# American Institute of Mathematical Sciences

October  2017, 13(4): 1901-1926. doi: 10.3934/jimo.2017024

## Analysis of a discrete-time queue with general service demands and phase-type service capacities

 SMACS Research Group, Department of Telecommunications and Information Processing, Ghent University, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium

* Corresponding author: Michiel De Muynck.

The reviewing process of the paper was handled by Wuyi Yue and Yutaka Takahashi as Guest Editors.

Received  October 2015 Published  April 2017

In this paper, we analyze a non-classical discrete-time queueing model where customers demand variable amounts of work from a server that is able to perform this work at a varying rate. The service demands of the customers are integer numbers of work units. They are assumed to be independent and identically distributed (i.i.d.) random variables. The service capacities, i.e., the numbers of work units that the server can process in the consecutive slots, are also assumed to be i.i.d. and their common probability generating function (pgf) is assumed to be rational. New customers arrive in the queueing system according to a general independent arrival process. For this queueing model we present an analysis method, which is based on complex contour integration. Expressions are obtained for the pgfs, the mean values and the tail probabilities of the customer delay and the system content in steady state. The analysis is illustrated by means of some numerical examples.

Citation: Michiel De Muynck, Herwig Bruneel, Sabine Wittevrongel. Analysis of a discrete-time queue with general service demands and phase-type service capacities. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1901-1926. doi: 10.3934/jimo.2017024
##### References:

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##### References:
Mean system content versus the mean service demand $\tau$ for Poisson arrivals with $\lambda=0.9$, shifted geometric service demands and various service-capacity distributions (as indicated), with mean $\mu = \tau$.
Mean customer delay versus the load $\rho$, for Poisson arrivals with varying $\lambda$, deterministic service demands with $\tau=11$ and various service-capacity distributions (as indicated), all with mean $\mu=10$.
Variance of the customer delay versus the load $\rho$, for Poisson arrivals with varying $\lambda$, deterministic service demands with $\tau=11$ and various service-capacity distributions (as indicated), all with mean $\mu=10$.
Dominant-pole approximation of the tail probabilities of the system content, for Poisson arrivals with $\lambda=3$, uniformly distributed service demands from 1 to 10 work units, and negative binomial service capacities with $\mu=10$ and various values of the parameter $m$, as well as deterministic service capacities.
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