Uncertainty in the Demand for Service - Faculty of Industrial ...

Note that (3) applies if and only if √ N(1 − ρN) converges to β (β>0). Indeed, formally the
Theorem of Halfin-Whitt reads:
As N ↑ ∞, P (W ait > 0) = E2,N → α (0 < α < 1)
iff √ N(1 − ρN) → β (0 < β < ∞)
(equivalently N ≈ RN + β √ RN) .
In practice, this rule makes the life of a call-center manager easier: he or she can actually
specify the desired delay probability and achieve it by following the square-root safety staffing
rule (3), simply choosing the right β.
2.2 Erlang-A: Summary of Some Results
Trying to make the M/M/N model more realistic and useful, the following assumption is added:
each customer has limited patience, that is, as the waiting time in the queue grows the customer
may abandon. We assume that patience is distributed exponentially with mean 1/θ. This
model is referred to as Erlang-A (A for Abandonment). It is also a Birth-Death process, and its
transition rate diagram is depicted below.
. . .
0 1 2 N-1 N N+1
2
N
N
Figure 2: Erlang- A as a Birth-Death Process
The steady-state distribution is found by solving of the following equations:
λπj = (j + 1)µπj+1
0 ≤ j ≤ N − 1
λπj = (Nµ + (j + 1 − N)θ)πj+1 j ≥ N
The solution of the steady-state equations is given by:
⎧
⎪⎨
πj =
⎪⎩
(λ/µ) j
π0,
j!
j
λ (λ/µ) N
Nµ + (k − N)θ N!
0 ≤ j ≤ N
π0, j ≥ N + 1
where
π0 =
n
j=0
k=n+1
(λ/µ) j
j! +
∞
j
j=N+1 k=N+1
λ
(λ/µ) N
Nµ + (k − N)θ N!
N
. . .
2
−1
6
(5)