Selma(ID:7041/sel012)

Whenever SELMA is to be used in conjunction with QAS,
the command to run QAS must be given to the central time-sharing system before
SELMA is started. This restriction is necessary because the SELMA command
exchanger contains no provision for communicating with the command language
interpreter of the time-sharing system. SELMA may then be started by scheduling
and running the single task at the location whose address appears on the SELMA
binary tape. (This address is not specified here because it is subject to change
without notice.)

Just after SELMA is started, the light buttons shown in
Figure 2a appear on the screen. When these light buttons are present, SELMA is
said to be in the construction phase, and the light pen interpreter is adjusted
so that a diagram of a queueing model may be drawn or modified. The other two
phases indicated in Figure 2 are the results phase (Figure 2b) and the plot
phase (Figure 2c). The results phase allows the user to request results, whereas
the plot phase displays a graph of the last requested result and allows the user
to expand sections of the graph and to request labels for points of interest on
the graph.

Transitions between SELMA phases are depicted by Figure
3. In this figure, each phase is represented by a circle, and the light buttons
which produce transitions between phases are indicated by labels on directed
paths between phases.

element that so with appears cross tracking A type. its denotes which 2a)
(Figure screen of corner right-hand upper in button light selecting by created
is An created. must one least at performed, can construction steps other Before
model. new a begin user'started it when phase placed >SELMA>The various
elements which may be created are shown in Figure 4. The interpretation of each
of these elements has been described previously

. Each element symbol includes one or more ports, i. e.,
attachment points for connections to other elements. In addition, an element
symbol may have a body and one or more numerical parameters. Since no value is
assumed for the parameter of an element when the element is created, the
position of each parameter in a newly created symbol is indicated by a dot,
called a parameter dot.

The creation of new symbols is the only operation
required for the construction of a model which requires the use of light
buttons. All other operations are performed with the aid of the light pen
interpreter. As mentioned in the introduction, the action to be taken by the
light pen interpreter is determined from the part of the diagram which is
referenced and often from a recognition of subsequent patterns of motion of the
light pen. The part of the diagram which is referenced is easily determined from
the display structure (Section 4) through the use of subroutines available in
the executive system. However, since the executive system provides no facility
for recognizing patterns of light pen motion, this latter determination is made
completely within SELMA.

The method employed by SELMA to recognize patterns of
light pen motion may be described with the aid of Figure 5. Whenever a light pen
reference is made to a part of the diagram which requires that subsequent
patterns of motion of the light pen be recognized, tracking is started at the
coordinates of the light pen, and one or more threshold patterns, i. e.,
patterns of imaginary lines of the form shown in Figure 5, are placed on the
screen. The number of patterns and the coordinates and size of each pattern are
determined by the part of the diagram which is referenced. ( The size of each
pattern is specified by a parameter d. )

Whenever an element symbol is referenced with the light
pen while SELMA is in the construction phase (and the light pen interpreter has
not been modified by the TRANSLATE task or the push button interpreter), two
threshold patterns are placed at the coordinates of the symbol as shown in
Figure 6a, and a tracking cross is placed at the coordinates of the light pen.
If the symbol has connected ports, the fragment of the diagram which consists of
this symbol and all connections and symbols associated with it are moved with
the tracking cross. Otherwise, the symbol may be either moved or deleted,
depending on subsequent motion of the light pen. In Figure 6a, the significant
regions of the screen which are bounded by the lines of the threshold pattern
and/or the edge of the screen are labeled A, B, or M. The decision to delete or
move the symbol is made by recognizing transitions among these regions. The
symbol is deleted if the pen is stroked vertically across the symbol as shown in
Figure 6b. This motion is recognized as a sequence of five transitions between
the A and B regions without entering an M region. The symbol is moved with the
light pen if the light pen is moved into a M region. This latter motion is
depicted in Figure 3c.

