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series method in Abate and Whitt [2] (the algorithm Euler exploiting Euler summation), although alternative methods could be used The inversion algorithm is remarkably fast; computation for each time point corresponds simply to a sum of 50 terms We provide numerical examples in Examples 1762 and 1781 Example 1781 is of special interest, because the level-holding-time distribution there is Pareto In Section 175 we show that in some cases we can avoid the inversion entirely and treat much larger models We can avoid the inversion if we can assume that the level holding times are relatively long compared to the times of interest for control Then we can apply a single-transition approximation, which amounts to assuming that the Markov chain is absorbing after one transition Then the conditional mean is directly expressible in terms of the level-holding-time distributions.

// Write the new comms settings err = theSerialPort.SetConfig(cBuf); if(err == KErrNone) { TRequestStatus readStat, writeStat; TUint8 dataBuf[KBufSize]; TPtr8 dataDes(&dataBuf[0],KBufSize,KBufSize); // Read some data from the port serialPort.Read(readStat, dataDes); User::WaitForRequest(readStat); if((err = readStat.Int()) == KErrNone) { // Write the same data back to the port serialPort.Write(writeStat, dataDes); User::WaitForRequest(writeStat); err = writeStat.Int(); } } serialPort.Close(); return(err); }

Alternatively, we can perform a two-transition approximation, which only involves one-dimensional convolution integrals In Section 176 we describe the value of having more detailed state information, speci cally the current ages of levels With heavy-tailed distributions, a large elapsed holding time means that a large remaining holding time is very likely; for examples see Duf eld and Whitt [14, Section 8] for background, and Harchol-Balter and Downey [21] for an application in another setting In Section 177 we turn to applications to capacity planning The idea is to approximate the probability of an excursion in demand using Chernoff bounds and other large deviation approximations, then chart its recovery to a target acceptable level using the results on transience Interestingly, the time to recover from excursions suf ciently close to the target level depends on the level durations essentially only through their mean.

class Driver { // void compute(Problem problem) throws { int n = problemsize / granularity; CyclicBarrier barrier = new CyclicBarrier(n); Thread[] threads = new Thread[n]; // create for (int i = 0; i < n; ++i) threads[i] = new Thread(new Segment(barrier, )); // trigger for (int i = 0; i < n; ++i) threads[i]start(); // await termination for (int i = 0; i < n; ++i) threads[i]join();

Correspondingly, the conditional mean demand relaxes linearly from its excursion, at least approximately so, for suf ciently small times If the chance for a larger excursion is negligible (as determined by the large deviation approximation mentioned) then this simple description may suf ce An example is given in Section 178 In Section 179 we show how long-range dependence in the level process arises through heavy-tailed level-holding-time distributions Finally, we draw conclusions in Section 1710..

The main steps WSERV performs in processing character events are: Calls the key click plug-in Deals with capture keys (including WSERV capture keys) Determines who should receive the event Checks to see if the event has a long capture Checks to see if repeat timer should start Queues the event First, WSERV sends the event to the key click plug-in, if there is one The key translator object has already returned a ag to say whether the character event should be captured, so WSERV checks this and sets the destination for the event accordingly If the key has not been captured, then WSERV sends the event to the currently focused window If the event was captured but WSERV is currently displaying the password window, then WSERV only sends the event if it was captured by the same group window as the password window.

17.4 17.4.1

If the character event is one of WSERV s capture keys, then WSERV will have captured it itself In this case, WSERV will process the event immediately and not send it on to a client If the key is either: A long capture key, or There is currently no repeating key, and the key is allowed to be auto-repeatable then the repeat timer is started Long capturing is a feature that we added in Symbian OS v70 It allows a long press of a key to be treated differently from a quick tap The long-key-press event and the short-key-press event yielded by a single physical key press can differ both in their destination client and the actual character code generated.

Throughout this chapter, the state information on which we condition will be either the current level of each source or the current level and age (current time) in that level of each source. No state from the within-level variation process is assumed. Conditional on that state information, we can compute the probability that each source will be in each possible level at any time in the future, from which we can calculate the conditional mean and variance of the aggregate required bandwidth by adding.