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A SCHEDULING
^
·
PROGRAM FOR A
MULTIPROGRAMMED COMPUTER
by
HERBERT JAY BOROCK
B.S., Columbia University
(1963)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF
SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1967
Signature of Author...;.-.;....4;.
.......
........................
Alfred P. Sloan School of Management, May 19, 1967
Certified by.....
Thesis Supervisor
Accepted
by..... 4.......
C.
..................
,Accepted
byee....n....
........
Committee o..
on
Graduate Students
Chairman,
Deparlnental
RESTRICTED DISTRIBUTION
This thesis has been given limited reproduction with the permission of the Alfred P. Sloan School of Management.
is issued with the understandingsthat
However,
this copy
none of the data resulting
this investigation will be used for advertising
from
or publicity purposes,
and that the thesis is solely for the confidential use of the organization or individual to whom it is addressed.
A SCHEDULING
PROGRAM FOR A MULTIPROGRAMED
COMPUTER
by
HERBERT JAY BOROCK
Submitted to the Alfred P. Sloan School of Management on May 19, 1967
in partial fulfillment of the requirements for the degree of master of
science.
ABSTRACT
This thesis describes a scheduling program for an IBM System/360
computer operating in a multiprogramming environment provided by the
IBM System/360 Disk Operating System.
In this environment up to three
programs compete for the use of the computer's resources.
A scheduling rule is developed for computer center operations
managers in the form of a computer program.
The program saves substantial amounts of computer time since it provides more efficient
schedules for the multiprogramming environment than could be provided
by the operations manager without using the program.
Input to the
program consists of statements describing the computer, the installation, and the programs to be scheduled.
Output from the program
consists of a suggested schedule, showing program sequence and clock
time for the scheduled period, which is a week.
The problem of scheduling the computer is solved by dividing the
problem into two smaller scheduling problems--a job shop scheduling
problem and a flow shop scheduling problem.
In the job shop scheduling problem, three programs compete for the use of the computer's
central processing unit and input/output channels. Given each program's characteristics, the time required to complete each program
is calculated by simulating the operation of the computer.
In the flow shop scheduling problem, each program can use up to
three areas of computer storage for program execution.
This is an
n-job, 3-machine sequencing problem, where each program is a job and
each area of computer storage is a machine.
Previous algorithms for
the flow shop scheduling problem are used to sequence the programs
for the class of problems in which two machines cannot operate on the
same job in parallel. New branch and bound algorithms are developed
for the class of three-machine problems in which two of the machines
can operate on the same job in parallel.
Thesis Advisor:
Donald C. Carroll
Title:
Associate Professor of Management
May
19, 1967
Professor Edward N. Hartley
Secretary of the Faculty
Massachusetts Institute of Technology
Cambridge, Massachusetts
02139
Dear Professor Hartley:
In accordance with the requirements for graduation, I herewith
submit a thesis entitled "A Scheduling Program for a Multiprogrammed
Computer."
I want to take this opportunity
assisted me in my research.
to thank the many people who
The National Dairy Products Corporation provided the financial
Paul A. Strassmann, Director of Systems
assistance for this work.
and Procedures, suggested the topic. Robert I. Benjamin, Manager
of Installation Support offered many helpful suggestions.
My thesis committee, consisting of Professors Donald C. Carroll
and John F. Pierce, were a constant source of encouragement and were
always available for advice.
Computer time was supplied by the M.I.T. Computation Center and
the Computer Facility of the Alfred P. Sloan School of Management.
Mrs. Anita Ciminero typed the final draft.
Sincerely,
Herbert Jay Borock
- 5-
TABLE OF CONTENTS
Chapter
1. INTRODUCTION
.............
.
.
Previous Research.....................
.e..e.
o
.
.
9
..
ee....
..
ee.ee
The Problem...........................
The Solution ..........................
.
e
ee.
10
12
...........
..............
13
............................
Control Device.............13
The Scheduling Program as a Management
15
Chapter 2. SCHEDULING ENVIRONMENT.................................
15
The Computer Components...........................................
18
1.........8....
The Programs.....................................
19
The Weekly Schedule...............................................
The Job Shop Scheduling Problem ..................................20
22
The Flow Shop Scheduling Problem..................................
Chapter 3. SIMULATION METHOD.....
.........
Two Component Example...........
.....
Use with the Flow Shop Scheduling
...............
.......... ...............
eeeeee,.e-.
........................
®e eeee
...
lo
e24
30
Problem.
General Case......
.........
31
Example
.........
........................ 32
, 39
Chapter 4. PREVIOUS ALGORITHMS....................
................
The N-Job, 2-Machine Sequencing Problem..................... ..... 40
The N-Job, 3-Machine Sequencing Problem...........
Use of the Algorithms
in Scheduling the Computer..
......-..-.....-
41
44
ee....... .
- 6-
Chapter
BRANCH AND BOUND ALGORITHMS .............
5.
.............
45
46
.............
Parallel Operations on Input .......................
47
Parallel Operations on Output......................
................
.............
47
Case 1:
Input, Different Machines
Case 2:
Output, Different Machines ................
Case 3:
Input, Identical Machines ................
Case 4:
Output, Identical Machines ................
Case 5:
Input, Same Operation, Different Machines. .............53
Case 6:
56
Output, Same Operation, Different Machines .............
Chapter 6. SCHEDULING RULE.........................
.
.............
49
.............
49
.............53
.............60
Chapter 7. CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH...... 62
Appendix A. PROGRAM USER MANUAL................................. 4
Input Formats
.6
................... ............................... 65
............................... 75
Output Formats...................
Appendix B. PROGRAM LOGIC MANUAL.
............................
...
9
Program Organization............................................ 79
Storage Requirements
Flowcharts
........... ............................... 83
....................
FORTRAN Source Statements ........
...............................
BIBLIOGRAPHY
...............................
86
..............................
127
263
.7
- 7 -
LIST OF ILLUSTRATIONS
Figure
1.
Component use as a fraction of total program time....25
Figure 2.
Component use as a fraction of one time unit.......26
Figure 3.
Total computer use by program 2 during one time unit. 2 7
Figure 4.
Component use by program 2 during one time unit ..... 27
Figure
Component use by programs
5.
1 and 2 during one time
28
unit
...................
6.
Total computer use by program 3 during one time unit.29
Figure 7.
Component use by program 3 during one time unit ..... 29
Figure 8.
Component use as a fraction of total program time ....3 3
Figure 9.
Component use as a fraction of one time unit ........ 34
Figure
Figure 10. Total computer use by program 2 during one time unit.35
Figure 11. Component use by program 2 during one unit of time...36
Figure 12. Component use by programs
1 and 2 during one time
36
unit ...............................................
Figure 13. Total computer use by program 3 during one time unit.38
Figure 14. Component use by program 3 during one time unit ......38
Figure 15. Tree structure
for 4-job, 3-machine
problem ..........43
Figure 16. Tree structure for 4-job, 3-machine problem, with
parallel operations on different machines,
input...48
Figure 17. Tree structure for 4-job, 3-machine problem, with
parallel operations on different machines,
output..50
Figure 18. Tree structure for 4-job, 2-machine problem, with
two identical machines ....
.......................52
-8 -
Figure 19. Tree structure for 4-job, 3-machine problem, with
two identical machines,
input ..................... 54
Figure 20. Tree structure for 4-job, 3-machine problem, with
two identical machines,
output....................55
Figure 21. Tree structure for 4-job, 2-machine problem, with
identical operations
on different machines ....... 57
Figure 22. Tree structure for 4-job, 3-machine problem, with
identical operations on different machines, input.5 8
Figure 23. Tree structure for 4-job, 3-machine problem, with
identical operations
on different machines,.oUtput .59
Figure 24.
Overall program organization.
Figure 25.
Flowchart symbols .................................
......................
80
88
- 9 -
Chapter
1
INTRODUCTION
Scheduling a multiprogrammed computer has been recognized for
some time as analogous to scheduling the work
in a factory.
However,
little practical work has been done to assist computer center opera-
tions managers in scheduling programs run on a multiprogrammed computer.
Part of the reason for this is that until recently multi-
programmed computers were used in only large computer installations.
Further, when researchers tried to attack the problem they dealt with
hypothetical
computers.
multiprogramming
With the introduction
is being extended to smaller sized installations.
The typical business computer installation
lem
is now faced with the prob-
of scheduling work on a multiprogrammed
In most cases the installation
programming
of the IBM System/360,
computer.
chooses a manufacturer-supplied
system which provides no scheduling
job-to-job transitions.
All the programming
assistance,
except for
system does is free the
operator from some routine work so that he can devote more time to
other routine work.
No assistance is given in actually scheduling the
jobs, i.e., deciding which job sequence results
use of the computer.
in the most efficient
The purpose of this research
computer center operations manager with guidance
is to provide
in this area.
the
Using
the scheduling program developed here, the manager will be able to
obtain good or "near" optimal schedules.
- 10 -
Previous Research
Lynch
described the operation of a university computing center.
In that environment,
the majority of the programs use less than 5
minutes of computing time.
Less than 5% of the programs use any input/
output devices other than for card input and printer output.
the programs are limited by computing
time, rather than input/output
Use of the computer is allocated
time.
All of
to different groups of individ-
uals on a priority basis.
Hutchinson
described the simulation of an entire computing center,
including accounting-in and accounting-out operations, and the use of
a tape library.
The scope of his study is limited to large aerospace-
manufacturing computing centers. He varied manpower levels and equipment to determine the most efficient center. His measure of service
was the amount of time to process jobs.
McKenney
3
developed a system which defined computer jobs so that
they could be recursively
independent
1
segmented into independent subjobs.
An
subjob is defined by the source of its inputs, the definition
W. C. Lynch, "Description of a High Capacity, Fast Turnaround
University Computing Center," Communications of the ACM, (IX, February,
1966), 117-123.
2
George K. Hutchinson. "A Computer Center Simulation Project,"
Communications of the ACM, (VIII, September, 1965), 559-568.
3
James L. McKenney,
"Simultaneous
Processing of Jobs on an
Electronic Computer," Management Science, (VIII, April, 1962),
344-354.
- 11 -
of its outputs, and a relation between its inputs and outputs.
He
concluded that computer programs should not be restricted by arbitrary
sequence constraints,
so that they could be segmented into independent
subjobs, which could be processed by different computer components
in parallel.
Heller4 used a linear graph technique to indicate the use of each
computer component and the precedence
relationships
among subprograms.
He formulated the problem; he did not solve it.
Schwartz
5
studied the feasibility of restructuring
programs to
take advantage of multiple facility computers. He attempted to develop
a procedure by which the computer could restructure
then schedule them.
the programs and
He was concerned with the network properties
of
computer programs and the derivation of a precedence matrix.
Codd 6' 7 developed an algorithm
for a multiple
He considered two kinds of facilities--those
and those that were space-shared.
facility system.
that were time-shared,
The purpose of his algorithm was
to schedule a set of programs so that they could be executed
in a
minimum amount of time.
4J.
Heller, "Sequencing Aspects of Multiprogramming, !'Journal
of the Association
426-439.
for Computing Machinery,
(VIII, July, 1961),
5Eugene
S. Schwartz, "An Automatic Sequencing Procedure with
Application to Parallel Programming,"Journal of
the Association
for Computing Machinery (VIII, October, 1961), 513-537.
