3.2 Manufacturing Systems
In this section, we study some Markovian queueing models for manufacturing systems. We first consider reliable machine manufacturing systems and then discuss failure-prone machine manufacturing systems.
3.2.1 Reliable Machine Manufacturing Systems
We first consider a Markovian model of a one-machine manufacturing system. We then study the case of two machines in tandem. In both cases, the machines are reliable.
3.2.1.1 One-Machine Manufacturing System
In the one-machine manufacturing system, the production time for one unit of product is exponentially distributed with a mean time of1. The inter-arrival time of demand is also exponentially distributed with a mean time of1. The demand is served in a first come first serve manner. In order to retain the customers, there is no backlog limit in the system. However, there is an upper limitn.n 0/for the inventory level. The machine keeps on producing until this inventory level is reached and the production is stopped once this level is attained. We therefore seek the optimal value of n, called the hedging point or the safety stock, which minimizes the expected running cost. The running cost consists of a deterministic inventory cost and a backlog cost. In fact, the optimal value ofnis the best amount of inventory to be kept in the system so as to hedge against the fluctuations in demand. The notation is summarized as follows (Fig.3.1):
(a) I, the unit inventory cost (b) B, the unit backlog cost (c) n0, the hedging point
(d) 1, the mean production time for one unit of product (e) 1, the mean inter-arrival time of demand
μ λ n
μ λ n–1
μ λ
1 0
μ λ Fig. 3.1 The Markov Chain (M/M/1 Queue) for the manufacturing system
If the inventory level (negative inventory level means backlog) is used to represent the state of the system, one may write down the Markov chain for the system as follows. Here we assume that > , so that the steady-state probability distribution of the above M/M/1queue exists and has an analytic solution
q.i/D.1p/pni; iDn; n1; n2;
wherepD=andq.i/is the steady-state probability that the inventory level isi. Hence it is straightforward to write down the expected running cost of the system (sum of the inventory cost and the backlog cost) when the hedging point isn as follows:
E.n/DI Xn iD0
.ni/.1p/pi
!
CB X1
iDnC1
.in/.1p/pi
! :
The first term is the expected inventory cost and the second term is the expected backlog cost. The following proposition gives the optimal hedging pointh.
Proposition 3.1. The expected running costE.n/is minimized if the hedging point nis chosen such that
pnC1 I
ICB pn:
3.2.1.2 Two-Machine Manufacturing System
We study a two-stage manufacturing system [43]. The manufacturing system is composed of two reliable machines (no break down) producing one type of product.
Every product has to go through the manufacturing process in both stages. Here it is assumed that an infinite supply of raw material is available for the manufacturing process at the first machine. Again the mean processing time for one unit of product in the first and second machine are exponentially distributed with parameters11 and 21 respectively. Two buffers are placed immediately after the machines.
Buffer B1 of size b1 is placed between the two machines to store the partially finished products. Final products are then stored in BufferB2which has a maximum size of b2. The demand is assumed to follow a Poisson process of mean rate . A finite backlog of finished product is allowed in the system. The maximum allowable backlog of product ism. When the inventory level of the finished product ism, any arrival demand will be rejected. Hedging Point Production (HPP) policy is employed as the inventory control in both buffersB1andB2. For the first machine, the hedging point is b1 and the inventory level is non-negative. For the second machine, the hedging point is h; however, the inventory level of buffer B2 can be negative, because we allow a maximum backlog ofm. The machine-inventory system is modeled as a Markov chain problem. It turns out that the process is an irreducible continuous time Markov chain. We now give the generator matrix for the process.
3.2 Manufacturing Systems 81
We note that the inventory level of the first buffer cannot be negative or exceed the buffer sizeb1. Thus the total number of inventory levels in the first buffer isb1C1.
For the second buffer, under the HPP policy, the maximum possible inventory level ish.hb2/(Fig.3.2).
If we letI1.t/andI2.t/be the inventory levels of the first and second buffer at timet respectively thenI1.t/andI2.t/are going to take integral values inŒ0; b1 andŒm; hrespectively. Thus the joint inventory processf.I1.t/; I2.t//; t 0g can be shown to be a continuous time Markov chain process taking values in the state-space
S D f.z1.t/;z2.t//Wz1D0; : : : ; b1; z2D m; : : : ; h:g:
To obtain the steady-state probability distribution of the system, we order inventory states lexicographically, according toI1 first and thenI2. One can then obtain the following generator matrix governing the steady-state probability distribution
A1D If we can get the steady-state probability distribution p of the generator matrixA1, then many useful quantities such as the throughput of the system
0
@1 Xh jDm
p.0; j / 1 A2
and the mean number of products in buffersB1andB2(work-in-process)
b1
X
iD1
0
@ Xh
jDm
p.i; j / 1 Ai
and Xh
jD1 b1
X
iD0
p.i; j /
! j
can be obtained. We remark that the generatorA1 is irreducible, has zero column sum, positive diagonal entries and non-positive off-diagonal entries, so thatA1has a one-dimensional null space with a right positive null vector. The steady-state probability distribution p is then equal to the normalized form of the positive null vector. Similar to the previous chapters, we consider the following equivalent linear system
G1x.A1Ce1eT1/xDe1; (3.4) where e1 D .1; 0; : : : ; 0/T is the.b1C1/.mChC1/unit vector. Similarly one can show that the linear system (3.4) is non-singular and hence the steady-state probability distribution can be obtained by normalizing the solution of (3.4). We have the following lemma.
Lemma 3.2. The matrixG1is non-singular.
However, the closed form solution of p does not exist in this case. Thus classical iterative methods including Conjugate Gradient (CG) type methods can be applied to solve the problem. We remark that the idea of the Markovian model can be extended to the case of multiple machines in tandem. However, the computational cost for solving the steady-state probability distribution will increase exponentially with respect to the number of machines in tandem. A heuristic method based on Markovian approximation has been proposed in [47] to obtain an approximate solution.
3.2.1.3 Multiple Unreliable Machines Manufacturing System
Manufacturing systems ofm multiple unreliable machines producing one type of product have been studied in [37, 46, 49]. Again the arrival of demand is assumed to be a Poisson process and the machines are unreliable; when a machine breaks down it is subject to an exponential repairing process. The normal time (functioning time) of a machine and the processing time of one unit of product are both exponentially