**8. Conclusions**

The queue GI/M*<sup>a</sup>*, *<sup>b</sup>*/*c* was successfully investigated by using the two-dimensional embedded Markov chain. Simple and exact analyses to determine queue-length distributions are presented. An algorithm was derived for the analysis of the steady state behaviour of the system. Our recursive solution approach is not only very efficient, but also accurate by providing the exact queue-length probabilities at p.a.e. In a similar manner, we studied the queue-length distribution at r.e. and derived closed-form formulae in terms of the root *w* for evaluating the exact queue-length probabilities at r.e. We also obtained the probabilities of p.d.e. through the relations between r.e. and p.d.e. The results for this system were provided numerically by considering three inter-arrival time distributions—Erlang, deterministic, and uniform. The work on higher order moments and other distributions can be conducted similarly.

There are two special features in this work. The first is the effort to express the important results in closed form; the second is the development of the methodology and algorithms to efficiently derive accurate results. The models under consideration were validated by using MAPLE to obtain numerical results with sufficient accuracy and trivial computational costs.

**Author Contributions:** The results in this paper are based on J.G.'s Ph.D. thesis Chapter 3. Conceptualization, methodology, writing—review and editing, funding acquisition, J.G. and M.C.; software, validation, formal analysis, writing—original draft preparation, J.G.; resources, supervision, M.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was supported by the Royal Military College of Canada Professional Development Allocation.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Algorithm for Calculating p.a.e. Probabilities**

The method for determining the complete solution to the stationary queue-length probabilities at p.a.e. for the model GI/M*<sup>a</sup>*, *<sup>b</sup>*/*c* is described in the following steps:


(11))


#### **Appendix B. Proof of Equation** (25)

$$D\_R(z) = \sum\_{l=0}^{\infty} (l|c)\_R z^l = \frac{\rho b}{1 - z} [1 - d(c\mu(1 - z))].$$

**Proof.**

$$\begin{split} \sum\_{l=0}^{\infty} \langle l|c \rangle\_{R} z^{l} &= \sum\_{l=0}^{\infty} z^{l} \int\_{0}^{\infty} \frac{e^{-\varepsilon\mu t} (c\mu t)^{l}}{l!} dR(t) \\ &= \int\_{0}^{\infty} e^{-\varepsilon\mu t} \sum\_{l=0}^{\infty} \frac{(c\mu t z)^{l}}{l!} dR(t) \\ &= \int\_{0}^{\infty} e^{-\varepsilon\mu t} e^{c\mu t z} dR(t) \\ &= \lambda \underbrace{\int\_{0}^{\infty} e^{-\varepsilon\mu(1-z)t} \lambda(1-A(t)) dt}\_{=1/\varepsilon\mu(1-z)} - \lambda \int\_{0}^{\infty} e^{-\varepsilon\mu(1-z)t} A(t) dt \\ &= \frac{\rho b}{1-z} + \frac{\rho b}{1-z} \int\_{0}^{\infty} A(t) d e^{-\varepsilon\mu(1-z)t} \\ &= \frac{\rho b}{1-z} (1 - \underbrace{\int\_{0}^{\infty} e^{-\varepsilon\mu(1-z)t} dA(t)}\_{=a(c\mu(1-z))}. \end{split}$$