In the above discussion of deleting and moving symbols,
a reference to a symbol with the light pen was assumed to be a reference to any
part of the symbol other than an output port. When an unconnected output port is
referenced on any symbol (regardless of whether or not the symbol has connected
ports), the drawing of a connection line is begun as shown in Figure 7a. A
threshold pattern is established at the coordinates of the output port. The
connection line is continuously updated as shown until the light pen is moved
into either a U or D region, at which time a new vertical segment is added to
the connection line (Figure 7b). A smaller threshold pattern is then established
at the coordinates of the output port. The original threshold pattern is moved
to the newly created corner of the connection line. If the light pen is now
moved into the E region, the vertical segment of the connection line is deleted,
the smaller threshold pattern is deleted, the larger threshold pattern is moved
to the coordinates of the output port, and action proceeds again as depicted in
Figure 7a. If the light pen is moved into an L or R region, a second horizontal
segment is added to the connection line as shown in Figure 7c. The small
threshold pattern is moved to the next to last corner, and the larger threshold
pattern is moved to the newly created corner. Again, if the light pen is moved
into the E region, the last segment of the connection line is deleted and action
proceeds according to Figure 7b. If the light pen is moved into a U or D region,
a second vertical segment is added to the connection line. The operation
proceeds in this manner until the drawing of the connection line is terminated
by either of the following events:
1. The connection
line is deleted by loss of tracking,
2. The connection
line is used to form or modify a connection.

Connection lines may be used to form the following five
types of connections:
1. Simple connection,
2. Random branch,
3. Priority
branch,
4. Random merge, and
5. Priority merge.
The first of
these, the simple connection, associates an output port of an element with an
input port of either the same or another element. An item flows from the output
port through the connection to the input port whenever an item is available at
the output port and the input port can accept it. A branch associates one output
port with several input ports. Whenever an item is available at the output port
and at least one input port can accept it, it flows through the branch to one of
the input ports which can accept it. If the branch is a random branch, no input
port can accept the item unless all input ports associated with the branch can
accept it. Whenever all of these ports can accept the item, one of them is
selected randomly in accordance with probabilities assigned to each of them. If
the branch is a priority branch, an available item flows through the branch if
any input port can accept it, and priorities assigned to the input ports
determine which port receives the item. A merge associates several output ports
with one input port. Whenever the input port can accept an item and an item
becomes available at at least one output port, one of the available items flows
through the merge to the input port. If the merge is a random merge, an item is
not available at an output port unless items are available at all output ports
which are associated with the branch. Whenever an item is available at all of
these output ports and the input port can accept an item, one of the available
items is selected randomly in accordance with probabilities assigned to each
output port. If the merge is a priority merge, an available item flows through
the merge if the input port can accept it, and priorities assigned to each
output port determine which item is selected.

For both branches and merges, selection of ports on the
basis of probability is indicated in a SELMA diagram by a dot (. ), and
selection of ports on the basis of priority is indicated by a diamond ( <>
) . Some examples of branches and merges which are drawn with these symbols are
shown in Figure 8. (The arrowheads on each branch are actually connected input
ports, which differ in appearance from unconnected input ports. ) In Figure 8a,
whenever the server has finished servicing an item and neither of the queues is
full, the item from the server is placed in the top queue with probability 0.3
or in the bottom queue with probability 0.7. The server in Figure 8b cannot hold
an item after it has been serviced, for there is always an input port which can
accept it. If the top queue is not full and the server finishes servicing an
item, the item is placed in the top queue. Otherwise, if the top queue is full
and the bottom queue is not, the item is placed in the bottom queue. If both
queues are full, the item leaves the system through the sink. In Figure 8c,
whenever the server is empty and no queue is empty, an item is placed in the
server from the top queue with probability 0.2, from the middle queue with
probability 0.5, or from the bottom queue with probability 0.3. Whenever the
right-hand queue in Figure 8d is not full, an item is placed in it from the
bottom left-hand queue if the bottom left-hand queue is not empty, or from the
top left-hand queue if this queue is not empty and the bottom left-hand queue is
empty.

All connections which are drawn with SELMA are
originally generated as simple connections, which later may be extended to form
branches or merges. A simple connection is drawn by starting a connection line
at an unconnected output port and terminating it at an unconnected input port by
passing the light pen over the input port while drawing the connection line.
SELMA will then stop tracking the light pen and will convert the connection line
to a simple connection.

