On (s, S)-control in a manufacturing system

international journal of

production economics ELSEVIER

Int. J. Production

On (s, S)-control

Economics

35 (1994) 343-349

in a manufacturing

system

Peter K&he1 Tu Chemnitz,

FB Informatik.

PF 964, 09009 Chemnitz,

Germane

1. Introduction

2. The model

Queueing network models have recently been used to mode1 and study computer systems or flexible manufacturing systems (see [l] and [2]). Optimizing such systems has proven difficult. In this paper we consider a queueing network with finite multiprocessing factor and finite waiting area at the entrance to the network. The entrance is closed whenever the total number ofjobs within the system reaches S and opened again as soon as the number of jobs decreases to s < S. By introducing such a control policy, systems will reach steady state even if the original load exceeds the system’s processing capacity. Such models are of importance for the investigation of flexible manufacturing systems (see [3]), for production-inventory models and other applications. In the paper we restrict ourselves to problems of optimal (s, S)-control. The mode1 is described in Section 2. As the queueing network the centralserver mode1 (CSM), a basic mode1 for flexible manufacturing systems, is introduced. In Section 3, a simple numerical procedure for defining an optimal (s, S)-control policy is given. To this aim we first need to develop some results on M/M/l/Squeues with state-dependent service rate, given (s, S)-control policy and given cost and gain structure. Some directions for further research are given in the conclusion.

Our model consists of two parts - a queueing network as a model for the service system and a waiting area for jobs which are accepted for service but not yet admitted (cf. Fig. 1). We assume that jobs arrive according to a Poisson process. Simultaneously there can be at most K 3 1 jobs in the network. K is called multiprocessing jizctor. External arrivals will be immediately admitted to the network if the number of jobs in the network is lower than K. If there are K jobs in the network, arriving jobs will be collected in the waiting area with S - K places, S > K. Whenever the total number of jobs within the network and the waiting area reaches S the entrance of new arriving jobs is closed and they are lost. S is called the switch-of level. After reaching the switch-off level, when the number of jobs in the whole system decreases to s < S, the entrance is opened again. s is called the switch-on level. We assume s > K. For the queueing network we consider the CSM, which is widely used to mode1 flexible manufacturing systems (see [ 11). The CSM is a closed queueing network (CQN) with following assumptions (see [4]): (i) There are N + 1 service nodes. (ii) K is the number ofjobs which are permanently in the system (a completely served job will be replaced immediately by an external waiting job). There is a single job class.

0925-5273/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0925-5273(93)E0139-M

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Fig. 1. Scheme of the model

(iii) Each node consists of one server with exponential distributed service time and a local storage of at least K places with a FCFS queue discipline. (iv) The routing rule is given by a transition matrix P = (pij), where pij is the routing probability from node i to node j. It holds that (Pj 1 Pij = 0

For an interpretation of the CSM, see Fig. 2. In terms of FMS nodes 1, . . . , N are workstations (performing operations such as machining, assembly, inspection, testing, . . . ). The workstations are linked by a material handling system (e.g. transporters, carts, conveyors, etc.), which operates as a central server (node 0). With probability pj, j=l 1 . . . . N, a job will go from the material handling system to work at station j. The loop from node 0 to node 0 represents replacing of a completely served job by an external waiting job. The fixed number K of jobs in the CSM can be viewed as the total number of pallets available in the FMS. If we define constants

I po.pi/pi

for i = 1,

. , N,

(2.2)

x:*/G(K),

i=l

with

(2.3)

v;Ki!!ixYi’

G(K) is a normalizing constant and ni denotes the number of jobs in node i, i = 0, . , N. To define G(K) for fixed K, Buzen [4] gives an algorithm which works as follows: (I) Define

po + p1 + .‘. + pN = 1.

Xi =

P(n,,, . , nN) = fi

..+n,+?+,

with

for i = 0,

it follows (cf. [4]) that

G(K) =

for i=O andj=O, . . ..N. for i = 1, . . . , N and j = 0, else

c 1

from CQN-theory

(2.1)

g(l,O) = 1

for 1 = 0, . . . . K,

g(O,n) = 1

for n = 0, . . . , N.

(2.4)

(11) Compute g(I,n)forl= recursion

1, . . . . Kandn=

g(l,n) = g(l,n - 1) + x;g(l

1, . . . . Nbythe

- 1,n).

(2.5)

(III) Set G(K) = g(K, N). Using constants G(K) for K = 0, 1, . . . , it is easy to define performance measures. For instance, we have:

345

P. Kiichqlllnr. J. Production Economics 35 (1994) 343-349

P . h

Fig. 2. Scheme of the CSM.

l

for utilization

e,,(K) = G(K - 1)/G(K) @i(K) = Xi.Qo(K)y l

for throughput

In the following section a procedure for defining optimal (s, S)-control policy will be developed.

qi of node i:

i = 1,

an

and 3. Approach for model analysis

. ,N;

D,(K) of node i:

Do(K) = QOW).PO and D,(K) = QO(K).P~ ‘Pi, i = 1, . . ..N. We note that G(0) = 1 and G( - 1) = 0 by definition. The throughput D(K) of a CSM with K jobs can be defined as the average number of jobs which depart the central server in the direction of the central server, i.e., D(K) = po. D,(K) = po. po. G(K - 1)/G(K).

