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poisf - Journal of Applied Pr-267 1994 A Generalization of...

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Journal of Applied Probability 31 , 262-267, 1994 A Generalization of “Expectation Equals Reciprocal of Intensity” to Nonstationary Exponential Distributions Eugene A. Feinberg SUNY at Stony Brook June 1992 Revised: November 1992 and January 1993 Abstract An observer watches one of a set of Poisson streams. He may switch from one stream to another instantaneously. If an arrival occurs in a stream while the observer is watching another stream, he does not see the arrival. The experiment terminates when the observer sees an arrival. We derive a formula which states essentially that the expected total time that the observer watches a stream is equal to the probability that he sees the arrival in this stream divided by the intensity of the stream. This formula is valid independently of the observation policy. We also discuss applications of this formula. Keywords: Nonstationary exponential distribution, Poisson arrivals, intensity. AMS (MOS) subject classification: 60E05, 90B25, 90B30, 90C40. 1
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1. Introduction. If X is a random variable having an exponential distribution with parameter λ, then EX = λ - 1 . In this note we describe a generalization of this formula to nonstationary exponential random variables. Let us consider the following example. There are two Poisson streams with positive arrival intensities λ i , i = 1 , 2 . At any epoch t R + = [0 , ) , an observer watches one of the streams and may switch from one to another instantaneously. If an arrival occurs in stream i while he is watching stream j, j 6 = i, then the observer does not see the arrival. Such models arise in multi-armed bandit problems in continuous time; e.g., Presman and Sonin (1983, 1990). Let X be the time before the observer sees the first arrival. Let X i , i = 1 , 2 , be the amount of time during [0 , X ] when the observer watches stream i, so that X 1 + X 2 = X. He may see this arrival either while watching stream 1 or while watching stream 2. Let p i , i = 1 , 2 , be the probability that the observer sees the first arrival while he is watching stream i, p 1 + p 2 = 1 . If X 2 = 0 a.s., the observer watches stream 1 all the time. Then p 2 = 0 , p 1 = 1 , and X 1 is an exponential random variable. So, in this case, EX 1 = λ - 1 1 . Theorem 1 below implies that EX i = p i i , i = 1 , 2 , independently of the rule according to which the observer watches the streams. If there is a finite or countable number of streams, for- mula EX i = p i i remains valid. Theorem 1 below gives such a result for the case where the set of streams may be an arbitrary measurable space. In addition to multi-armed bandit problems, possible applications of Theorem 1 include continuous-time Markov de- cision processes, the analysis and optimization of production systems, and discrete event simulation. 2. Results. Let ( A, A ) be a measurable space, λ be a measurable real-valued function defined on A such that 0 λ ( a ) < , and φ : [0 , ) A be a measurable mapping.
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