Working vacation policy for a discrete-time Geo^X/Geo/1 retrial queue

Abstract

Due to rapid growth of today’s technology in fast speeding digital networks and industrial organizations, there is a need to develop a model which proves to be useful in handling designing issues of computer and telecommunication systems and many other related digital systems. To accomplish this, we have made an attempt to provide a remedy for modeling some discrete-time (digital) systems of day-to-day life viz. Broadband Integrated Services Digital Network (BISDN), Asynchronous Transfer Mode (ATM) and related computer communication technologies, wherein the models for continuous-time queues fail. This is due to the fact that the discrete-time systems are more appropriate than their continuous-time equivalents to model digital systems. In these systems, time is treated as discrete random variable and is measured in fixed size data units such as machine cycle time, bits, bytes, packets, etc. In this study, we analyze a Geo^X/Geo/1 retrial queue wherein the service facility may leave for more economical type of vacation schedule, called as working vacation. The inter-arrival-time, retrial time, service time and working vacation time are assumed to be geometric distributed in discrete environment. We have used matrix geometric method to compute various useful performance measures of interest. Further, we obtain joint optimal values of most sensitive parameters such as vacation returning rate (η) and service rate of the server during working vacation (μ_V) via direct search method based on heuristic approach. Numerical results are also facilitated to depict the performance of the developed model.

Appendix

1.1 A. Construction of steady state equations

Let us consider the first two equations formed by the system of equations πQ = 0:

$$ {\pi}_{0,0}={\overline{\lambda}}_V{\pi}_{0,0}+{\lambda}_V{c}_1{\pi}_{0,1} $$

(A.1)

$$ {\pi}_{0,1}=\left(\overline{\eta}{\overline{\lambda}}_V{\overline{\mu}}_V+{\lambda}_V{\mu}_V{c}_1\right){\pi}_{0,1}+{\overline{\lambda}}_V{\mu}_V{\pi}_{0,0}+\eta {\overline{\lambda}}_V{\overline{\mu}}_V{\pi}_{0,3}+{\lambda}_V{\displaystyle \sum_{m=1}^d{q}_k^V\left(\overline{\eta}{\pi}_{k,1}+\eta {\pi}_{k,3}\right)} $$

(A.2)

A shortcut method for constructing the steady state Eqs. (A.1)–(A.2) directly is by means of a flow balance procedure called stochastic balance. The analysis tells us that for a given state, the average total flow into the state be equal to the average total flow out of the state if steady-state conditions exists (cf. Gross and Harris [6]).

To illustrate this, we equate the total mean flow into each state with the total mean flow out of that state. For state (0,0), the total mean rate of flow into this state is λ _V c ₁ π _0,1 and the total mean rate of flow out from this state is λ _V π _0,0. Thus, equating the two rates, we get

$$ \begin{array}{l}{\lambda}_V{\pi}_{0,0}={\lambda}_V{c}_1{\pi}_{0,1}\\ {}\mathrm{or},\kern1em \left(1-{\overline{\lambda}}_V\right){\pi}_{0,0}={\lambda}_V{c}_1{\pi}_{0,1}\\ {}\mathrm{or},\kern1em {\pi}_{0,0}={\overline{\lambda}}_V{\pi}_{0,0}+{\lambda}_V{c}_1{\pi}_{0,1}\end{array} $$

, which is same as Eq. (A.1).

Similarly, equating the total mean flow rate in and total mean flow rate out for state (0,1), we get

$$ \begin{array}{l}\left(1-\overline{\eta}{\overline{\lambda}}_V{\overline{\mu}}_V-{\lambda}_V{\mu}_V{c}_1\right){\pi}_{0,1}={\overline{\lambda}}_V{\mu}_V{\pi}_{0,0}+\eta {\overline{\lambda}}_V{\overline{\mu}}_V{\pi}_{0,3}+{\lambda}_V{\displaystyle \sum_{m=1}^k{q}_n^V\left(\overline{\eta}{\pi}_{k,1}+\eta {\pi}_{k,3}\right)}\\ {}\mathrm{or},\kern.2em {\pi}_{0,1}=\left(\overline{\eta}{\overline{\lambda}}_V{\overline{\mu}}_V+{\lambda}_V{\mu}_V{c}_1\right){\pi}_{0,1}+{\overline{\lambda}}_V{\mu}_V{\pi}_{0,0}+\eta {\overline{\lambda}}_V{\overline{\mu}}_V{\pi}_{0,3}+{\lambda}_V{\displaystyle \sum_{m=1}^k{q}_k^V\left(\overline{\eta}{\pi}_{k,1}+\eta {\pi}_{k,3}\right)}\end{array} $$

, which is same as Eq. (A.2).