144 research outputs found
Note on the GI/GI/1 queue with LCFS-PR observed at arbitrary times
Consider the GI/GI/1 queue with the Last-Come First-Served Preemptive-Resume service discipline. We give intuitive explanations for (i) the geometric nature of the stationary queue length distribution and (ii) the mutual independence of the residual service requirements of the customers in the queue, both considered at arbitrary time points. These distributions have previously been established in the literature by either first considering the system at arrival instants or using balance equations. Our direct arguments provide further understanding of (i) and (ii)
Sojourn times in non-homogeneous QBD processes with processor sharing
We study sojourn times of customers in a processor sharing model with a service rate that varies over time, depending on the number of customers and on the state of a random environment. An explicit expression is derived for the Laplace-Stieltjes transform of the sojourn time conditional on the state upon arrival and the amount of work brought into the system. Particular attention is given to the conditional mean sojourn time of a customer as a function of his required amount of work, and we establish the existence of an asymptote as the amount of work tends to infinity. The method of random time change is then extended to include the possibility of a varying service rate. By means of this method, we explain the well-established proportionality between the conditional mean sojourn time and required amount of work in processor sharing queues without random environment. Based on numerical experiments, we propose an approximation for the conditional mean sojourn time. Although first presented for exponentially distributed service requirements, the analysis is shown to extend to phase-type services. The service discipline of discriminatory processor sharing is also shown to fall within the framework
Congestion analysis of unsignalized intersections
This paper considers an unsignalized intersection used by two traffic
streams. A stream of cars is using a primary road, and has priority over the
other, low-priority, stream. Cars belonging to the latter stream cross the
primary road if the gaps between two subsequent cars on the primary road is
larger than their critical headways. Questions that naturally arise are: given
the arrival pattern of the cars on the primary road, what is the maximum
arrival rate of low-priority cars such that the number of such cars remains
stable? In the second place, what can be said about the delay experienced by a
typical car at the secondary road? This paper addresses such issues by
considering a compact model that sheds light on the dynamics of the considered
unsignalized intersection. The model, which is of a queueing-theoretic nature,
reveals interesting insights into the impact of the user behavior on the above
stability and delay issues. The contribution of this paper is twofold. First,
we obtain new results for the aforementioned model with driver impatience.
Secondly, we reveal some surprising aspects that have remained unobserved in
the existing literature so far, many of which are caused by the fact that the
capacity of the minor road cannot be expressed in terms of the \emph{mean} gap
size; instead more detailed characteristics of the critical headway
distribution play a role.Comment: This paper appeared in the proceedings of the 8th International
Conference on Communication Systems and Networks (COMSNETS), 5-10 Jan. 2016.
A related but more extended paper which analyses a more general model than
the one in the present paper can be also found on arXiv:1802.0673
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