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# 3.1 Matrix geometric solutions for GI/M/1-type and QBD processes

In this section, we give a brief overview3.1 of the matrix geometric solution technique for GI/M/1-type and QBD processes. While QBDs fall under both the M/G/1 and the GI/M/1-type cases, they are most commonly associated with GI/M/1 processes because they can be both solved using the well-known matrix geometric approach [67].

Key to the general solution for the generator of Eqs.(2.21) and (2.18) is the assumption that a geometric relation3.2 holds among the stationary probability vectors of states in as follows:

 (3.1)

where, in the GI/M/1-type case, is the solution of the matrix equation
 (3.2)

and can be computed using iterative numerical algorithms. The above equation is obtained from the flow balance equations of the repeating portion of the process, i.e., starting from the third column of . One can solve a GI/M/1-type process starting from the flow balance equations corresponding to the first two columns of . By substituting for with their equivalents from Eq. (3.1), i.e., , and adding the normalization condition as

we obtain the following system of linear equations
 (3.3)

that yields a unique solution for and . The symbol '' indicates that we discard one (any) column of the corresponding matrix, since we added a column representing the normalization condition. For , can be obtained numerically from Eq.(3.1), but many useful performance metrics such as expected system utilization, throughput, or queue length can be expressed explicitly in closed-form using , , and only (e.g., the average queue length is simply given by ) [64].

In the case of QBD processes, Eq.(3.2) simply reduces to the matrix quadratic equation

while and are obtained as the solution of the following system of linear equations [65]:
 (3.4)

Subsections

Next: 3.1.1 Additional measures of Up: 3. Matrix-Analytic Methods Previous: 3. Matrix-Analytic Methods
Alma Riska 2003-01-13