This paper presents a technique for building robust telescope schedules that tend not to break. The technique is called Just-In-Case (JIC) scheduling and it implements the common sense idea of being prepared for likely errors, just in case they should occur. The JIC algorithm analyzes a given schedule, determines where it is likely to break, reinvokes a scheduler to generate a contingent schedule for each highly probable break case, and produces a 'multiply contingent' schedule. The technique was developed for an automatic telescope scheduling problem, and the paper presents empirical results showing that Just-In-Case scheduling performs extremely well for this problem

Bresina, John

Drummond, Mark

Swanson, Keith

English

NASA Technical Reports Server

Keith Swanson
NASA
N95- 23742
Robust Telescope Scheduling
John Bresina Mark Drummond
Recom Technologies Recom Technologies
AI Research Branch, Mail Stop: 269-2
NASA Ames Research Center, Moffett Field, CA 94035-1000
E-mail: {swanson, bresina, drummond}_ptolemy.arc.nasa.gov
Phone: 415-604-6016, Fax: 415-604-3594
KEY WORDS AND PHRASES
Automatic telescopes, execution errors,
Just-In-Case scheduling, robust execution, telescope
scheduling.
ABSTRACT
This paper presents a technique for building
robust telescope schedules that tend not to break.
The technique is called Just-In-Case scheduling and it
implements the common sense idea of being prepared
for likely errors, just in case they should occur. The
JIC algorithm analyzes a given schedule, determines
where it is likely to break, reinvokes a scheduler to
generate a contingent schedule for each highly
probable break case, and produces a "multiply
contingent" schedule. The technique was developed
for an automatic telescope scheduling problem, and
the paper presents empirical results showing that
Just-In-Case scheduling performs extremely well for
this problem.
INTRODUCTION
This paper presents and evaluates a technique
for generating schedules that have robust execution
behavior. The technique is called Just-In-Case
scheduling, or JIC, and it implements the common
sense idea of being prepared for likely execution
errors, just in case they should occur. Jlc handles
schedule execution errors that are due to the presence
of actions with uncertain durations. JlC was
developed as part of a larger telescope management
and scheduling project. This section outlines only key
aspects of the problem; more details are available
elsewhere [4; 6].
In this application domain, the telescopes are
land-based and fully automatic; a telescope control
computer opens the observatory at twilight and
collects data through the night without human
assistance (see Genet and Hayes [8] for details). We
are implementing an overall automated management
system [6] to enable participating astronomers to
submit observation requests and obtain results from a
remotely located telescope. This interaction occurs
via electronic communication networks, without the
necessity of human intervention. To ensure the
telescope load is balanced over weeks or months, the
system will also include a sophisticated long-term
loader [2]. The long-term loader will assign
observation requests to specific nights. Each night,
the assigned observations are given to a scheduler to
determine the specific times during the night they are
to be executed. Producing robust nightly schedules is
the role of JIC.
An observation request specifies both hard
constraints and soft preferences. The most important
hard constraint is the observing window. Each
observation request, or action, can begin execution
only in a specific interval of time within a night; this
interval is defined by the astronomer who submitted
the request.. In the remainder of this paper we refer
to each observation request as an action.
The scheduling problem is to synthesize a schedule
that satisfies all hard constraints and achieves a good
score according to an objective function based on the
soft preferences. A schedule is a sequence of actions,
each with an enablement interval assigned by the
scheduler. The assigned enablement interval of each
action is a subinterval of the action's
(astronomer-provided) observing window. A
scheduler assigns the enablement intervals to further
restrict when the actions will begin execution. This
paper does not address the problem of generating a
basic schedule (this is discussed in [7]) - we assume
the existence of a scheduler that, given a set of
requests and an objective function, produces a
feasible observing schedule with a reasonable score.
(Our current scheduler produces reasonable schedules
in less than one minute.)
In this domain, a typical action has a duration of
several minutes with action duration uncertainty
occurring due to mechanical slop in the telescope
drive train, software timing inaccuracies, and star
centering. The amount of time it takes to center a
star depends on how accurately the telescope is
pointed when it starts the search and how clear the
sky is while the centering is going on. Hence, it is
impossible to accurately predict the duration of an
observing action. All we can do is gather data from
actual executions and then calculate the mean and
standard deviation of each action's duration.
