Cumulative Diagrams: An Example
Consider Figure 1 in which the functions
O
(t)
and
P
(t)
denote, respectively, the
demand rate and the service rate (or “capacity”) over time at the runway system of an
airport
1
.
The demand rate is the expected number of demands per unit of time, i.e., the
number of requests for service at the runway system per unit of time.
Note that not all
these requests will be necessarily satisfied due to capacity limitations. Similarly the
service rate is the expected number of service requests that can be satisfied per unit of
time when the runway system is continually busy, i.e., it is the maximum throughput
capacity of the runway system (details in the next lecture of our course).
The units we
shall use for both the demand rate and the capacity are “aircraft movements (arrivals or
departures) per hour”, unless otherwise stated.
Figure 1 approximates a situation that one sees all the time at busy airports.
The
time axis shows the busy part of a day at the airport, e.g., the origin may correspond to
07:00 local time.
To simplify the analysis we assume the demand rate is constant
throughout the busy part of the day, i.e.,
(t)
=
n
movements per hour for all
t
.
While the
normal capacity
h
(for “high”) is greater than
n
, a weather event occurs between
t
=
a
and
t = b
(e.g., this could be a period when fog is present during a particular day) which
reduces the capacity to
l
(for “low”) movements per hour.
It will be convenient later on
to denote with
T = b – a
the interval of time during which the capacity is low.
We
would like to explore quantitatively the implications of this temporary loss of capacity on
delay levels at the airport.
1
We shall use throughout the Greek symbols
O#
and
to denote the expected “demand
rate” and expected “service rate” (or “capacity”); this is consistent with the standard
notation used in queuing theory.
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View Full DocumentWe shall make another important simplification in our model at this point.
Namely, we shall assume that demands occur at evenly spaced intervals (e.g., if
n =
60,
a demand occurs exactly every 60 seconds) and, similarly, for any value of the service
rate,
h
or
l
, the service times are constant (e.g., if
l
= 30 and the runway system is
continually busy, an aircraft movement is completed exactly every 2 minutes).
In
queuing theory, this is called a
deterministic
model, by contrast to
probabilistic
models in
which either the times between the occurrence of successive demands or the service times
(or, more usually, both) are random variables, i.e., may take on different values, each
value associated with a certain probability.
Clearly the situation shown in Figure 1 will result in some delays to aircraft
movements during the time period between
t
=
a
and
t = b
, at the very least.
In fact, it is
very easy to plot, as a function of time, the number of aircraft (landing or taking off)
which will be in the queue, waiting to use the runway system.
This is done in Figure 2.
There is no queue until
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 Fall '02
 IsmailChabini
 #, 1 L, $600,000, $940, cumulative flow diagrams

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