The first emptiness time of an infinite reservior with inputs forming a birth-death diffusion process and an output process (Q2730732)

Let \(X_t\), \(t\geq 0\), be a Markov process with state space \([0,\infty)\) and generator \({\mathbf g}\) of the form NEWLINE\[NEWLINE {\mathbf g}f(x)=\frac{a}f''(x)+bxf'(x) +\alpha (x)\int_{[0,x)}[f(x-y)-f(x)] dH(y) +\alpha (x)[f(0)-f(x)][1-H(x-)], NEWLINE\]NEWLINE where \(a>0\) and \(b\) are constants, \(H(x)\) is a distribution function and \(\alpha (x)\) is a nonnegative function. The value of the process at moment \(t\) is treated as the content level of a reservoir. Clearly, \(X_t\) is a diffusion process with infinitesimal mean \(bx\), variance \(2ax\) and with output release rate \(\alpha (x)\) having magnitudes distributed according to the distribution \(H(x)\). Assuming that \(\alpha (x)=\nu x+c\) the authors investigate the properties of the distribution and mean of the random variable \(\tau =\inf\{t>0:X_t=0\}\).