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for details The algorithm in 3 of Daigle (1991) is in fact a special case of the spectral expansion method discussed in full generality in Mitrani and Mitra (1992) This is a general method for computing the equilibrium probabilities of a Markov process whose state space is a semi-in nite strip in the two-dimensional plane and whose equilibrium equations can be represented by a vector difference equation with constant coef cients The solution of that equation is expressed in terms of the eigenvalues and eigenvectors of its characteristic polynomial Another generally applicable method to compute the equilibrium probabilities for the twodimensional Markov process with quasi-birth-death rates is the matrix-geometric method of Neuts (1981) This method requires solving a matrix quadratic equation This can be done by a probabilistic and numerically stable algorithm discussed in Latouche and Ramaswami (1993) The computational effort of this algorithm increases logarithmically when the server utilization gets larger The computational burden of the spectral method, however, is relatively insensitive to the server utilization of the analysed system Unlike the Latouche Ramaswami algorithm, the spectral method often becomes numerically unreliable when the server utilization gets very close to 1 For the practitioner, the geometric tail approach is much easier to apply than the other two methods This approach combines simplicity with effectiveness The two steps of the geometric tail algorithm are: (a) Compute the zero of a non-linear equation in a single variable (b) Solve a nite system of linear equations These steps are simple, and standard software can be used to perform them The nite system of linear equations is usually relatively small for practical examples In general it is not possible to use the above computational methods on twodimensional continuous-time Markov chain problems in which both state variables are unbounded An example of such a problem is the shortest-queue problem in which arriving customers are assigned to the shortest one of two separate queues each having their own server Special methods for this type of problem are the so-called compensation method and the power-series algorithm discussed in Adan et al (1993), Blanc (1992) and Hooghiemstra et al (1988) Example 412 (continued) Unloading ships with an unreliable unloader The continuous-time Markov chain in the unloader problem satis es Assumption 441 except that the Markov chain cannot take on state (0, 0) The unloader can only break down when it is in operation However, the assumption made in the foregoing analysis can be released somewhat Assume that for some integer N 1 the state space I = I1 I2 , where I1 = {(i, s) | i = N, N + 1, ; s = 0, , m} and I2 is a non-empty subset of {(i, s) | i = 0, , N 1; s = 0, , m} The conditions in Assumption 441 are only assumed for the states (i, s) with i N Further it must be assumed that the only way to enter the set I1 from the set I2 is through the states (N, s) Then a minor modi cation of the above analysis shows.

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that Theorem 441 remains valid with the same matrix A(z) For the particular case of the unloader problem, we nd that (449) reduces to the polynomial equation ( + z) ( z2 + ( + + )z) + z = 0 Special case of linear birth-death rates Suppose that the transition rates s , s , s and s have the special form s = b1 (m s) + c1 s, s = b 1 (m s) + c 1 s (4410)

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