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Systems of heterogeneous parallel processing are studied such as arising in parallel programs executed on distributed systems. A lower and an upper bound model are suggested to obtain secure lower and upper bounds on the performance of these systems. The bounding models are solved by using a matrix-geometric algorithmic approach. Formal proofs of the bounds are provided along with error bounds on the accuracy of the bounds. These error bounds in turn are reduced to simple computational expressions. Numerical results are included. The results are of interests for application to arbitrary forkjoin models with parallel heterogeneous processors and synchronization.

Tandem queue configurations naturally arise in multistage stochastic systems such as assembly lines in manufacturing or multiphase transmissions in telecommunications. As finite capacity or storage constraints (buffers) are usually involved, the celebrated closed product form expression is generally not applicable. In this paper, a new bounding methodology for nonproduct form systems is applied to finite single-server exponential tandem queues. The methodology is based on modifying the original system into product form systems that provide bounds for some performance measure of interest. The product form modifications given for this finite tandem queue propose a computationally attractive and intuitively obvious lower and upper bound for the call congestion and throughput. Numerical results indicate that the bounds are reasonable indicators of the order of magnitude. This can be useful for quick engineering purposes as will be illustrated by an optimal design example. A formal proof of the bounds is given. This proof extends standard techniques for comparing stochastic systems and is of interest in itself.

S hould service capacities (such as agent groups in call centers) be pooled or not? This paper will show that there is no single answer. For the simple but generic situation of two (strictly pooled or unpooled) server groups, it will provide (1) insights and approximate formulae, (2) numerical support, and (3) general conclusions for the waiting-time effect of pooling. For a single call type, this effect is clearly positive, as represented by a pooling factor. With multiple job types, however, the effect is determined by both a pooling and a mix factor. Due to the mix factor, this effect might even be negative. In this case, it is also numerically illustrated that an improvement of both the unpooled and the strictly pooled scenario can be achieved by simple overflow or threshold scenarios. The results are of both practical and theoretical interest: practical for awareness of this negative effect, the numerical orders, and practical scenarios in call centers, and theoretical for further research in more complex situations.