9.1.1. The simple feedback system

This is a system in which each request, at its end of processing, has a probability p of being presented again in the system. This model for example corresponds to a transmission system with a failure transmission probability per message p, which results in its retransmission. It may also correspond to a processing system having to perform macrotasks which will only be capable of being run in several basic tasks. This may for example be the case of running processing whose code is not totally present in local memory: the processing must be interrupted with a probability p to go and retrieve the rest in the central memory (problem of virtual memory and page miss, for example). Figure 9.1 represents the system.

Figure 9.1: The simple feedback system

We have:

? = ? ₀ + p ? , that is ? =

In the hypothesis of Poissonian arrivals with rate ?, and with a service time per task obeying an exponential distribution with mean 1/ ?, the results of the M/M/1 queue can be applied and we can thus write:

• mean time spent in the system

  (9-1)

• mean number of tasks in the system:

  (9-2)

We will find this type of model again in the case of a call repetition model (the two should not however be confused, as the variables and the use of the parameters are quite different).

9.1.2.