The Asta Property

ed to fairly general stochastic processes, entirely divorced from a queueing setting; any pair consisting of a real-valued stochastic process and a point process over a common probability space will do. In this general setting, the queueing terminology is modified: time-observed statistics become time averages, and job-observed statistics become event averages. At its most general, the ASTA setting and associated ASTA problem (to be described below) may be profitably viewed simply as a sampling problem. To fix the ideas, we couch the discussion in engineering-like terminology. Imagine some random system (e.g., a physical device or network) whose stochastic state evolves in time (continuous or discrete), with the proviso that its state or a function thereof may be observed (sampled) only at a subset of discrete time points. Suppose that a set of state functions has been selected and a class of sampling procedures, obeying this proviso, has been devised. Usually, no other restrictions are imposed on the sampling mechanism. Admissible sampling procedures are allowed to be random or deterministic, and they may or may not interact with the state of the system under observation; the term “interaction” will be used in informal discussion in lieu of stochastic dependence. Next, form the event averages of interest for each admissible sampling procedure, and compare each of them to the corresponding time averages (note that the latter is just another sampling procedure, except that it permits continual observation of the system, without time gaps, and therefore is excluded from the admissible class). Informally, the ASTA problem seeks to answer the following question: For which state functions and admissible sampling procedures, will the limiting event averages coincide with the corresponding limiting time averages? A more refined ASTA problem would seek information on the estimation bias (the difference between the event average and time average): Do the event averages overestimate or underestimate the corresponding time averages, and if so, by how much? To illustrate that the theoretical ASTA problem is of practical interest, consider a computer operating system which collects some useful statistics on its own operation (e.g., CPU utilization, execution times of jobs, etc.) For reasons of efficiency, the system is permitted to collect those statistics only at prescribed time points, and statistics collection is combined with other housekeeping tasks; otherwise, the attendant overhead may well degrade system performance. System designers would then like to know whether the event averages are good approximations of the “true” statistics, and whether the former are hopelessly biased with respect to the latter. A qualitative first-cut analysis of the ASTA problem can be given in terms of the interaction (dependence) between the sampling procedure and the observed state. Clearly, if the sampling process is stochastically independent of the observed state process, then no bias will ensue. This case is mathematically uninteresting, and may not be feasible in practice (see, e.g., the example above). Conversely, if the sampling procedure “interacts strongly” with the observed state, then biasing effects should be fully expected. This is, in fact, what happens in most queueing systems, where the sampling procedure observes the state at job arrival times. Unlike “outside observers” which are non-intrusive, “job observers” interact with the state of the queue, since each ends up joining the queue. This example makes it clear that the aforementioned requirement that “jobs may not see themselves” is designed to lessen that interaction, which would otherwise preclude ASTA in most cases. Thus, the ASTA problem can be informally viewed as seeking to delineate the “upper limits” of dependence between the sampling procedure and the observed state, within which the ASTA condition holds true. Remarkably, a host of ASTA conditions have been identified. The most celebrated instance of these is the PASTA Theorem, which is due to R. Wolff [31] (see also [15]). Furthermore, the general ASTA problem (lack of bias), as well as its refined variant (direction of bias), both have a surprisingly simple resolution (see Melamed and Whitt [20]). Ignoring secondary technical details,