Abstract
The M/G/1 theory is a powerful tool, generalizing the solution of Markovian queues to the case of general service time distributions. There are many applications of the M/G/1 theory in the field of telecommunications; for instance, it can be used to study the queuing of fixed-size packets to be transmitted on a given link (i.e., M/D/1 case). Moreover, this theory yields results which are compatible with the M/M/1 theory, based on birth–death Markov chains.
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Notes
- 1.
In the case of continuous-time processes, we have to consider Poisson (or compound Poisson) processes. Instead, in the case of discrete-time processes, we have to consider Bernoulli or Binomial arrival processes on a slot basis (in this respect, symbol M used to denote the arrival process at the queue has to be considered in a wider sense and as such it will be substituted by “M”).
- 2.
- 3.
At instant ξ i +, the queue is empty, n i = 0. Hence, during the (i + 1)-th slot no cell is transmitted and at the end of this slot (instant ξ i+1 −) the system contains the new requests a i+1 arrived. There is no service completion at instant ξ i+1.
- 4.
In the case of time-slotted systems, the application of the Little theorem entails to divide the mean number of requests in the queue by the mean number of packets generated per slot, which typically corresponds to A′(z = 1), also representing the traffic intensity.
- 5.
We have to use this expression for s = s(z) because of the compound arrival process and the imbedding instants at the level of packets. However, we should use the more simple formula s = λ(1 − z) if the imbedding points are at message transmission completion instants.
- 6.
Let us recall that this is true under the approximation concerning message arrivals at an empty buffer.
- 7.
This is a special case of the round robin scheme with threshold, which can be studied on the basis of what is written in Sections 7.3.1 and 7.3.3. Other schemes could also be considered here, like the exhaustive service or the gated service, as explained in Sections 7.3.1 and 7.3.3. These aspects are however beyond the scope of the present exercise.
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Giambene, G. (2014). M/G/1 Queuing Theory and Applications. In: Queuing Theory and Telecommunications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-4084-0_6
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DOI: https://doi.org/10.1007/978-1-4614-4084-0_6
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