(2.6)

The operation of the model is controlled by a (s, S)-acceptance-and-rejection policy (AR-policy) ?n the manner described in Section 2. Without such a policy our model is an usual open queueing network (OQN). At present there exist many results on queueing networks (see [2] for a review); however, for queueing models with an AR-policy an exact analysis is not straightforward. Therefore, we will use an approximate method which stems from Avi-Itzhak and Heyman [S] - the two-step decomposition method (see Fig. 3). In accordance to the present situation the two steps are: Step 1. Analyze the CSM to obtain the state dependent service rate p(K) = D(K) for the network with K jobs, K = 1,2, . . .

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process is a birth-and-death Z = (0, 1,

process with state set

. , s; (s + I)‘, (s + 2)‘, . . . ,(S - l)‘),

where n and n’ denote the number of jobs in the system with opened and closed entrance, respectively. The corresponding birth-and-death rates are obtained from model description in Section 2 (see Fig. 4). From Fig. 4 we derive in the usual way the system of equations for the steady-state probabilities p(z), z E Z (s < S - 1 is assumed): i.P(O) = p(l).p(l) [jl + p(i)] ‘P(i) = Lp(i - 1) + p(i + l)p(i + 1) 1, . . ..K

i=

-

1,

[i + p(K)] p(i) = Ap(i - 1) + p(K)p(i i = K, . . . . S - 2, Fig. 3. Two-step

decomposition

method

[A +

PWI P(S) =

ibP(s -

2. Using the state-dependent service rate obtained in Step 1, replace the CSM by a single server. Solve the aggregated model. In [6] it is shown that the two-step decomposition yields exact results for OQN with one job class and product-form solution for steady-state probabilities. A sufficient condition for the last is that jobs arrive according to a Poisson process, service times are exponentially distributed and all queues have unlimited waiting place with FCFS scheduling discipline. Consequently (cf. Section 2) the twostep decomposition yields exact results for our model. To obtain these results we consider the M/M/l/S-system with arrival rate 1, > 0 and statedependent service rate

1)‘)l;

[A + p(K)] p(S - 1) = 1. p(S - ‘2) AK).p(S) p(f)

=

=

i.p(S

p(S),

p(n) =

N4, D(K),

K < n 6 S

(3.1)

-

,(S - 1)‘.

In the case s = S - 1 the system with AR-policy degenerates to the classical M/M/l/S-system and we have Z= {O,l, . . ..S} and l.P(O)

= Al)P(l)t

[A + p(i)].p(i)

= l:p(i

i= I ,u(K).p(S)

= A.p(S

r(i) = A/p(i),

R = R(K),

-

1) + p(i + l)p(i + 1)

1, . . ..s-

1

(3.3)

1).

we introduce i = 1, . . . , K

R(i) = r(1) ... r(i),

for some 1 < K < S. The parameter n represents the number of jobs in the system. We further assume that the service process is controlled by a (s,S)AR-policy with K < s < S. Obviously, the service

I),

i’ = (s + l)‘,

For abbreviation l
i # s,

1) + AK) cP(s + 1) (3.2)

+ P((S +

Step

+ l),

i=

r = r(K),

1, . . . . K

R(0) = 1 and s-s- 1 C = C rm. m=O

(3.4)

P. Kiicheljlnt. J. Production Economics 35 (1994) 343-349

jl(1)

/L(2)

P(K)

P(K)

341

P(K)

p(K)

L (s+1)

.-jTY7te

’

F(K)

Fig. 4. Markov

From

graph

(3.2) follows

p(i) = r(i).p(i

rimK.p(K) r.p(i

i = 1, . . . ,K;

- 1) = R(i).p(O), = riPK.R.p(0),

i = K + 1, . . . ,s; r-s- 1 - 1) - p(S) = rieK.R.p(0) - p(S) 1 rm, m=O

i = s + 1, . . . ,S - 1; rSmK.R/C.p(0),

(3.5)

i = S.

With the normalizing

equation

xzsZ p(z) = 1 we get

S-l

5 R(i) + R/rK.

p(O) =

i=O

1

ri +RJC.rSPK

i=K+l -1

s-s-2

x

(S-s)-

C

[

(S-s-1-m).r”

.