6
E. F. Codd, "Multiprogram Scheduling:
Parts 1 and 2. Introduction and theory," Communications of the ACM, (III, June, 1960),
347-350.
7
E. F. Codd, "Multiprogram
Scheduling:
Parts 3 and 4. Scheduling
Algorithm and External Constraints," Communications of the ACM, (III,
July, 1960), 413-418.
- 12 -
The Problem
The computer we are concerned with is used by business concerns
for useful work on a continuing
basis.
The work in these installations
consists of high volume input and output.
It is not uncommon
for a
single program to include hours of card reading and printing.
Since
these input/output operations are slow relative to other processing,
systems programs have been developed
to provide a computer environment
where one job's processing can be overlapped with other jobs' input/
output operations.
In this way the work
one job is not delayed by excessive
load can be balanced so that
input, say, and another
is not
delayed by excessive processing.
The computer environment we are concerned with consists of three
programs being multiprogrammed together. One program is a processing
program, and the other two programs are input/output
majority
programs.
The
of the jobs on the computer use the processing program area of
storage and at least one of the input/output
Most jobs enter the computer
programs.
installation at the beginning of one
day and must be completed at the end of that day.
The remainder must
be completed within the same week they enter the installation.
Other
than having different due dates, there is no priority among jobs.
This environment
poses two problems.
each program take to run on the computer?
First, how much time does
The time is easy to cal-
culate if there is only one program using the computer,
since the input
for each program consists of a known number of transactions,
and the
processing time per transaction and input/output time per transaction
are known.
However,
in this environment
three programs are sharing
- 13 -
the timing problem.
the computer, thus complicating
Second, given that we know how long each program takes, how do
The
we schedule the programs to make efficient use of the computer?
scheduling
problem is easy if each program uses the computer
However, in this environment
once.
three times--once
only
each program may use the computer
for processing, and twice for input/output.
The Solution
The two problems are solved separately.
treated as a job shop scheduling
for the use of the computer's
problem where three programs compete
central processing unit and input/output
Given each program's characteristics
channels.
machine
The timing problem is
instructions per input record),
(e.g., number of
the time required to complete
each program is calculated by simulating the operation of the computer.
The scheduling problem itself is treated as a flow shop scheduling
problem where
a machine.
machine.
However,
each of three areas of computer
to
storage is analogous
Each time a program uses the computer it uses a different
This is the familiar n-job, 3-machine
there is a substantial
may perform the same function,
difference.
sequencing
problem.
Both input/output
areas
i.e., they may be identical machines,
or at the very least, a single program may use them in parallel.
New
branch and bound algorithms are developed for the parallel operation
cases.
The Scheduling Program as a Management
Control Device
The scheduling rule developed here is provided to computer center
operations managers
in the form of a computer program.
The program is
- 14 -
for the IBM System/360,
written
computer center.
the same computer
The program saves substantial
time since it provides more efficient
that is in the
amounts of computer
schedules for a multiprogramming
environment than could be provided by the operations manager without
using the program.
Input to the program consists of statements describing
puter, the installation,
and the programs to be scheduled.
the comThe pro-
gram descriptions, which are provided by the programmers who design
the programs, include such things as the usage of various input/output
devices.
Output from the program consists of a suggested schedule,
showing
program sequence and clock time for the scheduled period, which is a
week.
The operations manager has the opportunity
to revise the schedule
after which the schedule becomes his plan for processing
the work in
the computer center. The actual performance of the computer center and
its manager can then be compared
to the planned performance,
more efficient and effective performance
can be produced.
so that
- 15 -
Chapter
SCHEDULING
A scheduling
scheduling rule.
environment
2
ENVIRONMENT
consists
of resources, jobs, and a
The resources and jobs in this scheduling environ-
ment are the components of a computer and the programs run on the
computer, respectively. They are described in this chapter. The
scheduling rule is described
in Chapter 6.
The Computer ComponentsL
The computer for which this scheduling program is written
IBM System/360.
A System/360
processing unit
(CPU), up to six selector channels,
channel, and input/output
is the
consists of main storage, a central
devices attached
and data reside in main storage.
one multiplexor
to the channels.
Programs
The CPU is used to execute a program's
instructions and to transfer data between the input/output units and
main storage via the channels.
The channels are used to relieve the CPU of the task of communicating directly with the input/output units and permit instruction
execution to occur concurrently with data transfer.
transfers data to or from a single input/output
A selector channel
device at a time and
is capable of handling high speed input/output devices.
channel can operate in either of two modes:
multiplex
The multiplexor
mode for low-
speed devices, and burst mode for high speed devices.
1
For a more detailed discussion see IBM System/360 Principles
Operation (Poughkeepsie:
IBM Corporation, 1964).
of
16 -
In the multiplex mode, a large number of low-speed input/output
devices time-share the channel's single data path, and the channel
simultaneously
transfers data between all the devices and main storage.
In the burst mode, a single high-speed
device uses the channel's data
path, and that device does not relinquish the channel until its data
transfer
operation is complete.
The CPU can concurrently execute a machine
data to or from a single input/output
instruction, transfer
device on each of the selector
channels, and transfer data to or from either a single high-speed
device on the multiplexor
devices on the multiplexor
channel
channel (multiplex mode).
System/360 is designed to be multiprogrammed.
System/360
low-speed
(burst mode) or multiple
A multiprogrammed
is one in which two or more programs reside in main storage
and compete for the use of the CPU, the selector channels, and possibly
the multiplexor
channel.
They compete for the use of the multiplexor
channel. if at least one program wants
to use that channel
in the
burst mode; otherwise, all programs can use the multiplexor
channel
at the same time.
A programming
system is used to control the operation of the
computer and to allocate the use of the CPU and the channels among
the programs. The programming system for which this scheduling program
is written is the IBM System/360 Disk Operating System (DOS/360).
DOS/360 provides a multiprogramming environment where main
storage is divided into three areas, each of which can contain a
single program.
Programs are loaded into one of the areas of main
For a more detailed discussion see TRM System/360 Disk Operating
System: System Control and Service Programs (Endicott: IBM Corporation.
1966).
- 17 -
storage
(called the background area) by DOS/360 as they are needed,
Programs are loaded into the other
without operator intervention.
two areas of main storage (called foreground area one and foreground
area two) by the operator.
When they compete for the CPU and the channels, the program in
foreground area one has the highest
priority, the program in fore-
ground area two has the next highest priority, and the program in the
background area has the lowest priority.
Control of the CPU is given
to the highest priority program which is not waiting
of an input/output
instructions
operation.
for the completion
That program can then issue machine
and initiate input/output
operations.
If the program issues a multiplex mode input/output operation,
and the multiplexor
channel is operating
input/output operation
in the multiplex
is executed immediately.
If the program issues
a burst mode input/output operation and the multiplexor
or a multiplex mode input/output
mode, the
hannel is in use,
operation and the multiplexor
is operating in burst mode, or an input/output
operation
channel and the channel is in use, then the input/output
channel
to a selector
operation is
executed when the channel finishes executing the previous input/output
operation issued to it.
If the program cannot execute any more machine instructions until
a particular input/output
operation is complete, the program relinquishes
control of the CPU to a lower priority program which is not waiting
the completion
of an input/output operation.
When the input/output
operation is complete, the higher priority program seizes control of
the CPU from the lower priority program.
for
- 18 -
The Programs
programs
The programs run on the computer are either foreground
or background
programs, depending on the area of main storage in which
they are executed.
The foreground programs are used to transfer data between a lowspeed device on the multiplexor
channel and a high-speed
device.
Since
the operation of the low-speed device takes much longer than the instructions used to initiate the input/output
operations on both devices, only
a small portion of the total program time is used to execute instructions
A somewhat longer time is used to transmit data over the
on the CPU.
channel to which the high-speed
device is attached.
is used to transmit data over the multiplexor
mode.
This permits the background
channel in the multiplex
program to use the CPU and the chan-
nels while the foreground programs are waiting
input/output operations on the multiplexor
Foreground
Most of the time
for the completion
of
channel.
programs may be used as input or output transcription
programs in conjunction with background programs, or they may operate
independently
of the background
programs.
may be used for the same function
Both foreground
programs
(e.g., output transcription)
at the
same time.
For example, a typical background
input/output
program includes many low-speed
operations on the multiplexor
channel for input transcrip-
tion (e.g., card reading) and output transcription
If the background
program actually
(e.g., printing).
performed these operations,
stantial amount of the program time would consist of waiting
completion
a sub-
for the
of these operations as is the case with the foreground programs.
- 19 -
Instead of performing card reading and printing directly with the
background
program, a foreground program can transfer the input data
to a high-speed
program can then read the data
device, the background
from that device and put its output data on another high-speed
device,
which another foreground program can then transfer to the printer.
In this example, while one foreground program is reading input
for a subsequent background program, another foreground program is
printing the output for a previous background
ground program
program, and a back-
is being executed using data previously read by the
first foreground program and supplying data which will subsequently
be printed by the second foreground program.
Depending
on the rela-
tive volumes of input and output, both foreground programs may be used
for either input or output, one foreground
program may be used for
either input or output while the other is doing independent work, or
both foreground programs may be doing independent work.
Some sets of background
programs represent
unit of processing in terms of DOS/360.
a single accounting
The programs in each set
must be executed in sequence, with no other programs intervening.
These background program sets may operate in conjunction with foreground programs as described above.
The Weekly
Schedule
In the installations we are concerned with, most of the programs
enter the computer installation at the beginning of one day and must
be completed by the end of that day.
Other programs have
different
turnaround times, but all programs must be completed during the same
calendar week that they enter the computer
installation.
Thus, each
- 20 -
week a static schedule must be produced for a group of foreground
programs and background
programs.
Given descriptions
(CPU, channel, and input/output device performance
the installation
(computer configuration,
of the computer
characteristics),
number and normal use of
foreground areas, and work week), and the programs
(execution times,
input/output volumes, device usage, and possible foreground program
usage), the object is to produce an optimal or near optimal schedule
for the week.
in the computer center
During the week, conditions
may change, such that the original schedule
schedule:
is no longer the best
new programs are added to the schedule; the operators make
errors and programs have to be rerun; an
input/output
device goes
In that case, a new schedule would have to be produced, given
down.
the descriptions
of the computer, the installation,
and the programs
when the change occurs.
The problem of scheduling programs in this environment
divided into two smaller scheduling
can be
problems--a job shop scheduling
problem and a flow shop scheduling problem.
The scheduling rule is
in the form of a computer program which is run on the same computer
for which the schedule is produced.
As long as the computer is
capable of performing useful work, then no matter what kinds of
changes occur during the week, a new schedule can be produced.
The Job Shop Scheduling Problem
Each of the three programs being multiprogrammed
the use of up to nine computer components:
can require
CPU for instruction
execution, up to six selector channels for data transfer, multiplexor
channel for data transfer in burst mode, and multiplexor
data
ransfer
in multiplex mode.
channel for
At any one instant in time, only one
program can use the CPU for executing instructions, only one program
can use each of the selector channels
to transfer data between main
storage and an input/output device, and either only one program can
use the multiplexor
channel to transfer data in burst mode or all pro-
grams can use the multiplexor
channel to transfer data in multiplex
mode.
Given a program's description,
it is easy to calculate the time
to complete the program, if it is the only program on the computer.
