We will now use this procedure of converting ASPNs to CHnMMs in order to exemplarily specify two CHnMMs of realistic systems with partially observable
2For ASPNs with an infinite number of reachable markings no CHnMM can be built.
However, for these, exact behavior reconstruction as attempted in this work would not be possible in any case, since exact reconstruction would need to take into account all of the infinite number of markings, which is not feasible.
3One exception is that an infinite sequence of immediate transitions may fire after a single timed transition has fired and thus no tangible state is ever reached. In this case the ASPN is deadlocked, no tangible target state exists and the ASPN cannot be converted to a CHnMM.
However, such deadlocking ASPNs are not practically relevant since the deadlock means that the system can never progress beyond the point in time at which the deadlock occurred, which does not occur in real-life systems.
Machine 2
Tester Machine 1
Item by M2
~N(120; 20) Item by M1
~N(150; 25)
Previous Item by M2 Ok Defective Ok
Defective 0.9
0.1 0.05 0.95
Figure 3.1: The quality tester model, shown schematically (left) and as an augmented SPN (right).
behavior. These two CHnMMs will also be used repeatedly in the remainder of this work to experimentally test the algorithms developed.
The two systems were selected to show potential practical applications of CHnMMs and at the same time cover different kinds of CHnMMs: They differ in the model complexity, where one model has only a single marking, whereas the other one has about 5000 reachable markings. And they differ in the amount of information present in the observations: In one model, different transitions emit symbols with different probabilities, so that the type of emitted symbol contains information on what activity is more likely to just have been completed. In the other one, every transition emits the same single symbol so that the information of the observation sequence lies only in the times of the symbol emissions, but not in the symbols themselves.
3.5.1 The Quality Tester
The “Tester” example model represents a partially observable part of a pro-duction line (cf. Figure 3.1): Two machines are imperfectly producing indis-tinguishable items, causing some items to be defective. Producing an item takes a random amount of time and the two machines have different continuous probability distributions to describe this randomness, because they are different models.
The items are then put on conveyor belts to be fed to a single automatic quality tester for quality control. This tester logs the time of each quality test and the test result (“ok” or “defective”). Since the time that the items spend on the conveyor belts is known, constant, and identical for both machines, the times at which each item was produced can directly be derived from the test protocol.
This yields a production protocol containing the times of item production and the quality predicate (“ok” or “defective”) of each item produced.
In this scenario, the owner or the operator of the production facility may be interested in finding out which of the two machines has a higher ratio of defective items. For safety reasons, the area between the two machines and the tester is unobservable and thus the only source of even partial observations of the system behavior is the quality test protocol. Determining the source of defective items thus requires behavior reconstruction and thus prior to that the
modelling of the system as an ASPN and a conversion to an CHnMM.
The simple schematic version of this model (cf. left hand side of Figure 3.1) can almost directly be converted into an ASPN, since the system has only a single discrete state and two distinct activities (the production of items by the two machines) that are always active. The only obstacle is that independently of which machine has produced an item, the system is alwas in the same mark-ing, which violates one of the limitations of ASPNs. As a remedy, the procedure developed to circumvent theSCone limitation of CHnMMs can be applied (cf.
Section 3.2): by storing the source of the previously produced item the single marking of this system is duplicated and the firing of the state transition rep-resenting the two machines are made to reach different clone markings. The resulting ASPN can be seen on the right-hand side of Figure 3.1).
With the procedure outlined in Section 3.4, this ASPN can then be converted to the following CHnMM:
• States: Sprev f rom1, Sprev f rom2
• Symbols: Vok, Vdef ective
• Activities:
– T RM1: (∼N(150,25),0, Vok→0.9;Vdef ective→0.1, f alse) – T RM2: (∼N(120,20),1, Vok→0.95;Vdef ective→0.05, f alse)
• State transition matrixA=
T RM1 T RM2
T RM1 T RM2
• Initial probability vector Π = (1,0)
With only two discrete states and two concurrent activities, this model is close to the most simple CHnMM imaginable. It can thus be used to illustrate the upper limit of computational performance for the CHnMM algorithms to be developed, and is used as an application example for those algorithms.