ItI=0

(3.6) In an analogous way we derive from (3.3) for the case 0 < K Q s = S - 1 that c

p(i) = R(i). p(O),

i = 1,

p(i) = ripK. R.p(O),

. , K,

i = K, . . ..S. (3.7)

p(o) = 1 i

i$oR(i) + R/r’. ‘[

5 i=K+

ri] 1

for the service process.

Thus for given K, s, S, 1 and p(n) the steady-state probabilities p(z), ZEZ, can be computed. These probabilities are the basis for defining AR-policies which are optimal in accordance to a given criterion. Here we restrict ourselves to the criterion maximum profit under the following cost and gain structure: W denotes the waiting cost factor, i.e., the cost which arise per time unit for one job staying in the waiting area; V denotes the service cost factor, i.e., the cost which arise per time unit for one job staying in the service system; A denotes the rejection cost per rejected job; G denotes the gain per served job. Let F (A, p, K, s, S) denote the expected profit per time unit for a M/M/l/S-system with arrival rate A, state-dependent service rate p in accordance to (3.1) and (s, S)-AR-policy. Clearly, we have F(i,p,K,s,S)=c-

w--

v-2,

where G, w, v and 2 denote the expectation per time unit for gain, waiting, service and rejection, respectively. We have,

c?= G. c

D(z). p(z)

348

P. Kiichrl/Int.

D(K) -

= G.

J. Production

; [D(K) - D(n)]. p(n)

(3.8)

n=O

i.e. the expected system’s throughput C,,zD(z) p(z) is equal to the maximum throughput D(K) minus the expected reduction of system’s throughput by the not fully loaded service system. For I? the following holds:

w=w.[ nzg+l(n - Jv.p(n) (S-

The expected

v=v.

K[

K).Pk)

1

.

) i(K-n).p(n)

(3.9)

(S-

A = A. 2. p(S) +

(3.10)

1

which can be interpreted (3.8). Finally, we have

c

n=o+

in an analogous

way as

1)’

p(n) 1)’

1

= A . i. (S - s) p(S),

(3.11)

where (S - s).p(S) denotes the steady-state probability for closed entrance under given (s,S)-ARpolicy. In principle with function F( .) we can solve the above-mentioned optimization problem: F* (2, p, K) =

343-349

considered numerical examples) the following procedure for defining an optimal (s, S)-AR-policy can be proposed: (I) Compute D(M) for M = 1, . , K by (2.6). (II) Set S equal to K. Repeat (11.1) S + S + 1; (11.2) for s = K, . ,S - 1 solve the corresponding M/M/l/S-system with service rates (3.1) and (s, S)-AR-policy; compute F (L P, K s, S); (11.3) choose the maximizing value s* = s*(S) until F* (A, p, K) is achieved.

4. Conclusion

service costs per time unit are

n=O

[

35 (1994)

1)’

c (n n=(s+l)'

+

Economics

max F (& PL,K s, S). I, sj:K < s< s

Optimization can be extended to parameters 1, p and/or K. However, since the steady-state probabilities depend on all parameters in a complex way, the best, we can expect is a numerical solution. Therefore, qualitative properties for function F (.) are of interest. For instance, if function F( .) possesses the discrete concavity property regarding s and S (which holds for all

We have investigated a service system controlled by a (s, S)-AR-policy. The paper contains only first results on the problem. There is a variety of open questions for further work: (1) How to solve generalized optimization problems with (a) more general cost and gain functions (b) other policy sets and (c) more decision parameters? (2) Which is the quality of the two-step decomposition in our model for systems without productform solution? on the system’s para(3) Do simple conditions meters exist implying s* = K or s* = S - l? overall optimization by analytical (4) Since methods probably is impossible, the combination of simulation and analytical methods should be applied. How to do it?

Acknowledgements The author would like to thank H.-J. Girlich and an anonymous referee for their careful reading and valuable suggestions.

References

[l]

Buzacott, I.A. and Yao, D.D., 1986. On queueing network models of flexible manufacturing systems. Queueing Systems, 1: 5-27.

P. Kiicheljlnt.

J. Production

[2] Lavenberg, S.S., 1988. A perspective on queueing models of computer performance, in: “Queueing Theory and Its Applications-Liber Amicorum for J.W. Cohen”, Amsterdam, North-Holland, pp. 59-94. [3] Buzacott, LA. and Shanthikumar, J.G., 1980. Models for understanding flexible manufacturing systems. IEEE Trans., 12: 3399350. [4] Buzen, J.P., 1973. Computational algorithms for closed

Economics

35 (1994) 343-349

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queueing networks with exponential servers. Commun. ACM, 16: 527-531. [S] Avi-Itzhak, B. and Heyman, D.P., 1973. Approximate queueing models for multiprogramming computer systems. Oper. Res., 21: 1212-1231. [6] Chandy, K.M., Herzog, U. and Woo, L., 1975. Parametric analysis of queueing networks. IBM J. Res. Dev., 19(l): 36-42.