Abstract
In this chapter the reader will be shown how the arrival process can be linked to the statistics of the service time, in order to achieve a model for the queuing system as a whole. The Markov Chain is the formal tool that can help solving this sort of problems in general. Here we will focus on a specific subset of Markov Chains, the so-called birth–death processes, which well match with the memoryless property of the Poisson process and of the negative exponential distribution. The general model described in this chapter will be re-used in the remainder of the book to characterize the specific queuing systems suitable for different application scenarios.
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Notes
- 1.
Unless otherwise specified, in this book we will refer to systems that can be described with a scalar state variable k, which can be assumed, without loss of generality, to take values in the natural number set \(k \in \mathbb {N}\). Usually k = 0, 1, …, n, ….
- 2.
Recall that the servers are identical from the customer perspective, therefore they all behave the same.
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Callegati, F., Cerroni, W., Raffaelli, C. (2023). Formalizing the Queuing System: State Diagrams and Birth–Death Processes. In: Traffic Engineering. Textbooks in Telecommunication Engineering. Springer, Cham. https://doi.org/10.1007/978-3-031-09589-4_3
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DOI: https://doi.org/10.1007/978-3-031-09589-4_3
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