All one has to do is calculate its time on the CPU and on each of the
channels, and then choose the longest of these times as the total program time, since the operations on these computer components are
overlapped.
However, when more than one program is being run at the same
time, the scheduling problem is more complex.
is assigned
to the same resource
mode) at the same time.
More than one program
(multiplexor channel in multiplex
More than one resource
(e.g.. CPU and a
channel) is assigned to the same program at the same time.
gram uses a resource
A pro-
(CPU or any channel) more than once.
It would be unwise to keep track of all the changes in resource
assignment
in order to calculate the time for each program
multiprogramming
each machine
environment.
in the
To do so would require simulating
instruction and input/output
operation.
Such a sim-
ulation would require more time to execute on a computer than would
the schedule itself.
- 22 -
The approach taken here is first to calculate the total CPU time
assuming that each program
and channel time for each of the programs,
Then, using the DOS/360 program
is the only one on the computer.
priorities, and the fact that the CPU time of one program can be overlapped with the channel time of that program and the other two programs,
The simulation method used to cal-
new program times are calculated.
culate these times is described
in Chapter 3.
The Flow Shop Scheduling Problem
Given the program times calculated
in solution to the job shop
scheduling problem, the problem of scheduling
computer reduces to a flow shop scheduling
the week's work on the
program.
In the flow shop scheduling problem, each area of main storage
represents a machine.
on these machines.
The program times represent operation times
Jobs consist of from one to three operations, with
no more than one operation on each machine.
A three operation
job
consists of a background program run in conjunction with two foreground programs which read the background
program's input and/or print
its output.
If both foreground programs are used in conjunction with the
background programs, then we have an n-job, 3-machine sequencing
problem.
If only one foreground
program is used in conjunction with
the background programs, then we have an n-job, 2-machine
problem, since the other foreground
sequencing
program can always operate in
parallel with both the first foreground program and the background
program.
- 23 -
There are four distinct sequencing
1.
problems:
The n-job, 2-machine sequencing problem
(background programs
plus a foreground program for either input or output).
2.
The n-job, 3-machine
sequencing problem
(background programs
plus one foreground program each for input and output).
3.
The n-job, 3-machine
sequencing problem
(background programs
plus two foreground programs for input).
4.
The n-job, 3-machine
sequencing problem (background programs
plus two foreground programs
for output).
Algorithms which provide optimal solutions to problems 1 and 2
have been developed previously.
They are described in Chapter 4.
As
far as I know, algorithms which provide optimal solutions to problems
3 and 4 have not been developed previously.
for problems 3 and 4 are described
Branch and bound algorithms
in Chapter 5.
-4 -
Chapter
3
SIMULATION IETHOD FOR THE JOB SHOP SCHEDULING
Up to three programs may be multiprogrammed
being simulated.
PROBLEM
in the environment
For each program, the total CPU time and the total
channel time for each channel are calculated, assuming that each
program is the only one on the computer.
The maximum of these times
for each program is the total program time for that program in the
non-multiprogrammed
environment,
puter components are overlapped.
since the operations on these comThe times on these computer
com-
nonents are then expressed as fractions of the total program time for
each program.
Two Component Example
Suppose there are three programs--program
1, program 2, and pro-
gram 3, each of which uses only two computer components--the
the multiplexor
channel in the multiplex
mode.
When
CPU and
it is the only
program using the computer, program 1 uses the CPU for 10 minutes and
the multiplexor
channel for 100 minutes; program 2 uses the CPU for
10 minutes and the multiplexor
channel for 50 minutes; and program
3 uses the CPU for 45 minutes and the multiplexor
minutes.
Then in the non-multiprogrammed
channel for 150
environment,
gram times for programs 1, 2, and 3 are 100 minutes,
the total pro-
50 minutes, and
This is true for most of the programs we are concerned with.
Some programs, however, do have non-overlapped operations. The
scheduling rule developed here provides for such programs by including
the percentage of non-overlapped operations in the input to the
scheduling program.
- 25 -
150 minutes, respectively,
since the time on the multiplexor
channel
is the maximumtime for each program.
Program 1 uses the CPU 10/100
or .1 of the total time it uses
the computer, and program 1 uses the multiplexor channel 100/100 or
1.0 of the total time it uses
the
computer.
Similarly,
uses the CPU .2 of the time, and the multiplexor
program
channel 1.0 of time;
and program 3 uses the CPU .3 of the time, and the multiplexor
1.0 of the time.
These times are shown in Figure
________
_______
________
________
2
channel
1.
…_______
_______
________
_______
Programs
CPU/Total
time
Multiplexor
Figure 1.
Channel/Total
time
1
2
3
.1
.2
.3
1.0
1.0
1.0
Component use as a fraction of total program time
We now know how long each program uses each computer component
during one time unit, when the programs are not multiprogrammed
gether.
to-
We have to calculate how long each program uses each computer
component during one time unit, when the programs are multiprogrammed
together.
During one time unit, however, a program uses each computer
component in the same proportion that it uses the component over the
total program time.
Therefore, we have to calculate how long each
program uses the computer during each time unit.
Let CPU/MPX(i).
represent the fraction of the total program time
that program i uses both the CPU and the multiplexor
MPX(i) represent
channel.
Let
the fraction of the total program time that program
i
- 26 -
uses only the multiplexor channel.
Then CPU/MPX()=.l and MPX(1)2-9,
since program l's total program time is 100 minutes, and during 10
the
minutes of this time it uses both the CPU and
channel;
multiplexor
and during the remaining 90 minutes it uses only the multiplexor channel.
CPU/MPX(2)=.2and MPX(2)-.8; and CPU/MPX(3)s.3 and MPX(3)-.7.
Similarly,
These fractions are shown in Figure 2.
I
IL
I
Figure 2.
CPU/MPX(l)
.1
[
MPX(l) =.9
2
i
MPX(2)-.8
I
MPX(3)-.7
,CPU/MPX(2)=.
CPm/MPX(3)-.3
t
1
Component use as a fraction of one time unit
Now suppose that the three programs are multiprogrammed
together,
and that program 1 has the highest priority, program 2 has the next
highest priority, and program 3 has the lowest priority.
During one time unit, program 1 uses the computer for the entire
time unit, since it can seize control of the CPU whenever
it needs it.
Program 2 can use the CPU only when program 1 is not using it, and
program 3 can use the CPU only when programs
1 and 2 are not using
it.
When program
1 is using both the CPU and the multiplexor
program 2 can also use the multiplexor
operations on the multiplexor
channel, since multiplex
mode
channel can be overlapped with each other.
During any time period, program 2 uses the multiplexor
for a fraction MPX(2)=.8
channel,
of the time period.
channel alone
During the time period
- 27 -
CPU/MPX()=.l time
unit,
program 2 uses the multiplexor channel for
(.8)*(.l1)=.08 time unit.
During the time period MPX(1).9
either the multiplexor
time unit, program 2 can use
channel alone, or both the multiplexor
channel
In any case, program 2 uses the computer during the
and the CPU.
The total time that program 2 uses the computer
entire .9 time unit.
during one time unit is .08+.9-.98 time unit, as shown in Figure 3.
I
CPU/MPX(l)=.l
[
tMPX(2)=.08
I
MPX(1)=.9
t
[
MPX(2)+CPU/MPX(2)=.9
i
Total computer use by program 2 during one time unit
Figure 3.
However,
during any time period, program 2 uses each computer
component in the same proportion
total program time.
that it uses the component over the
Specifically,
CPU/MPX(2)=.2
and MPX(2)=.8.
But program 2 is active for only .98 time unit, so in one time unit,
time unit, and MPX(2)=-(.8)*(.98)=.784
CPU/MPX(2)=(.2)(.98)=.196
time unit.
These times are shown in Figure 4.
Note that .704+.08=
.784=MPX(2).
i
I
cPo/MPX(l)=.
. MPX(2)-.08
MPX(2)=.08
Figure 4.
,
MPX(1) =.9
MPX(2)+CPU/MPX(2)=.9
CPU/MPX(2).196
[ MPX(2)-.704
Component use by program 2 during one time unit.
.__--______________
____________ __________________…......-_
--- -------
I
- 28 -
Let
CPU/MPX(l,2) represent
the CPU and the multiplexor
the fraction of a time unit when both
channel are being used by either program
Let MPX(1,2) represent
program 2, or both programs I and 2.
tion of a time unit when only the multiplexor
either
the frac-
channel is being used by
and 2.
program 1, program 2,or both programs
these fractions as derived from Figure 4.
1.
Figure 5 shows
CPU/MPX(l,2)=.2~6
and MPX(1,2)
=.704.
CPU/MPX(1)=.l I
I MPX(2.08
MPX()=.
ICPU/MPX(2)=.
196
I
I
CPU/MPX(1,2)=.296
Figure 5.
MPX(2)=.704
MPx(1,2)=.704
Component use by programs I and 2 during one time unit
When programs
1 and 2 are using both the CPU and the multiplexor
channel, program 3 can also use the multiplexor
plex mode operations
channel, since multi-
can be overlapped with each other.
time period, program 3 uses the multiplexor
MPX(3)=.7
I
of the time period.
During any
channel alone for a fraction
During the time period CPU/MPX(1*2)=.296
time unit, program 2 uses the multiplexor
channel for (.7)*(.29 6 )2072
time unit.
During the time period MPX(l1,2)=.704 time unit, program 3 can use
either the multiplexor
and the CPU.
channel alone, or both the multiplexor
channel
In any case, program 3 uses the computer during the
entire .704 time unit.
The total time that program 3 uses the computer
during one time unit is .2n72+.704=.?112
time unit, as shown in Figure 6.
- 29 -
I
CF/MPX(1,2)=.296
I
MPX(3)=.2
0
i
I
72
MPX(1,2)=.704
1
MPX(3)+CPIJ/MPX(3)=.7n4
Total computer use by program 3 during one time unit
Figure 6.
However,
during any time period, program 3 uses each computer
component in the same proportion that it uses the component over the
total program time.
Specifically,
CPU/MPX(3)=.3
But
and MPX(3).7.
program 3 is active for only .9112 time unit, so in one time unit,
CPU/MPX(3)=(.3)*(.9112)=.27336
time unit and MPX(3)=(.7)*(.9112)
These times are shown in Figure 7.
.63784 time unit.
Note that
.43064+. 2072=. 63784=MPX(3).
II
CPU/MPX(1,2)=.296
PX(3)=-.2072
,
iMPX(3)+CPU/MPX(3)=.704
MPX(3)=.2072
Figure 7.
I
MPX(1,2)=.704
CPU/MPX(3)=.27336IMPX(3)=.43064
Component use by program 3 during one time unit
To summarize, during 1 time unit, program 1 uses the computer
for 1.0 time unit, program 2 uses the computer for .98 time unit,
and program 3 uses the computer for .9112 time unit.
are divided into the total program times calculated
programming
environment to determine how
multiprogramming
environment.
These fractions
for the non-multi-
long each program takes in the
In this example, the total time for
- 3
-
program 1 is 100/1.0=100 minutes; the total time for program 2 is
50/.98=51 minutes; and the total time for program 3 is 150/.9112=
165 minutes.
Use with the Flow Sho p Scheduling Problem
The program with the minimum total time (in this case program
2) is the first program to complete processing.
