DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

y
k
.The decision rule or policy function says that longer queues are served with higher priority.
Two possibilities considered are:
1) All-or nothing decision: the server decides to spend all its time on the activity corresponding to
the longer queue. Hence Φ is a Heaviside function given by
Φ( x − y)
= 1 if x ≥ y
=0 if x < y .
2) Mixed Solutions: the server decides to spend most of its time to the activity corresponding to
the longer queue. For this decision function a S-shaped logistic function is used as given by:
1
Φ ( x − y)
= .
k ( x−
)
1+
e
y
The parameter k tunes the “steepness” of the S-shape.
With these decision functions the new queue lengths and are given equations
xk+1
y k+ 1
xk+ 1
= xk
+ α − Φ(
xk
− yk
) ,
yk+ 1
= yk
+ β − Φ(
xk
− yk
) .
Using the constraints α + β = 1 and Φ
X
+ ΦY
= 1, it is sufficient to consider the dynamics of
the map in order to study the behavior of the system
f ( x)
= x + α − Φ(2x
− 2) .
Dynamical Behavior
For 0