ADCAA98IF07 GEORGE WASHINGTON UNIV WASHINGTON 0 C PROGRAM IN LOG--ETC PFN: 12/1NUMERICAL METHODS FOR TRANSIENT SOLUTIONS OF MACHINE REPAIR PRO--ETC(URJAN 81 H ARSHAM. A R B1ALANA, 0 GROSS NO 17G -C _1729
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STUDENTS FACULTY STUDY RESEARCH DEVELOPMENT FUJURE CAREER CREATIVITY CCMMUNITY LEADERSHIP TECF-NOLOGY FRONTIF IGNENGINEERINGAP NGEORGE WASHINI
SCHOOL OF ENGINEERINGAND APPLIED SCIENCE
MIS1 DOCUMENT WIS USS AlIPRON R VOODOOS i I~~
NUMERICAL METHODS FOR TRANSIENT SOLUTIONSOF MACHINE REPAIR PROBL24S
by
Hossein ArshamArturo R. BalanaDonald Gross
Serial T-4365 January 1981
DTIC
MAY 131981
The George Washington UniversitySchool of Engineering and Applied Science
Institute for Management Science and Engineering
Program in Logistics
Contract N00014-75-C-0729Project NR 347 020
Office of Naval Research
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16. SUPPLEMENTARY NOTES
IS. KEY WORDS fCmlnu.w an reverse aide if neesary AndIdenify by block rnuber)
FINITE SOURCE QUEUESMACHINE REPAIR PROBLEMS
- I TRANSIENT QUEUES
20. AS TRACT (Continue an reverse oid* It necessary and iden~tify by blook rmrne)
This paper reviews and compares three numerical methods of computingtransient probabil'~ties of finite Markovian queues (particularly themachine repair problem). A brief review of each method is followed by anumerical example cf a moderate size machine repair problem (two-stagecyclic queue).
DD IOR A14 7 1473 01@NO NV S @OIT NONES/N 012.0144401 SCURITYv CLASIFICAION OF TWIS PA** (1111" eta 1L~
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THE GEORGE WASHINGTON UNIVERSITrSchool of Engineering and Applied Science
Institute for Management Science and Engineering
Program in logistics
Abstracto fA e s s i o n o r
Serial T-436 NTIS •RA&I5 January 1981 DTIC TAB
Unannounced
Justification
yDistribution/
NUMERICAL METHODS FOR TRANSIENT SOLUTIONS Availability CodesOF MACHINE REPAIR PROBLEMS "---- -
.
s v a i I a d / o - - i
ist special• by
Hossein ArshamArturo R. Balana
Donald Gross
This paper reviews and compares three numerical methods of com-puting transient probabilities of finite Markovian queues (particularlythe machine repair problem). A brief review of each method is followedby a numerical example of a moderate size machine repair problem (two-stage cyclic queue).
Research Supported by
Contract N00014-75-C-0729Project NR 347 020
Office of Naval Research
I
THE GEORGE WASHINGTON UNIVFRSIT'bchool of Engineering and Applied Science
Institute for Management Science and Engineering
Program in Logistics
NUMERICAL METHODS FOR TRANSIENT SOLUTIONS
OF MACHINE REPAIR PROBLEMS
by
Hossein ArshamArturo R. Balana
Donald Gross
1. Introduction
The classic machine repair problem with spares is a typical
example of a finite state space queueing problem and consists of a fixed
number of identical machines of which initially M a:e operating and Y
. are spares. The M machines are in parallel and are independent. When
one fails, it is instantaneously replaced by a spare if a spare is
available; if not. less than M machines will operate until a repaired
machine becomes avilable. Simultaneously, the failed machine goes
instantaneously into a repair facility.
L1.1 Assumptions and Problem Statement
The following assumptions are made concerning the machine repair
example.
(a) The syitem failure rate is proportional to the number of
operating machines.
(b) Each mazhine has exponential failure time with mean 1/X
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(c) There are c parallel servers in the repair facility.
(d) Each server has exponential service time with mean l/p.
Thus, the machine repair problem is Markovian and the states of this
Markov process can be described by a single number i , where i rep-
resents the number of machines in the repair station. The intensity
matrix of this vrocess is given by
SX0 XO 0 0
0 0
where
(0 < i < Y)
=(M-i+Y) (Y < i < Y+M)0 (i> Y+M)
ij , (0 < i < c)
c1i, (i > c)
For this problem, steady state solutions in closed form are readily at-
tainable [see, for example, Gross and Harris (1974)].
