The analysis of the M/M/1 queue with two vacation policies (M/M/1/SWV+MV)

ABSTRACT

We consider an M/M/1 queue with two vacation policies which comprise single working vacation and multiple vacations, denoted by M/M/1/SMV+MV. Using two methods (called R-matrix method and G-matrix method), we obtain the stationary distribution of queue length (including the customer being in service) and make further analysis on the stationary numbers of customers in the working vacation and vacation period, respectively. The stochastic decomposition results of stationary queue length and the sojourn time of a customer are also derived. Meanwhile, we show that a simple and direct method of decomposition developed in Liu et al. [Stochastic decompositions in the M/M/1 queue with working vacations, Oper. Res. Lett. 35 (2007), pp. 595–600] is also applicable to our model. Furthermore, busy period is analysed by the limiting theorem of alternative renewal process. Finally, some boundary properties and numerical analysis on performance measures are presented.

KEYWORDS:

• Vacation
• working vacation
• matrix-geometric solution
• stochastic decomposition
• sojourn time
• busy period

2010 AMS SUBJECT CLASSIFICATION::

• 60K25
• 68M20

Disclosure statement

No potential conflict of interest was reported by the authors.