W e compare the performance of seven methods in computing or approximating service levels for nonstationary M t /M/s t queueing systems: an exact method (a Runge-Kutta ordinary-differential-equation solver), the randomization method, a closure (or surrogate-distribution) approximation, a direct infinite-server approximation, a modified-offered-load infinite-server approximation, an effective-arrival-rate approximation, and a lagged stationary approximation. We assume an exhaustive service discipline, where service in progress when a server is scheduled to leave is completed before the server leaves. We used all of the methods to solve the same set of 640 test problems. The randomization method was almost as accurate as the exact method and used about half the computational time. The closure approximation was less accurate, and usually slower, than the randomization method. The two infinite-server-based approximations, the effective-arrival-rate approximation, and the lagged stationary approximation were less accurate but had computation times that were far shorter and less problem-dependent than the other three methods.

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An efficient parallel algorithm is reported for determining all bound rovibrational energy levels for the HO 2 molecule for nonzero angular momentum values, Jϭ1, 2, and 3. Performance tests on the CRAY T3D indicate that the algorithm scales almost linearly when up to 128 processors are used. Sustained performance levels of up to 3.8 Gflops have been achieved using 128 processors for J ϭ3. The algorithm uses a direct product discrete variable representation ͑DVR͒ basis and the implicitly restarted Lanczos method ͑IRLM͒ of Sorensen to compute the eigenvalues of the polyatomic Hamiltonian. Since the IRLM is an iterative method, it does not require storage of the full Hamiltonian matrix-it only requires the multiplication of the Hamiltonian matrix by a vector. When the IRLM is combined with a formulation such as DVR, which produces a very sparse matrix, both memory and computation times can be reduced dramatically. This algorithm has the potential to achieve even higher performance levels for larger values of the total angular momentum.

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The discrete variable representation (DVR) method has been modified in three major ways to produce a more efficient scheme for calculating the rotational−vibrational energies of van der Waals molecules. First, the implicitly restarted Lanczos method (IRLM) of Sorensen (SIAM J. Matrix Anal. Appl. 1992, 13, 357) is used to determine the eigenpairs of interest. Second, Chebychev polynomial preconditioning is applied to make it easier to project out unwanted eigenfunctions and thus speed up the convergence of the IRLM. Finally, a very efficient matrix−vector algorithm is introduced that makes maximum use of the underlying sparsity of the DVR Hamiltonian. Calculations for a protypical system Ar−HCl are reported. For the 20 lowest Ar−HCl eigensolutions corresponding to angular momentum J = 1 and using only a single processor of a Cray YMP, the modified DVR approach is about 6 times faster than the original DVR method and 14 times faster than the collocation method of Peet and Yang (J. Chem. Phys. 1989, 91, 6598). As the total angular momentum is increased, the relative performance of the modified DVR approach improves dramatically. For instance, with J = 5 the modified method is about 45 times faster than the original DVR and 100 times faster than the collocation method on a single processor of a Cray YMP. This modified DVR approach also runs significantly faster on a range of workstations such as DEC Alpha and the IBM RS/6000. Application of this method is also shown to be effective in obtaining an improved interaction potential for the A 2Σ+ state of Ar−HO.