Whenever a simple connection is to be converted to a
branch, the type of branch (i. e., random or priority) must be established. This
is accomplished by recognizing a pattern of light pen motion. In addition, a
simple connection may be deleted by stroking vertically across it in the same
manner that one would use to delete an element symbol. The threshold patterns
which are involved are shown in Figure 9. When the light pen is aimed at a
connection, a dot appears on the connection at the coordinates at which the
light pen was detected. If the connection is a simple connection and the light
pen is then moved into an M region without entering a W, an X, a Y, and a Z
region or moving back and forth five times between A and B regions, a connection
line is started at this point, and the dot remains to indicate that the
connection is to be converted to a random branch. The conversion is performed
when the connection line is completed by passing the pen over an unconnected
input port. This operation is shown in Figure 10a. If the light pen enters a W,
an X, a Y, and a Z region without entering an M region or moving back and forth
five times between A and B regions, the dot is converted to a diamond to
indicate that the connection is to be converted to a priority branch. When the
light pen is subsequently moved into an M region, a connection line is started
in the center of the diamond. This operation is indicated in Figure 10b. If the
light pen is moved back and forth five times between A and B regions without
entering an M region or a W, an X, a Y, and a Z region, the connection is
deleted. This action for deleting simple connections is the same as that for
deleting symbols, and it applies to branches and merges as well.

A merge, like a branch, is also formed by adding a
connection line to a simple connection. However, the connection line to be added
is drawn from an output port to the simple connection, rather than from the
simple connection to an input port. The type of merge (i. e., random or
priority) is determined by recognizing light pen motion as the connection line
is terminated. The thresholds involved in this operation are shown in Figure 11.
If the light pen enters the W, X, Y, and Z regions without entering an M region,
tracking of the light pen is terminated by SELMA, and a priority merge is
formed. If the user removes the light pen near the simple connection without
entering an M region, a random merge is formed. These operations are shown in
Figure 12. If the light pen is moved into an M region, the threshold pattern is
removed, and drawing of the connection line continues as through no simple
connection had been encountered.

Once a branch or a merge is formed, more element ports
may be associated with it by drawing additional connection lines. A connection
line to be added to a branch is drawn from the branch to an input port, whereas
a connection line to be added to a merge is drawn from an output port to the
merge. No threshold pattern is required to determine whether a dot or diamond is
to be placed at the intersection of the new connection line with the branch or
merge, since the type of branch or merge is already established.

The values of the probabilities and priorities which are
associated with branches and merges, as well as the values of the parameters
which are associated with certain symbols, are assigned through the use of push
buttons. Push buttons 0 through 9 represent the decimal digits 0 through 9,
respectively. Push button 10 represents a decimal point, and push button 11 is
used to delete erroneous parameter values. A parameter value is specified by
supplying a sequence of from one to four characters. The value is displayed in
the lower right-hand corner of the screen as it is inputted, and the light pen
interpreter is modified so that all parameter dots and parameter values, but no
other parts of the diagram, are sensitive to the light pen. If the light pen is
now aimed at a parameter dot or parameter value, the parameter dot or parameter
value is replaced with the new parameter value which was obtained from the push
buttons, and the light pen interpreter is restored to its original state.

Two modes of parameter values may be assigned: integer
and real. The function of each parameter value in the diagram determines its
mode, e. g., a queue length must be an integer, a probability must be a real
parameter, etc. SELMA refuses to accept a parameter value of the wrong mode.

Occasionally, a model will be so complex that its
diagram will not fit onto the display screen. For this reason, the diagram is
considered to be drawn on a 75"x 75" piece of "paper" which may be moved to
various positions under the screen. The TRANSLATE light button is provided in
the lower left-hand corner of the screen in the construction phase to provide
this function. When this light button is referenced with the light pen, the
light pen interpreter is put into a translate mode, and the TRANSLATE light
button is modified to indicate this fact. A subsequent reference to this light
button will restore the light pen interpreter and the TRANSLATE light button to
their original states. (The creation of a new element will also produce this
effect. ) When the light pen interpreter is in the translate mode, a subsequent
reference I to a point on the diagram causes tracking to be started and the
corresponding point on the "paper" to be affixed to the tracking cross. The
"paper" may then be moved by moving the light pen.

When a diagram is thought to be complete, SELMA may be
put into the results phase (Figure 2b) via a reference to the RESULTS light
button. When SELMA is in this phase, the user may request a steady-state
probability density function either for any single element I or for the entire
model. The light pen interpreter is modified when this phase is entered so that
the only function which can be performed I by referencing the diagram without
further modifying the light pen interpreter (via the TRANSLATE light button or
by assignment of parameter values) is the identification of elements.

