Thinning of point processes—covariance analyses

Abstract

Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

The asymptotic variance rate of the output process of finite capacity birth-death queues

“…[15]) and using formulas similar to (10) that appear in that reference. For the simple case of the M/M/1/1 queue, an explicit expression was obtained for the limiting correlation coefficient between the outputs and the overflows in [6]. We now extend this result:…”

“…[6]). This is a 2 state CTMC and it is the only M/M/1/K queue that has a renewal output process (cf.…”

“…The overflow rate is of course λ − λ * . Berger and Whitt, [3] in their equation (6) derive the SCV for the inter-overflow times. Multiplying these we obtain the asymptotic variance rate of the overflows 2 :…”

Bernoulli, multinomial and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes

“…But since R (t, 0) =°for all t then, asymptotically, we have a stationary renewal process being thinned to produce two uncorrelated processes. But by Chandramohan et al (1985) this implies F is exponential with parameter A and hence H(t) = At. The technical details are to be found in Theorem 3.1 of Chandramohan et al Thus it only remains to show that G(t) = 1 -exp( -At).…”

“…In a recent paper, Chandramohan et al (1985) studied the cross-correlation structure between the two processes obtained by thinning a point process under various assumptions. Specifically, it was shown that when a stationary or ordinary renewal process is thinned by an independent Bernoulli process, the resulting processes will be uncorrelated if and only if the original process is Poisson.…”