LIDS-P-1528
JANUARY 1986
DELAYS IN ACYCLICAL DISTRIBUTED DECISIONMAKING ORGANIZATIONS
by
Victoria Y-Y Jin
Alexander H. Levis
Pascal A. Remy
ABSTRACT
Distributed decisionmaking organizations with synchronous protocols
are represented using Petri nets. An algorithm for computing time delays
is developed. Starting with a matrix representation of the organizational
structure, all possible information processing paths are scanned and the
When the decision
time delay associated with each one is computed.
system can be
overall
the
of
delay
strategies are known, the expected
description
matrix
flow
the
on
based
formulation
obtained. An alternative
also
presented.
of the Petri net is
at the MIT Laboratory for Information
*This work was carried out
provided by the Office of Naval
support
with
Systems
Decision
and
Research under Contracts N00014-84-K-0519 (NR 649-003) and N00014-85-K-0782
(NR 964-001).
Laboratory
02139.
for
Information
and
Decision
Systems,
MIT,
Cambridge,
MA
DELAYS IN ACYCLICAL DISTRIBUTED DECISIONMAKING ORGANIZATIONS*
Victoria Y-Y Jin
Alexander H. Levis
Pascal A. Remy
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139, USA
ABSTRACT
with
organizations
decisionmaking
Distributed
synchronous protocols are represented using Petri
nets. An algorithm for computing time delays is
developed.
Starting with a matrix representation
of the organizational structure, all possible
information processing paths are scanned and the
time delay associated with each one is co=puted.
known,
the
When the decision strategies are
expected delay of the overall system can be
obtained. An alternative formulation based on the
flow matrix description of the Petri net is also
presented.
KEYWORDS
theory,
Organization
decisionmaking.
1.
Petri
nets,
Distributed
INTRODUCTION
Decisionmakers in a distributed organization have
access to specified information sources and control
some specified resources. Usually, even in simple
organizations, there are many alternative paths
be
processed.
which
information
can
through
Internal decisions, which affect the sharing of
information and the issuance of directives or
commands, determine the paths information will
decision3aking
a
distributed
In
take.
organization, each member makes his own internal
decisions on the basis of decision rules which
depend on the task, his role in the organization,
his style, and his workload.
A useful measurement of organizational performance
is the time interval between the moment data about
received
by the organization and
an event are
the moment a response can be made. This time delay
to respond
reflects the organization's ability
to external events in a timely manner. In earlier
work (Boettcher and Levis, 1983) another performance measure, the accuracy of the response, was
organizations.
and
compare
used
to
evaluate
Together, accuracy and timeliness of response
evaluating
basis
for
a
better
provide
organizational structures.
The focus of this paper is on the computation of
given
characterizes
a
delay
that
the
time
organizational form. A distributed decisionmaking
organization (DDMO) is often a large scale system
(DMs) and
which
contains many decisionmakers
decision support systems (DSSs) that interconnect
evaluate the organization's response
them. To
*This work was carried out at the MIT Laboratory
for Information and Decision Systems with support
provided by the Office of Naval Research under
and
(NR
649-003)
N000014-84-K-0519
Contracts
N00014-S5-K-0782 (NR 964-001).
time, all possible information processing paths
the delay associate
must be identified and then
An algorithm for (a)
with each path computed.
scanning all paths and identifying concurrent paths
the
(b) computing
in acyclical organizations,
conditional probability of each path given the
decision strategies of the organization members,
and (c) computing path delays and the expected
This
delay has been developed and implemented.
algorithm is applicable when the protocols that
events
within
the
specify
the
sequence
of
interactions
and
the
admissible
organization
between members are synchronous.
In the next section, Petri nets are introduced to
third
represent organizations
forms.
In
the
section, the algorithm for scanning all paths is
described and the analytical formulation of the
Once the paths have been
problem is given.
identified, the computation of the path delays is
and
The
analytical
4).
(section
direct
computational approaches described in this paper
are illustrated with a two-person organization: the
results have also been applied to the comparison of
a hierarchical
a two three-person organization and a parallel one.
2.
PETRI NET REPRESENTATION
OF ORGANIZATIONAL FORMS
Petri nets were used by Tabak and Levis (1985) to
represent organizational forms because they show
explicity the interactions between decisionmakers
and the sequence of operations within an organizaThe basic elements of a Petri net, a
tion.
directed bipartite graph (Peterson, 1981) are the
circle node or place that represents a signal, and
the bar node or transition that represents a
presence of
The actual
process or function.
information in a place is denoted by one or more
A transition is enabled, if
tokens in that place.
