A person waiting in line has a waiting time $\tau$. Define $$E[H(\tau)] = \rho H(0) + \int_{0}^{\infty} H(t) f(t) dt \tag{*},$$ for all functions $H$ for which $(*)$ is defined. I wish to find the conditions on $\rho$ and $f$ for which this formula is an expectation on the sample space $\Omega_{\tau}$, and to give an interpretation of this formula. There are three axioms which I must verify:
(1) $E(c_1 X_1 + c_2X_2) = c_1E(X_1) + c_2E(X_2)$.
- By the linearity of the integral, this property immediately follows.
- $E[\sum_{i=1}^{2} c_i X_i] = \sum_{i=1}^{2} c_i(\rho X_i(0) + \int_{0}^{\infty} X_i(t) f(t) dt$)
(2) $E(1) = 1$
- $E(1) = \rho + \int_{0}^{\infty} f(t) dt = 1$
(3) $X \geq 0 \Rightarrow E(X) \geq 0.$
- $E(X) = \rho X(0) + \int_{0}^{\infty} X(t)f(t) dt \geq 0$
I do not see what information about $\rho, f$ I can extract from properties $(2)$ and $(3)$, or how to give a physical interpretation of the formula $(*)$.