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Queuing Theory and Telecommunications pp 319–365Cite as

Markov Chains and Queuing Theory

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Abstract

Telecommunication systems are characterized by the transmission of data on wired or wireless links. In these cases, we have that different “messages” contend for the use of the same transmission resources. Typical examples can be as follows:

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Notes

  1. 1.

    Agner Krarup Erlang was a Danish engineer who worked for the Copenhagen Telephone Company. He was a pioneer of the queuing theory with his paper published in 1909. Note that the traffic intensity is a dimensionless quantity, but CCIF (a predecessor of ITU-T) decided in 1946 to adopt the ‘Erlang’ as the unit of measurement of the traffic intensity in honor of the Erlang’s work.

  2. 2.

    Processor Sharing (PS) is an ideal service discipline where the server is equally shared among all customers in the queue. Let us consider a single-server queue of the M/M/1−PS type. Surprisingly, it is possible to shown that even in this special case, the mean delay T is insensitive to the service time distribution. Hence, the following queuing systems are “equivalent” in terms of mean delay T and mean number of requests N: M/M/1−FIFO, M/M/1−LIFO, M/M/1−PS.

  3. 3.

    The insensitivity is lost for some special queue disciplines; this happens when, for instance, the service order is determined by the duration of the service itself, as in the SPT case [12].

  4. 4.

    For the sake of simplicity, let us assume here that it is possible to receive an empty message (i.e., a message without packets). Otherwise, we should rescale the binomial distribution to exclude the empty message case. Of course, the solution method of this exercise does not depend on the distribution adopted for the message length.

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Authors and Affiliations

  1. Department of Information Engineering and Mathematical Sciences, University of Siena, Siena, Italy

    Giovanni Giambene

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  1. Giovanni Giambene
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Giambene, G. (2014). Markov Chains and Queuing Theory. In: Queuing Theory and Telecommunications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-4084-0_5

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  • DOI: https://doi.org/10.1007/978-1-4614-4084-0_5

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