# American Institute of Mathematical Sciences

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January  2019, 15(1): 37-58. doi: 10.3934/jimo.2018031

## Delay characteristics in place-reservation queues with class-dependent service times

 1 SMACS Research Group, Department TELIN, Ghent University, Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium 2 Department of Industrial Systems Engineering and Product Design, Ghent University, Technologiepark 903, 9052 Zwijnaarde, Belgium

* Corresponding author: Sabine Wittevrongel

The reviewing process of this paper was handled by Yutaka Takahashi and Wuyi Yue

Received  March 2017 Revised  July 2017 Published  February 2018

This paper considers a discrete-time single-server infinite-capacity queue with two classes of packet arrivals, either delay-sensitive (class 1) or delay-tolerant (class 2), and a reservation-based priority scheduling mechanism. The objective is to provide a better quality of service to delay-sensitive packets at the cost of allowing higher delays for the best-effort packets. To this end, the scheduling mechanism makes use of an in-queue reserved place intended for future class-1 packet arrivals. A class-1 arrival takes the place of the reservation in the queue, after which a new reservation is created at the tail of the queue. Class-2 arrivals always take place at the tail of the queue. We study the delay characteristics for both packet classes under the assumption of a general independent packet arrival process. The service times of the packets are independent and have a general distribution that depends on the class of the packet. Closed-form expressions are obtained for the probability generating functions of the per-class delays. From this, moments and tail probabilities of the packet delays of both classes are derived. The results are illustrated by some numerical examples.

Citation: Sabine Wittevrongel, Bart Feyaerts, Herwig Bruneel, Stijn De Vuyst. Delay characteristics in place-reservation queues with class-dependent service times. Journal of Industrial & Management Optimization, 2019, 15 (1) : 37-58. doi: 10.3934/jimo.2018031
##### References:

show all references

##### References:
Insertion of 4 packets arriving during the same slot under reservation-based scheduling
Sample path of the queueing model
An $M\times M$ switch with output buffers
Mean packet delays versus load $\rho$, for $M = 16$, $\mu_1 = \mu_2 = 3$ and various values of $\alpha$
Mean packet delays versus load $\rho$, for $M = 16$, $\mu_1 = 3$, $\mu_2 = 20$ and various values of $\alpha$
Standard deviations of packet delays versus load $\rho$, for $M = 16$, $\mu_1 = 3$, $\mu_2 = 20$ and various values of $\alpha$
Tail probabilities of the packet delays, for $M = 16$, $\rho = 0.8$, $\mu_1 = 3$, $\mu_2 = 20$ and various values of $\alpha$
Tail probabilities of the packet delays, for $M = 16$, $\rho = 0.8$, $\alpha = 0.15$, $\mu_2 = 4$ and various values of $\mu_1$
Tail probabilities of the packet delays, for $M = 16$, $\rho = 0.8$, $\alpha = 0.15$, $\mu_2 = 4$ and various values of $\mu_2$
Mean packet delays versus the standard deviation of the class-1 service times $\sigma_1$, for $M = 16$, $\rho = 0.8$, $\mu_1 = \mu_2 = 3$ and various values of $\alpha$
Mean packet delays versus the standard deviation of the class-2 service times $\sigma_2$, for $M = 16$, $\rho = 0.8$, $\mu_1 = \mu_2 = 3$ and various values of $\alpha$
Mean values and standard deviations of packet delays versus traffic mix $\alpha$, for $M = 16$, $\rho = 0.8$, and $\mu_1 = \mu_2 = 3$
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