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DYNAMICS
• Grades N1..N4
• Personnel exhibit one of the following behaviors:– get promoted– quit, causing a vacancy
that is filled during the next promotion period
– remain in grade– get demoted
STATE SPACE
•S = {N1, N2, N3, N4, V}
• V for Vacancy
• Every time period, the employee moves according to a probability
MODELED AS A MARKOV CHAIN
• Discrete time periods
• Stationarity– transitions stay constant
over time– transitions do not depend
on time in grade
PROBABILITY TRANSITION MATRIX
1 2 3 4 V
1 0.1 0.6 0.3
2 0.1 0.5 0.3 0.1
3 0.1 0.6 0.2 0.1
4 0.1 0.8 0.1
V 1
= P
MEASURES OF INTEREST
• Proportion of the workforce at each level
• Expected labor costs per year
• Expected annual cost of Entry-level training
• PDF of passage from N1 to N4
TRANSITION PROBABILITY CALCULATION
• Start with employee in N1• a0 = [1, 0, 0, 0, 0]• a1 = a0 * P• a1 = [0.1, 0.6, 0, 0, 0.3]• a2 = a1 * P
37.0
1
1
1
2
1,
1,22
1,11
1
VV
N
N
N
Pa
Pa
Pa
a
STEADY STATE PROBABILITIES
• a0 * P * P * P * P * ....
• P is singular (rank 4)
• P is stochastic– rows sum to 1
is the stationary probability distribution
N1 is the proportion of the time spent in state N1
COMPUTATION STRATEGY
PN1N2 N3 N4
V
• Substitute stochastic equation for first component of P
• Solve Linear System via Gaussian Elimination
...more COMPUTATION STRATEGY
• Start with arbitrary a0
• calculate a1, a2, a3, ...
• will converge to
ENTRY-LEVEL TRAINING
• 12% of the time we are in state V
• Cost of ELT = – 12%– times the Workforce size– times the cost of training
LABOR COSTS
• Salaries– CN1 = $12,000
– CN2 = $21,000
– CN3 = $25,000
– CN4 = $31,000
• Total Workforce = 180,000
• Cost = 180K * (C * ) = $3.7B
EXCURSION
• Promotion probabilities unchanged
• Allow attrition to reduce workforce– PV,N1 = 0.6 results in
workforce of 108,000
• How much $ saved?
• How fast does it happen?
LABOR COSTS
3.35
3.4
3.45
3.5
3.55
3.6
3.65
3.7
100000 120000 140000 160000 180000 200000
WORKFORCE
$B
CONVERGENCE TO 75% WORKFORCE (135K)
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10
ITERATION
N1
N2
N3
N4
V
CONVERGENCE TO 60% WORKFORCE (108K)
0
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 10
ITERATION
N1
N2
N3
N4
V