It was
there are tokens in all its input places.
convenient to extend the basic Petri net formalism
to include a special type of transition to represent the internal decisionmaking by an organization
member. In the model of the decision-maker (Levis,
1984) the internal decisiornmaking is repre-sented
by a switch and a decision rule that specifies the
position of the switch. In the case of a DM, this
means which algorithm (from a finite set) is to be
used to process the data at some stage.
The concept of transitions being enabled allows the
explicit representation of organizational protocols
that dictate under what conditions decisions may be
made or information processed. The decision switch
-=h its associated rules allows the introductfon
of decision rules that determine which one of
several enabled transitions (or processes) is to be
activated or fired.
The Petri net representing a simple two-person DDMO
(Boettcher and Levis, 1983) is shown in Figure 1.
tes
sFigure
ut 1p.
oPetri
net representation of a two-pers
re onitions
ti ,
t
formulatd
be
an
net representing
organizationr
rotocol
of this
event, denoted by p
The dcann
continue,
ot accorn
A (Peterson,
the
are fl ow m atrix
represented in termsofhat
the assessment P
n
un lti
sit,on DM receives
organiza
in turn, selects its
local operations. The a plactter,
ingle
situation
hich
is produ ce by a sd
abfromDMthe w
condi
tions,
but within
transition.
Then,
the
markelement
nt
edi
by transition t.
assessment algori
thm represe
are
matrix isdefine stablished by headquarters.
situation
o
ass,
determines whichP and Plgorithm,
Th etw
(IF)produce
in together
the information
fusion
fr t4 will be used
processed
then by the sing
isHowever,
p,.
assessment
the situation which
protoduc
e DMO
this
canresponse
selecntinueon
acclgording tohmthe to
not pac
caseivesoutput
the
doassessme
respgaonse
p.
In this
ecituation
a
to the
s
envi
he
roent,
is
but command
fron
t
If there were no decision switches, then the Petri
connect
e directl
d
y
Anet
represe=
if P
and
t
no
are t
(Peterson,
represented in terms Of a flow matrix
s oan utputte
a place and j denote at.
1981). Letif iP
p
and situation
P , are
assessments,
The two situation
decisionmaker.
is the revised
transition
tI; the output
is defined
as anfollows
matrix -1,
if Pi is
input place to transition tj
thrane
sintions.
c
hoosessed
the responseby
t,,
tch which in turn,
swi
Pi and
are not connected directly
i
th 0, i
response selection algorithm isthatto produce DM sss
switches
can
be
d etricision
net
with
resituation
p asseIn
ssme,. case,this
the output
t
decision
1, by a set
P of etri nets withou
Pecon
represented
but is a command to thesignal
on the environmesent
rules that
of and by the set
or t
to switches
produce DMdecision
the
decis
processed either by t
Each one of
reproduced
control
the
these
switches.
organization's
output
Synchrougnizh
aoccurs
tionat
three
points
in
this
switche
a at
specific setting - thus
eliminatis
there
then areof
eah with
umber
positions,
the ropcont
cessing
is
sate time decisito both
Ms
transitions.
the info
turn, chooses the respion
se
synwitchron at
ized
Rhe
thatecommand
i
nterprocetation
tselecti
and, onceagain, atthm
be
switches
can
with decision
Petri
P
net A P
assessment,
t.
Pf 2of,
(Note thature
transituation
without decision
r
synchronization poinnetsof
explicit depiction
andthe
switches and by the set of decision rules that
are very usefutl in reproduce the
is prone
reason eithey
the switches. Each one of these reduced
control
organization'sal protoput.
f the
obtained bythesettin. For
eah onxample,
reducedis Petri
nets
four
specific
nizat setting - thus elimains
swiet for the orga
occurganizs
ational form could be u sed
T
his
very simplezation
One isobtained by setting switches,
s hip
reduced nets.
data,
between a central
d to
esribe
the iterreationput
and switch are
t, there
to t;
to
select
tran
positions 3the
h
eadquartering
is
anda regi onal office,or a corporate
f
ises.
ranchMarke
dataion
t
syand
chron
ized
at the informatiny
R
rspretation
national
a nced localgain,
are receiveommand inheadquarte
tr(DM)
andsition
t. local
h
operatiunre
e
Of Petri nets, namctively, P th
s
r
The explocal
opicit
depicrtion ofesynch ronizseation points,
about local conditions to headquarters; the latter,
on
the
basis
of both
national and
local
then the resulting matrix A has the same numerical
values as A,
but place 5 replaces place 4 in the
Figure 2.
