Operations Management · OITN-6-E
Queueing Analysis
Waiting times due to random variability — Sakasegawa approximation
ρ = λ / (S·μ)
L
q
= ρ^√(2(S+1)) / (1−ρ) · (CV²_A + CV²_S) / 2
W = W
q
+ t
S
L = λ · W
A
Arrival Characteristics
Mean inter-arrival time (t
A
)
min
arr/hr
λ = 60/t
A
Coefficient of variation of arrivals (CV
A
)
1.00
0 = deterministic · 1 = Poisson (random) · >1 = chaotic
S
Service Characteristics
Mean service time (t
S
)
min
srv/hr
μ = 60/t
S
Coefficient of variation of service (CV
S
)
0.50
Number of servers (S)
10
ρ
Utilization
—
0%
■ safe
■ caution
■ critical
100%
Calculating…
ρ ≥ 1 — Queue grows without bound. Add capacity or reduce arrivals.
KPI
Performance Indicators
L
q
Avg. queue length
—
items waiting in line
W
q
Avg. wait in queue
—
before service begins
L
Avg. system length
—
queue + in service
W
Avg. time in system
—
wait + service time
Little's Law:
L
=
λ
×
W
·
L
q
=
λ
×
W
q
∑
Calculation Trace (Sakasegawa)
Step-by-step
—
∿
L
q
vs. Utilization (current parameters)
!
Managerial Insights