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Future directions

Interesting future directions include establishing a theoretical guarantee (bound) on the accuracy of DR-based analyses (DR, DR-PI, and DR-CI) and increasing the accuracy or DR-based analyses. Although we validate the accuracy of DR-based analyses numerically, no theoretical guarantee is proved. Note that the accuracy of DR-based analyses depends on how the busy period is approximated by a collection of PH distributions. The moment matching algorithm developed in Chapter 2 enables matching the first three moments of the busy period for the first time, but matching more moments may increase the accuracy. DR can also be made as accurate as desired by setting the ``aggregation'' level $\kappa$ higher (recall that DR approximates the set of states in levels $\geq\kappa$ by a collection of PH distributions). Increased accuracy of DR would be particularly needed in computing the higher moments or distribution function of various performance measures, as we have observed that the error tends to increase for higher moments computed via DR.

Another interesting future direction is an extension of DR for analyzing more complex multiserver systems. Although we apply DR in the analysis of the RFB/GFB processes in this chapter, the idea in DR has a broad applicability beyond the RFB/GFB processes. Below, we provide examples of possible extensions of DR.

For example, throughout this chapter, we assume that service demands have PH distributions, but in some cases, they can be extended to general distributions. For example, in the analysis of the system with threshold-based policy for reducing switching costs in cycle stealing, the donor's service demand and switching back time can be extended to general distributions [151,152]. This extension relies on a (well known) analysis of the busy period in an M/GI/1 queue with a setup time. In fact, the idea in DR can be applied to the RFB/GFB processes with a quite broad class of background processes. The analysis of the RFB/GFB processes relies on the analysis of the ``busy period'' in a QBD process via Neuts' algorithm; however, Neuts' algorithm applies to a much broader class of Markov chains, namely the M/G/1 type semi-Markov process. Fully utilizing Neuts' algorithm, DR could be applied to the RFB/GFB process with much less restrictions on the background process.

Restrictions on the foreground process can also be relaxed. Our restrictions on the foreground process guarantee that the 1D Markov chains reduced from the RFB/GFB processes are QBD processes. However, the QBD process is not the only Markov chain that allows analytical tractability. For example, the M/G/1 and G/M/1 type processes can also be analyzed efficiently via matrix analytic methods [111]. Thus, the restrictions on the foreground process can be relaxed accordingly.

Further, the GFB process is limited to 2D Markov chains, but this can be extended to higher dimensions, as we extend the FB process to the RFB process. Finally, we remark that the multidimensional Markov chain that we analyze via DR need not have multidimensionally infinite state space. DR can reduce a Markov chain that have a large number of states in multiple dimensions into a Markov chain that have a small number of states in each dimension, since matrix analytic methods and Neuts' algorithm apply to QBD processes with finite number of levels as well.


next up previous contents
Next: Performance analysis of multiserver Up: Concluding remarks Previous: Concluding remarks   Contents
Takayuki Osogami 2005-07-19