TABLE 1
2
Flow Matrix A
PZ\t'
p
1
\
3
5
6
7
8
0
0
0
0
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
7 gure
-1
1
0
0
0
0
-1
-1
1
0
0
0
0
-1
1
8
9
10
11
12
13
15
16
0
1
0
0
0
0
0
0
0
0
0
0
0
0
01
0
0
0
0
0
0
0
0
connection matrix ae very useful in constructing
tests
for
capturing
data
errors.
The
?1
interconnection
matrix
for
the
two-person
organization of Fig. 1 is shown in Table 2.
2 I3
3P6
t2
1
2
3
4
6
Reduced net #2
Flow Matrix
0ABLE
0
0
-1
-1
0
1
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
-1
0
1
0
0
9
0
0
0
0
0
01
0
0
0
0
-1
-1
10
0
11
0
0
0
0
0
-1 00
0
0
0
0
0
0
-1
1
3.
The algorithm that is based on the interconnection
matrix is presented first (Jin, 1985). The problem
is to find all possible paths from an input place
(source) to an output place
(seful ink).
The
transition
thatf
follows
immediath e
source
place becomes
the main root of the tree that represents the
paths: every path forms a branch of this tree.
A\t
fourth row and transition t4 replaces transition t3
in the third column.
It
is
to
constructom
alternative
representation
the Petri net in terms of an
interconnection
9
1
matrix,
C, that
is
based
on
(columns) and ontransitions
lnks (rows) raof
than places.
The elements 0Ci
this -1 inter0
connection
are
15
0 matrix
0
0 defned
0
0 as folows:
0
0
1
1
row, ifcolumn.
link i leav
infourth
the third
es tra
+1, if link
-
jnsition
It
=
o, if link i is not conrnected to
representansitionofmatrix,
the Peti C, net
of
interconnection
thatin is termbased
arrives at transitions)
PATH IDENTIFICATION
an
on
rather
The matrix is constructed row by row by considering
each
link
that
leaves a transition
(-1) and
indicating the transition to which it is directed
(+1).
Since each row represents a single link, it
must
have
and
oneto -1
a
C.. ifexactly
r, link i cne
is not connected
co
transequently,
t
ne
sum of its elements must be zero.
This is not the case, of course, for the links that
connecti
plac
rrives)
transitions
and
the matrix is input
constructed
row by torow
by considering
the
links the
thattransition
connect totransitions
output
indicating
which it is to directed
placthe
links
that connect
transitions
are recogzed
to outputforer
by a
row with a single +1, while the latter by a row
with a single -1. These properties
of the inter-
TABLE 2
The Interconnection Matrix C
1
3
2
4
5
6
7
8
9
10
11
1
1
0
0
0
0
0
0
0
0
0
0
2
-1
1
0
0
0
0
0
0
0
0
0
3
0
1 that is based on the interconnection
4 -1
is presented
matrix
1
first
in,0 0
0
0
0
0problem
5 to find
0an -1all possible
1place0
0
0
0
0
0
0
6of
0
11
0
00
000 -1
00
00
11
00
00
00
00
0
12
0
0 -10-1
1
0
0
0
7
0
0
0 -1
1
0
0
0
0
0
0
8
0
0
0
0ther
-1
1
represents the
9
0
1
0
0
0
10
0
03
0
0
0
C0
1
2
4Interconnection
5
6
7
8 atrix
9
10
110
133
14
15
167
1
0
0
0
00
11
0
0
0O
O
0
0
0
0O
O
118
180
O
O
O
0
O
0 0
0
0
0
O
O
0
0
01
0O - 0
10
0
0
0
0
0O
0O
O
0 0 0
0
I
0
1
0 -1
0O -10
O
0
0
1
-0
O
0
0
0
1
O
0
0
O
0
00
The Algorithm
The elements of C can be interpreted as follows:
(a) If C
= -1 and Ck = 1
then transition
aond, 0l
a
i1
_seCee
connedted
tran s
on
t
0
0
0
0
0
0
0
(b) If there are more than one (-1) in a column J,
then transition t is a root or a subroot of
he element
Algotree.of C can be interpreted as follows:
The
() If there are q (+1) in a column j, then q
paths coalesce after they reach transition tw.
flow matrices Ai, as described in Section 2.
