Abstract
To be able to perform quantitative analysis of a queuing system, a related mathematical model is required. In this chapter we will discuss the issue of modeling both customer service requests and customer service time when they are randomly distributed. The chapter will mostly focus on the Poisson model and on its memoryless property. After reading this chapter, the reader will know how to describe a customer arrival process, will learn the mathematical model to describe a Poisson process and how to calculate its relevant quantitative characteristics, and will learn about the memoryless property of an exponentially distributed service time. Finally, some examples of application of these results will be provided.
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Notes
- 1.
To simplify the notation used in this section, we represent with T ⋆ and T both the time intervals and their respective sizes.
- 2.
As usual the symbol o(P a) means that the quantity considered is negligible when compared to P a.
- 3.
It is interesting and important to note that the dimension of λ is a frequency (i.e., seconds−1). It will be called average arrival rate in the following and is the physical quantity that characterizes the Poisson process.
- 4.
This case resulted from a method that was used in the old analog telephone network, where it was not possible to provide a specific counter per call and a single counter was available for a given central office, triggering counting steps periodically at 𝜗 t pace. In that situation the number of tokens per call was given by the number of counting steps of the general counter that happened during the call, such that the first counting step occurred randomly after the start of the call, that is a purely random quantity between 0 and 𝜗 t. For instance, in Italy this token generation algorithm was required by the law and is described in art (10 of the Decree of the Minister of Telecommunications of 28/2/97).
- 5.
This is what is usually done in typical tariffs for mobile phones or fixed phones in the countries where this is applicable.
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Callegati, F., Cerroni, W., Raffaelli, C. (2023). An Introduction to Queuing System Modeling. In: Traffic Engineering. Textbooks in Telecommunication Engineering. Springer, Cham. https://doi.org/10.1007/978-3-031-09589-4_2
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DOI: https://doi.org/10.1007/978-3-031-09589-4_2
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