1.2 Transient Solutions
It is desired to find the transient solutions for the machine re-
pair problem. If the problem has N states (N = M+Y+l) , the inten-
sity matrix provides N equations
H I Mt = l(t ) • Q ,(1
where 1(t) is a N = M+Y+l component vector whose elements are
7i(t) , the unconditional probability that the system is in state i at
-2--. -
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time t , and iP'(t) a vector of derivatives of if (t) . Finding solu-
tions in closed form [see, for example, Marlow (1978)] is extremely dif-
ficult and in most cases impossible (unless M+Y is very small).
However, a variety of procedures is available which can yield numerical
solutions to the differential equations. In this paper we will discuss
several of these numerical procedures and attempt to apply each proce-
dure to a machine repair problem with the following parameters:
M-4 , number of machines initially operating
Y- , number of spares
c=2 , number of service channels
=.15 ,machine failure rate
V=.5 , service rate.
Under the assumptions mentioned in the preceding section one can obtain
the following initialized first order system of differential equations.
Assuming at trO all machines are up:
it Th W T 6 .6 0 0 0 0
Irl Wt7 Wt .5 -1.1 .6 0 0 0
nt(t) T2(t) 1 -1.45 .45 0 0
t) W3(t) 0 0 1 -1.3 .3 0
n t) M r4(t) 0 0 0 1 -1.15 .15
L orIW 0 0 1 -
with initial value 1(0) = [1,0,0,0,0,01
2. Randomization Techniques
The randomization procedures give a method of calculating the
transition probability matrix P(t) , i.e., the order (M+Y+l) square
matrix whose eleme its are Pi (t) , the probability that the system is
in state j at tire t given that it started in state i (0 < ij <
M+Y) . From P(t) , by using the initial probability vector 1(0) ,
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H(t) can be calculated from 11(t) = 1(O)P(t) . We denote the (i,j)th
element of Q by qij , and define a scalar 3 as
8 max I . (2)i
Now define a matiix P with elements pij as
P {p I Q
where I is an identity matrix having the same order asQ
The matrix P is stochastic, and it has been shown [see Cohen
(1969)] that the elements of P(t) can be obtained by
W n nf t nt (n)
Pij(t) e n= Pij (3)
_(n)where pin is the (i,j)th element of matrix P raised to the nth
ij
power. This method is called randomization because it can be inter-
preted as a discrete time Markov chain with transition probabilities
Pij and transition time generated by a Poisson process at rate 8
To compute pij(t) involves raising the matrix P to the nth
power for n=0,1,2,.... The numerical procedure truncates n at some
appropriate value, say m , so
p (t) = eBt I -nPn +R (4)ij 0 n ij m
where R is the error due to truncation.m
2.1 Barzily and Gross Method
Barzily and Gross (1979) proposed a criterion to truncate n
such that for a given E>0 , the smallest m is chosen such that the
error RM obeys
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R < I - ea t ml(t)n < E (5)R m n ! On=O
Their paper shows that such an m exists. For a machine repair problem10- 3
with parameters M - Y = C = 1 and C = 1 , the result of applying
this criterion has been shown to be quite satisfactory, in fact accurate
up to the second decimal place. A complete description of the Barzily-
Gross method together with the study of the transient effects and the
speed of convergence to steady state for machine repair problems can be
found in the above cited reference. The algorithm has been coded in
FORTRAN to run on the IBM 3031 computer, under the program name WONG.
The following section is a brief description of progiam WONG.
2.2 Program WONG
WONG, a FORTRAN code originally designed to find Lhe spares in-
ventory level and number of repair channels necessary to guarantee a
prespecified service level for a machine repair problem, included in it
a program to provide transient solutions for the system state probabil-
ities. This portion of the program was separated fron the original and
updated to stand alone as a provider of transient solutions to machine
repair problems. Input requirements for program WONG are given in
Appendix 1. Results of applying program WONG to the sample problem of
Section 1.2 are given in Section 5.
3. The QUE Package
Grassmann and Servranckx (1979) developed a FORTRAN based package
for finding transient solutions for moderate sized queueing networks (up
to ten state variables). The method adopted in this package is in fact
based on the randomization procedure discussed in the preceding sections.
The truncation criteria are somewhat different and are desctibed in-the
following section. We have adopted the sample problew of Section 1.2 to
the specifications of this package and the results are presented, along
with those of program WONG, in Section 5.