The
above procedure, the determination of the minimal
support T-invariants, must be applied to each one
of the matrices Ai.
In this simple case, the null
space
of
each
Ai
is
one-dimensional
and,
consequently, there is only one firing sequence
that transfers a token from p,
to p,, in each
reduced net. For example, for the reduced net of
Fig. 2 and the flow matrix of Table 1, the only
minimal
T-invariant
supportis
j =
1
2
5
3
6
7
8
9
51
52
53
and the delay from to to t, is also equal to 3r.
The delay of a simple path consisting of n subpaths
is the sum of the delays
f the subpaths.
For
example, from Table 3, simple path Z1 is composed
of subpaths t(9,2) and (11l,1).
Then the delay of
Z: is given by
11
D =1 D
1 12 1+ D1,1
1 =
9,2
11,1
[(1+1+0)
+ (1+0) - O]
Each minimal support T-invariant of T represents a
complete path. In this example, each Ai yields one
decision switch is counted twice.
complete path for a total of four for
wi.
completepathfor a total
The of
delay
fourfor
of a complete path that
This procedure is very efficient for determining
complete paths (a set of concurrent simple paths).
If it is desired to determine the individual simple
paths, from a given source to a given sink then the
S-invariants of each Tz must be determined.
Again
(Memmi and Roucairol, 1979) the minimal support
i
invariants of A are sufficient to determine the
simple paths.
For the example of Fig. 2, there are four minimal
support S-invariants with 1 at the first place and
the sixteenth place (Pl and p:,). They are,
0
1
1
1]
from
the
previous
transition,
tJ,
conditional probability p(ti/tj) is 1.
7
8
9
10
11
12
13
~: =[1
1
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
a 5tll
go
-El
calculate
expected
delay,
probabilities
associated with each path need to be calculated
first.
Probabilities
are
usually
given
as
conditional- probabilities associated with each
transition,
If transition ti has only one input
11
6
0
To
1
4
0
delay is used to compare alternative organizational
forms.
I
3
4s -El
After all delays are computed,
the paths with
minimal time delay and those with maximal delay are
easily identified.
Often, however, the expected
16
2
then
the
If ti is a
0
0
0
0
0
0
1
0
1
0
1
1
1]
transition of a n-way decision switch t , a set of
conditional probabilities
(ti/t j ) < . will be
0
0
0
0
0
0
1
0
0
1
0
1
1]
assigned such that
If each marking representing a simple path is
expressed as a set of places, then the complete
path is the union of the four sets that correspond
to pi' i = 1,2,3,4.
There are four simple paths
that correspond to each Al, therefore there are 16
simple paths and four complete paths.
However,
only 10 Of these simple paths are distinct.
This
only 10 of these simple paths are distinct. This
is the same result obtained using the algorithm
based on the interconnection matrix C.
4.
n
p(tiJt
tj)
1
The
probability that
k
dk
k
(D
(3)
Di2
Di3
will
n-1
where k ranges over the indices of the transitions
in the subpath. Let D
Di, and D
be the delays
of three subpaths, t(i,fi,
(i,
and t(i,3)
ending at transition ti.
Then the delay up to that
transition is the maximum of the three delays,
i.e.,
Di = max
information processing
follow a certain simple path X with n transitions
is given by
CALCULATION OF TIME DELAYS
The algorithm for the computation of delays is
based on the following two propositions (Jin,
1985).
Consider the m-th
subpath
ending in
transition tsiand let the delay associated with a
Then the delay of the
transition tk be denoted d
subpath t(i m) is
kt
~D
=' 3
(5)
i=1
p(X) = p(tl)
im
of
concurrent simple paths, is the delay of the simple
path with the maximal delay.
For example, the
complete path of Fig. 2 which consists of simple
paths Z,, Z,, Z., and Z, has delay equal to 6T, the
delay of path Z, or Z,.
15
1
i =
consists
3v
(4)
Consider again the two-decisionmaker organization.
There are three subpaths that end at transition ts,
t(5,1), t(5,2), and t(5,3) as shown in Table 3. If
the delay at each transition is equal to a unit of
time, r, and if the decision switch is assumed to.
have negligible delay, then
nj P(tilti-1)
i=2
(6)
where X is the path number J, ti is some transition
on the path, and t
is the transition preceding
ti .