When a density function is to be requested for a
particular element, that element may be identified by referencing it with the
light pen. A box of dotted lines is placed around the element and a
one-character alphabetic name is assigned to the element as shown in Figure 13.
If a different element is subsequently selected before the result request is
completed by referencing either the PLOT or TYPE light button (described below),
the box is moved to that element, and the name associated with the box is
changed to the name of the new element. A reference to the box with the light
pen will remove it from the diagram. When no box appears around any element, the
entire model, rather than any particular element, is identified.

The probability density function for the item (i. e.,
either the entire model or a particular element) identified in this manner may
be obtained in either of two forms: (1) as a histogram on the display screen, or
(2) as a table on the teletype. A histogram is obtained through a reference to
the PLOT light button, and a typed table is obtained through a reference to the
TYPE light button.

When a result is requested, it will not be returned if
the diagram is not complete or if contradictory parameter values exist. The
diagram is not complete under either of the following conditions:
(1) At least one port in the diagram is not connected, or

(2) At least one parameter dot remains on the diagram.

Contradictory parameter values exist if either of the
following conditions is true:
(1) Two equal priorities
are assigned to one priority branch or merge, or

(2) The probabilities assigned to a random branch or
merge do not sum to 1.0.
If any of these four
conditions is true, SELMA is forced into the construction phase, a large
flashing arrow is placed on the screen to indicate the location of the error,
and an appropriate comment is typed on the teletype. The flashing arrow
disappears from the screen when the error is corrected. However, if the error
involves contradictory parameter values, the arrow may not point to the value
which the user decides to change to correct the problem. In this event, the
parameter which is being indicated should be reassigned the same value to delete
the arrow and thus avoid confusion on subsequent errors.

If the result which was requested was a histogram and
none of the above errors is present in the diagram, SELMA enters the plot phase
(Figure 2c) . The diagram is removed from the screen, and axes for the graph are
displayed, together with a label for the abscissa. For the example in Figure 13,
this result is shown in Figure 14b.

Only the maximum and minimum values of the abscissa and
ordinate which could be represented without scaling the graph are represented on
the final plot. However, the user may request numerical values from QAS for any
particular bar in the histogram by pointing to that bar with the light pen.
(Subsequent references to other bars will erase previous values so that only the
last one is displayed. ) The result of pointing to the fourth bar in Figure 14b
is shown in Figure l5a.

In addition to facility for obtaining numerical values
for points on the graph, facility is also provided for looking at various
sections of the graph. This facility is necessary because a maximum of 50
abscissa values may appear in one graph (limited by local storage and resolution
of the display) and some results require more than 50 abscissa values to plot in
their entirety. It is also a convenience, since detailed examination of a graph
with a wide range of ordinate values or a large number of abscissa values is
more difficult without it. Various sections of the graph may be displayed
through the use of the RIGHT, LEFT, and DETAIL light buttons. If the RIGHT light
button is referenced, and then the bar at a particular abscissa is referenced,
the plot is modified so that that abscissa, together with all abscissas of
greater value (up to a total of 50 abscissa values) is plotted. In similar
manner, a plot involving an abscissa, together with all abscissas of smaller
value (up to a total of 50 values) may be obtained through the use of the LEFT
light button. For example, the graph obtained by applying the LEFT light button
to the abscissa 2 in Figure 15a is shown in Figure 15b. The DETAIL light allows
the user to specify the maximum and minimum abscissas he wishes to plot. After
the DETAIL light button is referenced, these two abscissas are specified by
referencing the two bars of the graph which correspond to them. For all of these
operations, the fact that a particular light button has been selected is
indicated by the removal of all other light buttons from the screen until the
new graph is completely specified.

As indicated by the above discussion, all of the very
common operations which are involved in the specification of a queueing model,
with the exception of parameter value input, are performed with the light pen
only. This type of operation was chosen because the user can usually generate
faster input with the light pen than with any other input device. (input of
parameter values with the light pen was found to be awkward, so push buttons
were used for this purpose. ) However, there are some operations which are
performed very infrequently, and, should they be performed accidentally,
compensating for their effect would be difficult. In particular, destroying the
current model in order to begin a new model, and terminating SK: LMA are such
operations. These operations are performed through the use of the keyboard,
which is a relatively inaccessable device to the user. In this way, accidental
performance of these operations is less likely than it would be if these
operations were invoked with the light pen.

All operations performed from the keyboard are initiated
by commands which are invoked by a single character. The keyboard interpreter
responds to each command by typing a word or phrase which is a more complete
description of the command. A command may be deleted before it is effective by
typing a rubout in place of any subsequent character. The command which destroys
the current model in preparation for beginning a new one is the following:
CLEAR? OK

(The underlined characters are those typed by the user.)
Although the character "C" would be sufficient for specifying this operation, a
confirmation (accomplished by typing "O") is required to help avoid accidental
performance of this operation. The fact that the confirmation is required is
indicated by the question mark (?) . If the user wishes not to confirm the
operation, he may type "N" for "NO". Similarly, the following command, which
terminates SELMA (and QAS), requires confirmation:
ESCAPE ? OK

Two other keyboard commands which represent infrequent
operations which cannot easily be reversed once they are performed are saving
the current model on a file at the IBM 360/67 and retrieving a queueing model
from a file. These two commands are the following:
SAVE
ON FILE ____
GET FILE _____
Each command is completed by typing a file name, or a file
name followed by a space, in turn followed by the word "ALL". (The command is
terminated by a carriage-return typed on the keyboard. ) In the former case, any
results which have been computed are not saved, but in the latter case they are
saved. The user is prevented from performing any other operations while a model
is being saved or retrieved. Retrieval of a model effects a "CLEAR" command
before the model is actually retrieved.

Although saving or retrieving a model is costly to
perform accidentally, no provision for explicit confirmation is provided. The
file name which is required to complete each command serves as a confirmation.
If an illegal file name is specified, a comment to this effect will be typed,
and QAS will ignore the command.

Some users have found the keyboard, rather than the push
buttons, to be a more suitable device for inputting parameter values. For this
reason, a command to substitute the teletype for the push buttons and a command
to specify a parameter value from the keyboard are provided. The device from
which parameter values are to be obtained is specified
by the command


  
  
    
    
  
    

VALUES FROM	PUSH BUTTONS
	TELETYPE

(The default device
is the push buttons. ) While the teletype is the input device for parameter
values, the following command may be given to specify a parameter value:
PARAMETER: ____
This command is
completed by typing a number which consists of from one to eight characters,
followed by a carriage return. All of the characters are sent to QAS, but only
the first four are displayed if more than four are specified. Real values are
restricted to be less than or equal to 999.9999 to avoid misrepresentation of
the decimal point on the display.

Two other keyboard commands are provided for purposes of
debugging SELMA and/or CAS. However, these commands are not normally available
to the user, since they deal with dataphone commands between SELMA and QAS. They
are described in Section 5, following a detailed description of the command
exchanger in Section 3 and a description of the data structure used to represent
the topology of the network in Section 4.

3. The Command Exchanger

In order to be effective, the frequently used operations described in the
previous section must provide rapid response to user inputs. If all programmed
operations were performed in sequence, the low data rate of the dataphone would
pose a threat to this response time in that the transmission of commands to
inform QAS of certain inputs might delay responses to subsequent inputs. For
this reason, the response task for each user input which requires communication
with QAS merely places the commands to be sent to QAS into a buffer. A separate
task, the command exchanger, then reads commands from this buffer and sends them
to QAS. Since the command exchanger and response tasks for user inputs are
executed asynchronously, the rate of dataphone transmission does not seriously
affect the rate of response to user inputs.

Since the dataphone under consideration is a half-duplex dataphone, SELMA and
CAS must not transmit simultaneously. For this reason, SELMA and QAS alternately
transmit records, with QAS sending the first record. Since CAS sends the first
record, QAS need not be executing before SELMA is started. Since a record
directed to the SELMA command exchanger is not acknowledged by the SEL executive
system until the command exchanger has removed it from the dataphone input
buffer, SELMA need not be executing before execution of QAS begins.

Each dataphone record which is sent from SELMA to QAS or from QAS to SELMA
contains exactly one command. The general format of a command is shown in Figure
16. The command is delimited by bytes whose value is FF₁₆, and all
other bytes in the command have values less than FF₁₆. (Although the
beginning and end of the record are sufficient to delimit the command, the two
delimiting bytes were included in the format to facilitate a possible future
program modification to permit transmission of multiple commands per record. )
The first byte after the initial delimiter specifies the "phase" of the command,
and the byte which follows it identifies the command within that phase. (The
command phase is not related to the phase of SELMA. ) The combination of the
phase and command bytes identifies the command, and, consequently, specifies the
format of the data bytes.

