It is commonly supposed that,L=\lambdaWapplies to "almost any" queueing system with\lambdasome average customer entrance rate,Lthe asymptotic expectation or time average of the number of customers in the system andWsome average of the sojourn time. This formula is studied for irreducible positive recurrent Markov queueing systems whose state vectorZconsists of entries representing queue lengths at the respective service stations; blocking, finite capacities, batch arrivals, and variable rates of arrival and service are consistent withZ. Sojourn times are defined inan augmented Markov modelY=(Z,U), where the customer marking processUdescribes the service discipline in sufficiently general terms to include most possibilities of interest. It is shown thatL=\lambda Wis universally applicable, if properly interpreted to take account of state-varying entrance rates, batch arrivals, and multiple customer classes.L,\lambda, andWmay each be equivalently viewed as time averages, means over a regeneration cycle, or expectations with respect to the asymptotic probability structure ofZ. Indeterminate forms ofL=\lambda Ware possible within the scope of Markov queueing networks (MQN); as is shown by some examples, these may take the form\infty \times 0for\lambda W, or of\infty= \inftyforL=\lambda W.

Modern communication theory and practice are heavily dependent on the representation of continuous parameter signals by linear combinations, involving a denumerable set of random variables. Among the best known and most useful is the cardinal seriesf_{n} (t) = \sum^{+n}_{-n} f(k) \frac{\sin \pi (t - k)}{ \pi ( t - k )}for deterministic functions and wide-sense stationary stochastic processes bandlimited to(-\pi, \pi). When, as invariably occurs in applications, samplesf(k)are available only over a finite period, the resulting finite approximation is subject to a truncation error. For functions which areL_{1}Fourier transforms supported on[-\pi + \delta, + \pi - \delta], uniform trunction error bounds of the formO(n^{-1})are known. We prove that analogousO(n^{-1})bounds remain valid without the guard band\deltaand for Fourier-Stieltjes transforms; we require only a bounded variation condition in the vicinity of the endpoints- \piand+ \piof the basic interval. Our methods depend on a Dirichlet kernel representation forf_{n}(t)and on properties of functions of bounded variation; this contrasts with earlier approaches involving series or complex variable theory. Other integral kernels (such as the Fejer kernel) yield certain weighted truncated cardinal series whose errors can also be bounded. A mean-square trunction error bound is obtained for bandlimited wide-sense stationary stochastic processes. This error estimate requires a guard band, and leads to a uniformO(n^{-2})bound. The approach again employs the Dirichlet kernel and draws heavily on the arguments applied to deterministic functions.

The notion of alias-free sampling is generalized to apply to random processesx(t)sampled at random timest_n; sampling is said to be alias free relative to a family of spectra if any spectrum of the family can be recovered by a linear operation on the correlation sequence\{r(n)\}, wherer(n) = E[x(l_{m+n}) \overline{x(t_m)}]. The actual sampling timest_nneed not be known to effect recovery of the spectrum ofx(t). Various alternative criteria for verifying alias-free sampling are developed. It is then shown that any spectrum whatsoever can be recovered if\{t_n\}is a Poisson point process on the positive (or negative) half-axis. A second example of alias-free sampling is provided for spectra on a finite interval by periodic sampling (fort \leq t_oort \geq t_o) in which samples are randomly independently skipped (expunged), such that the average sampling rate is an arbitrarily small fraction of the Nyquist rate. A third example shows that randomly jittered sampling at the Nyquist rate is alias free. Certain related open questions are discussed. These concern the practical problems involved in estimating a spectrum from imperfectly known\{ r(n) \}.