Normally
it will be
replaced by a new program, and the process of calculating multiprogramming times will continue. However, if the foreground programs
are being used in conjunction with the background
programs,
the same
foreground programs will always be using the computer, and the only
programs that will replace one another are the background
programs.
Since the flow shop scheduling problem exists only when at least one
of the foreground programs is used in conjunction with the background
(We still have to
programs, we only have to consider that case now.
simulate in the other cases, but only when we have found a place in the
schedule for the program in question.
However, we need to simulate the
programs in the flow shop scheduling problem before we place them in
the schedule.)
When the foreground programs are being used in conjunction with
the background programs, the fractions calculated
for each foreground
program will always be the same, since the programs do not change.
only time fractions change is when a new background
into storage.
The
program is loaded
Given a set of programs which constitute a job, with
one program to be executed in each of the three areas of storage, the
foreground program times can be calculated by dividing the non-multi-
programmed program times by the relevant fractions, which had previously
- 31 -
been computed. The background program time can be calculated only after
a new fraction, unique to this background
program, is calculated.
General Case
Programs usually use more than two computer components. For example,
a foreground
transcription program uses the CPU, the multiplexor
and a selector channel.
gram times is
channel,
The method for calculating multiprogrammed
pro-
extended in this section to cover the general case where
each program uses more than 2 computer components.
Let program 1, program 2, and program 3 be foreground program 1,
foreground program 2, and a background program, respectively.
cl/ci.../cn(i)
represent
Let
the fraction of the total program time that
program i uses computer components
cl, c2, ... , c at the same time.
Let
c1 /c2/.../c
2
(1,2)
represent
the fraction of a time unit when all
n
the computer components
cl, c2, ... , c are being used by either
n
gram 1, program 2, or both programs
Then
the
1 and 2.
total time that program
time unit is 1.0 time unit, since
1 uses the computer during
program
CPU and the selector channels whenever
1 can
seize
it needs them.
them, but it can use some of them while program 1 is
one
control of the
Program 2 can
use the CPU and the selector channels only when program 1 is
not using
using the others.
Program 3 can use the CPU and the selector channels only when
1 and 2 are not using them,
pro-
programs
but it can use some of them while programs
1 and 2 are using the others.
The total time that program 2 uses the computer is calculated as
follows:
- 32 -
1. Consider each time period c/c
2
/...c
n
(1), the fraction of the
total time that program 1 uses components
cl,
c2, ....
,
cn at the
Calculate the time that program 2 uses the computer
same time.
during each of these time periods.
2. Sum the
times
calculated for program 2.
The total time that
program
3 uses the computer is
calculated
as
follows:
i.
Consider each time period cl/c2/.../c
the total time that
grams
either program
1 and 2 use components
(1,2), the fraction of
1, program 2, or both pro-
cl, c2,
... ,
c
at
the same time.
Calculate the time that program 3 uses the computer during each
of the time periods.
2.
Sum the times calculated
for program 3.
Example
For example, in the non-multiprogramming
environment,
suppose
program 1 uses the CPU for 10 minutes, selector channel 1 for 30 minutes,
and the multiplexor
channel in the multiplex mode for 100 minutes; pro-
gram 2 uses the CPU for 10 minutes,
and the multiplexor
selector channel 2 for 30 minutes,
channel in the multiplex mode for 100 minutes; and
program 3 uses the CPU for 20 minutes,
and selector channel 2 for 100 minutes.
selector channel 1 for 50 minutes,
Then in the non-multiprogramming
environment,
the total program times for programs 1, 2, and 3 are all
100 minutes,
since the time on the multiplexor
time for programs
channel is the maximum
1 and 2, and the time on selector channel 2 is the
maximum time for program 3.
- 33 -
Program 1 uses the CPU 10/100 or .1 of the total time it uses
the computer, program 1 uses selector channel I 3/1in
or .3 of the
total time it uses the computer, and program 1 uses the multiplexor
channel 100/100 or 1.0 of the total time it uses the computer.
Simi-
larly, program 2 uses the CPU .1 of the time, selector channel 2 .3
of the time, and the multiplexor
channel 1.0 of the time; and program
3 uses the CPU .2 of the time, selector channel 1 .5 of the time, and
These times are shown in Figure 8.
selector channel 2 1.0 of the time.
Programs
1
2
3
.1
.1
.2
Selector Channel I/Total Time
.3
--
.5
Selector Channel 2/Total Time
--
.3
1.n
1.0
1.0
--
CPU/Total
Time
Multiplexor
Figure 8.
Channel/Total
Time
Component use as a fraction of total program time
Let CPU, S,
S2, and MPX represent the CPU, selector channel 1,
selector channel 2, and the multiplexor
CPU/S1/MPX(1)-.1,
SI/MPX(1)=.2,
total program time is 100 minutes,
channel. respectively.
and MPX(1)=.7,
since program
Then,
l's
and during 10 minutes of this time
it uses the CPU, selector channel 1, and the multiplexor
channel to-
gether; during 20 minutes of this time it uses both selector channel 1
and the multiplexor
channel; and during 70 minutes of this time it
uses only the multiplexor
channel.
Similarly,
CPU/S2/MPX(2)=.1,
S2/
- 34 -
MPX(2).2,
.5.
and MPX(2)=.7; and CPU/S/S2(3)-.2,
S/S2(3)=.3,
and S2(3)=
These fractions are shown in Figure 9.
I
I
CPU/sl/MPX(l).1
|
1
CPU/S2/MPX(2)=.
! S2/MPX(2)=.2
MPX(2)=.7
CPU/S1/S2(3)=.2
I S1/S2(3)-.3
S2(3)'.5
Figure 9.
Sl/MPX(l)=.2
MPX().7
Component use as a fraction of one time unit
When program 1 is using the CPU, selector channel 1,
multiplexor
channel together,
program 2 can use either the multiplexor.
channel alone or both the multiplexor
channel and selector channel 2.
During any time period, program 2 uses the multiplexor
for a fraction MPX(2)=.7 of the time period.
CPU/Sl/MPX(l).l1
and the
channel alone
During the time period
time unit, program 2 uses the multiplexor
channel for
(.7)*(.1)=.07 time unit.
During any time period, program 2 uses both the multiplexor channel
and selector channel 2 for a fraction S2/MPX(2)-.2 of the time period.
During the time period CPU/SI/MPX(1)=.l time unit, program 2 uses both
the multiplexor
channel and selector channel 2 for (.2)*(.1)-.02 time
unit.
During the time periods S/MPX(l)-.2
time unit and MPX(1)=.7
unit, program 2 can use either the multiplexor
multiplexor
channel alone, both the
channel and selector channel 2, or the multiplexor
selector channel 2, and the CPU together.
time
channel,
In any case, program 2 uses
- 35
-
the computer during the entire .2+.7-.
time unit.
The total time
that program 2 uses the computer during one time unit is .07+.02+.9=
.99
time unit, as shown in Figure 10.
i MPX(l)=.7
CPU/Sl/MPX(1).
SIMPX
1)-.2
(
MX(2)=.o07S2/PX(2)
MPX(2)+2/MPX(2)+CPu/2/MPX(2)= .9
Figure 10. Total computer use by program 2 during one time unit
However, during any time period, program 2 uses each computer
component in the same proportion that it uses the component over the
Specifically,
total program time.
and MPX(2)=.7.
in one
CPU/S2/MPX(2).1,
S2/MPX(2)=.2,
But program 2 is active for only .99 time unit, so
time unit, CPU/S2/MPX(2)=(.1)*(.99)099
time unit, S2/MPX(2)
=
(.2)*(.99)=.198time unit, and MPX(2)-(.7)*(.99)-.693 time unit.
These times are shown in Figure 11.
Note that .02+.04+.138-.198-
S2/MPX(2), .07+.138+.485=.693-MPX(2),and .022+.077=.099-CPU/Sl/
MPX(2).
Now CPU/SI/MPX(1,2)=.08,CPU/Sl/S2/MPX(1,2)-.042, S/S2/MPX
(1,2).04,
S/MPX(l,2)=.138,
CPU/S2/MPX(1,2)=.0 7 7 , S2/MPX(1,2)-
.138, and MPX(1,2)=.485.
Figure 12 shows these
times as derived from Figure 11.
When programs 1 and 2 are using both the multiplexor channel
and selector channel
1, program 3 can use selector channel 2.
During
- 36-
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-37 -
any time period, program 3 uses selector channel 2 alone for a fraction
S2(3)-.5 of the time period.
During the time period S1/MPX(1,2)-.138
time unit, program 3 uses selector channel 2 for (.5)*(.138)=.069
unit.
Similarly, during the time period CPU/Sl/MPX(1,2)-.08
time
time unit,
program 3 uses selector channel 2 for (.5).*(.08)-.04 time unit.
During the time period MPX(1,2)-.485
time unit, program 3 can
use either selector channel 2 alone, both selector channel 2 and selector
channel
, or selector channel 2, selector channel I, and the CPU together.
In any case, program 3 can use the computer during the entire .485 time
unit.
Program 3 cannot use the computer during any of the remaining
time periods that programs
and 2 are using the computer,
since during
those time periods selector channel 2 is in use, and whenever
program 3
uses the computer it must use selector channel 2.
the total
Therefore,
time that program 3 uses the computer during one time unit is .069+.04+
.485-.594 time unit, as shown in Figure 13.
However, during any time period, program 3 uses each computer
component in the same proportion that it uses the component over the
total program time. Specifically, CPU/Sl/S2(3)-.2, S/S2(3)m.3, and
S2(3)-.5.
But program 3 is active for only .594 time unit, so in
one time unit, CPU/Sl/S2(3)=(.2)*(.594)m.1188
(.3)*(.594)=.1782
time unit, and S2(3)=(.5)*(594)-.297
These times are shown in Figure 14.
.594-,S2(3).
time unit, S1/S2(3)=
time unit.
Note that .069+.04+.188+.297a
-38-
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- 39 -
Chapter 4
PREVIOUS ALGORITHMS FOR THE FLOW SHOP SCHEDULING PROBLEM
was the first to try to develop optimal sequencing
Johnson
rules for the two- and three-machine
scheduling problems.
able to develop a rule for the two-machine
unable to develop one for the three-machine
He was
problem, but he was
problem, except in
special cases.
The three-mcahine problem remained unsolved, except for enumeration, until two separate studies produced branch and bound
algorithms which
lead to optimal
schedules.
technique was first used by Little, Murty,
The branch and bound
Sweeney, and Karel
for
solving the traveling salesman problem. The branch and bound technique consists of breaking a problem up into smaller and smaller
subsets and calculating a lower bound for the best alternative
in.
The bounds guide the partitioning
identify an optimum.
there-
of the subsets and eventually
The subsets are represented as nodes of a tree
and the process of partitioning
as a branching
of the tree. 3
S. M. Johnson, "Optimal Two- and Three-Stage Production Schedules
with Setup Times Included," Chapter 2 John F. Muth and Gerald L.
Thompson, editors, Industrial Scheduling, (Englewood Cliffs: Prentice-Hall, 1963), 13-20.
2John
D. C. Little, Katta G. Murty, Dura W. Sweeney, and Caroline
Karel, "An Algorithm for the Traveling Salesman Problem," Operations
Research, (XI, November-December, 1963), 972-989.
3Ibid., p. 974.
