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Operations Notes – Waiting and Service Quality

Variability causes waiting
⇒ When variability is present, a queue will result even if avg. arrival rate (λ) < avg. service rate (μ)

λ = average arrival rate
μ = average service rate
ρ = flow intensity of the system
Wq = average waiting time
Lq = average queue length
L = average number in the system
W = average time in the system
U = utilization (1-U = idle time)
M = the number of service providers

ρ = λ/μ
Wq = Lq
Lq = Wq * λ
L = Lq + ρ
W = Wq + 1/μ = Lq/λ + 1/μ
U = λ/Mμ

There is a trade-off between cost and waiting times. Returns from adding capacities are significantly higher when utilization is brought down from 95% to 90% than from 85% to 80% (set Utilization to less than or equal to 90%):

Good News: initial improvements are easy
Bad News: diminishing marginal returns

Decision variables when designing a queuing system:

Channels are the number of parallel windows open (M)
Lines are the number of waiting lines
Stages are the number of windows a customer gets serviced at

Reduce/Improve waiting times by: