washington univ washington 0 c pfn: 12/1 machine …

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

This document has been approved for publicsale and release; its distribution is unlimited.

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THE GEORGE WASHINGTON UNIVERSITYAEAWOKNINUESPROGRAM IN LOGISEICSWASHINGTON, DC 20052

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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~

MA"

THE GEORGE WASHINGTON UNIVERSITrSchool of Engineering and Applied Science


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



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

T-436

(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--. -

T-436

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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T-436

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.

i-5-

L7 -

T-436

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

T-436

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- --

T-436

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.

-8-

-- 8 --

I~T-436

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.

-9-

-- 9 --

T-436

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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T-436

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.

-12-'1

T-436

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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I' T-436

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


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)


1-2 12 Maximum value of state variable 1

4. Event title cards (one for each event)Function: for documentation purposes only


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


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


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


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)


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


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

- 17-

T-436

TABLE A2. i--continued


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

-18 -

T-436


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

-19-

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)

-20 -

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.

- 21 -

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-

ential equations. Information Processing 68. (A.J.H. Morrell,

ed.). North-Holland, Amsterdam, 187-193.

GEAR, C. W. (1971a). DIFSUB for solution of ordinary differential equa-

tions. C~mm. ACM, 14, 185-190.

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

and pedagogy. Proc. 8th Symp. Interface, Los Angeles, California,

206.

MARLOW, W. H. (1978). Mathematics for Operations Research. Wiley, New

York.

NEUTS, M. F. (1975). Algorithms for the waiting ti-ne distribution under

various queue disciplines in the M/G/l queue with service time

distribution of phase type. Department of Statistics, Mimeograph

Series, No. 420.

- 27 -

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