- 40 -
Ignall and Schrage4 and Lomnicki5 showed how the branch and
bound technique could be used to solve the three-machine
flow shop
scheduling problem. They developed identical branch and bound techniques.
Ignall and Schrage's notation is used here.
The N-Job, 2-Machine Sequencing Problem
Johnson's algorithm for the 2-machine
1,
Let a
be the operation
problem is as follows:
time for the ith job on the first
machine, and bi the time on the second machine.
List the
a's and b's in two vertical columns.
2.
Scan all the operation times for the shortest one.
3.
If it is for the first machine, place the corresponding
job first.
4.
If it is for the second machine,
place the corresponding
job last.
times for the job from the list.
5.
Remove both operation
6.
Repeat steps 1-5 until all jobs are sequenced.
4
Edward Ignall and Linus Schrage,
"Application of the Branch
and Bound Technique to Some Flow-Shop Scheduling Problems," Operations
Research, (XIII, May-June, 1965), 400-412.
5
Z. A. Lomnicki,
"A Branch-and-Bound
Algorithm
for the Exact
Solution of the Three-machine Scheduling Problem," Operations Research
Quarterly, (XVI, March, 1965), 89-100.
6
Johnson, pp. 16-17
- 41 -
For emple,
consider the following:
i
a.
b.
1
13
3
2
7
12
3
26
9
4
2
6
Then the optimal sequence is (4, 2, 3, 1), and the total operation
time is 51.
The N-Job, 3-Machine Sequencing Problem
Ignall and Schrage's algorithm
for the 3-machine
problem is as
follows:7
1.
Let J
r
be a sequence of size r out of a total of n jobs.
Let TIMEA(Jr), TIMEB(Jr),
and TIMEC(Jr) be the times when
machines A, B, and C, respectively, complete processing
the last of the r iobs.
Let a i , bi, and c
be the processing
times of the ith job on machines A, B, and C, respectively.
Let J
be the set of n-r jobs that are not in the sequence
Jr
Then a lower bound for the sequence Jis
J
· Then a lower bound for the sequence J
r-1i
TIMEA(J )+
a.+min (b +ci)
r
LB(Jr )--max
TIMEB (Jr)+
Jr
b +min
J
r
TIMEC(Jr)+ E
r -ic
r
7
Ignall and Schrage, pp. 401-403.
ci
J
r
is:
- 42 -
2.
the lower bound LB(J )
Assign
to the node in a tree which
corresponds to the sequence J .
r
Keep the lower bounds in a
list.
3.
a null
Start the tree with a single node which represents
sequence.
4.
Remove the first node from the list.
5.
Create a new node for every job that the just removed node
has
6.
not yet scheduled.
Compute the lower bounds for the new nodes and insert them
ranked on the list.
7.
If a node that has scheduled all jobs is first on the list,
that node's sequence is an optimal one.
8.
If an optimal sequence has not been found, go to step 4.
Consider the example shown in Figure 15.
node 1 is 55, since that is the maximum
and 36 (12+24).
The lower bound for
of 55 (13+35+7), 31 (3+27+1),
Nodes 1 and 2 are at the top of the list initially,
since the only other nodes are 3 and 4, which have higher
Suppose 1 is ahead of 2 on the list.
lower bounds.
Then node 1 is removed
from the
list, and nodes 12, 13, and 14 are created, and their lower bounds
are computed.
All of the newly created nodes have higher lower bounds than
node 2, so node 2 is the next node removed
from the list.
Nodes 21,
23, and 24 are created, and their lower bounds are computed.
Node 21
goes to the top of the list, since it has the smallest lower bound.
Node 21 is then removed from the list and nodes 213 and 214 are created.
- 43 -
11
11%
2
64
71
63
1
i
2
ai 13---- ,Z
i
Ci 12
1.
16
3
2
4
-- 2
7
1
An optimal sequence is 2-3-1-4
Figure 15. Tree structure for 4-job, 3-machine problem.
- 44 -
Their lower bounds are greater than that of node 12, which is removed
from the list.
Nodes 123 and 124 are created and are put into the
list behind node 23.
Node 23 is removed from the list and nodes 231
and 234 are created.
Node 231 has a lower bound than any other node
in the list.
Since node 231 has scheduled all jobs and is first on the
list, that node's sequence is an optimal one.
Use of the Algorithms in Scheduling the Computer
Johnson's algorithm and Ignall and Schrage's algorithm are used
to produce optimal daily schedules
for the following main storage
configurations:
1.
Johnson's algorithm
is used when one foreground program
used in conjunction with the background
other foreground
programs, and the
program does independent work.
If the
foreground program is used to read the background
input, then the foreground program
background program is machine
B.
is used to produce the background
is
program's
is machine A and the
If the foreground program
program's output, then
the background program is machine A and the foreground program
is machine
2.
B.
Ignall and Schrage's algorithm
is used when one foreground
program is used in conjunction with the background program
for input, and the other foreground program is used in conjunction with the background
program for output.
foreground program is machine A, the background
machine B, and the output foreground
The input
program is
program is machine C.
- 45 -
Chapter
5
BRANCH AND BOUND ALGORITHMS FOR THE
N-JOB, 3-MACHINE, 2-PARALLEL-OPERATION
FLOW SHOP SCHEDULING PROBLEM
The algorithms in Chapter 4 can be used to produce optimal daily
schedules
with.
for many of the storage configurations
However,
we are concerned
up to now no algorithm has been developed
optimal sequences for the class of three-machine
problems
to produce
in which
two of the machines can operate on the same job in parallel.
This
is a very common problem in the context of scheduling the computer
described here.
Often both foreground programs are used for back-
ground program output, or both are used for background program input.
There are three separate cases for both the input and output
problems. The foreground programs may be using different input/output devices
(e.g., a card reader and a paper tape reader on input,
or a printer and a plotter on output).
may be using identical devices
Both foreground programs
(e.g., two 1000 card per minute card
readers on input, or two 1100 line per minute printers on output).
The foreground programs may be performing the same operation, but
on different
input/output device models
(e.g., one 1000 card per
minute card reader and one 500 card per minute card reader, or one
1100 line per minute printer and one 600 line per minute printer).
Using Ignall and Schrage's notation,
as follows:
the problems can be stated
- 46 -
Parallel Operations
On Input
Let J r be a sequence of size r out of a total
.li
TIMEA(Jr), TIMEA*(Jr),
Let
and TIMEB(Jr) be the times when machines A,
A*, and B, respectively,
Let ai, a,
of n jobs.
times of the ith job on machines
and b i be the processing
L
LetJ r be the set of n-r jobs that are
A, A*, and B, respectively.
not in the sequence Jr.
the last of the r jobs.
complete processing
Then a lower bound for the sequence Jr is:
r
7-
.
_
_
TIMA(Jr)
+ 7 amin bi
max
a*+min
TIMEA*(J)+
LB(J )max
r
r
rr
r
TIMEB(Jr)+
r Ti
Since TIMEA(J )+
a =Za
r
i
bi
b
and TIMEA*(Jr)+
i
_a*"7a*,
i
the
problem
i
r
can be expressed
in the following form:
i
M a min_b
Jr
max
7 a*+min
i
LB(Jr) -max
r
I
_ .. _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
..
,_
TIMEB(J )+
T bi
E
bi
r
i_
- 47 -
Parallel Operations on Output
be a sequence of size r out of a total of n
Let J
obs.
Let
r
TMEA(Jr), TIMEB(J ),
and B*, respectively,
Let ai, bi, and b*
and TIMEB*(Jr) be the times when machines A, B,
complete processing
be the processing
times of the ith job on machines
LetJ r be the set of n-r jobs that are not
A, B, and B*, respectively.
in the sequence J .
the last of the r jobs.
Then a lower bound for the sequence J
r
is:
r
.
min
b
b>b*
i
J I
TIMEA(J )+
r
£
j-
LB(J )=max
r
a.+min
r
TIEB(Jr)+
i
minb*, b*>b
Jr
i
i
b
r
maxr
TIMEB*(Jr)+Z
b*
Jr i
r
As in the previous problem, TIMEA(J )+
Case 1:
i
_ a -a
r
Parallel Operations on Input, Different Operations
Sum the a 's and the a* 's and choose the larger sum.
i
lower bounds as described above.
for assigning
new nodes.
Choose
I
Follow the algorithm
in Chapter 4
lower bounds to nodes, discarding nodes, and creating
When a node which has scheduled all jobs is at the top
of the list, an optimal solution has been found.
the use of this algorithm.
Figure 16 illustrates
-48-
r, /I
49
55
49
4
55
59
60
i
-------------
i
An optimal sequence is 2 - 1 - 3 - 4
Figure 16.
Tree structure for 4-job, 3-machine problem, where 2
different operations are performed on different machines
in parallel before the third operation.
- 4
Case 2:
Parallel Operations
Sum the a.'s.
-
on Output, Different Operations
From the operation
times for the remaining
add to the sum the minimum of the following:
the minimum b,
jobs.
where
1
bi>b?,
Choose lower bounds as
or the minimum b*, where b*>bi.
1I
i
i'
described above. Follow the algorithm in Chapter 4 for finding an
optimal solution.
Case 3:
Figure 17 illustrates
Parallel Operations
the use of this algorithm.
on Input, Identical Machines
In this case we have two parallel operations
for each job, where
each operation can be put on either machine, and the machines have
identical processing characteristics
(i.e., each operation will take
the same amount of time no matter which machine
processes which operation.)
Before we can include the b. operations in the sequencing problem,
we have to solve the parallel machine
a' represent
the processing
sequencing problem.
Let a. and
times for the two parallel operations
for
i
the ith job.
We want to assign the a.'s and the a''s to machines A and
1
i
A*, such that we minimize total processing
we have a minimum, we know which operations
on which machines.
time for the operations.
for each job should appear
The sequencing of the jobs will not affect this
minimum, since a machine can begin processing one
finishes processing
no operation
sequence J
r
the previous job.
precedes operations
is:
Once
ob as soon as it
(This is so because for any job,
a i and a.)
A lower bound for the
- 5
-
L
54
L
9
3.
a
13
7
26
2
b
3
12
9
6
12
16
7
1
b*
An optimal sequence is 2-1-4-3
Figure 17.
Tree structure for 4-job
3-machine problem where 2
different operations are performed on different machines
in parallel after the third operation.
-
51
+
(a ,a') ar>a*r
TIEA(J
r
)
i ,i
(a. a'),
J.L
r
]-
a 4a*
r r
LB(J )max
r
A_ min(ai,a),
a*a
r
TIMEA*(J ),
r
Z max(ai,) , a*<a
Jr
:
I1
r r
I
I
Figure 18 illustrates the use of this algorithm. Choose any
job to start.
1.
In Figure 18, job 1 was chosen as the first job.
Compute the lower bounds as described above.
Keep a sep-
arate tree for each machine.
2.
Extend the trees by adding both possible operations to both
trees.
3.
Computethe lower bounds for the new nodes.
Group the new nodes in pairs, each pair representing
feasible schedule
machine A*; 12'
4.
a
(e.g., 12 for machine A, and 1'2' for
for machine A, and 1'2 for machine A*).
Compute the lower bound for each pair of nodes as described
above.
5.
Keep the minimum of the two lower bounds, and compare it with
the previous lower bound.
6.
