm, Erlang A

• l(2,2,2,2), m5, 70/20 SL, g1 (av. patience)
• Greedy s(14,14,14,0), total 42
• Optimal s(10,10,10,9), total 39
• What is wrong?
• The algorithm
• or
• The problem??

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About service levels

• Mathematics when objective needs to be concave,
  then direct reward "should" also be concave
• But direct reward I(W(s) ? t) is 0/1 function
• Practice if P(W(s) ? t) is SL definition then
  calls that wait gtt seconds do not count
• ? "rational" call center manager ignores these
  calls
• "Better" SL definition E(W(s)-t)

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New SL definition

• Erlang C, E(W(s)-t) again concave in s
• Erlang A, E(W(s)-t)
• concave in s if g ? m
• no results (yet) for g gt m (ongoing work)
• Techniques dynamic programming / coupling

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Extension to shift scheduling

• Output of staffing is input of scheduling
• Peaked staffing requirements lead to inefficient
  scheduling
• Solution integrate staffing and scheduling
• Average SL requirement
• Simple case shifts of equal lengths without
  breaks
• What is the optimal schedule?
• (joint work with Erik vd Sluis, IIE Tr 2003)

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Staffing and shift scheduling

• Difficult problem no longer additive objective,
  because shifts span multiple intervals!
• Optimization problem on m-dim grid
• How to find optimal solution without exhaustive
  search?
• Solution Use some kind of convexity property
  multimodularity

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Multimodularity

• Define Vv1,,vm with viei-ei1, iltm, and
  vmem-e1
• f is MM if for all v¹ w2 V
• f(sv)f(sw) f(s)f(svw)
• -1 x Average SL is MM (each vi is a translation
  of a shift)
• Crucial for proof concavity of performance

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Local search

• Schedule s is optimal if no improvement in
• sy yåi2 Uvi, U½ V
• Defines (long) local search procedure for a
  certain number of agents improve schedule by
  moving shifts
• Finds optimal solution quickly
• Takes long to prove optimality (2m points to
  search)
• Compare with all neighboring grid point 3m,
  exhaustive Sm (greedy m)

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Numerical results
Very good heuristic in case of breaks!

• Per interval to average SL requirement from 28
  to 24 agents

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Other monotonicity problems

• Up to now varying s, typical for WFM
• Economies of scale varying s and l, typical when
  merging call centers or hospital wards
• Is this better? Not always
• Counterexample for standard SL, transient model
  (depends on wrong SL definition)
• In case of different priorities, decrease of SL
  for high priority customers/patients ?
  introducing thresholds solves this

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More monotonicity problems

• Waiting costs makes blocking attractive, at
  threshold level n
• 2-dim optimization problem
• Performance as function of s for best n (for that
  s) not concave, not even unimodal!
• Other optimal linear search procedure

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Thanks for your attention!

• More stuff www.math.vu.nl/koole and
  www.math.vu.nl/obp/callcenters .

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