We present a series of basic modules for buffers and machines and the interconnections between them. Thanks to the modularity of PNs, the modules in this section can be composed in a bottom-up fashion. A job-shop example has been presented in [18] to well clarify this.
3.4.1 BUFFERMODELS
Let q be the number of different part classes in the system. An MC buffer bi is modelled with q continuous places prbi, for r = 1,. . ., q, whose marking represents the buffer content of parts of class r.
Let Iin,biq and Iout,biq be the set of indexes of machines that may, respectively, deposit or take from the buffer parts of class r. The arrival of parts of class r from machine Mj, where j∈Irin,bi, is modelled by a continuous transition labelled tMj,bir inputting into prbi. The routing of parts of class r to machines Mj, where j∈Iout,bir , is modelled by a continuous transition labelled trbi,Mjoutputting from prbi. These transitions represent the interfaces among machines and buffers: if no constraint is associated with these flows, the MFS of these transitions are taken to be∞.
If the buffer has a finite capacity Dbi, then a continuous place pbiwill also be present in the buffer model. This new place will have arcs Pre[pbi,ã] =q
r=1Post[prbi,ã]and Post[pbi,ã] =q
r=1Pre[prbi,ã].
The initial marking of place pbiis chosen as mpbi(τ0)=Dbi−q
r=1mprbi(τ0). Thus, for any reachable marking m holds mpbi+q
r=1mprbi=Dbi.
The FOHPN model of a finite capacity MC buffer is shown in Figure 3.3. The initial marking shown assumes that the buffer is initially empty. For each MC buffer, the following set of equations will be included inS(N, m):
⎧⎪
⎨
⎪⎩
j∈Iin,bir
vrMj,bi ≥
j∈Iout,bir
vrbi,Mj if mprbi =0 (a)
q r=1
j∈Iout,bir
vrbi,Mj ≥ q
r=1
j∈Iin,bir
vrMj,bi if mpbi =0 (b) (3.8)
For each class r, we use a constraint of type (3.8a) if the buffer does not contain parts of this class, that is, mprbi=0. We use constraint (3.8b) if the buffer is full, that is, mpbi =0. The number of constraints associated with an MC buffer depends on the current marking m and may vary from 0 (when the buffer is not full but contains parts of all classes) to r (when the buffer is full of parts of only one class).
If we want to model a single-class buffer bi, we need only two continuous places, denoted pbi
and pbi.
t1Mh,bi
p1bi pbi pqbi 0
class q class 1
Buffer i
0 Dbi
t1Mk,bi
∞ ∞ tqMh,bi ∞ tqMk,bi ∞
t1bi,Mu ∞ t1bi,Mw ∞ tqbi,Mu ∞ tqbi,Mw ∞
FIGURE 3.3 The model of an MC finite buffer.
3.4.1.1 Zero–Capacity Buffer
Let us now show how to impose synchronization constraints among continuous transitions. As an example, we want that the overall flow of transitions t1, t2 and t3 is equal to the overall flow of transitions t4and t5, that is, we want
v1+v2+v3=v4+v5.
This can be done, as in Figure 3.4, introducing a zero-capacity buffer, represented by the empty continuous places p and p. These two empty places enforce two constraints of the form given in Equation (3.6d) on the IFS of their input and output continuous transitions:
v1+v2+v3 ≥ v4+v5
v4+v5 ≥ v1+v2+v3
Finally, note that the zero-capacity buffer introduces an empty cycle. Thus, if we want to use the definition of David and Alla [31], we have to assign anε-marking to this cycle (see Section 3.3.5).
3.4.2 MACHINEPRODUCTIONMODELS
The production of an MC machine Mi processing q classes of products is modelled with q single-class subnets. For each subnet associated with each class r, for r = 1,. . ., q, we have a continuous transition tMir that represents the processing of parts of class r, and one input and one output zero-capacity buffer—represented by continuous places prin,Mi, pin,Mir and prout,Mi, pout,Mir —that impose that for parts of class r the total input flow is equal to the processed flow and to the total output flow.
Let Iin,Mir and Iout,Mir be the set of indexes of, respectively, input and output buffers of machine Mifor parts of class q. The interfaces among machine and buffers are represented by continuous transitions tbj,Miq (for j∈Iin,Mir ) and tqMi,bj(for j∈Irout,Mi), as already discussed in the model of the buffers.
We assume that the production of any part class is not singularly bounded, that is, the MFS of each trMiis∞, but we assume that the machine has an overall production rate, modelled by the firing of transition tMi that is bounded by VMi. Thus, the continuous transition tMi is synchronized with all tMir by the zero-capacity buffer represented by continuous places pMiand pMi: this ensures that vMi=v1Mi+ ã ã ã +vqMi.
