Preliminaries

We first define the transition probability matrix, $\mathbf{P}(x)$, such that its $(s,t)$ element, $({\mathbf{P}}(x))_{s,t}$, represents the probability that (i) the sojourn time at state $s$ is $\leq x$ AND (ii) the first transition out of state $s$ is to state $t$. (Note that state $s$ denotes a state ($i,\ell$), where $s = i+\sum_{\ell'=0}^{\ell-1}n_{\ell'}$. State $t$ is defined analogously.) By the following Lemma, which follows from the memoryless property of the exponential distribution, $({\mathbf{P}}(x))_{s,t}$ is also the product of the probabilities of events (i) and (ii).

Lemma 7   Let $X_1$ and $X_2$ be independent exponential random variables. Let $W = \min(X_1, X_2)$. Then

$$\Pr( W < x \mbox{ AND } X_1 < X_2 ) = \Pr( W < x ) \cdot \Pr( X_1 < X_2)$$

Matrix $\mathbf{P}(x)$ has the same structural shape as ${\mathbf{Q}}$, although its entries are different, and thus it can be represented in terms of submatrices, analogous to those in ${\mathbf{Q}}$, indicating backward transitions, local transitions, and forward transitions, as follows:

$$\mathbf{P}(x) = \left(\begin{array}{cccccc} \mbox{\boldmath ... ...\cal{F}$}}^{(2)}(x)\\ & & \ddots & \ddots \end{array}\right)$$

Specifically, the $(i,j)$ element of $\mbox{\boldmath {$\cal{L}$}}^{(\ell)}(x)$ is the probability that the sojourn time in state $(i,\ell)$ is $\leq x$ AND the first transition out of state $(i,\ell)$ is to state $(j,\ell)$. Likewise, the $(i,j)$ element of $\mbox{\boldmath {$\cal{F}$}}^{(\ell)}(x)$ (respectively, $\mbox{\boldmath {$\cal{B}$}}^{(\ell)}(x)$) is the probability that the sojourn time in state $(i,\ell)$ is $\leq x$ AND the first transition out of state $(i,\ell)$ is to state $(j,\ell+1)$ (respectively, to state $(j,\ell-1)$).

Next, we define the $r$-th moment of submatrices, $\mbox{\boldmath {$\cal{B}$}}^{(\ell)}(x)$, $\mbox{\boldmath {$\cal{L}$}}^{(\ell)}(x)$, $\mbox{\boldmath {$\cal{F}$}}^{(\ell)}(x)$, as follows:

$$\mbox{\boldmath {$\cal{B}$}}^{(\ell)}_r \equiv \int_0^\infty... ...v \int_0^\infty x^r d\mbox{\boldmath {$\cal{F}$}}^{(\ell)}(x)$$

for $r=1,2,3$, and $\ell\geq 0$, where an integral of a matrix $\mathbf{M}$ is a matrix of the integrals of the elements in $\mathbf{M}$. For the repeating part, we define $\mbox{\boldmath {$\cal{B}$}}_r \equiv \mbox{\boldmath {$\cal{B}$}}^{(\hat{\ell})}_r$, $\mbox{\boldmath {$\cal{L}$}}_r \equiv \mbox{\boldmath {$\cal{L}$}}^{(\hat{\ell})}_r$, and $\mbox{\boldmath {$\cal{F}$}}_r \equiv \mbox{\boldmath {$\cal{F}$}}^{(\hat{\ell})}_r$, omitting the superscript.

We now define the limits as $x \rightarrow \infty$ of $\mbox{\boldmath {$\cal{B}$}}^{(\ell)}(x)$, $\mbox{\boldmath {$\cal{L}$}}^{(\ell)}(x)$, and $\mbox{\boldmath {$\cal{F}$}}^{(\ell)}(x)$ as follows:

$$\mbox{\boldmath {$\cal{B}$}}^{(\ell)} = \lim_{x \rightarrow ... ... \rightarrow \infty} \mbox{\boldmath {$\cal{F}$}}^{(\ell)}(x);$$

for $\ell\geq 0$. For the repeating part, we define $\mbox{\boldmath {$\cal{B}$}} \equiv \mbox{\boldmath {$\cal{B}$}}^{(\hat{\ell})}$, $\mbox{\boldmath {$\cal{L}$}} \equiv \mbox{\boldmath {$\cal{L}$}}^{(\hat{\ell})}$, and $\mbox{\boldmath {$\cal{F}$}} \equiv \mbox{\boldmath {$\cal{F}$}}^{(\hat{\ell})}$, omitting the superscript.