Poisson Flows with Alternating Intensity and Their Application

Abstract

Poisson flow of arrivals is considered in a case when flow intensity is an inconstant value. It is supposed that a random environment exists in which the flow operates. The environment is described as alternating Markov chain. The sojourn times in the alternating two states are independent random variables having the exponential distributions with known parameters. The intensity of the flow equals λ_i, i = 1, 2, if the i-th state of the random environment takes place. The following indices of the flow are considered: the distribution, the expectation and the variance of the number of arrivals on a given interval; the correlation of the numbers of arrivals for two adjacent intervals, etc.

APPENDIX

Lemma. Let X, Y, and Z are independent random variables, Z be a Boolean variable, P{Z = 1} = q; X and Y have the expectation E(X) and E(Y), the variances Var(X) and Var(Y), the densities of distributions ${{h}_{X}}\left( t \right)$ and ${{h}_{Y}}\left( t \right)$ correspondingly. Then random variable $S = ZX + \left( {1 - Z} \right)Y$ has the variance

$$Var\left( S \right) = qVar\left( X \right) + \left( {1 - q} \right)Var\left( Y \right) + q\left( {1 - q} \right){{\left( {E\left( X \right) - E\left( Y \right)} \right)}^{2}}.$$

Proof. Because Z ² = Z, E(Z ²) = q, $E\left( {Z\left( {1 - Z} \right)} \right) = 0$, then

$$Var\left( S \right) = Var\left( {ZX + \left( {1 - Z} \right)Y} \right) = E\left( {{{{\left( {ZX + \left( {1 - Z} \right)Y} \right)}}^{2}}} \right) - {{\left( {E\left( {ZX + \left( {1 - Z} \right)Y} \right)} \right)}^{2}}$$

$$ = ~\,\,E\left( {{{{\left( {ZX} \right)}}^{2}}} \right) + E\left( {{{{\left( {\left( {1 - Z} \right)Y} \right)}}^{2}}} \right) + 2E\left( {ZX\left( {1 - Z} \right)Y} \right) - {{\left( {E\left( {ZX + \left( {1 - Z} \right)Y} \right)} \right)}^{2}}$$

$$\begin{gathered} = qE\left( {{{X}^{2}}} \right) + \left( {1 - q} \right)E\left( {{{Y}^{2}}{{\;}}} \right) - {{\left( {qE\left( X \right) + \left( {1 - q} \right)E\left( Y \right)} \right)}^{2}} \\ = qE\left( {{{X}^{2}}{{\;}}} \right) \mp qE{{\left( {X{{\;}}} \right)}^{2}} + \left( {1 - q} \right)E\left( {{{Y}^{2}}} \right) \mp \left( {1 - q} \right)E{{\left( Y \right)}^{2}} \\ \end{gathered} $$

$$ - \left( {{{{\left( {qE\left( X \right)} \right)}}^{2}} + {{{\left( {\left( {1 - q} \right)E\left( Y \right)} \right)}}^{2}} + 2qE\left( X \right)\left( {1 - q} \right)E\left( Y \right)} \right)$$

$$ = \left( {qE\left( {{{X}^{2}}{{\;}}} \right) - qE{{{\left( {X{{\;}}} \right)}}^{2}}} \right) + \left( {\left( {1 - q} \right)E\left( {{{Y}^{2}}} \right) - \left( {1 - q} \right)E{{{\left( {Y~} \right)}}^{2}}} \right)$$

$$~ + \,\,\left( {qE{{{\left( {X{{\;}}} \right)}}^{2}} - {{{\left( {qE\left( X \right)} \right)}}^{2}}} \right) + \left( {\left( {1 - q} \right)E{{{\left( {Y{{\;}}} \right)}}^{2}} - {{{\left( {\left( {1 - q} \right)E\left( Y \right)} \right)}}^{2}}} \right)$$

$$ - \,\,~2q\left( {1 - q} \right)E\left( X \right)E\left( Y \right) = qVar\left( X \right) + \left( {1 - q} \right)Var\left( Y \right) + q\left( {1 - q} \right){{\left( {E\left( X \right) - E\left( Y \right)} \right)}^{2}}.$$

The Lemma is proved.

Setting Z $~ = 1,\,~{\text{if}}~\,J\left( t \right) = i,~$ and $Z = 0,~$ otherwise; $X = ~N\left( {t,t{\text{*}}} \right)~$ with the condition $J\left( t \right) = i;~Y = N\left( {t,t{\text{*}}} \right)~$ with the condition $J\left( t \right) \ne i,$ we have formula (19).