2.5.2 GI/M/1-type processes

Consider a single server queue where failures can happen during service. The occurrence of failures flushes the queue. Suppose that the arrivals are Markovian with rate $\lambda$ and the service process is two exponential stages with rates $\mu_1$ and $\mu_2$ respectively. Failures occur exponentially with rate $f$. The state transition diagram of this process is illustrated in Figure 2.3.

Figure 2.3: The state transition diagram of a GI/M/$1$ process with failures.

The block partitioned infinitesimal generator for the process is

$${\mathbf{Q}}_{GI/Hypo2/1} = \left[ \begin{array}{c c c c c c... ...vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right] ,$$ (2.19)

where the matrices $\widehat{{\mathbf{L}}}, \widehat{{\mathbf{F}}}, {\mathbf{F}}, \L , {\mathbf{B}}$ and ${\mathbf{B}}^{(j)},~~j\geq 1$ are defined as follows:

$$\begin{array}{c c c c} {\mathbf{F}}= \left[ \begin{array}{c ... ... & 0 \\ f & 0 & 0 \end{array}\right], ~~~j\geq 2, \end{array}$$ (2.20)

where $\alpha_j \stackrel{\rm def}{=}\lambda + \mu_j + f,~j=1,2$.

The process with state transition diagram in Figure 2.3 is an example of a GI/M/1-type or skip-free to the right process. The block partitioned infinitesimal generator ${\mathbf{Q}}_{GI/M/1}$ of a GI/M/1-type process resembles the infinitesimal generator of the GI/M/1 queue:

$${\mathbf{Q}}_{GI/M/1} = \left[ \begin{array}{c c c c c c c} ... ...vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right] .$$ (2.21)

${\mathbf{Q}}_{GI/M/1}$ is a lower Hessenberg type matrix, i.e., its blocks above the main diagonal, but the first one, are all zero matrices. As illustrated in Figure 2.3, examples of GI/M/1 type processes include systems that allow the jobs to be served in bulks and systems that capture failure of service nodes [34]. We elaborate on solution techniques for GI/M/1-type process in Chapter 3.