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Grassmann (1977) has also shown that the truncation error, Rm
can be bounded with reasonable accuracy; that is,
SM-1 e,~n-at
R < 1- e ()(6)n--O n(
For small at , ,he sum in the RHS of (6) can easily be evaluated for
fixed m , and thus m can be determined such that R will be below
a prescribed value c > 0 , as was done by Barzily and Gross (1979). For
large at one can approximate the Poisson distribution by the normal
distribution. In the QUE package, m is set equal to
m - t + 4/t + 5 . (7)
Using Poisson tables for 8t < 20 , or normal tables otherwise this
procedure guarantees that such an m yields an R less than 10
3.1 Formulation of the Sample Problemfor the QUE Package
To solve the machine repair problem by using the QUE package,
one must formulate the problem to fit the network structure input
requirements. The output statistics are then obtained from the program.
One way of formulating the problem to fit the QUE package requirements is
shown in Figure 1.
Spares inventory facility
Figure l.--Formulation of the samplemachine repair problem forthe QUE package.
6
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It is necessary to define state variables aud event descriptions
together with type, conditions, net effect, and ratp parameters. These
are shown in Table 1.
TABLE 1
THE EVENT DESCRIPTION OF THE PROBLEM FOR THE QUE PROGRAM
Events Type Rate Condition Net Effect
1. Arrival into repairstation
a. when no machineis in repairstation 1 .6 X1=O (+1)
b. when the-e aresome machinesin the repairstation 1 .75 - .15X1 1 < Xl < 4 (+1)
2. Service: when onlyone repairman isbusy 1 .5 Xl=l (-1)
3. Service: when bothrepairmen are busy 1 1 2 < Xl < 5 (-1)
The state variable is described by Xl H number of machines in
repair station, 0 < X1 < 5 .
Each event has a rate function which associates the rate of each
transition with the starting state. The rate functior may be constant or
a function of the state variable. Types of events are classified as type
one, having finite rate event, or as type two, having infinite rate
event. The state space of the system is defined by general conditions
represented by linear inequalities involving the state variables. The
net effect is the %alue of the state variable after the event occurs and
is determined by inirementing or decrementing the value by a constant
prior to the event's occurrence. A more detailed explanation can be
found in Grassmann and Servanckx (1979). Appendix 2 shows the QUE pro-
gram input requirements for the sample problem.
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j ____---7- --
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4. Numerical Integration Methods
Numerical integration methods can be employed to solve a system
of ordinary differential equations described by
yi(t)" f 1(Yl, ... ,y p ;t) -
y(t)
Y(t -E f(Yt) (8)
yP,(t) f p ... •' ,y;t)
L _j
with known initial value Y(t O) The standard techniques are generally
variations of either Runge-Kutta (R-K) or predictor-corrector (p-c) methods.
Runge-tutta methods are based on formulas that approximate the
Taylor series solutions
Yi(t+h) = Yi(t) + h~y (t) +hy"(t) + ... +h h (k)
i=l,...p . These methods use approximations for the 3econd and higher-
order derivatives. Euler's method is a special R-K method, with k-l
These methods have been used by several authors [e.g., Bookbinder and
#Iartell (1979), Grissmann (1977), Liitschwager and Ames (1975) and Neuts
(1975)] to find transient solutions in queueing systems.
The predictur-corrector methods require information about several
previous points in order to evaluate the next point. These methods in-
volve using one formula to predict the next Y(t) value, followed by the
application of a more accurate corrector formula. Unlike the R-K
methods, p-c methods are not self-starting; hence, they use the R-K
method to obtain the first Y(t) value.
Predictor-ccrrector methods can provide an estimate of the local
truncation error at each step in the calculations, in contrast to the R-K
methods, which cannot obtain such an estimate.
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-- 8 --
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Predictoc-corrector methods include Milne's method, Hamming's
method, and Adams' methods. Methods based on the Adams formulas have
performed very well in test problems [see Hull, Enright, Fellen, and
Sedgwick (1972)] even for nonstiff systems or when function evaluations
are relatively expensive. Hull et al. also conclued that R-K methous
are not competitive, although fourth or fifth order methods are best for
problems in whinh function evaluations are not very expensive and accura-
cy requirements are not very stringent.
Predictor. corrector methods have been used by Ashour and Jha (1973)
for queueing problems.
A variety of routines is available for solving a system of ordi-
nary differential equations. They include RKGS, DRKGS (fourth order R-K
formulas), HPCG, DHPCG, HPCL, DHPCL (Hamming's Methods), all from the IBM
Scientific Subro'itine Package [IBM (1968)]; and DVERK (Verner's fifth and
sixth order R-K formulas) and DVOGER (Gear's Method) in the International
Mathematical and Statistical Libraries (IMSL) package [IMSL (Ed. 6)]. One
routine based on extrapolation methods is DREBS, also in the IMSL
package, which uses the Bulvisch-Stoer method.
