We consider the sums of a random number of random variables or random sums $S=X_1+X_2+\ldots+X_N$, where $X_1$, $X_2$, $\ldots$ are independent and identically distributed random variables. These frequently encounter in stochastic models. In spite of this frequent usages, some literature dealing with queueing systems have presented the distribution of S which is wrong.
Wrong results are found when $\mathnormal{N}$ depend on $X_i's$. These are due to Wald's Equation, because most of the performance measures in queueing models are consists of expected values. In these cases, we cannot easily derive the results in contrast with the cases that \mathnormal{N}$ is independent of $X_i's$.
Begin with the random sums of independent cases, those of independent cases are dealed. The mistakes in the literature will be corrected and why can be wrong will be discussed. Moreover, many examples of the random sums arising in queueing systems will be presented. All these examples satisfy Wald's Equations and these must have brought the similar mistakes.
