Simple Bounds for Finite Single-Server Exponential Tandem Queues

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.