4.1 Gear's Algorithm
C. W. Gear (1971a, 1971b) proposed a variable-order integration
method based on Adams' predictor-corrector formulas of orders one through
seven. It uses an order one formula to start and, for this reason, must
start with very small step size when the error tolerance is stringent.
Gear's algorithm includes a special approach for dealing with
stiff differential equations.
A stiff system of ordinary differential equations is character-ized by the property that the ratio of the largest to the smallesteigenvalue is much greater than one.
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-- 9 --
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The Adams' formulas fall under two general categories--open and
closed formulas. The Adams-Bashforth pth order formulas (open formula)
can be written as
p I 1I
Yn = Yn-1 + KK , (9)
K I
where yi Y(t , i = ih , f(y,,ti) . The order of the method
is one less than the order of the truncation error per step. The Adams-
Moulton pth order formulas (closed formulas) can be written as
p-IYnm- n1+ a~f(nmhI *IK(0~n~m ~n o 8 -f(nlltn) K=1 K n-K(0
The coefficients 8 and M* are given by Henrici (1962). Equation (9)
K K
is used as the first approximation in Equation (10). Thus (9) is used as
the predictor eqiation and (10) as the corrector equation. Whenever (10)
converges (as is t:-ue when h is small and f is smooth), the trunca-
tion error introdLced at the nth integration step is
CiA hP+ y (p+l)(t n) + o(hP'2) , where y(K) is the kth derivative of
y 9 and CA. are constants [see Henrici (1962)].
The predictor equation (9) is equivalent to fitting a pth degree
polynomial through the known quantitites Y hyn' "" 'Yn-l' n-I n-p
For more details of the algorithm, see Gear (1971a, 1971b).I4.2 DVOGER Subroutine
DVOGER is a FORTRAN routine based on Gear's algorithm designed to
solve a set of first order ordinary differential equations. The algo-
rithm chooses the 3rder of approximation such that the step size is in-
creased, thereby decreasing the solution time. The option of using a
particular method is done through a switch variable (MTH). Results of
using DVOGER on the sample problem are also given in section 5. Appendix
3 shows the input and programing requirements for exercising DVOGER.
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5. Conclusions and Numerical Results
We have nresented three numerical methods in computing the tran-
sient probabilities for finite Markovian queues. Transient probabilities
often provide a realistic picture of actual queueirg systems, and some-
times it is desirable to know how fast they converge to steady state
[Barzily and Gross (1979)].
The methods considered in this paper fall into two categories,
namely, randomization and predictor-corrector numerical integration. In
general these methods give reasonably accurate results for a moderate
sized problem. Table 2 shows the output of these programs fort-1,3,5,7,9,12. The QUE package and DVOGER show almost equal results,
while program WONG deviates from the other two by at most 3x10 4 , which
is reasonably compatible.
In terms of set-up effort, program WONG gives the least degree of
difficulty since it was written primarily for machine repair problem.
The biggest concern with respect to the QUE program was the huge core
storage requirement, which exceeds the current daytime capacity of 384K
bytes of the IBM 3031 at the GWU Center for Academic and Administrative
Computing. Future modification by redimensioning is suggested. In using
DVOGER, one must carefully choose applicable parameter values, as in the
step size. Total running times of the programs are 2.35 seconds for pro-
gram QUE, 6.13 seconds for program WONG, and 162.62 seconds for DVOGER.
The reason for the length of the latter is that with step size fixed at
-4lXl0 , DVOGER m-ist be called 120,000 times to integrate for each time
point from t-0 t3 t-12 . There is a need to explore further the best
options of DVOGER to find those which might reduce running time consider-
ably.
, -11i-
iF,
________
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TABLE 2
PROGRAM OUTPUT OF WONG, QUE, DVOGER FOR THE TRANSIENT SOLUTIONOF A MACHINE REPAIR PROBLEM WITH PARAMETERS M-4, Y-1, C-2,
A-.15, P-.5 and H(O)-(l,0,0,O,0,0) AT t-1,3,5,7,9,12
Time Program 7r 0 (t) 7r1(t) 7r2(t) r3(t) ff4(t) n 5 (t)
1 WONO .6235 .2946 .0708 .0096 .0007 .0000
QUE .6237 .2949 .0709 .0097 .0007 .0000
DVOGMR .6237 .2949 .0709 .0097 .0007 .0000
3 WONG .3944 .3686 .1722 .0536 .0099 .0008
QUE .3945 .3688 .1723 .0536 .0099 .0008
DVOGER .3945 .3688 .1723 .0536 .0099 .0008
5 WONG .3331 .3663 .2005 .0781 .0192 .0022
QUE .3333 .3665 .2006 .0782 .0192 .0022
DVOGEa .3333 .3665 .2006 .0782 .0192 .0022
7 WONG .3119 .3618 .2091 .0886 .0244 .0033
QUE .3122 .3620 .2093 .0887 .0244 .0033
DVOGER .3122 .3621 .2093 .0887 .0244 .0033
9 WONG .3038 .3595 .2123 .0931 .0269 .0038
QUE .3039 .3597 .2124 .0932 .0269 .0038
DVOGER .3039 .3597 .2124 .0932 .0269 .0038
12 WONG .2995 .3580 .2138 .0955 .0283 .0042
QUE .2997 .3582 .2140 .0956 .0284 .0042
DVOGER .2997 .3582 .2140 .0956 .0284 .0042
6. Acknowledgments
The authors are most appreciative of the comments and suggestions
of Professors Z. Barzily and W. H. Marlow, which were incorporated into
the final version of this paper.
