We consider a single observation window [0, T] and compute long-run flow metrics over this finite horizon.
Let A(t) be the
cumulative number of arrivals up to time
t
and D(t) be the
cumulative number of departures up to time
t for t ∈ [0, T]
From this we derive:
Sample Path The sample path is the instantaneous number of items present in the system at time t: N(t) = A(t) − D(t), for t ∈ [0, T]
Cumulative Presence
H(T)
The total “item-time” accumulated by all
active items during [0, T]. This is the
area under the sample path over
[0,T].
H(T) = ∫₀ᵀ N(t)
dt
L(T) — Average
Work-in-Process (WIP)
The time-average of items present in the
system over [0, T].
L(T) = H(T) ⁄ T = (1 ⁄ T) ∫₀ᵀ N(t)
dt
Λ(T) —
Cumulative Arrival Rate
Let A(T) be the
cumulative number of arrivals by time
T. Then: Λ(T) = A(T) ⁄
T
w(T) — Average
Residence Time
The average time items that arrived
over [0, T] spend in the system.
w(T) = H(T) ⁄
A(T)
Together, these quantities satisfy the finite-window form of Little’s Law:
L(T) = Λ(T) · w(T) for all t ∈ [0, T]