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

4 Our Proposed Research
In order to demonstrate how uncertain arrival rate may influence staffing decisions, we first
consider an M/M/n+M queue with model-parameter uncertainty characterized by a random
arrival rate Λ. We let the mean service time and the mean patience time be 1, i.e. µ = θ = 1,
and we seek service level β which yields a time-stable delay probability α.
Assume Λ is a discrete random variable with finite or countably infinite set of values KΛ, and a
probability distribution pλi = P (Λ = λi). Let Ri = λi/µ denote the offered load in case Λ = λi,
and R the expected offered load
By the SLLN and the Garnett function (6):
α = lim
k→∞
where,
k
j=1 Wj
k
j=1 Aj
= lim
k→∞
1 k k j=1 Wj
1 k k j=1 Aj
R =
pλi · Ri .
KΛ
= EW
EA =
KΛ pi · Ri · αi
KΛ pi · Ri
=
KΛ pi · Ri · αi
R
Wj=Number of waiting customers at day j;
Aj=Number of arrivals at day j;
N−Ri
αi = P (W ait > 0|Λ = λi) = 1 − φ (since µ = θ; otherwise αi is the Garnett function
(6)).
√ Ri
Therefore, given α, we would like to find N. Then, assuming a square-root staffing rule, we
calculate the QOS parameter β = N−R √ .
R
Because of the analytical tractability of the Erlang-A model, using R, one can calculate the
value of β for each α according to (6). For analytically intractable models, one can use the ISA
simulation developed by [4] to obtain values of performance measures.
13
(8)