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APPENDIX 1: PROGRAM WONG
A. Input, Reqiuirements
TABLE Al.1
PARAMETER CARD INPUT
[CO1W rs Format Input Name of Explanation oprtn
1-5 15 M Number of machines initially operating
6-10 15 IC Number of service channels
11-15 5 IY Number of spares
16-27 F12.7 RLAM Machine failure rate (Poisson mean)
28-39 F12.7 RMU Service rate (Poisson mean)
40-49 F1O.6 EPS Tolerance value
TABLE A1.2
TIME INPUT
Columns Format Input Name Explanation*
1-7 F7.3 TDEL Time T
*Time at which transient probability isrequired is format free, but it must be codedstarting from column one, and a separate cardis required for every time desired.
B. Numerical Example Input
We shall illustrate the use of program WONG on the sample problem
given in Section 1.2. The cards for the sample problem, with C 10 3
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and t 3,5 are shown in FigureAl.l. The output obtained for this prob-
lem is tabulated in Table A1.3.
Card Type Card Image
Job // STANDARD JOB CARD
JCL // EXEC FORT2
JCL //FORT.SYSIN DD *
Program Deck [Program WONG deck]
JCL //GO.SYSIN DD *
Parameter 4 2 1 .15 .5 .001
Time 3
Time 5
JCL //
Figuie Al.1--Card input program WONG foi sampleproblem.
TABLE A1.3
THE OUTPUT OF PROGRAM WONGFOR THE SAMPLE PROBLEM*
t Iro t 0l t 0 T1 W n2 (t) n 3(t) 7 4(t) I5(t)
3 .3944 .3686 .1722 .35-6 .0099 .0008
5 .3331 .3663 .2005 .0781 .0192 .0022
*Note: the initial distribution of the system isassumed to be it0(0)=1, 71 (0)-0, 1>1
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APPENDIX 2: PROGRAM QUE
A. Input Deck Format Specification
TABLE A2.1
INPUT DECK FORMAT FOR PROGRAM QUE
1. Problem title card:Function: for documentation only
Columns Format Field description
1-80 20A4 Problem title
2. Problem specification card:Function: describes the number of state variables, number ofevents, and the number of general system conditions
Columns Format Field description
1-2 12 Number of state variables
3-4 12 Number of events
5-6 12 Number of state space restrictions
3. Maximum vector card (one card for each state variable--formachine repair problems only one is required)Function: to describe the highest possible value of each statevariable (a maximum of ten state variables)
Columns Format Field description
1-2 12 Maximum value of state variable 1
4. Event title cards (one for each event)Function: for documentation purposes only
Columns Format Field description
1-80 20A4 Event title
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TABLE A2.1--continued
5. Event specification card (one for each event)Function: indicates type of event and the rate of this event
Columns Format- Field description
1 Ii Print flag for transitions (1 - Yes)
2 Ii Event type (1 for machine repair problem)
3 Ii Number of specific conditions (0 Oor machine repairproblem)
4 Il Flag for new minima and maxima of the state variable(U - Yes)
5-10 F6.2 Rate of this event
11-12 12 State variable on which rate depends (if zero, rateis a constant)
13-18 F6.2 Increase of rate
6. New maxima and minima vector card (one for each event)Function: resets the maximum and minimum values for the statevariable
Columns Format Field description
1-2 12 New maximum for state variable Xl
3-4 12 New minimum for state variable Xl
7. Net effect -ardFunction: defines the function f(x) which converts the startingstate into the target state
Columns Format Field description
1-3 13 Net effect for state variable Xl
8. Trailer cardFunction: delimiter card used to indicate the end of eventssection of the system's input. The first four bytes of therecord must contain the string "END"
Columns Format Field descriptior
1-4 A4 Control field (value is "END")
5-80 19A4 Ignored
-16-
________
T-4 36
TABLE A2.1--continued
9. Probability specification cardFunction: gives the number of nonzero initial probabilities(only the nonzero ones need to be entered)
Columns Formot Field description
1-2 12 Number of nonzero initial probabilities
3-8 F6.2 The starting time of the system
10. Initial probability cardFunction: specified initial probability and the state to whichit pertains (one for each state variable--onl one required formachine repair problem)
