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5.3 LOGICAL UNIVERSALITY Certain families of thin-layer reaction diffusion chemical media can implement sensible transformation of initial (data) spatial distribution of chemical species concentrations to nal (result) concentration pro le [1,45]. In these reaction diffusion computers, a computation is realized via spreading and interaction of diffusive or phase waves. Specialized, intended to solve a particular problem, experimental chemical processors implement basic operations of image processing [5,28,39,40], computation of optimal paths [5,9,46], and control of mobile robots [5]. A device is called computationally universal if it implements a functionally complete system of logical gates, for example, a tuple of negation and conjunction, in its space time dynamics. A number of computationally universal reaction diffusion devices were implemented: the ndings include logical gates [42,48] and diodes [17,29,34] in BelousovZhabotinsky (BZ) medium, and xor gate in palladium processor [2]. All the known so far experimental prototypes of reaction diffusion processors exploit interaction of wave fronts in a geometrically constrained chemical medium; that is, the computation is based on a stationary architecture of medium s inhomogeneities. Constrained by stationary wires and gates, chemical universal processors pose a little computa-

2.5 SYSTEMATIC PROCEDURE FOR SOLVING FLOWGRAPHS To solve a complicated owgraph such as Figure 2.1 requires several steps for reducing the series, parallel, and loop components. We illustrated this in Section 2.4 for the reversible illness death model of Figure 2.22. For such complex structures we use a procedure based on Mason s rule (cf. Mason, 1953) to solve owgraphs. Mason s rule was developed in the context of graph theory for solving systems of linear equations. In its original implementation, Mason s rule did not involve probabilities or MGFs. However, owgraph models can be solved by applying Mason s rule to the branch transmittances. This gives a systematic procedure for computing the transmittance of the owgraph from any state A to any state B for any nite system. When the system is certain to pass eventually from state A to state B, the transmittance is the MGF of the waiting time distribution. Practical use of Mason s rule entails identifying all of the paths from A to B and the loops involved in those paths. It requires computing the transmittance for every distinct path from the initial state to the end state and adjusting for the transmittances of various loops. We illustrate using the cancer example of Figure 2.1. Suppose that our initial state is I and that our waiting time of interest is simply death, D1 or D2 . Path transmittances are simply products. There are four paths from I D1 and I D2 : I A D1 , I A D2 , I D2 , and I A R D2 .

Another way to see the effect of diminishing returns is to note the increasing costs for producing output. If you pay workers 6 per hour to pick apples, your cost to have 300 apples picked in the first hour is 2 pence per apple, as shown in Table 3-1. The second hour yields only 200 apples, costing you 3 pence per apple (because you still have to pay the worker 6 for that hour s work). Only 120 apples get picked in the third hour, so the labour cost per apple rises to 5 pence. Eventually, the effects of diminishing returns drive prices so high that you stop devoting further labour resources to picking additional apples. Virtually all production processes show diminishing returns, and not just for labour. Additional amounts of any particular input usually result in smaller and smaller increments of output, holding all other inputs constant. (In some special cases this doesn t hold or it can look for a while as if it doesn t, but this is pretty much the case for almost all production.)

The general idea of implementing asynchronous requests with threads and synchronous requests is very much the same as with the deferred synchronous request implementation. The difference is that it is the request thread that initiates the synchronization rather than the main thread. As soon as the request is completed, the new thread invokes a callback operation in order to pass on the result and then joins the main thread as in Example 7.4. Note that Example 7.4 does not synchronize the two threads. The child thread dies after the callback has been completed. This is appropriate in this example as the callback prints some information on the console and does not modify the state of the client. If at some stage the client needs to access the result of the request or review whether the request has been completed, synchronization would be necessary. This can be done using the thread synchronization primitives that are available in Java. The client, for instance, could suspend its execution and then the callback could resume it, or the parent thread simply suspends its execution. Other options include the j o i n operation, which makes the parent thread wait for the child thread to die, or the use of s y n c h r o n i z e d methods. Although we have explained the use of threads for asynchronous requests using Java/RMI, the considerations are again directly applicable to C O RBA and C O M. If we used a Java programming language binding to C O RBA/IDL or MIDL, we would just have to replace the remote method invocation with a call of a client stub generated by the IDL compiler.