As parts of different classes may require for their processing different service times, let us denote withθr the average service time of parts of class r, and letθ˜ = minr{θr}. Thus, we can assume
t1
p 0 p 0
t2 t3
t4 t5
FIGURE 3.4 The model of a zero-capacity buffer.
t1bh,Mi
p1in,Mi pqin,Mi
pqout,Mi
pqin,Mi
pqout,Mi tqMi
p1out,Mi p1out,Mi
p1in,Mi α1
α1 pMi
pMi tMi VMi
αq
αq t1Mi
0 0
0
0
0 0
0 0
t1bk,Mi tqbh,Mi tqbk,Mi
∞ ∞
∞
∞
∞
∞
Class 1 Class q
Machine i
0 0
t1Mi,bu ∞ t1Mi,bw ∞ tqMi,bu ∞ tqMi,bw ∞
FIGURE 3.5 The production model of an MC machine.
VMi = ˜θ−1, and definingαr = θrVMi, for r = 1,. . ., q, we obtain the FOHPN model shown in Figure 3.5. For such an MC machine, we have the following set of(2q+2)equations:
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
vMi≤VMi (a)
vMi=q
r=1αrvrMi (b)
vrMi=
j∈Iin,Mir vrbj,Mi (∀r=1,. . ., q) (c)
vrMi=
j∈Iout,Mir vrMi,bj (∀r=1,. . ., q) (d)
(3.9)
Equation 3.9b derives from the zero-capacity buffer pMi and pMi. The 2q equations (3.9c) and (3.9d) derive from the input and output zero-capacity buffers of each class r subnet.
It is possible to simplify the machine model assuming either a single–class machine or only one input buffer. In the case of a single–class machine, we have just one of the sub-nets represented within the continuous line boxes in Figure 3.5. Clearly, the specification of the class is needless:
the unique continuous transition of this sub-net will be simply labelled tMi and have an MFS VMi. The zero-capacity buffer pMi and pMi will be removed. In the case where part of class r is coming from just one input buffer bh, the zero-capacity buffer represented by places prin,Miand prin,Mimay be removed, directly connecting transition trMito the buffer bh. Similar reasonings can be applied when the machine is putting processed parts into a single buffer.
3.4.3 MACHINEFAILUREMODELS
Let us now consider machines that are unreliable. Two different failure models are presented: a time-dependent failure (TDF) and an operation-dependent failure (ODF) model.
The TDF model assumes that a machine fails after a given time has elapsed since the previous repair operation. On the other hand, an ODF model assumes that a machine fails after a given production volume has been processed since the previous repair operation. As discussed in [22], the ODF model is more appropriate than the TDF model when dealing with manufacturing systems.
However, TDF models are suitable when programmed maintenance is adopted.
An FOHPN representing a TDF model is shown in Figure 3.6a. This model is similar to the one presented in [3]. The firing of the continuous transition tMicorresponds to a continuous production at rate vMi≤VMi. The machine will be working until the place pup,iis marked. The firing of transition tF,icorresponds to the machine failure and this event occurs after a random delay exponentially distributed with parameterλF,i. When tF,ifires, the token in pup,imoves to place pdown,i; hence, tMiis disabled and cannot fire. Analogously, the firing of transition tR,icorresponds to the machine repair.
pdown,i pdown,i
tdown,i
pup,i pup,i
pR,i
tMi VMi tMi VMi
tR,i tR,i
λF,i
tF,i γi 0pF,i
γi γi
λR,i λR,i
(a) (b)
FIGURE 3.6 (a) TDF model. (b) ODF model.
Stochastic transitions used in Figure 3.6a to represent the fail/repair events may be substituted by deterministic transitions without changing the behaviour of the model.
An FOHPN representing an ODF model is shown in Figure 3.6b. The continuous place pR,i is initially marked withγithat represents the production volume that will be processed by the machine Mi before failing. After machine Mi has processed the fluid quantityγi, the continuous place pF,i
will be marked by wi. Then, the immediate transition tdown,iis enabled and fires, emptying place pF,i
and removing 1 token from place pup,i, thus disabling transition tMi. At the same time, 1 token is added to the discrete place pdown,i, thus enabling the stochastic transition tR,i. Place pdown,irepresents the condition of the machine under repairing because when it is marked transition tMi is disabled, that is, vMi=0. The machine will be down until the repair event occurs, that is, transition tR,ifires, bringing the net back to the initial state.
The ODF model exploits one of the hybrid features of FOHPN: the transformation of fluids into discrete tokens and vice versa through discrete transitions.