Columns Format Field description
1-6 F6.5 Initial probability
7-8 12 State variable Xl
11. Time speciication cardFunction: to indicate the time for which tran31ent solutions arerequired, to indicate what measures are to be printed, and togive a criterion whether or not to continue calculating thetimes on the following card
Columns Format Field description
1 11 Number of times for which solutions are required(5 or less)
2 11 Print flag; if value is 1, joint distributions areprinted
3 I1 Print flag; if value is 1, marginal distributions
are printed
4 Il Print flag; if value is 1, expectations are printed
5 Il If value is 1, other cards follow (having the sameformat as this one)
6-11 F6.5 Stopping criterion (accuracy desired)
12. Time cardFunction: gives each time (a maximum of five) for which resultsare desired
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TABLE A2. i--continued
Columns Format Field description
1-5 F5.2 First timet
6-10 F5.2 Second timet
A the times t1,t2,. must be input in increasing order of magnitude
2....
B. Numerical Examiple Input
Input data for the sample machine repair problem is shown in Table
A2.2.
TABLE A2.2
INPUT CARDS FOR PROGRAM QUE
CardType Card Input
1 TRANSIlENT SOLUTION FOR MACHINE REPAIR PROBLEM
2 010400
3 05
4 ARRIVAL INTO REPAIR STN: (A) NO MACHINE
5 1101000.60
6 COO
7 001
4 ARRIVAL INTO REPAIR STN: (B) 1 MACHIAJE
5 1101000.7501-00.15
6 0401
7 001
4 SERVICE WHEN ONE REPAIRMAN IS IDLE
5 1101000.50
6 0,01
*7 -01
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TABLE A2.2--continued
CardType Card Input
4 SERVICE WHEN BOTH REPAIRMEN ARE BUSY
5 1101001.00
6 0502
7 -01
8 END OF EVENT SECTION
9 01000.00
10 1.000000
11 41111.0001
12 90.2500.5000.7501.00
11 41111.0001
12 01.2501.5001.7502.O0
11 41111.0001
12 02.2502.5002.7503.00
11 41111.0001 '
12 03.2503.5003.7504.00
11 41111.0001
12 04.2504.5004.7505.00
11 41111.0001
12 05.2505.5005.7506.00
11 41111.0001
12 06.2506.5006.7507.00
11 61111.000112 07.2507.5007.7508.00
11 41111.0001
12 06.2508.5008.7509.00
11 41111.0001
12 09.2509.5009.7510.00
11 41111.0001
12 10.2510.5010.7511.00
11 4.111.0001
12 11.2511.5011.7512.00
11 00000
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T-436
APPENDIX 3: DVOGER SUBROUTINE
A. Input and Options
This sec.tion discusses the input requirements for DVOGER. Since
DVOGER is a libriry subroutine, one must write a computer program in
order to use it. The advantage is in the flexibility of inputting the
parameter values, as well as in the choice of output variables, frequency
of printing the solutions, and so forth.
The input structure is as follows.
1. Job and JCL cards. See Section B for the standard and job
control language cards.
2. Main program. The main or the calling program is to be
writtei in FORTRAN. The proper dimensioning of arrays, the
input mode of parameter values, the number of calls to DVOGER,
and the frequency of printing the solution must be determined
by the user. Moreover, an external subrontine DFUN is to be
written by the user to compute functional values F(y,t) or
the Jacobian of F(y,t)
The parameters needed for the main program include:
N - number of first order differential equations
M - order of Jacobian (H-N)
T - initial value of independent veriable (e.g., time)
MTH - method indicator
0, predictor-corrector (Adams) method 11, variable-order method, suitable for stiff
- systems (partial derivatives provided by user)
2, variable-order method (partial derivatives com-puted by numerical differencing)
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T-436
Y(l,N) - an input array of initial solutions at T ; arrayY is to be declared 8xN
YHAX(N) - an input array of maximum absolute value of solu-tion
HMIN - smallest step size allowable
HMAX - maximum step size allowable
H - step size to be attempted on the next step; thisis to be used if it does not cause a larger errorthan requested
-1, repeat the last step with a new H
DVOGER)
1, take a new step continuing from the last
EPS = maximum error criterion such that the single steperror estimates divided by YMIX(I) are less thanEPS in norm.
