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

to as the Erlang-C formula. It is denoted E2,N and is given by
E2,N =
j≥N
πN = (λ/µ)N
N−1 (λ/µ)
N!(1 − ρ)
i=0
i
−1 (λ/µ)N
+
i! N!(1 − ρ)
The Poisson distribution of arrivals has an important and useful consequence, known as PASTA
(Poisson Arrivals See Time Averages): it implies that the probability E2,N is in fact also the
probability that a customer is delayed in the queue (as opposed to being served immediately
upon arrival).
Using formula (2) and its relatives, one can calculate staffing levels N for any desired service
level, given the arrival rate and average service time. This can be easily done even with
a spreadsheet. However, such a solution does not provide any insight on the dependence of
N on model parameters, for example, how should N change if the load was doubled. Such
insight comes out of a staffing rule that goes back to as early as Erlang [3], where he derived
it via marginal analysis of the benefit of adding a server. (Erlang indicated that the rule had
been practiced actually since 1913.) This is the square-root safety- staffing rule, which we now
describe.
Let R = λ/µ denote the average offered load. Then the square root safety-staffing rule states
the following: for moderate to large values of R, the appropriate staffing level is of the form
5
(2)
N = R + β √ R (3)
where β is a positive constant that depends on the desired level of service; β will be refereed to
as the Quality-of-Service (QOS) parameter: the larger the value of β, the higher is the service
quality. The second term on the right side of (3) is the excess (safety) capacity, beyond the
nominal requirement R, which is needed in order to achieve an accepted service level under
stochastic uncertainty.
The form of (3) carries with it a very important insight. Let ∆ = β √ R denote the safety
staffing level (above the minimum R = λ/µ.) Then, if β is fixed, an n-fold increase in the
offered load R requires that the safety staffing ∆ increases by only √ n-fold, which constitutes
significant economies of scale.
What does (3) guarantee as far as QOS is concerned? For Erlang-C, this is the subject of
the seminal paper by Halfin and Whitt [7], where they provided the following answer:
Theorem 1: Consider a sequence of M/M/N queues, indexed by N=1,2. . . As the number of
servers N grows to infinity, the square-root safety-staffing rule applies asymptotically if and only
if the delay probability converges to a constant α (0 < α < 1), in which case the relation between
α and β is given by the Halfin-Whitt function
α = [1 + β/h(−β)] −1
here h(x) = φ(x)/(1 − Φ(x)) is the hazard rate of the standard normal distribution N(0,1).
(4)