A general description is given of the power-series algorithm for the performance evaluation of multi-queue systems which can be modeled as multi-dimensional quasi-birth-death processes. The complexity of the algorithm when applied to polling systems, consisting of several stations where jobs are generated and a single server which attends to these stations by turns, is studied for various service disciplines. The algorithm is applied to a general class of polling systems with Poisson arrival streams, with Coxian service and switching time distributions, with infinite buffers, with a fixed periodic visit order, and wíth a Bernoulli schedule for each visit to a station.by means of the epsilon algorithm, cf. Wynn [25], especially when the occupancy of a system is high, and a broader sCope of applications for the L j-1 J J J-1 J J J~O ,~tl P(n.h,0,~t1) I{~~~0}

This paper deals with the time-dependent behaviour of two queueing systems in series, the simplest example of a Jackson network. The Laplace transform of the probability p 0(t) that the tandem system is empty at time t is obtained by reducing the functional equation for the generating function of the joint queue length distribution to a Riemann-Hilbert boundary value problem. From this Laplace transform the relaxation time of p 0(t) is determined for all cases, and the first term of the asymptotic expansion ofp 0 (t)-p 0(oo) as too is found in the ergodic and in the null recurrent cases.

An iterative numerical technique for the evaluation of queue length distribution is applied to multiserver systems with queues in parallel in which customers join (one of) the shortest queues upon arrival. The technique is based on power-series expansions of the state probabilities as functions of the load of the system. The convergence of the series is accelerated by applying a modified form of the epsilon algorithm. The shortest-queue model lends itself particularly well to a numerical analysis by means of the power-series algorithm due to a specific property of this model. Numerical values for the mean and the standard deviation of the total number of customers and the waiting times in stationary symmetrical systems are obtained for practically all values of the load for systems with up to ten queues and for a load not exceeding 75% for systems with up to 30 queues. Data are also presented for systems with four queues and unequal service rates.

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