Applied Discrete-Time Queues by Attahiru Alfa

By Attahiru Alfa

This book introduces the theoretical basics for modeling queues in discrete-time, and the fundamental strategies for constructing queuing versions in discrete-time. there's a concentrate on purposes in smooth telecommunication systems.

It presents how such a lot queueing versions in discrete-time should be manage as discrete-time Markov chains. innovations similar to matrix-analytic equipment (MAM) which could used to research the ensuing Markov chains are incorporated. This e-book covers unmarried node structures, tandem method and queueing networks. It exhibits how queues with time-varying parameters might be analyzed, and illustrates numerical concerns linked to computations for the discrete-time queueing structures. optimum keep watch over of queues is additionally covered.

Applied Discrete-Time Queues pursuits researchers, advanced-level scholars and analysts within the box of telecommunication networks. it's compatible as a reference publication and will even be used as a secondary textual content e-book in desktop engineering and machine science.  Examples and routines are included.

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0 0 0 ··· ··· θ .. 0 0 ··· .. ⎤ ⎥ ⎥ 0 ⎥ ⎥. ⎥ . ⎦ 0 ··· 1−θ If we define a random variable A = {0, 1, 2, · · · , n}, for which we have Pr{A = i} = αT i−1 t, i = 1, 2, · · · , n. One sees that this random variable has the negative binomial distribution. For example, Pr{A = j} = n−1 (1 − θ )j−1 θ n−j . 3 Arrival and Service Processes 31 These are just simple examples of representation of some well known discrete distributions using the matrix approach. The key here is that we have demonstrated, through examples, that at least some well-known discrete distributions can be represented using matrix form.

So this is a queueing system in which the server is available only intermittently and the customer (secondary user) has to find which channel (server) and when it is available, keeping in mind that other secondary users may also be attempting to use the same channel. These are just a few of the queueing systems that have historically been associated with telecommunication systems. Trying to present material in this book specifically for telecommunication systems would be limiting the capabilities of the analysis techniques in it.

E. 0 is failure and 1 is success. e. Pr{X = 0} = q and Pr{X = 1} = p = 1 − q be the probability that it is a success. The mean number of successes in one trial is given as E[X] = 0 × q + 1 × p = p. Let θ ∗ (z) be the z−transform of this random variable and given as θ ∗ (z) = q + pz. Also we have E[X] = d(q + pz) |z→1 = p. 9) Suppose we carry out an experiment which has only two possible outcomes 0 or 1 and each experiment is independent of the previous and they all have the same outcomes X, we say this is a Bernoulli process,and θ (z) is the z−transform of this Bernoulli process.

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