Observation actions are typically executed many
times over a period of weeks or months, so such
statistics are easily available.
The telescope used in this application is fully
automatic and runs unattended; thus, unlike many
scheduling domains where printing a schedule is the
final goal, our system must be able to automatically
execute a schedule. Schedule execution proceeds by
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https://ntrs.nasa.gov/search.jsp?R=19950017322 2020-06-16T08:44:23+00:00Z
executingeachactionin thescheduledsequence.
Afteranactionfinishesexecution,if thecurrentime
isoutsideof thenextaction's(scheduler-assigned)
enablementinterval,thentheschedulebreaksand
executionhalts.
Schedulebreakageis thecentralproblem.The
predictedstart timeof anactioninascheduleis
basedonthesumoftheestimatedurationsof the
actionsthatprecedeit. Hence,thefurtherintothe
futureanactionoccursin theschedule,thegreater
theuncertaintysurroundingits actualstarttime.
Giventhewaythat uncertaintygrows,it ispossible
for ascheduleto breakduesolelyto accumulated
durationpredictionerrors.
Thereis asimplesolutionto theproblemof
durationpredictionerrors:makethestarttimeof
eachactionequalto a worstcaseestimateof the
previousaction'sfinishtimeandintroducea
busy-waitin casethepreviousactionfinishesearly.
Unfortunately,introducingsuchbusy-waitswill waste
observingtime.Ourgoalis to avoidschedulebreaks
withoutsacrificingschedulequality.
Schedulescanalsofail forreasonsotherthan
durationpredictionuncertainty.Cloudsor windcan
makestaracquisitionimpossible,resultingin
unavoidableschedulebreaks.In oursystem,whenthe
schedulebreaks,thetelescopeinvokesthescheduler
to generatea newoneforthecurrentsituation.
Hence,whileweathercancauseabreakinschedule
execution,thesystemisrobustenoughto
dynamicallyrescheduleandtry again.Dynamic
reschedulingcouldalsobeused,in placeofJlC,to
handlebreaksdueto durationpredictionerrors.
However,theproblemwithdynamicreschedulingis
that it wastesvaluableobservingtimewheneverthe
telescopecontrolleriswaitingforaschedule.Thereis
limitedobservingtimeavailableduringthenight,and
wedonotwantto wasteit!
JlCproactivelymanagesdurationuncertaintyby
identifyinghighprobabilityschedulebreaksand,for
eachone,generatinganalternativeschedulejust in
casethebreakoccursduringexecution.This
proactivemanagementcanuseoff-linetimeduring
thedayto computeandstorealternativeschedulesin
orderto reduceon-linereschedulingtimeduringthe
night.JICproducesa "multiplycontingent"schedule
that specifieswhatactionsthetelescopecontroller
shouldtake,conditionedbythecurrentsituation.
Thus,if accumulateddurationpredictionerrorsforce
thetelescopeintoasituationforwhichthenominal(i.e.,theinitial) scheduleis inapplicable,thenan
appropriatecontingentschedule(if available)is
automaticallyselectedandexecutioncontinues.If an
appropriatescheduleisnotavailable,thesystem
resortsto dynamicrescheduling.
JUST-IN-CASE SCHEDULING
In overview, the JIC algorithm accepts a
schedule as input and robustifies it as follows. First,
using a model of how action durations can vary, the
temporal uncertainty at each step in the schedule is
estimated. Second, the most probable break due to
this uncertainty is determined. Third, the break point
is "split" into two hypothetical cases: one in which
the schedule breaks and one in which it does not.
Fourth, the scheduler is invoked on a new scheduling
subproblem to produce an alternative schedule for the
break case. Fifth, this alternative schedule is
integrated with the initial schedule producing an
updated multiply contingent schedule. This completes
consideration of one break case; if there is more time
before schedule execution begins, then the JIC process
can be repeated with the current multiply contingent
schedule as the new input.