Table 4 shows the conditional probability
matrix Ps for the organization of Fig. 1. Table 5
shows the probabilities associated with the simple
paths and the complete paths for this organization.
The expected delay can be calculated
by the
following equation:
ng q
r
E
r
D
i=l
where Pi and D(i) are the probability and time
delay associated to the i-th complete path and r is
the total number of complete paths in a system.
There are 4 complete paths j; path tl has delay of
6r and probability of 0.372.
In this particular
example, because the delay of all complete paths is
6v, the expected delay is 6r.
TABLE 4
P
o.o
1.0
o.o
o.o
o.o
o.o
1.0
o.o
o.o
o.o
o0.
0.0
0.0
0.6
0.4
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
o0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
1.0
0.0
0.0
1.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
o.o
0.7
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
TABLE 5
has
128
organization
parallel
switches,
the
The corresponding number
distinct complete paths.
of complete paths for the hierarchical organization
is sixty four.
Probability. Matrix P, of Example
As a result of choosing a constant time delay of
one unit for each process or transition and zero
delay for the decision switches, the delay of each
complete path of the parallel organization is six
units, while the delay of the paths of the
units.
is
eight
organization
hierarchical
Consequently, the expected delays are six and eight
units, respectively.
The detailed analysis of this
found in Jin (1985).
Probabilities Associated With Simple Paths
and Complete Paths
Simple Path
Conditional Probability
1 x
1 x
1lx
1 x
1 x
1 x
1 x
1 x
1 x
1 x
Z,
Z,
Z
Z.
Z,
Z,
Z,
Z.
Z,
Zo
Complete Path
6.
1 x
1 x
1 x
1 x
1 x
0.4
0.6
1 x1
0.4
0.6
1 x 0.7 =
0.7 = 0.7
1 x 0.3 =
0.3 = 0.3
1 x 1 x 1
x 1 x 1 x
x 1x x 1
x 1 xx
x 1 x 1 x
x 1 x 1 x
0.7
0.3
x
1
x
x
1
i
0.3
x 1
I
0.7
x 1
x 1
=
x
x
=
x
x
0.3
0.3 = 0.12
0.3 = 0.18
0.7
0.7 = 0.28
0.7 = 0.42
- _- _
Probability
0.372
0.558
f,
t
0.028
44
0.042
_ _
Two three-person organizational forms have been
used to test the approach and the algorithm for
Both organizations have to
computing delays.
perform the same set of tasks: monitor an airspace
and handle objects that are within it. The first
illustrates
one,
parallel
the
organization,
is
airspace
The
decisionmaking.
distributed
divided into three sectors with each decisionmaker
Each DM monitors his
assigned to one sector.
sectors and makes decisions about the objects in
it. Since the objects can cross fron one sector to
another (e.g., in air traffic control, aircraft
move from one sector to another) decisionmakers #1
and #3 who monitor the outside sectors must
communicate (share information) with the middle DM,
#2. No DM issues commands or restricts the options
of the others.
In the hierarchical organization, the airspace is
divided into two sectors, with each one assigned to
a single DM. Since the resulting workload will be
high for each DM, a central region is defined that
A
supervisor is
two sectors.
the
stradles
not observe the airspace
introduced who does
directly, but receives information about objects in
the central region from both DMs. He processes the
received information (not raw data) and allocates
these objects to either one of the two DMs (command
inputs) depending on the trajectory of the object.
This would represent the action of a supervisor who
resolves conflicts and allocates some of the load
so that neither of the subordinates is overloaded.
The interconnection matrix for each organizational
The parallel organization
form was constructed.
had sixteen simple paths while the hierarchical had
Because of the interconnections between
twenty.
organization members and the existence of decision
can
be
CONCLUSION
In this paper, Petri nets have been used for
modeling distributed decisionmaking organizations
of
problem
The
protocols.
synchronous
with
computing organizational delays was formulated and
two approaches were described for its solution.
One is based on an algorithm and an associated
interconnection matrix, while the other is based on
the flow matrix and the associated S and TWhich approach is preferable depends
invariants.
on the number of decisions switches and the number
of positions of each switch. For a small number of
simple switches, the analytical method is effective
For more complex
insight.
__and yields additional
is
approach
algorithmic
the
organizations,
preferable.
Current research is addressing (a) the problem of
computing delays when asynchronous protocols are
used and (b) the design of asyncrhonous protocols
to meet delay specifications.
7.
5. APPLICATIONS
application
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