If the previous lower bound is smaller, its nodes define
an optimal sequencing for the two machines.
7.
If the new lower bound is equal to or less than the previous
lower bound, than the new lower bound is used to continue
- 52 -
3
A
38
A*d~
i
-1
2
3
4
a.
13
7
26
2
a'
3
12
9
6
b. 12
16
7
1
1
An optimal sequence is 1-2'-3'-4' & 1'-2-3-4
Figure 18.
Tree structure for 4-job, 2-machine problem, where
2
identical operations are performed on identical machines
in parallel.
- 53 -
branching with step 2, unless the new lower bound has
scheduled n-l jobs, in which case it defines an optimal
sequencing
Now the b
for the two machines.
operations
can be included in the sequencing problem.
"Parallel Operations
Compute a lower bound as described under
Follow the algorithm in Chapter 4 for finding an optimal
Input."
Figure 19 illustrates the use of this algorithm.
solution.
Case 4:
on
Parallel Operations on Output,
Identical Machines
As in case 3, we have two parallel operations
for each job,
where each operation can be put on either machine, and the machines
have identical processing characteristics.
We want to use the lower bound formulas listed under
Operations
on Output."
b
To do so, compute
J
and
i
for case 3.
b*
in the
Ji
r
r
way that ai and a* sums were computed
T
"Parallel
After computing
i
the lower bounds using the algorithm
"Parallel Operations
in case 3 and formula under
on Output," we can follow the algorithm in
Chapter 4 for finding an optimal solution.
Figure 2
illustrates
case 4.
Case 5:
Parallel Operations
on Input, Same Operation, Different
Machines
In this case we have two parallel operations
each operation
for each job, where
can be put on either machine, but the machines have
different operating
characteristics
to complete on one machine
(i.e., each operation
than it does on the other).
takes longer
- 54 -
48
_62
54
set
2
7
a'
3
bi
12
3 "4
'26
2
1-2 ----9
6
7
1
16
An optimal sequence is 2-1-3-4.
Figure 19.
Tree structure for 4-job, 3-machine problem, where 2 identical
operations are performed on identical machines in parallel
before the third operation.
-
55 -
59
1
2
3
4.
13
7
26
2
3
12
9
6
12
16
7
1
____________________
a
b
i ------------
bl
An optimal sequence is 2-1'-4-3 & 2'-1-4'-3'
Figure 20. Tree structure for 4-job, 3-machine problem, where 2 identi-
cal operations are performed on identical machines in
parallel after the third operation.
- 56 -
The technique used is similar to that for case 3, except that
two pairs of trees are maintained
choose any job to start.
instead of one pair.
As in case 3,
As shown in Figure 21, assign one operation
to machine A in one tree pair, and to machine A* in the other tree
pair.
For each tree pair follow the algorithm
an optimal sequence for that tree pair.
in case 3 to obtain
Compare the operation times
for the two sequences, and choose the smaller as the optimal sequence.
Now include the b i operations
in Figure 22.
in the sequencing problem, as shown
Compute a lower bound as described under "Parallel
Operations on Input."
Follow the algorithm
in Chapter 4 for finding
an optimal sequence.
Case-6:Parallel Operations on Output. Same Operation Different Machines
As in case 3, we have two parallel operations for each job, where
each operation can be put on either machine, but the machines have
different operating characteristics.
We want to use the lower bound formulas listed under "Parallel
Operations on Output."
To do so, compute
b
Jr
way the ai and a
1.
sums were computed for case 5.
i
ijB
and
b* in the
r
After the lower bounds
are computed in this manner, we can follow the algorithm
for finding an optimal sequence, as shown in Figure 23.
in Chapter 4
- 57 -
52
31
A
AA
A*
Jobset
i
a
A
1 2
13
i
7
. .
26
a* 1 - ..
---- §-
i
a'
An optimal sequence is 1'-2-3-4' on
machine a & 1-2'-3'-4 on machine a*
3
*
2
12
9
6
16
7
1
5
i
b
12
i
Figure 21.
Tree structure for 4-job, 2-machine problem, where 2 identical
operations are performed on different machines in parallel.
- 58 -
51
1
5
UJ
H"
-?
3
4
26
2
%
IA.
f)
'4
a'
3
12
6
af*
5
14
11
8
12
16
7
1
i
b
An optimal
Figure 22.
Tree
i
sequence is 2-1-3-4
structure for 4-job, 3-machine problem, where 2
identical operations
are performed on different machines
in parallel before the third operation.
-59-
,I
60
59
0
9
9
0
7
60
i
ai
1
2
3
13
7
26
2
12
9
6
An optimal sequence is 2' - 4' -
on machine b & 2 - 4 - 1' - 3' on machine
b*
3
b
i
b*
--
4
- - - - - -
5
14
11
8
b'
12
16
7
1
b'*
14
18
9
3
i
Figure 23.
Tree structure for 4-job, 3-machine problem, where 2 identical'
operations are performed on different machines in parallel
after the third operation.
- 3
- 60 -
Chapter
SCHEDULING
6
RULE
We can now combine the tools developed in the preceding three
chapters to schedule the programs run on the computer.
period is a week.
The schedule
Most jobs enter the computer center at the be-
ginning of one day and must be completed by the end of that day.
Other jobs have different turnaround times up to a maximum of one
week, but all jobs must be completed during the week they enter the
computer center.
for the week
The foreground areas are assigned a normal use
(e.g., one area for background program input, the other
for background program output), but there may be other foreground
programs to schedule.
Some background programs form a program set
which must be executed in sequence with no other programs intervening.
Some background programs need devices which one or both of the foreground programs use.
Such background
programs cannot be multipro-
grammed with other programs, if it results
in a conflict of this
sort.
1.
Remove from the list of programs to be scheduled those
background programs which cause conflicts, those foreground programs which are not assigned to a foreground
area as the normal program for that area, and those programs
whose entrance and completion times are not the beginning
and end of the same day.
2.
Combine background program sets into a single conceptual
program by summing the processing times and foreground
program times across programs.
- 61 -
3.
For the remaining programs
background
(the normal foreground programs;
programs which do not cause conflicts and whose
entrance and completion times are the beginning and end of the
same day) compute optimal daily schedules as follows:
a.
Compute the individual program times using the simulation method in Chapter 3.
b.
Sequence the programs in each day using the appropriate
algorithm in Chapter 4 or 5.
4.
For each day's schedule, starting with the first day, and
proceeding day by day until the end of the week, do the
following:
a.
Adjust the part of the schedule which includes the background program sets.
This is required, since we can
now overlap one program's processing with other programs'
input or output transcription
b.
operations.
Insert the daily programs which have been removed from
the list because of device conflicts, foreground area
usage, or entrance and completion times.
c.
Insert as many programs as possible that have longer than
one day turnaround times, starting with those with the
nearest due date.
This scheduling rule is in the form of a computer program.
A describes the program's input and output formats..
the program flow charts and FORTRAN coding.
of pr-"
'"-. '.. rurf orutfut.
Appendix
Appendix B contains
::.,d;i;C contain
exap!.s1
- 62 -
Chapter
CONCLUSIONS
7
AND SUGGESTIONS
FOR FUTURE
STUDY
A scheduling program has been developed which provides good or
"near" optimal schedules for a set of programs run on a multiprogrammed
computer. The program
provides more efficient schedules for
the com-
puter than could be provided by a computer operator without using the
program.
The scheduling rule developed here consists of a simulation
method which is used to calculate the running times for programs in
the multiprogramming environment, and sequencing algorithms which are
used to schedule the programs after the running times have been cal-
culated. New branch and bound algorithms have been developed for
n-job, 3-machine
operate
sequencing problem, where 2 of the machines
the
can
on the same job in parallel.
Future research in computer scheduling should be directed towards
more complex environments than the one studied
here,
and towards
pro-
viding accurate data for use with the scheduling program.
The environment
studied here is a static one.
That is, all the
programs to be scheduled during a period are knownbefore the schedule
is made. Within the context of the IBMSystem/360 Disk Operating System,
research
should be directed
Such a scheduling
rule
towards developing a dynamicscheduling
rule.
would consist
of a computer programwhich is
always in the computer's main storage,
new program enters the computer center.
ready to be executedwhenevera
This scheduling
programwould
- 63 -
relieve the operator of the burden of producing a new schedule whenever
the scheduling environment changed.
Instead, the scheduling program
would keep track of the then current schedule, dispatch programs when an
area of main storage is free, and modify the schedule when there are changes
in the environment.
The ability of the scheduling program to produce good schedules
is dependent upon the accuracy of the input data which describes
program
characteristics. Before more complex scheduling rules are produced, research should be directed towards developing accurate data on program
characteristics.
- 64-
Appendix A
PROGRAM USER MANUAL
This appendix contains the information needed to use the schedprogram is an efficient means for pro-
uling program.
The scheduling
ducing a weekly
schedule for programs run on the IBM System/360 under
control of the IBM System/360 Disk Operating
provides a multiprogramming
System (DOS/360).
DOS/360
environment where main storage is divided
Input
into three areas (a background area and two foreground areas).
to the scheduling program consists of statements describing
puter, the installation
the com-
(including normal foreground area assignments),
and the programs to be scheduled.
Output from the program consists of
a suggested schedule, showing program sequence and clock time for the
scheduled week.
The programs to be scheduled are divided
into three categories:
weekly programs, partial working day programs, and full working day
programs. Each program category may consist of background programs,
foreground programs, or both background
and foreground programs.
Weekly programs are ready for processing on one day of the week, but
do not have to be completed until some day later in the week.
working day programs must be completed
Partial
the same day they are ready for
processing, but in a shorter time span than full working day programs.
Full working day programs are ready for processing at the start of a
day and must be completed by the end of the same day.
The scheduling program checks all input statements
syntax.
for proper
If there are any errors in a statement, the program prints
-65-
an error message,
prints the statement
The statements
a schedule.
in error, and does not produce
in error can then be corrected and the
scheduling program can be rerun.
Input Formats
The input to the scheduling
statement, hardware
program consists of a dictionary
statements, installation
statements, and job
statements.
Dictionary
Statement:
The dictionary
statement is used to define
the syntactic characters used by the scheduling program.
be the first statement in the input stream.
It must
The format of this
statement is:
Columns
Contents
1-24
25-36
37-80
ABCDFHIJOPRSTUXZ01234567
Must be blank
Should be blank
Hardware Statements:
characteristics
The hardware
of the computer.
statements describe the physical
They consist of an HC statement,
one or more HD statements, and an HZ statement.
The HC statement describes
central processing unit.
the physical characteristics
This statement must immediately
If the HC statement
of the
follow the
is in error, the run is
dictionary
statement.
terminated
immediately, since the error may have been caused by the
absence of the dictionary
is:.
statement.
The format of the HC statement
- 66-
Contents
Column(s)
1
H
2
C
3-6
Central processing unit model number
(e.g., 2030)
Average instruction time in tenths of
a microsecond
Interference time per byte on selector
channels in hundredths of a microsecond
Interference time per byte on multiplexor channel in tenths of a
microsecond.
This time may be
overridden for individual devices
by use of the HD statement.
Should be blank
7- 9
10- 12
13- 15
16- 80
The HD statements describe
input/output devices.
HC statement.
the physical characteristics
of the
These statements must immediately follow the
There must be at least one HD statement.
number of HD statements is determined when the scheduling
compiled.
