DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

D
k
: Outflow from high priority queue in the kth interval
λ : Inflow to low priority queue from the outside in the kth interval
k
The system is characterized by the following parameters:
µ
a
: Rate per unit of the schedule cycle at which the low priority queue can be served
µ : Rate per unit of the schedule cycle at which the high priority queue can be served
b
l: The feedback interval in units of the schedule cycle
The following four equations then completely describe the evolution of the system:
A
C
B
k + 1
= Ak
+ λk
− Ck
(1)
Dk
= min( Ak
+ λk
, µ
a
(1 − ))
(2)
µ
k
b
k +1
= Bk
+ Ck
−l
− Dk
(3)
D = min( B + C , µ )
(4)
k
k
k −l
b
Equations (1) and (3) are merely conservation rules, while equations (2) and (4) model the
constraints on the outflows and the interaction between the queues. This model while
conceptually simple, exhibits surprisingly complex behaviors.
Dynamical Behavior
The analytic approach to solve for the flow model under constant arrivals (i.e. λ
k
= λ for all k)
shows several classes of solutions. The system is found to batch its workload even for perfectly
smooth arrival patterns. Following are the characteristics of behavior of the system:
1) Above a threshold arrival rate ( λ ≥ / 2 ), a momentary overload can send the system
µ b
into a number of stable modes of oscillations.
2) Each mode of oscillations is characterized by distinct average queuing delays.
3) The extreme sensitivity to parameters, and the existence of chaos, implies the system at a
given time may be any one of a number of distinct steady-state modes.
The batching of the workload can cause significant queuing delays even at moderate occupancies.
Also such oscillatory behavior significantly lowers the real-time capacity of the system. For
details of application of this model in supply chain context, refer to (Kumara et al. 2003).
5.1.2 Managerial Systems
Decision-making is another typical characteristic in which the entities in a supply chain are
continuously engaged in. Entities make decisions to optimize their self-interests, often based on
local, delayed and imperfect information.
To illustrate the effects of decisions on the dynamics of supply chain as a whole, we consider a
managerial system, which allocates resources to its production and marketing departments in
accordance with shifts in inventory and/or backlog (Rasmussen and Moseklide 1988). It has four
level variables: resources in production, resources in sales, inventory of finished products and
number of customers. In order to represent the time required to adjust production, a third order
delay is introduced between production rate and inventory. The sum of the two resource variables
is kept constant. The rate of production is determined from resources in production through a
nonlinear function, which expresses a decreasing productivity of additional resources as the