The formats of the commands which are accepted by QAS are indicated by Figure
17, and the formats of the commands which are accepted by SELMA are indicated by
Figure 18. The abbreviations for various groups of data bytes are interpreted as
follows:

  

CN
  

Connection name. A number between 129 and 254 which identifies a
  particular connection. (1 byte)

  

  

CPN

  

Connection port number. A number between 1 and 254 which specifies a
  particular port of the connection specified by an associated connection name.
  (1 byte)

  

CT

  

Connection type. A number between 129 and 254 which specifies the type of
  connection (e.g., simple
  

connection, random branch or merge, priority branch or merge). (1
  byte)

DW

In addition to those features described in Section 3, a special feature
is included in the command exchanger to facilitate testing of future
modifications to SELMA which affect the command repertoire. Whenever SELMA is
started, but the dataphone is not connected to another party, the teletype is
substituted for the QAS program. This is accomplished by the addition of a
command to the keyboard interpreter to permit simulated input from QAS and by
the routing of all command output to the teleprinter (to be typed in hexadecimal
format).

The command which is added to the keyboard interpreter is of the following
form:
QAS: [ ____ ]
The command is begun by typing the letter "Q" on the
keyboard. A command is then typed (except for delimiter bytes) on the keyboard
in hexadecimal notation. If a carriage return is typed instead of the command, a
null command is assumed. Otherwise, an even number of hexadecimal digits must be
typed, followed by the carriage return. (In all cases, the right bracket is
typed in response to the carriage return. ) If a phase byte is explicitly
inputted, a carriage return will be ignored until the command byte is inputted.
The command may be deleted at any time before it is completed by typing a null
character .

Because the command exchanger actually interprets the QAS keyboard command,
commands between the user and SELMA via teletype are subject to the same
restrictions that govern commands between QAS and SELMA. The first command via
teletype must be inputted by the user, and only one command is typed by SELMA
until the next command is inputted via keyboard. For example, assume that SELMA
is started with the data set disconnected. Only after the user'supplies the
command
QAS: [ 0000 ]
does SELMA respond by typing
SELMA: [ 0005 ]

to indicate that QAS should be initialized. (The zeros in the QAS keyboard
command were not underlined here because it is assumed that all null commands
are inputted simply by typing a carriage return.) If the user then again inputs
a null command, no response from SELMA is produced until the user performs some
action which generates one or more commands. The first of these generated
commands is then typed, and the others are retained until more QAS commands are
inputted.

Since the only communication between SELMA and QAS is in the form of
commands, the feature described above permits complete testing of new versions
of SELMA without the use of the IBM 360/67. This feature of the command
exchanger is intended only for this purpose, and should not be used by users who
are attempting to solve queueing models.

6. Foreseeable Modifications

The system described in this report was designed to demonstrate the
  usefulness of graphical input for the specification of queueing networks to a
  computer. It was not intended to be a tool for the queueing analyst. However,
  with a few modifications, the system could serve the latter purpose as well.
  

The elements which are available in SELMA
  (i.e., queue, server, source, and exit) may not be sufficient
  for the specification of many models. In fact, various users may disagree on
  what elements are sufficient. Since a universally acceptable set of elements
  is difficult, if not impossible, to derive, a definition capability for
  elements is desirable in SELMA. Since the ,"menu" of elements in
SELMA is table-driven, the inclusion of
  this facility in SELMA would
  not be difficult. However, a corresponding definition facility for QAS might
  be difficult to implement.
  

Another type of definition facility could allow definition of elements
  which would be equivalent to existing fragments. This definition facility
  could be supported exclusively by the display terminal, since the creation of
  a copy of such an element could be described to QAS by the sequence of
  commands required to generate the corresponding fragment. Each element defined
  in this way could be represented by a symbol such as one which is used to
  represent an element in the current version of SELMA. Much larger models than can presently be
  accommodated in the display terminal could then be accommodated because the
  detail of each fragment which represents a composite element would not have to
  be stored locally once the equivalent element was defined.
  

For SELMA/ QAS to be a
  useful tool, results should be available in forms other than probability
  distributions for individual state variables. A suitable generalization of
  this type of result might be a conditional probability distribution for an
  algebraic function of state variables, where the algebraic function and the
  conditions which affect the distribution are specified by the user. Such
  distributions could be further processed by QAS to produce conditional
  expectations of functions of state variables.