Since the number of the machines are finite, the machine repair system falls into ...... Kleinrock, L. Queueing Systems, Vol II: Computer Applications. Wiley, 1976.
Continuous approximations repair system
of the machine
Ho Woo Lee and Seung Hyun Yoon Department of Industrial Engineering,
Sung Kyun Kwan University, Su Won, Republic of Korea
Soon Seok Lee Switching Method Section, ETRI,
Tae Jon, Republic of Korea
This paper considers two dtfjiision approximations to the queueing systems with jnite calling population, with special emphasis on the machine repair problem. We consider two boundary policies: the elementary return boundary (ERB) and the instantaneous return boundary (IRB). We obtain approximate solutions to the probability distribution of the number of inoperative machines for each of the boundary policies. Keywords: diffusion repair system
approximation,
elementary
return boundary,
1. Introduction Many production systems have machine failures. How well the manager of the system can handle the failures and repairs of the machines greatly affects the system throughput. That is why the machine failure problems have been studied by numerous researchers. In such systems we usually encounter an optimization problem to determine the number of operating machines, the number of spare machines, and the number of repairmen, etc. This has been a classic problem named the machine repair problem or the machine interference problem. As far as the repairs are concerned, we have a queueing system in which the failing machines are customers and the repairmen are servers. Since the number of the machines are finite, the machine repair system falls into the category of the queueing system with Jinite calling population.
Most of the solutions of the machine repair problem were concerned with finding the exact solutions, many of which were expressed in complicated mathematical forms such as transforms. Accordingly, extracting the necessary information from the exact solutions posed another big problem to the industrial managers. It is inevitable to devise a scheme that provides them with an easier access to the solution of problems. This paper concerns developing two approximate solutions that
This research was supported by the Korea Science and Engineering Foundation (KOSEF). Grant 901-0915~6-2. Address reprint requests to Dr. Ho Woo Lee, Department of Industrial Engineering, Sung Kyun Kwan University, Su Won, South Korea 440-746. Received 28 June 1994; revised 16 November 1995
1994; accepted
Appl. Math. Modelling 1995, Vol. 19, September 0 1995 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010
17 March
instantaneous
return boundary,
machine
make it possible to obtain the solutions with a simple desk calculator. Benson and Cox’ and Benson’ studied the case of exponential life and repair times. Benson2 also studied the case of Erlang lifetimes and showed that the probability distribution of the number of machines in the repair shop is identical with the case of exponential lifetimes. Later this invariance was proved by Bunday and Scraton3 This invariance property states that if the repair time follows the exponential distribution, the probability distribution of the number of failed machines is invariant irrespective of the lifetime distributions. Sztrik4 used the supplementary variables to model the system. The assumption of the exponential repair time, on which most of the above-mentioned studies were made, is hard to accept in real-world systems. Goheen’ studied Erlang lifetime and repair time distributions. But since an Erlang random variable can be represented by a sum of identical and independent exponential random variables, it is hard to see that his work overcame the Markovian barrier. Readers are urged to see Stecke and Aronson6 for a detailed description of the problem and related works. Also invaluable is the book written by Takagi,7 which contains a chapter for the M/G/1J/N system to which the machine repair system belongs. He made a comprehensive use of imbedded Markov chains and supplementary variables. For the case of general lifetime and repair time distributions, it is almost impossible to obtain the exact solutions. One could resort to a simulation to obtain the estimates of the system performance measures, but simulation does not provide the insight the mathematical solutions possess. The best one can do would be to develop approximate solutions. 0307-904x/95/$10.00 SSDI 0307-904X(95)00052-L
Continuous
approximations
Bhat’ classified the approximation in queueing theory into three categories: system approximation, numerical approximation, and process approximation. The system approximation represents the system characteristics (such as service times, repair times, etc.) approximately. The numerical approximation includes, for example, the simplification of the system equations or the numerical inversion of transform solutions. The process approximation uses a totally different method to approximate a discrete process. This approximation usually includes the fluid approximation and the diffusion approximation.’ The fluid approximation considers a queueing system as a water tank where water flows in and out. The diffusion approximation uses a continuous-time continuous-state process called the diffusion process to approximate a discrete stochastic system. This paper adopts this diffusion approximation technique. The theories on diffusion processes are deep and can be found in many books and papers.‘s’5 Based on these rich mathematical developments, active researches on the diffusion approximation began in the 1970s. Feller” classified the boundary behaviors of the diffusion process into three types: the reflecting boundary (RB), the instantaneous return boundary (IRB), and the elementary return boundary (ERB). The diffusion process with RB reflects itself as soon as it touches a boundary. The diffusion process with IRB jumps into the diffusion region as soon as it touches a boundary. Thus the probability that the diffusion process with IRB stays at a boundary is zero. The diffusion process with ERB is absorbed into the boundary for a random amount of time and then jumps into the diffusion region. Gaver16 and Gaver and Shedler” used the RB. Haryono and Sivazlian18 and Sivazlian and Wang23 used RB to approximate the repairman problem. ERB was first used by Gelenbe*’ and then by Kimura21 and Kimura and Ohsone. Chiamsiri and Moore23 showed that ERB provides better approximation for batch arrival queues. Kimura24 used ERB for the lower boundary and RB for the upper boundary to approximate the state-dependent queues. In this paper we use ERB and IRB to approximate the system depicted in Figure 1. Section 2 describes the system under study. Section 3 describes the diffusion process with ERB and derives the approximation by using ERB. Section 4 seeks the diffusion approximation by using both ERB and IRB.
of the machine
repair system
H. W. Lee et al. :PAlR SHOP
4-
Figure
1.
