2.5.4 GI/G/1-type processes

Now consider a queueing system where both arrivals and service may occur in bulk. Using a similar example as the ones in Figures 2.4 and 2.3, the state transition diagram of the embedded Markov chain of such process is illustrated in Figure 2.5.

Figure 2.5: The state transition diagram of a GI/G/1 queue.

The processes with similar patterns in the embedded Markov chain are known as GI/G/1-type processes and are generalizations of the M/G/1 and GI/M/1-type processes. Their infinitesimal generator matrix ${\mathbf{Q}}_{GI/G/1}$ can be block-partitioned as:

$${\mathbf{Q}}_{GI/G/1} = \left[ \begin{array}{c c c c c c} \L... ...\vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right].$$ (2.23)

The infinitesimal generator ${\mathbf{Q}}_{GI/G/1}$ of a GI/G/1 type process is a full matrix. However the matrix is structured in repeating blocks. As depicted in Figure 2.5, GI/G/1-type processes usually characterize systems with bulk arrivals and possible failures.