Initially, the maximum number is ten.
statements
is:
The maximum
program is
The format of these
Contents
Column:(s)
1
H
2
D
Device type (e.g., 2311)
Model number; used to distinguish devices with the same device type, but
3-6
7- 8
with different physical characteris-
9
tics. The field must be used to
distinguish (1) the punch and read
units of a card read punch, (2) 7track and 9-track tape drives, and
(3) devices with the same device
type that have different speeds.
Device class--D for disk, T for tape,
U for unit record
10-16 (D or T in 9)
10-12 (U in 9)
Device speed in bytes per second
Record size in bytes
-67-
13-16
Device speed in records per minute.
This speed may be overridden for
individual files by use of JBIO
and JFIO statements.
Device set-up time in tenths of a
minute
Average latency time for device in
This time may be
milliseconds.
overridden for individual files
by use of the IF, JBIO, and JFIO
statements.
Start-stop time for device in tenths
of a millisecond.
Should be blank
Average seek time for device in milliThis time may be overseconds.
ridden for individual files by use
of the IF, JBIO, and JFIO statements.
Rewind time for device in tenths of a
minute
Should be blank
Device density in bytes per inch
Should be blank
Interference time per byte on multiplexor channel in tenths of a
microsecond, if desired to override time on HC statement.
Otherwise should be blank.
Should be blank
(U in 9)
17- 18
19-21
(D in 9)
19-21
(U in 9)
19-21
22-24
(T in 9)
(D in 9)
22-24
(T in 9)
22- 24 (U in 9)
25-27 (T in 9)
25-27 (D or U in 9)
28-30
31- 80
The HZ statement
It must immediately
indicates the end of the hardware
follow the last HD statement.
statements.
The format of this
statement is:
Column
Contents
1
H
z
2
Installation Statements:
The installation
statements describe
input/output and main storage configurations,
consist of one or more ID statements,
ment, and an IZ statement.
the
and the work week.
They
two IF statements, an IS state-
-68-
The ID statements relate input/output unit numbers to device
descriptions.
statement.
These statements must immediately
There must be at
follow the JZ
east one ID statement.
The maximum
number of unit numbers is determined when the scheduling program is
compiled.
Initially the maximum number is 20.
statements
is:
Column(s)
Contents
1
I
2
D
3-6
7-8
Device type
Model number.
The device type and
model number must be the same as a
device type and model number that
appears on a valid HD statement.
Number of unit numbers in this statement, an integer from 1 to 23
Unit numbers in hexadecimal, each
unit number taking up three columns
Any non-blank character, if there are
more ID statements,
Otherwise,
blank.
9-10
11-79
80
The IF statements describe
areas.
ment.
The format of these
the normal usage of the foreground
These statements must immediately
There must be two IF statements,
The format of these statements
follow the last ID state-
one for each foreground area.
is:
Column(s)
Contents
1
I
2
F
3
1 for the first statement (foreground
area one) and 2 for the second statement (foreground area two).
R for card reader transcription program, P for printer transcription
program, TI for telecommunications
input transcription program, TO for
telecommunications output transcription program, or blank if no program
assigned to this area.
4-5
-69-
6-13 (4-5 non-blank)
14-16 (4-5 non-blank)
17-19 (4-5 non-blank)
20-22 (4-5 non-blank)
23-24 (4-5 non-blank)
Program name
Unit number for peripheral operation.
The device class of this unit number
must be U.
Unit number for intermediate storage.
The device class of this unit number
must be D or T.
Number of instructions executed per
record
Blocking factor on intermediate storage
device.
25-27 (4-5 non-blank)
28-30 (4-5 non-blank)
6-30 (4-5 blank)
31-80
Average latency time for intermediate
storage device (device class D),
if desired to override time on HD
Otherwise, blank.
statement.
Average seek time for intermediate
storage device (device class D),
if desired to override time on
Otherwise, blank.
HD statement.
Should be blank
Should be blank
The IS statement describes the work week in terms of the time
the installation is active during each day.
This statement must
immediately follow the second IF statement.
The format of this
statement is:
Contents
Column(s)
1
I
2
S
3
Number of days in week, from 1 to 7
Begin time and end time for each day
in the form HHMMHHMM, where HH is
an hour from 00 to 23 and MM is a
minute from 00 to 59.
Should be blank
4-59
60- 80
The IZ statement indicates the end of the installation
It must immediately
ment is:
follow the IS statement.
statements.
The format of this state-
- 70-
Column
Contents
1
Z
3-80
Should be blank
Job Statements:
scheduled.
I
2
The job statements describe
Each background
program description consists of a JB
statement, one or more JBD statements,
or more JBIO statements.
the programs to be
two JBF statements,
and one
Each foreground program description
consists
of a JF statement, one or more JFD statements and one or more JFIO
statements.
A JZ statement follows the last program description.
The descriptions
for weekly programs must precede the descriptions
for daily programs. The daily program descriptions must be ordered by
day.
Within each day, the partial working day program descriptions
must precede the full working day program descriptions.
Within each
program category (weekly, partial working day for each day, and full
working day for each day), background
cede foreground program descriptions.
program descriptions
If a background
of two or more job steps, the program descriptions
job consists
for the steps must
appear in sequence with no other program descriptions
the job name and begin- and end-times
must pre-
intervening, and
for each step must be the same.
The maximum number of programs in each category
when the scheduling program is compiled.
Initially,
is determined
the maximum
number
of weekly programs is 10, the maximum number of partial working day
programs is 5 for each day, and the maximum number of full working day
programs is 5 for each day.
-71-
The JB and JF statements contain
the names of the programs to be
scheduled and describe the scheduling constraints
for each program.
The first of these statements must immediately follow the IZ statement.
Additional
statements of this type must immediately
JBIO and JFIO statement for the previous program.
statements
The format of these
is:
Contents
Column(s)
J
Blank for weekly
1
2
from
programs or an integer
1 to 7 corresponding
to a day
on the IS statement for daily programs.
B for background programs or F for foreground programs
Should be blank
3
4-5
6-13
follow the last
(B in 3)
6-13 (F in 3)
14-21 (B in 3)
14-21 (F in 3)
22-26
Job name
Program name
Step name
Should be blank
Time when program is ready for processing,
in the form
DHHMM,
where
D is
a day from 1 to 7, HH is an hour
from 00 to 23, and MM is a minute
from
27-31
32-36
00 to 59.
Time by which program must be completed,
in the form DHHMM, where D is a day
from 1 to 7, HH is an hour from 00;
to 23, and MM is a minute from 00 to
59. For weekly programs, column 22
does not equal column 27. For daily
programs those columns are the same.
For full working day programs, columns
23-26 and 28-31 must be the same as
the begin-time and end-time for day D.
For partial working day programs at
least one of those times must be different than the corresponding times
on the IS statement.
Fixed number of instructions executed
by program regardless of the volumes
of its files.
-72-
37-41
42-49
(B in 3)
42-49 (F in 3)
50-51
52-80
Number of instructions executed per
record for files indicated on JBIO
or JFIO statements
For jobs which consist of two or more
steps, the step name of the precedOtherwise,
ing program description.
should be blank.
Should be blank
The percentage of machine instructions which cannot be overlapped
with input/output operations
Should be blank
The JBD and JFD statements relate file names to unit numbers.
These statements must immediately
follow the JB and JF statements.
There must be at least one of these statements for each program
description.
The maximum number of files for each program is deter-
mined when the scheduling program is compiled.
ber is 20.
The format of these statements
Initially this nunr
is:
Contents
Column(s)
2
J
Same as first statement for this
3
Same as first statement for this
4
D
Should be blank
Number of file names on this statement,
an integer from 1 to 7
One or more relations between a file
name and a unit, each consisting
of a 7 character file name and a
3 hexadecimal digit unit number.
The unit number must be the same
as one that appears on a valid ID
statement.
Should be blank
Any non-blank character, if there are
more of these statements for this
2
program description
program description
5
6
7-76
77- 79
80
program description. Otherwise,
blank.
-73-
The JBF statements define
processed by the foreground
IF statements.
the unit record files which can be
transcription
programs described by the
These statements must immediately follow the last
JBD statement for each background
be two JBF statements
program description.
for each background
format of the statements
There must
program description.
The
is:
Column(s)
Contents
1
J
2
Same as first statement for this prOgram description
3
B
4
5
6-7
F
Should be blank
R if file can be processed by card
reader transcription program, P
if file can be processed by printer
transcription program, T if file
can be processed by telecommunications
input transcription program, TO if
file can be processed by telecommunications output transcription program,
or blank if there is no file name
on this statement.
File name. Must be the same as a file
name that appears on a valid JBD
statement for this program description. The device class of the unit
number for this file must be U.
Should be blank
Should be blank
8-14 (6-7 non-blank)
8-14 (6-7 blank)
15-80
The JBIO and JFIO statements describe
the characteristics
of
The JBIO statements must immedi-
each file processed by the program.
ately follow the second JBF statement for each program description.
The JFIO statements must immediately
each program.
follow the last JFD statement for
There must be one JBIO statement for each file name that
appears on a valid JBD statement.
Similarly,
there must be one JFIO
- 74-
statement for each file name that appears on a valid JFD statement.
The format of these statements is:
Contents
Column(s)
1
2
J
Same as first statement for this pro-
3
Same as first statement for this pro-
gram description
gram description
4
5
I
0
6- 12
13-20
21-25
File name
Number of records in file
If the device
Record size in bytes.
class of the unit assigned to this
file is U, then this field is ignored and the device's record size
26-27
If the device class
Blocking factor.
of the unit assigned to this file
is U, then this field is ignored,
and the blocking factor is set
28
X if the number of instructions executed by this program is a function
of this file's size. Otherwise,
blank.
If X, then the number of
records in this file is multiplied
by the number in columns 37-41 of
the JB or JF statement, and the
product is added to the fixed number
of instructions.
A character to indicate valid device
classes for the unit assigned to
this file. A if device class can
be D, T, U; D if device class can
be D; H if device class can be D
or T; T if device class can be T;
or U if device class can be U.
Average latency time for device (device class D), if desired to override time on HD statement.
Otherwise, blank.
Average seek time for device (device
class D), if desired to override
Otherwise,
time on HD statement.
blank.
is used.
equal
29
30-32
33-35
to 1.
- 75-
Factor in thousandths (device class U),
if desired to modify device speed
on HD statement.
Otherwise, blank.
If a number appears here, the device
speed is multiplied by it.
Should be blank
Any non-blank character, if more statements of this type. Otherwise, blank.
36-39
40-79
80
The JZ statement indicates the end of the job statements.
must immediately follow the last statement
tion.
The format of this statement
Column
It
for the last program descrip-
is:
Contents
1
J
2
Z
Output Formats
Output from the scheduling
program restrictions,
a summary of the input statements, error
messages, and the schedules
Program Restriction
program maximums
program consists of a summary of
Summary:
for each main storage area for each day.
The first page of output contains
for number of device types, unit numbers,
the
files,
full working day programs per day, partial working day programs per
day, and weekly programs.
Input Summary:
Summaries are printed for the central processing
unit, each device type and model number, both foreground area assignments, the weekly schedule, and each program.
The central processing unit summary consists of a line for each
of the fields on the HC statement.
