Written by a prominent Russian mathematician, this concise monograph examines aspects of queuing theory as an application of probability. The three-part treatment begins with a study of the stream of incoming demands (or `calls,` in the author's terminology). Subsequent sections explore systems with losses and systems allowing delay. Prerequisites include a familiarity with the theory of probability and mathematical analysis. A. Y. Khinchin made significant contributions to probability theory, statistical physics, and several other fields. His elegant, groundbreaking work will prove of substantial interest to advanced undergraduates, graduate students, and professionals in the fields of statistics, probability, and operations research.
The general theory of streams (of uniform events) might reasonably begin with abstract definitions of the main components of such streams. However, any such general approach will be postponed till Chapter 2 where it will be dealt with to the extent necessary. Meanwhile, we prefer to introduce the reader into the field of concrete investigations of streams of the simplest type in order to achieve from the start a clear picture of the basic ways of thought, of the mathematical models in the theory of mass service, and of the form of this science as a mathematical discipline. When these concrete explanations have been sufficiently mastered, the study of a more abstract general theory should not seem difficult.
We also add that the simple type of stream to be studied in this chapter has, in course of time, shown itself to be almost the only one useful in application; only in comparatively recent years has the necessity been clearly shown for the study of streams of a more general type. Meanwhile, even nowadays the great majority of applications of the theory of mass service (in particular in telephone applications) arise from the assumption that an incoming stream of demands (calls) belongs to this simple type. Explanations, particularly technical ones, of the theory of the simple stream have in the past few decades developed to the extent that even elementary courses in the theory of probability nowadays usually include a special chapter devoted to this theory.
1 Definition and statement of the problem
We will call a stream of uniform events âsimpleâ if it possesses the following three characteristics â
(1) Stationariness. For any t > 0 and integer k
0, the probability that during the period of time (a, a + t) there will occur k events is the same for every a
0 (and so depends only on t and k); hereafter we shall indicate this probability by Ï k(t). Throughout the book we shall deal only with streams in which, during a finite period of time with probability 1, there occur only a finite number of events. Thus, for any t, we shall always have
The stationariness of a stream expresses the invariability of i...