The call to DVOGER is done by the statement,
CALL D"OGER (DFUN, Y, T, N, MTH, MAXDER, JSTART, H, HMIN,
HMAX, EPS, YMAX, ERROR, WK, IER).
3. Subroutine DFUN. DFUN is user-supplied and is to be declared
by an EXTERNAL DFUN statement in the main )r calling program.
DFUN specifies the problem for DVOGER. It provides the system
of equations and the Jacobian. The parameters include:
YP(l,N) - vector of solution TP
TP - present time
M - order of Jacobian
I 0, DFUN computes F(YPTP)
SIND - , DFUN computes Jacobian of F evaluated at
A ( (YPITP)
YP is to be declared as an 8xN array.
B. Numerical Exa le Input
We shall illustrate the solution of the sample problem using the
DVOGER subroutine. The structure of a single job run is as follows.
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T-436
(a) Job and JCL cards
// STANDARD JOB CARD
/1 EXEC FORG2
//F0RT.SYSIN DD *
[Main Program and Subroutine]
//G0.SYSLIB DD
I/ DD DSN=GWU.IMSL6.LM0D.D,DISP=3HR
1/ DD DSN=GWU.IMSL6.LMOD.S,DISP-SHR
//GO.SYSIN DD *
(b) Figure A3. 1 shows the main program and subroutine DFUN, and the
input and our options, where the transient solutions are re-
quired for times T-1,2, and 3
(c) Program Output. The step size h was fixed at 0.0001 by
specifying HMAX=HMIN-H=0.0001 . The method used was the
predictor-corrector method based on the Adams formulas
(MTH-0) . The output of transient solutions was printed out
at every time increment of .01 starting at t-O to t-3
-7The error tolerance was set at 107'
The program below was written for this specific problem, although
a more general program can be written to handle any problem with arbi-
trary parameter values.
The authors have not tested this subroutine for options which give
minimum execution time, as this was not the purpose here.
22
- 22 -
.~.S~ - q
T-436
DIMEINSION VqK(1I40.2O),ERRP( IJ),C( 10)D)(UBLIE PR~ECIS ION YPDOU3iLl PRKTIS ION' ,,EHP ,Y R 3frFSEPiIi'.HA D
*ENO2 ,EIJQ3, i AX -, ~I' IE- , diHi.iioLD,'fou.f, Y'iAY,* ~~~~ERROR, PACUA , ViK , XK,Z E.'O, HAL.F. (J.~~*~A
EXp:-RNAL. DF'jrKK-ON=6M=6
YI1,6)=U.ODOY(1I,5)=0.OD0Y(19 4)=O.Oi)0Y(1I,3)=0.0OY(1,2)=0.O000Y(1,1)=1.ODOYMAX(I1)= .01)0YMAX (2)=I1 .0OfYMAX(3)=I .01)0YMAX(4)=I .0DJI YMAX(5)= .000YMAX(6)=I.OU0JSTARTI=0I ND=OMTH-OH14AX=I .OD-4HM[.N='I .00-4
'H1=1.00-4EPS=I .OD-7WRITE(69 100)
100 FORFAT('' 99X, PT*, 12X, P'O"12X,PI'1,.I2X, P21,12X, 'P3',*12X,-'P4-',12X,'P'5-)WRITE(6,200) T,Y( I, I),Y(1I, ).Y( I 3).Y( I 4),YC I 5),Y( 1.6)Do JO K=i,30000CALL DVGEDFU,Y,T,N,TH,AAXDER,JSTARTH.HIIJ.IAXEDS.*YMAX, ERROR, NK, IER)KK=KK+I-
I IF(KK.NE.100) GOTO 250* ~ ~ ~~RITE(6,200) T,Y( 1,1 ),Y(1I,2),Y(1I,3),Y(1I,4),Y( 1,5) ,Y(1,6)
200 FORMAT("',7(2X,FIO.8))* KK-O
250 CONTFINUEH-I .00-2INDmOMTH0OIF(MTH.NE.I) GOT() 10
Pbl(192)-.5PVi( I,3)-O.O
Figure A3. l-Pzogran listing to call DVOGR for the machine repairproblem, t-1l,2 and 3
I' -23-
j T-436
Py~( 10,)=0.0Pv,,( I ,6")=0.O
Pd(2,I)=1.