3.5.2 The Car Rental Agency
The Car Rental Agency model (cf. Figure 2.2 on page 21) that models the behavior of customers and the only employee in said rental agency has already been introduced in Section 2.5 as an example of ASPNs. In this section, we only present the so far omitted representation of this system as a CHnMM.
In this model, a marking consists of the numbers of ordinary and premium customers currently standing in line, as well as the type (ordinary or premium) of the customer who is currently being served. With the system allowing for up to 50 customers per queue the resulting CHnMM will have more than 5000 discrete states and the state transition matrix consequently more than 25 million entries. In order to give a feasible representation of the state space we use the set-builder notation. And we only list those elements of the state transition matrixA that are not equal to∅.
Discrete States We name states using a tuple (X, i, j) where X ∈ {O, P} denotes whether the currently served customer is an ordinary (O) or a premium (P) customer, i denotes the number of waiting ordinary and j the number of waiting premium customers. The set of discrete states of the Car Rental Agency model is then
{Sidle} ∪ {SO,i,j|i, j∈ {0. . .50}} ∪ {SP,i,j|i, j∈ {0. . .50}}
Set of Symbols The set of externally observable symbols contains only a sin-gle symbolVDoor, since all discrete state changes (arrival and service completion of a customer) cause the door of the rental station to open.
Activities The only activities than occur in this model are the arrivals and service completions of ordinary and premium customers, respectively:
• T RpremiumArrive: (∼Exp(1/45),0, VDoor→1, true)
• T RpremiumService: (∼W eibull(16,2),1, VDoor →1, true)
• T RordinaryArrive: (∼Exp(2/45),2, VDoor →1, true)
• T RordinaryService: (∼W eibull(12,2),3, VDoor →1, true)
State Transition Matrix Since it is impractical to given the whole state transition matrix with its more than 25 million entries, we only list those entries that represent possible state changes, i.e. matrix entries that do not contain the special symbol∅.
• SO,0,0⇒Sidle:T RordinaryService
• SP,0,0⇒Sidle:T RpremiumService
• Sidle⇒SO,0,0:T RordinaryArrive
• Sidle⇒SP,0,0:T RpremiumArrive
• ∀i∈ {0. . .50}, j∈ {0. . .49}:SO,i,j⇒SO,i,j+1:T RpremiumArrive
• ∀i∈ {0. . .49}, j∈ {0. . .50}:SO,i,j⇒SO,i+1,j :T RordinaryArrive
• ∀i∈ {0. . .50}, j∈ {1. . .50}:SO,i,j⇒SP,i,j−1:T RordinaryService
• ∀i∈ {1. . .50}:SO,i,0⇒SO,i−1,0:T RordinaryService
• ∀i∈ {0. . .50}, j∈ {0. . .49}:SP,i,j ⇒SP,i,j+1:T RpremiumArrive
• ∀i∈ {0. . .49}, j∈ {0. . .50}:SP,i,j ⇒SP,i+1,j :T RordinaryArrive
• ∀i∈ {0. . .50}, j∈ {1. . .50}:SP,i,j ⇒SP,i,j−1:T RpremiumService
• ∀i∈ {1. . .50}:SP,i,0⇒SO,i−1,0:T RpremiumService
Initial Probability Vector The initial probability vector Π contains the probability 1.0 for the initial marking (no customers are present in the rental station and the clerk is thus idle) and 0.0 probability otherwise:
Π :πi=
(1.0 ifSi=Sidle
0.0 otherwise
This model has an extensive discrete state space 2500 times the size of the Tester state space. And the model has three concurrent activities (two arrival activities and the service of either an ordinary or a premium customer) in most discrete states. Thus, computation time and memory consumption will likely be far higher for this model than for the Tester model. The Car Rental Agency model is therefore used in this work to test the limits of practical applicability of the algorithms developed.