Each action Ai has a duration mean Pi and
standard deviation ai. One of the preconditions of
each action is the interval of time during which it can
begin execution; let Pi be this precondition interval
for Ai. (The precondition interval for observation
requests is provided by an astronomer.)
A schedule is a sequence of actions, where each
action is associated with an enablement interval, El,
assigned by the scheduler: (A0, E0);... ; (A,, En),
such that for i = O,..., n, Ei Q Pi. During schedule
execution, as soon as action Ai-1 is finished
executing, action Ai is selected for enablement
testing; Ai is enabled if the current time is within Ei.
If Ai is enabled, then it is immediately executed; else,
the schedule breaks.
A multiply contingent schedule can be thought of
as a set of alternative schedules; to save space, our
implementation uses a tree to represent this set of
schedules. Let/3(i) be defined such that Az(i) is the
predecessor of Ai in the schedule, if one exists. For
simplicity, we assume that A0 is the unique first
action in a multiply contingent schedule.
Using a duration uncertainty model discussed
below, JIC estimates the temporal uncertainty at each
step in the schedule by starting at the beginning of
the schedule and propagating uncertainty forward.
This process involves estimating the time at which
each action in the schedule will start and finish
executing. The start interval, Si, is the set of possible
execution start times for action Ai. Similarly, the
finish interval, Fi, is the set of possible execution
finish times for action Ai. Let So denote the interval
during which schedule execution can start. For our
telescope application, schedule execution always
starts exactly at twilight; hence, So is the degenerate
interval [twilight, twilight].
Ai cannot start executing at a time outside its
enablement window. Hence, if Al3(i ) finishes executing
at a time outside of Ei, then either an action in an
alternative contingent schedule will be executed or
the schedule will break. Thus, Si is computed to be
F_(_) N Ei.
Given that Ai's start interval, Si, is [tx,t2], its
finish interval, Fi, is computed to be [tl + tti - ai,
t2 + tq + ai]. The current duration uncertainty model
simply uses one standard deviation of the mean when
computing each finish interval - this has worked well
348
in practice,asshownby the empirical results in the
next section.
The break probability of an action is a function of
the enablement probability of that action and of all
preceding actions. Let p(enable(Ai)) be the
enablement probability for action Ai; that is, the
probability that Ai will be enabled when selected. It
is computed to be the proportion of the previous
action's finish interval during which Ai is enabled.
p(enable(Ai)) - ]Ft_(i) n Eil
For simplicity, this computation is based on the
erroneous assumption that all of an action's possible
finish times are equi-probable (i.e., that Fp(i) has a
uniform probability distribution) and, hence, is only
an estimate of the true enablement probability.
Let p (select(A_)) be the selection probability for
action Ai; that is, the probability that Ai will be
selected for enablement testing. An action will be
selected if the preceding action was both selected and
enabled; the schedule's first action will always be
selected.
For i = 0: p(select(Ai)) = 1.0.
For i > 0: p (select(Ai)) = p (select(A_(i))) ×
P (enable(As(i))).
Let p(break(A_)) be the break probability for action
Ai; that is, the probability that the schedule will
break at Ai when it is selected for enablement testing.
p (break(A/)) = p (select(Ai)) × [1 - p (enable(Ai))].
Note that the computation of break probabilities is
similar to the computation of the conditional
probabilities characterized by a Markov chain.
After determining the action with the highest
break probability, JlC "splits" the associated
uncertainty time interval into two subintervals. One
subinterval will be the intersection of the finish
interval F_(i) with the enablement interval Ei. The
remaining subinterval (not part of the intersection) is
split off as a break case and a new scheduling
subproblem is formed using a point in the subinterval
as the start time. JIC then invokes the scheduler on
this subproblem and incorporates the returned
alternative schedule into the original schedule.
EMPIRICAL EVALUATION
To evaluate the performance of JIC we performed
an experiment using real telescope scheduling data.