The machine
repair system.
operating group if there are fewer than M operating machines. Otherwise it joins the standby group. (e) Machines are identical with individual failure rate 1. (fJ We assume cold standby, that is, the failure rates of the machines in the standby group are zero. (g) The repairmen are identical with repair rate p. The system is depicted in Figure 1. It is to be noted that as far as the repair shop is concerned, it is the queueing system that can be denoted by GI/GIRIN/N. 3. Diffusion approximation
by elementary return
boundary Feller” introduced the diffusion process having the elementary return boundary (ERB). Let r, and rz be the lower and upper boundaries of a diffusion process. The diffusion process with ERB is absorbed into the boundary as soon as it touches a boundary. Since the diffusion process is Markovian, the staying time at a boundary after the absorption should follow the exponential distribution. Let e, and e2 be the mean staying times at each boundary. According to Feller,” the diffusion process with ERB is governed by the following equation on diffusion region (r,, r2):
2. The system
In this paper, we consider characterized as follows:
a machine
repair system
(4 There is an operating group that consists of M machines at most.
@I There is a standby group that consists of S spares.
Thus we have a total of N = A4 + S machines. As soon as an operating machine fails, it becomes (4 inoperative and is attended by one of the R repairmen. If all repairmen are busy, the machine waits for the first free repairman. (4 As soon as a machine is repaired, it joins the
where f(t, x) = probability density function of the position of the diffusion process at time t, IIJt) = probability that the process is on ri at time t (i = 1, 2),
Appl.
Math.
Modelling,
1995,
Vol. 19, September
551
Continuous
approximations
of the machine
repair system:
@(x),or(x)= infinitesimal mean and variance defined by p(x) = lim a(x) = lim
3.2 Solution of the system equations From equations (l)-(3), we have the following set of steady-state system equations:
E[X(t + h) - X(t)IX(t) = x] h
h+O
H. W. Lee et al.
Vur[X(t + h) - X(t)(X(t) = x] h
h-0
TI~I -d = __
e, = mean absorption time once it is absorbed into r-i, 7; = probability of jumping into the diffusion region (rl, r2), not onto the other boundary rj (j # i) given that it jumps from ri, P,(x) = distribution function of the position of the diffusion process after a jump given that it jumps from ri into the diffusion region. Also, we have the equations for the behavior at each boundary:
$Il,(t) = -__
nl@) el
p,,
+ e,
7'2n2 Pi(X) - ~ -d dx e2 dx
e,
P2b)
The boundary conditions are25 lim f(x) = 0
n2w
(2)
x-0
lim f(x) = 0 X-N
+ x_I, lim
i
h(t) $ n,(t) = - __
Ca(x)fk 41 - PWk
4
I
{
1 a Tj T& C4xm
To approximate a queueing process by a diffusion process, we first need to express the diffusion parameters by queueing parameters. We get the following representation of the diffusion parameters: MA
(6) (7)
$ PI(X) = 6(x -
1)
- (Iv - 1)]
0
otherwise
(15) with x = 0 and using equation
- &Mx)
which is equal to the left-hand side of equation Thus we get
=“1
(16) (11).
(17)
Modelling,
(15) with x = N and using equation
Since equation (18) is equal to equation (12), we need to solve equation (15) by using C* given by equation (17). Then we obtain
(9)
6(.) in equations (8) and (9) is the Dirac delta density function. Naturally we have r, = 0 and r2 = N( = M + S).
Math.
1 ifx>i,
c* = fz ; $ c4wx)l
From equation (17), we have
T2 = 1
Appl.
CJ(x - i) =
(15) and U(x - i) is
e1
1 e2 = RP Tl = 1
r;‘,PZ(X) = S[x
U(x - N + 1) + c*
e2
where C* is the integration constant the unit step function defined by
p
1
- 1) - 3
el
From equation (13), we get
3.1 Representing diflusion parameters by queueing parameters
552
= _-=1 U(x
41 - Bc4.m x) I
(3) where I’&(t) is the probability that the process is on ri at time t, and rij (i # j, i, j = 1 or 2) is the probability that the process Jumps onto the other boundary given that it jumps from rti
e, =-
Integrating equation (10) once, we have
PI, + e, n,(t)
e2 - x_,2 lim
5;
1995,
Vol. 19, September
=MLII,[l-U(x-l)]-QrI,U(x-N+l) (19)
Continuous
approximations
5 2 SAX)= C,n,,
.fi(X) =
C1111Cexp(2xP11aJ -
11
(26)
b1
From equation (21), if 1 < x 5 2 and & # 0, then h(x) =
C2
wVB2xla2)
From smoothing thus
O