PW~2,4)=I .0Pi4(2,5)=0.O
Pii(2,6)=0.0P4d( 3.1) =0.0Pii(3,2)=.6
Pi( 3,4)=l C,
Ps!(3,6)=0.QP,( 6,) =0. 0Pi4(4,2)=O.O
P, - (4,5)=I 0Pii(4, 6) =0 0PVW(5. 1) =0. 0Pi('5,2)=0 CPh(5,3)=0.0
VY -j, 4 ) =.3PW 5, 5) =-I1 15Pi't5, 6) =I .0Pii(6,1)=0 0PlqC6,2)=0.oPW(693)=0.0PW(6,4)=0.0P;4(6,5)=. 15
10 CONTINUESTOPE NOSUBROUT1INE DFUN(%YP,TP,i,DY,PVJ, IND)DIMENSION P~J( 0, 10),YP(8, I0),DY( 10)DOUB3LE PRECISION YP,TP,DY.,IF(INL.E0.0) GoT() 5
PVJ(1I,2)=.5Pil(1,3)=0.O
PkV( ,40=0.0
Pv4(2,2)=-1.lI
Pa(2.3)=0.0
PWJ(2,4)=0.0
PAs(3 9j )=Q*QPWi(392)-.6
* PN(3.3)=-1.45Figure A3.1--co;ttinued
-24-
T-436
PMd3,4)=I *(PK(3,5,)=0.OPW( 3,6)=0.0PNl(4,1I)=0.OPW(4,2)=0.O
P~i ( 4,4) =-1 .0P4( 4,6)=0.0
P,4(5, I )=O.OPvid5,2 )Q*(.P4( 5. 3)=0.PWJ(5,4)=O.3PV4(55)=-I1. 15P6(5,6)=I .0PWV(6, 1)=O.OPV4( 6,2)=0.0PI'( 6,3 )=0.OPW(694)=0.0P4J(6,5)=. 15Pbl(6,6)=-I .0OO 10
5CONTINUEDYC I )=-.6*Y'P(1I I)+.5*YP( 1,2)DY(2.)=.6*Y.t I, I)-j*I*YP( 1,2)+YP( 193)L)Y(3)=.6*YP( 1,2)-I .45*YP( I,3)+YP( 194)DY( 4)=. 45*YP( 1,3 )-I.3*YP( 1,4) YP1,5)DY(5)=.3*YP(1,4)-1.15*YP(1,5)+YP(I,6)DY(6)=. 15*YP(1I,5)-YP( 1.6)
10 RETURNEND)
Figure A3.1--continued
-25-
1-436
REFERENCES
ASHOUR, S. and 1. D. JHA (1973). Numerical transient-state solutions of
queueing systems. Simulation, 21, 117-122.
BARZILY, Z. and P. GROSS (1979). Transient solutions for repairable? item
provisioning. Technical Paper Serial T-390, Program in Logistics,
The George Washington University.
BOOKBINDER, J. H. and P. L. MARTELL (1979). Time-dependent queueing
approach to helicopter allocation for forest fire initial attack.
INFOR, 5g-70.
COHEN, J. W. (1959). The Single Server Queue. North-Holland, London and
Wiley, New York.
GEAR, C. W. (1969). The automatic integration of stiff ordinary differ-
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ed.). North-Holland, Amsterdam, 187-193.
GEAR, C. W. (1971a). DIFSUB for solution of ordinary differential equa-
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GEAR, C. W. (1971b). The automatic integration of ordinary differential
equations. Comm. ACM, 14, 176-179.
GRASSMANN, W. (1977). Transient solutions in Markovian queueing system.
Computers and Operations Res., 4, 47-56.
GRASSMANN, W. and J. SERVRANCKX (1979). The QUE pacKage. University of
Saskatchewan.
GROSS, D. and C. M. HARRIS (1974). Fundamentals of Queueing Theory.
Wiley, New York.
HENRICI, P. (1962). Discrete Variable Methods in Ordinary Differential
Equations. Wiley, New York.
HULL, T.; W. ENRIGHT; B. FELLEN; and A. SEDGWICK (192_). Comparing
numerical methods for ordinary differential equations. SIAM
Journal on Numerical Analysis, 9, (4), 603-637.
26
T-436
IBM CORPORATION (1968). System/360 Scientific Subroutine Packagi- Version
Ill. White Plains, New York, IBM Corporation.
IMSL. International Mathematical and Statistical Library, IBM Edition 6.
LIITSCHWAGER, J. and W.F. AMES (1975). On transient queues--practice
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MARLOW, W. H. (1978). Mathematics for Operations Research. Wiley, New
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various queue disciplines in the M/G/l queue with service time
distribution of phase type. Department of Statistics, Mimeograph
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