(Additional experimental results and algorithm
details are available elsewhere [3; 5].) The observation
actions were provided by Greg Henry of Tennessee
State University [9; 11]. The scheduler used in this
experiment deterministically hill-climbs on a
domain-specific heuristic [1]. The experiment required
collecting data from thousands of schedule executions;
since this is impractical on a real telescope, we
developed a simulator of the telescope controller's
z
100
90
80
70
60
5o
0
Mean Performance
i i i i i
5 10 15 20 25
Cases Covered
3O
Figure 1: Mean performance, measured as night per-
centage, vs. cases covered.
schedule executor. The simulator computes an
action's execution duration by using a random
variable with a normal (Ganssian) probability
distribution whose mean and standard deviation are
exactly those characterized by statistics obtained from
a number of nights of actual execution on a telescope
at the Fairborn Observatory (Mt. Hopkins, Arizona).
The experimental question is: given real telescope
scheduling data, can JlC provide a useful increase in
schedule robustness within a reasonable number of
contingent cases? To answer this question we varied
the number of break cases considered and measured
how far into the night a multiply contingent schedule
would execute without dynamic reseheduling. The
experimental procedure is as follows.
First, the scheduler is used to find a single nominal
schedule. This schedule is executed 1000 times in the
simulator; for each execution run we note the
percentage of the night that the schedule executes
before halting, either due to a break or schedule
completion. Next, we allow JIC to find and fix what it
deems to be the next most probable break case. We
then run the augmented schedule through the
execution simulator (again, 1000 times). In this
manner, we allow JXC to cover up to thirty break
cases.
Figure 1 illustrates the resulting performance. The
independent variable is the number of break cases
covered by JIC. The dependent variable is the
percentage of the night that the schedule executes
before halting, averaged over 1000 runs. It clearly
shows that the mean percentage of the night executed
increases with the number of cases considered by JIC.
The performance increase is most dramatic early on,
as we had hoped. After only ten cases, the schedule
executes, on average, through 96% of the night.
Although not shown, experimental results also
indicate that schedule size (measured as the total
number of actions contained in the multiply
contingent schedule) increases linearly with the
number of cases, as one might expect.
349
CONCLUSION
Thispaperhaspresentedanalgorithmfor
.lust-hi-Casescheduling.Usingalmostanyscheduler
andsimplestatisticalmodelsof durationuncertainty,
thealgorithmproactivelymakesanominalschedule
morerobust.Despitesomeratheregregiousmodeling
assumptions,thealgorit.hmworksextremelywellfora
r,_al telescope scheduling problem. Traditional
intuitions surrounding the inanagement of uncertain
action outcomes suggest the inevitability of large
search spaces and intractable reasoning. Using a
"splitting" technique, our algorithm makes action
outcome distinctions only when necessary (see [tanks
[10] for background to this idea). Further, most of the
likely schedule breaks are covered in a few iterations
of JIC.
While Jlc works extremely well for our particular
telescope scheduling problem, it will not necessarily
fare so well on all domains. We have analyzed the
nature of the schedule breaks in our domain in order
to characterize the general conditions under which JIc'
achieves useful robustness increments in a few
iterations. The results are suggestive, but not yet
mathematically precise. Essentially, JlC appears to
work well when the following three conditions hold.
First, there must be room for improvement. If the
prior probability of successful execution of the
schedule is close to 1.0, there is not much JIC can add.
Second, there must be a small number of schedule
breaks responsible for most of the total break
probability mass. If this is so, then each break case
covered by Jm stands a good chance of increasing the
probability of executing the entire schedule. Third,
each contingent, schedule found nmst be no worse in
its break characteristics than the nominal schedule.
In some sense, this is simply a recursive application of
the first two conditions; it requires that each
contingent schedule be as easy to robustify as the
initial one.
Finally, we recognize that a number of interesting
issues remain outstanding regarding the applicability
of J[c to other domains and how JlC compares to, or
might integrate with, other existing scheduling
techniques. This is an excellent area for further work.
ACKNOWLEDGEMENTS
Significant thanks to Will Edgington for helping
make the telescope management and scheduling
software a reality.
REFERENCES
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M., Swanson, K., Crawford, D., Genet, D.,
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[2] Bresina, J. 1994. Telescope Loading: A Problem
Reduction Approach. In Proc. of i-SAIRAS 94.
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[3] Bresina, J., Drummond, M., Swanson, K. 1994.
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[4] Bresina, J., Drummond, M., Swanson, K., and
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