- 76-
Each device and model number summary consists of a line for each
field on the HD statement followed by the unit numbers related
to the
device type and model number via the ID statements.
The foreground area summaries consist of a line for each field
on the IF statements.
The weekly schedule summary contains
the begin and end times
for each day in the week.
Each program summary consists of three parts.
The first part
contains a line for each field on the JB or JF statement.
part contains a line for each file assigned
The second
to a unit number as shown
on the JBD or JFD statement, except for those background
that can be processed by the foreground transcription
program files
programs.
If
the file cannot be processed by the foreground program because of device conflicts,
the unit number printed
the JBD statement.
is the one that appeared on
If the file can be processed by only one fore-
ground program, the unit number printed is the intermediate storage
device used by the foreground
program.
If the file can be processed
by both foreground programs, no unit number is printed here.
Instead
the unit number assigned is printed just before the schedule is
printed.
The third part of the program summary contains a header
line for each file, followed by a line for each field on the JBIO or
JFIO statements.
Error Messages:
message,
Error messages
consist of a heading, followed by a
(usually) followed by the input statement
in the following example.
in error, as shown
Some errors do not refer to a specific
- 77-
statement and, therefore, no input statement follows the message.
**********ERROR**********
UNIT NUMBER 28I NOT DEFINED BY AN ID STATEMENT
J1BD 1 MASTER 28I
Schedules:
A separate schedule is printed for each main storage
area for each day.
These schedules are followed by a list of programs
that could not be included in the schedule due to time constraints.
The format of the background area schedule is:
Contents
Columns
1-2
3
4-5
Hour on
Blank
Minute on
6
Blank
7-8
Hour off
9
10- 11.
12
13- 20
21
22-29
30
31-32
33
.34
35
36-39
Blank
40
Minute off
Blank
Job name
Blank
Step name
Blank
Set-up time in minutes
Blank
Rewind time in minutes
Blank
Central processing unit work time.
in the form HHMM, where HH is hours
and MM is minutes
Blank
41-44
Central processing unit wait time,
in the form HHMM, where HH is hours
MM is minutes
45
46-48
49
50- 53
Blank
54
Blank
time, in the form HHMM,
Unit wait
where HH is hours and MM is minutes
Blank
Unit address in hexadecimal
Blank
Unit work time, in the form HHMM,
where
55- 58
59
60- 128
HH is hours and MM is minutes
Additional unit addresses,
work times,
and wait times.
If additional lines
are needed, they are printed start-
ing with Column 46.
-78-
The format of each foreground area schedule is:
Contents
Column(s)
1- 2
3
Hour on
Blank
4-5
Minute on
6
7- 8
9
Blank
Hour off
Blank
10-7l
Minute off
12
Blank
13-20
21
22-29
Program name
Blank
Job name, if program is a transcription program.
Otherwise, blank.
Blank
Step name, if program is a transcripOtherwise, blank.
tion program.
Blank
Set-up time in minutes
Blank
Rewind time in minutes
Blank
Central processing unit work time,
in the form HHMM, where HH is hours
and MM is minutes
Blank
Central processing unit wait time,
in the form HHMM, where HH is hours
and MM is minutes
Blank
Unit address in hexadecimal
Blank
Unit work time, in the form HHMM, where
HH is hours and MM is minutes
Blank
Unit wait time, in the form HHMM, where
HH is hours and MM is minutes
Blank
Additional unit addresses, work times,
and wait times.
If additional lines
are needed, they are printed starting
30
31-38
39
40-41
42
43
44
45-48
49
50-53
54
55-57
58
59-62
63
64-67
68
69-123
with Column 55.
- 79~;
Appendix
B
PROGRAM LOGIC MANUAL
This appendix is a guide to the logic of the scheduling proIt is intended for persons who want to modify the program or
gram.
diagnose
program malfunctions.
Included are descriptions
gram's major components and their storage requirements,
of the pro-
the program
flow charts, and the FORTRAN source statements.
Program Organization
The scheduling program consists of a main program and eighteen
Figure 24 shows how these components are related.
subprograms.
Figure 24, vertical
they call.
In
lines connect calling programs with the programs
For example, MAIN (the main program) calls subprogram
IGNSCH which calls subprograms PARALl, PARAL2, and ERRPRT.
The following text briefly describes
the function of the main
program and each subprogram and lists the subprograms that each calls.
Main Program:
Reads and checks syntax of all input statements except
the JBD, JFD, JBIO, and JFIO statements.
Writes all program output
except those error messages written by subprograms. Adjusts schedules
for inclusion of programs which are not scheduled by use of the
algorithms.
Subprograms Called:
UNITX, DISKX, TAPEX, FRCTN, TIME23, FILE,
FILEIO, TOD, SWITCH, JOHNSN, IGNSCH, ERRPRT.
-80-
0
4.J
-N
0Si
0co
t)
1-4
1-
co
wr
F0
81-
Subprogram UNITX:
Computes device, channel,
CPU, and set up times
for unit record devices.
Subprogram Called:
Subprogram DISKX:
CHNCUU
Computes device, channel, CPU, and set up times for
disk devices.
Subprogram Called: CHNCUU
Subprogram TAPEX:
Computes device, channel, CPU, set up and rewind
times for tape devices.
Subprogram Called:
Subprogram CHNCUU:
CHNCUU
Determines channel and interference rate for a
device.
Subprograms Called:
Subprogram FRCTN:
None
Separates resource uses into multiplexor mode,
burst mode, and CPU portions.
Subprogram Called:
Subprogram FRCTBL:
FRCTBL
Completes computation
of resource uses and then
sorts resource uses.
Subprogram
Called: None
Subprogram TIME2: Calculates
program.
Subprograms Called:
None
multiprogramming time for second priority
-
82-
Calculates multiprogramming
Subprogram TIME3:
time for third priority
program.
Subprograms Called:
None
Reads and checks syntax of
Subprogram FILE:
JBD and JFD statements.
Subprogram-Called: ERRPRT
Subprogram FILEIO:
Reads and checks syntax of JBIO and JFIO statements.
Subprogram Called: ERRPRT
Subprogram TOD:
Checks validity of begin and end times on JB and JF
statements.
Subprogram Called:
Subprogram SWITCH:
ERRPRT
Moves program description
from one storage area to
another storage area.
Subprogram Called:
Subprogram JOHNSN:
Solves two-machine
Subprograms Called:
Subprogram
IGNSCH:
None
two-operation
sequencing poblem.
None
Solves three-machine
three-operation
sequencing
problem.
Subprograms Called:
Subprogram PARAL1:
PARAL1, PARAL2, ERRPRT
Solves parallel operation sequencing problem for
same operations and same machines.
Subprograms
Called:
None
- 83-
sequencing problem for
Solves parallel operation
Subprogram PARAL2:
same operations and different machines.
Subprograms Called:
None
Calculates multiprogramming
Subprogram TIME23:
times for second priority
and third priority programs.
Subprograms Called:
TIME2, TIME3
Prints error heading and increments error count
Subprogram ERRPRT:
Subprograms Called:
None
Storage Requirements
The main storage required for the main program and for several
the main program.
in
to certain variables
subprograms depends upon the values assigned
The variables are:
MDEV, the maximum number of device types and model numbers in
the installation.
MCUU, the maximum
number of unit numbers
in the installation
MFILE, the maximum number of file names per program description
MJOB, the total number of weekly program descriptions (MWJOB),
partial working day program descriptions
MWJOB), and full working
per day
day program descriptions
(MDJOB per day
(MJOB - MDJOB).
Values are assigned
gram is compiled.
whose dimensions
to
these
variables
when the scheduling pro-
The following text lists the names of those arrays
should be changed when the values assigned
to the
- 84--
above variables are changed.
In addition,
the main storage for sub-
program IGNSCH depends upon the value assigned to the variable NLIST,
which is defined
in subprogram IGNSCH.
Main Program:
MDEV
MCUU
DEVTYP CUUPNT
DEVMOD CUU
MFILE
FILPNT
FILNAM
DON
HON
DOFF
DEVCLS
HOFF
MON
MOFF
TOTAL
TCPU
TNOVLP
FLFGPT
DEVSPD
DEVROD
DEVSTP
DEVLSS
DEVSRW
DEVDEN
DEVINT
FLNMPT
TSEL1A
TSEL1W
TSEL2A
TSEL2W
TSEL3A
TSEL3W
TSEL4A
TSEL4W
TSEL5A
TSEL5W
UNITX:
None
DISKX:
None
TAPEX:
None
CHNCUU:
FRCTN:
FRCTBL:
None
None
Nndn
TIME2:
None
TIME3:
None
MJOBMDJOB
MIBB
TSEL6A
QL
TSEL6W
QN
TMPXMA
TMPXBA
QA
QB
QC
TMPXBW
QDX
TSETUP
TWNDUP
QDP
JOBNAM
QD
TMPXMW
STEPNM
FILEFG
FRACM
FRACB
FRACC
FRACMX
FRACBX
FRACCX
ISCHED
SCHED
QDPX
MJOB & MCUU
TDEV
SDEV
- 85-
FILE:
MDEV
MCUU
MFILE
DEVCLS
CUUPNT
CUU
FILPNT
FILNAM
A
FILEIO:
MDEV
MCUU
DEVCLS
CUUPNT
FILPNT
FILNAM
DEVLSS
DEVSRW
TOD:
MFILE
None
SWITCH:
None
JOHNSN_
MJOB-MDJOB
A
B
N
IGNSCH:
The main storage for this subprogram depends upon the value
assigned to NLIST, which is the number of nodes created.
If N is the
number of full working day jobs, then NLIST must be between
and N!
(.5*N* (N-l) ) + 1
The maximum number of full working day jobs that can be scheduled is
9, except for case 4 and case 6, where the maximum
MJOB - MDJOB
is 8.
NLIST
A
B
C
M
PA
PB
BOUND
LIST
TIMEA
D
DX
DP
PD
PDX
PDP
TIMEB
TIMEC
POPER
DPX
L
PDPX
PL
N
-86-
PARALl:;-
MDJOB-MDJOB
A
B
AX
BX
L
LMAX
LMIN
PARAL2:
MDJOB-MDJOB
A
B
C
D
AX
BX
L
Li
LMAX
LMIN
TIME23:
ERRPT:
None
None
Flowcharts
The following charts can be used to locate the FORTRAN source
statements that perform specific program functions.
identifies the flowchart symbols.
Figure 25
To locate the statements
that
perform a specific function use the flowcharts to find the flowchart
symbol or symbols that describe
the function.
The FORTRAN statement
number of the first source statement used to perform a function appears
above the upper left hand corner of the symbol that defines
Flowcharts
highest
for the main program have two-digit identifiers.
level charts have zero as their first digit.
the subprograms
the function.
have two-letter
identifiers.
Flowcharts
The
for
If there is more than
-87-
one chart for a subprogram, all the charts have the same first letter.
This letter corresponds
to the number which appears in columns 74-75
of the source statements for the subprogram.
- 88-
Terminal
Predefined process.
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123
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FORTRAN Source Statements
The following pages contain the source statements for the
scheduling program.
Basic FORTRAN
The program is written
IV Language and has been tested on a System/360
Model 165 using the Operating System/360
compiler°
in the IBM System/360
(Release 11) FORTRAN G
The input and output data set reference numbers are 5
and 6, respectively.
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