A short note on the queueing system
Introduction
In the past, several authors have discussed the analytical and computational aspects of the discrete-time GI/Geo/1 queue. Bruneel and Kim (1993) discuss in detail various theoretical results for discrete-time queueing models and their applications in communication systems. Using the imbedded Markov chain technique (IMCT), Hunter (1983) discusses in detail some discrete-time queues. Chaudhry et al. (1996) derive relations among the steady-state system-length distributions at prearrival and random epochs as well as at an outside observer’s epoch for the late arrival system with delayed access (LAS-DA) and the early arrival system (EAS) for the GI/Geo/1 queue. However, they do not discuss the system-length distribution at an intermediate epoch nor do they obtain the system-length distribution at a post-departure epoch for the GI/Geo/1 queue. The purpose of the present paper is fourfold:
- (i)
obtain the system-length distribution at intermediate epochs for both LAS-DA and EAS systems;
- (ii)
derive the system-length distribution at post-departure epochs for both LAS-DA and EAS systems;
- (iii)
present an alternative derivation of the system-length distribution at an outside observer’s epoch given in Chaudhry et al. (1996);
- (iv)
examine the relationship between our discrete-time system and its continuous-time counterpart.
Essentially, this note complements the work that has been presented in Chaudhry et al. (1996).
Section snippets
Model description and analysis
A detailed description of the discrete-time GI/Geo/1 queue is given in Chaudhry et al. (1996) and Hunter (1983). We give a brief account of the model for the sake of completeness. Assume that the time axis is slotted into intervals of equal length with the length of a slot being unity, and it is marked as . In LAS-DA, a potential arrival occurs late in the slot, i.e., just before the slot boundary and a potential departure occurs just after the slot boundary, whereas in EAS a potential
The continuous-time case
Here, we study the relationship between the discrete-time queue and its continuous-time counterpart. For the continuous-time queue, we assume that the interarrival times of two successive arrivals are independent and identically distributed with probability distribution function , Laplace–Stieltjes transform and mean . Let the time axis be slotted into intervals of equal length so that , , and , where and ,
Acknowledgments
The first and second authors received partial support from the Foundation for Science and Technology (FCT). The second author was supported by FCT research grantSFRH/BPD/64372/2009. The third author was supported partially by NSERC.
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