In this section, we consider plants that have naturally discrete time models. That is, these systems may not even have a differential equation model. Computing systems are ideal examples of this category. Indeed, a digital control text book devoted to the control of only computing systems has recently appeared [22]. In this section, we present the example of an IBM Lotus Domino server and an example from the supply chain area.
2.3. Naturally Occurring Discrete Time Systems 15
2.3.1
IBM Lotus Domino Server
IBM Lotus Domino server is an email server. In this discussion, we will abbreviate it as simply the server. Clients may be considered as the conduits through which the end users access the server. Clients access the database of emails maintained by the server through Remote Procedure Calls (RPCs). The number of RPCs, denoted as RIS, has to be controlled. If the number of RIS becomes large, the server will be overloaded, with a consequent degradation of performance. RIS should not be made small either. If RIS is less than what can be handled, the server is not being used optimally.
Unfortunately, it is not possible to regulate RIS directly. Although RIS is closely related to the number of users, the correspondence could be off at times, because some users could just be idling. Regulation of RIS may be achieved by limiting the maximum number of users (MaxUsers) who can simultaneously use the system.
Based on experience in operating the server, it is possible to come up with an optimal RIS to achieve acceptable performance. If the actual RIS is smaller than the reference value of RIS, MaxUsers may be increased and vice versa.
It is difficult to come up with an exact analytical model that relates MaxUsers and the actual RIS in the server. This is because the server could have different types of administrative loads, such as memory swapping and garbage collection at different times. Moreover, it is difficult to predict how many users will idle at a given time.
A possible way to arrive at a model is through identification, which consists of the following steps: experimentation, data collection and curve fitting, to be explained in detail in Chapter 6. Hellerstein et al. [22] report an experiment in which the parameter MaxUsers is varied about the operating level of MaxUsers = 165 and the corresponding variation of RIS about the operating level of RIS = 135. Defining
x(k) = RIS(k)− RIS
u(k) = MaxUsers(k)− MaxUsers (2.29)
and carrying out an identification exercise, they arrive at the following relationship:
x(k + 1) = 0.43x(k) + 0.47u(k) (2.30)
which is in the form of Eq. 2.2a. Discrete time models are a natural choice in the field of computing systems.
2.3.2
Supply Chain Control
Another field in which discrete time modelling is on the upswing is the area of supply chains. The main reasons for the increased activity in this field are increasing competition and availability of data. We now present an example from this field.
If a manufacturer produces less than what is required, they may lose market share. On the other hand, if they produce more than required, they will incur losses through inventory costs, interest, etc.
We will now present an example from the general area of supply chain management [58]. The idea is that if we have a good estimate of the inventory and the demand for goods, we can decide how much to manufacture. The system inventory is made up of the inventories at the manufacturer and the distributor, as well as the goods in transit, corrected for estimated arrival times.
16 2. Modelling of Sampled Data Systems
The functions of the production ordering and inventory management system include demand forecasting, customer order fulfilment, production ordering (deter- mining the production release quantities), and the production process. The forecasted demand FD of the products is based on first order exponential smoothing of the customer sales rate SALES, with a smoothing constant ρ and a sampling interval δ:
FD(n) = FD(n− 1) + ρδ(SALES(n − 1) − FD(n − 1)) (2.31)
The sampling interval δ or the integration time step is said to correspond to the frequency at which the information is updated within the system. The inventory level (INV) accumulates the difference in the production rate (PRATE) and the sales rate:
INV(n) = INV(n− 1) + δ(PRATE(n) − SALES(n)) (2.32)
The term production release refers to the quantity ordered for production. The work in process (WIP) level accumulates the difference in the production release (PREL) and production rate (PRATE):
WIP(n) = WIP(n− 1) + δ(PREL(n) − PRATE(n)) (2.33)
The production release quantities (PREL) are determined using the ordering rule in Eq. 2.33, based upon the forecasted demand, the difference between the desired level of WIP and the current WIP level, and the difference between the desired level of inventory and the current inventory level:
PREL(n) = FD(n− 1) + α(L × FD(n − 1) − WIP(n − 1))
+ β(FD(n− 1) − INV(n − 1)) (2.34)
where α is the fractional adjustment rate for WIP and it describes how much of the discrepancy between the desired and current levels of WIP are to be added to the production release order. Similarly, β is the fractional adjustment rate for inventory and it describes how much of the discrepancy between the desired and current levels of inventory are to be added to the production release order. These variables are to be chosen so as to achieve a good performance. This topic is discussed in more detail in Example 4.20 on page 88.
Based on the Little’s law, the desired WIP in the system is set to yield the desired throughput (set equal to forecasted demand), given the lead time, L [58]. To provide adequate coverage of inventory, the manufacturer seeks to maintain a desired level of inventory set equal to the forecasted demand. In the following equation, the production process is typically modelled as a fixed pipeline delay L:
PRATE(n) = PREL(n− L) (2.35)
An objective of this model is to determine production release as a function of sales. In this section, we have presented two models that have been posed directly in discrete time. For similar models in diverse fields, such as banking and the criminal justice system, the reader is referred to [4].